1 Online Appendix to “Portfolio Liquidity and Security Design with Private Information” † Peter M. DeMarzo (Stanford University) David M. Frankel (Melbourne Business School and Iowa State University) Yu Jin (Shanghai University of Finance and Economics) This Revision: May 5, 2020 1. Introduction This document is the online appendix to “Portfolio Liquidity and Security Design with Private Information” by Peter M. DeMarzo, David M. Frankel, and Yu Jin (2020), henceforth “DFJ”. It is organized as follows. Section 2 formally states the technical results that are previewed in section 3.5.2 of DFJ. Section 3 proves these results as well as the results that are stated without proof in section 3 of DFJ. Section 4 relaxes ASSUMPTION A (monotonicity) in the case of 2 assets and 2 seller types. Section 5 checks the robustness of our results to the alternative asset sale procedure of Maskin and Tirole (1992). 2. Technical Results We first formally state the technical results that we discuss in section 3.5.2 of DFJ. We begin by specifying the continuous and discrete models. The continuous model, denoted “model ∞”, is as follows. The issuer's type ~ t G has full support [0,1] and the final asset value Y has conditional and unconditional support 0, y and conditional distribution function H that satisfies the Hazard Rate Ordering property and Lipschitz-H (defined in DFJ, section 3.5.2). † DeMarzo: Stanford, CA 94305-5015; [email protected]. Frankel: 200 Leicester St., Carlton, VIC 3053, Australia; [email protected]. Sadly, our coauthor Yu Jin died in 2018.
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1
Online Appendix to “Portfolio Liquidity and
Security Design with Private InformationӠ
Peter M. DeMarzo (Stanford University)
David M. Frankel (Melbourne Business School and Iowa State University)
Yu Jin (Shanghai University of Finance and Economics)
This Revision: May 5, 2020
1. Introduction
This document is the online appendix to “Portfolio Liquidity and Security Design with
Private Information” by Peter M. DeMarzo, David M. Frankel, and Yu Jin (2020),
henceforth “DFJ”. It is organized as follows. Section 2 formally states the technical results
that are previewed in section 3.5.2 of DFJ. Section 3 proves these results as well as the
results that are stated without proof in section 3 of DFJ. Section 4 relaxes ASSUMPTION A
(monotonicity) in the case of 2 assets and 2 seller types. Section 5 checks the robustness
of our results to the alternative asset sale procedure of Maskin and Tirole (1992).
2. Technical Results
We first formally state the technical results that we discuss in section 3.5.2 of DFJ. We
begin by specifying the continuous and discrete models. The continuous model, denoted
“model ∞”, is as follows. The issuer's type ~t G has full support [0,1] and the final asset
value Y has conditional and unconditional support 0, y and conditional distribution
function H that satisfies the Hazard Rate Ordering property and Lipschitz-H (defined in
We now define a sequence of discrete models i = 1,2,... that converge to model ∞. Let
( )1i i
N
= and ( )
1i iN
= be any two increasing sequences of positive integers. In model i, let
the gaps between adjacent types t and shocks Z be 1i iN = and i iy N = , respectively.
That is, t lies in 0, , ,1 ,1i i iS = − and Z lies in 0, , , ,i i iy yS = − . By
construction, both gaps i and i converge to zero as i goes to infinity. Let the conditional
distribution of Y in model i be the restriction of the continuous distribution function H to
types in iS and shocks in iS . Similarly, the distribution of t in model i is the restriction
of G to types t in iS .79
Let iE and E denote the expectations operators in models i and , respectively. Let
( ) ( ), min , |i iv D t E D Y t= denote the expected payout of simple debt with face value D
in model i given a type it S . Let ( ) ( ), min , |v D t E D Y t = denote the expected
payout of the same security in model given a type [0,1]t .
Fix equilibria in models i and in which the issuer’s security is simple debt. Let i
tD and
tD be the equilibrium face values of these securities for a given type t. The equilibrium
price of this security in model i, denoted pi(t), is simply the security’s expected payout
( ),i i
tv D t . And the issuer’s expected issuance profit, denoted ( )iu t , is simply the gains
from trade ( ) ( )1 ,i i
tv D t− as competition drives investors’ payoffs to zero. Similarly, in
the continuous model the price ( )p t of the security equals the expected payout
),( tv D t and the issuer’s profit ( )u t
equals the expected gains from trade
( ) )1 ,( tv D t − .
79 That is, in model i, the probability that the type does not exceed some t in
iS is G(t), while the probability,
conditional on a type t, that the cash flow Y does not exceed some y in iS is ( )|H y t .
3
In the present notation, equation (13) is written as:80
THE CONTINUOUS INITIAL VALUE PROBLEM (CIVP).
0D y= and, for each t, ( )
( )2
0|1
0.1 1
tD
yt
t
H y t dydD
dt H D t
==
− −
Our first result shows that CIVP and its discrete analog have unique solutions:
PROPOSITION 11. Assume Hazard Rate Ordering and Lipschitz-H.
1. There exists a unique function D that satisfies CIVP for v v= . This
function is decreasing and differentiable in the type t, and takes values
in (0, ]y . The associated price and profit functions, p and u , are
decreasing and continuous in the type t.
2. For each discrete model i=1,2,..., there exists a unique, decreasing
function iD with 0
iD y= and satisfying (12) with iv v= for all it S
and .i =
The proof for CIVP runs roughly as follows. The Picard-Lindelöf theorem is the usual tool
for proving the existence and uniqueness of the solution to a differential equation.
Unfortunately, we cannot apply this theorem directly because the differential equation in
CIVP is not Lipschitz continuous in D : it approaches negative infinity as D
approaches y .
We sidestep this difficulty in the following way. We define upper and lower bounds on
D using a modification of CIVP that is Lipschitz continuous with constant k. We then
show that as k grows, these upper and lower bounds approach the same limit, which
satisfies CIVP and thus must be its unique solution D .
80 Integrating by parts, ( ) ( )0
, min , |y
yv D t y D dH y t
== equals ( )
0|
D
yD H y t dy
=− whence
( ) ( )2 20
, | .D
yv D t H y t dy
== − As ( ) ( )| ,H D t D t= by Lipschitz-H, CIVP is equivalent to (13).
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Having shown the existence of unique face value functions in both models, we now show
that the face value function in the discrete model converges to that of the continuous model
uniformly in the issuer’s type t. As the equilibrium of the discrete model uniquely satisfies
the intuitive criterion, this supports the use of the continuous equilibrium when convenient.
PROPOSITION 12. Assume HRO and Lipschitz-H. For any 0 there is an i∗ such
that for all models i > i∗ and all types t in [0,1], i
t tD D − .81
The idea of the proof is as follows. For any model i = 1,2,... and constant k > 0, we first
show that any solution iD must lie between fixed upper and lower bounds i
kD and i
kD ,
where these bounds are Lipschitz continuous with constant k. Moreover, these bounds
converge to the aforementioned upper and lower bounds on D as i grows. By the prior
intuition, these bounds on D converge in turn to D as k grows. Thus, by taking i and
k to infinity simultaneously, i
kD and i
kD - and thus iD which lies between them – must
converge to the unique solution D of model .
Our next result gives conditions under which the convergence of the discrete model to the
continuous model is uniform in various parameters. This property can be useful in
applications in which the security design game is preceded by some interaction in which
the issuer also chooses an optimal action, as in Frankel and Jin (2015). In such settings,
the result can help establish that the issuer’s optimal choices in the discrete model are well-
approximated by her optimal choice in the continuous model.
Henceforth, we assume the distribution of types G is continuous with a strictly positive
density g that satisfies the following technical condition:
LIPSCHITZ-G (L-G). There are constants ( )3 4, 0,k k such that for all types t and
t′ in [0,1], ( ) 3 g t k and ( ) ( ) 4g t g t k t t −− .
81 Technically, the function Di is defined only for types t in the discrete set Si. We extend it to all types t in
[0,1] by evaluating it at the highest type in Si that does not exceed t.
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We now show that the key functions of the model converge uniformly in the distributions
G and H, the type 0,1 ,t and the cash flow parameter y .82 These key functions are the
face value function iD , price functionip , and the conditional profit function iu . We also
show uniform convergence of the issuer’s unconditional expected issuance profits
( )i i iEu E u t= to its continuous analogue, ( )[ ]Eu E u t = . Finally, let ( )i t denote
the issuer's conditional total profits in model i: the sum of issuance profits ( )iu t and the
conditional expected portfolio return |iE Y t . Let ( )i i iE E t = denote
unconditional expected total profits.83 We show that these converge uniformly to their
continuous counterparts, ( ) ( ) |t u t E Y t = + and ( )=E E t .
PROPOSITION 13. Fix constants k₀, k₁, k₂, k₃, k₄, and y , all in (0,∞). Let H be the
set of conditional distribution functions ( )|H z t that satisfy Hazard Rate Ordering
and Lipschitz-H with constants k0, k1, and k2. Let G be the set of distribution
functions G that satisfy Lipschitz-G with constants k3 and k4. For all ε > 0 there is
an i∗ such that for all models i > i∗, G in G, H in H, y in 0,y , and t in [0,1],
|ωi(t)-ω∞(t)| is less than ε for ω equal to D, p, u, and Π; and |Eωi-Eω∞| is less than ε
for ω equal to u and Π.84
We next show a useful comparative statics property: the issuer chooses a higher face value
when the rate of change function in (13) (which is negative) is smaller in absolute value.
PROPOSITION 10 is essentially a special case of this result (and its proof relies on this result).
PROPOSITION 14. Let the discount factors and lie in ( )0,1 . Assume the
conditional distribution functions H and H satisfy HRO and Lipschitz-H.
suppose that for any given D and t, the rate of change function
82 Uniformity in t does not apply to the expected profit functions Eu and EΠ, as they do not depend on t. 83 This last quantity is especially important in applications: if there is a pregame period, the issuer will act
so as to maximize the sum of EΠi (perhaps multiplied by a discount factor) and any pregame payoff. 84 As in Proposition 14, we extend these functions to all types t in [0,1] by evaluating them at the highest type
in Si that does not exceed t.
6
( )
( )2
0|1
1 1
D
yH y t dy
H D t
=
− −
in (13) is no larger in absolute value when ( ), H equals
( )ˆ ˆ, H than when it equals ( ), H . Let the face value functions ˆtD and tD solve
equation (13) for ( ), H equal to ( )ˆ ˆ, H and ( ), H , respectively. Then for all t,
ˆt tD D : the issuer does not choose a lower face value under ( )ˆ ˆ, H than under
( ), H .
Finally, we show a homogeneity property that can simplify the analysis of models in which
security design is embedded (e.g., Frankel and Jin (2015)).
COROLLARY 15. The face value functions Di and D , the price functions pi and
p, the profit functions iu , i , u , and
, and the issuer’s expected profits ,iEu
iE , Eu, and E , defined above, are all homogeneous of degree one in the
cash flow parameter y .
3. Proofs
We now give the omitted proofs from section 3 of DFJ, as well as the proofs of the results
of section 2 of this online appendix.
PROOF OF PROPOSITION 6. We first define the Intuitive Criterion in the GSD game.
Consider an equilibrium and any interim asset vector I. If type t sticks to her continuation
strategy, she will get ( ) ( ) ( )( )| , , | ,d
t tu t I U I P I I t p= . If instead she chooses some other
ex-post action ( ),P , for her to lose from this deviation it suffices that her equilibrium
payoff ( )|u t I exceeds her maximum payoff ( ) |I
PE W Y t − from the deviation – or,
equivalently, that
( )| ( ) | .I
Pu t I E W Y t + (16)
7
The right hand side of (16) is thus the minimum revenue that type t requires to be willing
to deviate to ( ),P .
The Intuitive Criterion for the GSD game states that on seeing the deviation ( ),P
following I, investors must put zero probability on type t if she is never willing to choose
( ),P following I but some other type might be: if condition (16) holds for t but fails for
some other type s. That is:
THE INTUITIVE CRITERION (GSD GAME). An equilibrium of the GSD game with
posterior belief function and outcome u is intuitive if, on seeing any action
( ), ,I P , investors’ posterior probability ( )| , ,t I P is zero for any type t that
satisfies (16) as long as there is some type s for which the inequality is reversed:
for which ( )| ( ) | .I
Pu s I E W Y s +
Let ( ) ( )( )ˆ, , , ,E I P p = be any equilibrium of the GSD game. We will show that the
following equilibrium ( ) ( )( ), , , ,E I P p = is also an equilibrium, and is intuitive if
E is. Let ( )# V denote the length of a vector V.
1. Securitization. In E , the issuer’s interim asset vector ( )1I I= consists of a single
security whose payout equals her cash flow: ( )1I Y Y= . She then chooses an ex-
post action as follows for any given t and interim asset vector I. First, if she deviated
in the prior stage (so that I I ), she chooses the ex-post action ( ) ( )( ),t tP I I that
she would choose in E after choosing I. Else she issues an ex-post action that
consists of her equilibrium revenue cap ( )ˆt I in E together with an ex-post asset
vector consisting of a single security whose payout equals the aggregate payout
( ) ( )( )( )( )ˆ#
1
ˆ ˆtP I j
tjP I I Y
= of her equilibrium ex-post security vector ( )ˆtP I in E.
8
2. Beliefs. Let ( ), |T P I denote the set of types whose strategies in E instruct
them to choose the ex-post action ( ),P conditional on having chosen some
interim asset vector I. Upon seeing an action ( ), ,I P that is expected in E , the
investors’ posterior probability ( | , , )t I P equals the probability that the issuer’s
type is t conditional on her type being in ( ), |T P I .85 That is, it equals
( ) ( )( )
1
, |s T P Ig t g s
−
if ( ), |t T P I and zero otherwise. Upon seeing an
action ( ), ,I P that is unexpected in E , ( | , , )t I P equals ( | , , )t I P : its value
in E.
3. Pricing. Prices are given by equation (8) with substituted for : for any action
( ), ,I P , the resulting price vector is ( ) ( )( ), , [ | ] ( | , , ).t
p I P E P I Y t t I P =
Pricing and Beliefs ensure that E satisfies Competitive Pricing and Rational Updating.
As for Payoff Maximization, the payoff of an issuer of type t from taking an action ( ), ,I P
that is unexpected in E equals her payoff from taking this action in E, since, by Pricing
and Beliefs, investors respond with the same price vector. As for expected actions in E ,
we rely on the following claim, whose proof appears below.
CLAIM. Fix an action ( ), ,I P that is expected in E . Let denote the set of
equilibrium actions ( )ˆ, ,I P that are taken in E by any type whose prescribed
action in E is ( ), , .I P 86 Then an issuer of any given type gets the same payoff
in E from taking the action ( ), ,I P as she gets in E from taking any action
( )ˆ, ,I P in .
85 An action ( ), ,I P is expected in an equilibrium if it is selected with positive probability in that
equilibrium.
86 By construction of E , a given type t chooses the same equilibrium revenue cap ( )ˆt I in E as in E.
Thus, every equilibrium action in must involve the same revenue cap that is prescribed in E .
9
By this Claim, no action in E affords an issuer of a given type more than her equilibrium
payoff in E; and taking her prescribed action in E gives her the same payoff that she gets,
in equilibrium, in E. Thus she will take her prescribed action in E : Payoff Maximization
holds, whence E is an equilibrium as claimed.
PROOF OF CLAIM. Let ( )ˆ, ,I P be any action in . Note first that ( )ˆ, ,I P has
the same aggregate payout function ( ) ( )( )1 1W Y P I Y= as ( ), ,I P and thus the
same aggregate expected payout conditional on the issuer’s type, ( ) | .E W Y t
Hence the Claim holds if ( )ˆ, ,I P raises the same revenue in E as ( ), ,I P raises
in E . We first show that all actions in raise the same revenue in E:
LEMMA. In E, following the equilibrium interim asset choice I , if there are two
ex-post actions that are expected given I and that have the same revenue cap and
aggregate payout function, they must raise the same revenue.
PROOF OF LEMMA. Suppose not: there are types t and t (possibly equal) who,
after choosing I and seeing their types, choose ex-post actions ( ),P and
( ), ,P respectively, that have the same revenue cap and aggregate payout
function, such that ( ),P yields less revenue than ( ),P :
( )( )
( )( )# #
1 1, , , ,
P Pj j
j jp I P p I P
= = . Since the aggregate payout
functions are identical, their conditional expectations are the same conditional on
the issuer’s type being t: ( ) ( )ˆ ˆ
| |I I
P PE W Y t E W Y t = . But then by (7), type t’s
payoff ( )ˆ, , | ,U I P t p from ( ),P exceeds her payoff ( )ˆ, , | ,U I P t p from
( ),P which she is therefore unwilling to choose - a contradiction.
By this Lemma, each action in raises the same issuance revenue R in E. And
if, in E, the issuer chooses action ( )ˆ, ,I P in then, by (6) and (8), investors’
10
willingness to pay ( ) ( )( )#
1
ˆ ˆ, , , ,d
P j
E jWTP I P p I P
= = for her ex-post assets equals
their estimate ( ) ˆ| ( | , , )tE W Y t t I P of the aggregate payout given what the
action ( )ˆ, ,I P reveals about the issuer’s type. The issuance revenue R in E is
then the minimum of ( )ˆ, ,EWTP I P and the cap . Since, by the above Lemma,
this issuance revenue is the same for any action ( )ˆ, ,I P in , it follows that
either (i) ( )ˆ, ,EWTP I P for every such action or (ii) ( )ˆ, ,EWTP I P for
every such action. Moreover, in case (ii), ( )ˆ, ,EWTP I P must take a common
value ( )EWTP for any action ( )ˆ, ,I P in .
Likewise, if the issuer chooses action ( ), ,I P in E then the amount ( )1 , ,p I P
that investors are willing to pay for her ex-post security is just their estimate
( ) | ( | , , )tE W Y t t I P of this security’s payout conditional on the action
( ), ,I P ; the resulting issuance revenue is the minimum of ( )1 , ,p I P and the cap
.
But for each t, by Beliefs, ( | , , )t I P can be written as ( ) ( )
( ) ( )
, |
, |
1
1
t T P I
s T P Is
g t
g s
. And
the sets ( )ˆ, |T P I of types who, in E, take actions ( )ˆ, ,I P , is a partition of
the set ( ), |T P I of types who take the action ( ), ,I P in E . Thus we can
rewrite ( | , , )t I P as a weighted sum ( )( )
( )ˆ, ,ˆ, ,
ˆ| , ,I P
I P
t I P w
where the
weight ( )
( ) ( )
( ) ( )
ˆ, |
ˆ, ,
, |
1
1
s s T P I
I P
s T P Is
g sw
g s
=
is the probability that the issuer would have
chosen ( )ˆ, ,I P in E conditional on her having chosen ( ), ,I P in E . Hence we
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can rewrite ( )1 , ,p I P as ( ) ( ) ( )( )
ˆ, ,ˆ, ,
ˆ| | , ,tI P
I P
w E W Y t t I P
or,
equivalently, as ( ) ( )( )
ˆ, ,ˆ, ,
ˆ, ,EI PI P
w R I P
. Since, the weights ( )ˆ, ,I P
w
sum to one,
the revenue ( ), ,E
R I P raised by the action ( ), ,I P in E equals in case
(i) above (where the revenue raised in E by each ( )ˆ, ,I P in is also ) and the
common value ( )EWTP in case (ii) (when the revenue raised in E by each
( )ˆ, ,I P in is also ( )EWTP ). This proves the Claim.
It remains to show that if E is intuitive, so is E . Suppose E is intuitive. Let ( )|u I and
( )|u I denote the equilibrium payoffs in E and E , respectively, of an issuer of type t who
chooses the interim asset vector I and then follows her continuation strategy in the given
equilibrium. Consider any action ( ), ,I P for which there are types t and s satisfying:
( )| ( ) |I
Pu t I E W Y t + and ( )| ( ) | .I
Pu s I E W Y s + (17)
First assume ( ), ,I P is unexpected in E . As previously noted, the payoff of an issuer
of type t from taking the action ( ), ,I P in E equals her payoff from taking this action in
E. It follows that ( ) ( )| |u v I u v I= for ,v s t= . Hence ( )| ( ) |I
Pu t I E W Y t + and
( )| ( ) |I
Pu s I E W Y s + whence, since E is intuitive, ( )| , ,t I P is zero. But since
( ), ,I P is unexpected in E , Beliefs implies that ( )| , ,t I P equals ( )| , ,t I P
which thus also is zero as required.
Now suppose instead that ( ), ,I P is expected in E : ( ), |T P I is nonempty. By (17),
( )|u t I exceeds ( ) |I
PE W Y t − which is the highest payoff type t can expect from
( ), ,I P . Hence t is not in ( ), |T P I whence, by Beliefs, ( )| , ,t I P is zero. We
conclude that E is intuitive as claimed.
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PROOF OF PROPOSITION 7. We first define the notions of equilibrium, intuitive beliefs,
and fair pricing for an EPSD game. Each definition is the natural restriction of the
analogous definition for a GSD game.
EPSD EQUILIBRIUM. A perfect Bayesian equilibrium of the EPSD game is a
security design tS M and price cap t + for each type t, as well as a price
function ˆ ( , )p S and belief function ˆ ( | , )t S , with the following properties:
1. Payoff Maximization: for all t, the issuer’s choice ( , )t tS solves
,ˆmax ( , ) ( ) |S p S E S Y t − subject to S .
2. Competitive Pricing: for any monotone security S and price cap , the
price function ˆ ( , )p S equals ˆmin , [ ( ) | ] ( | , )tE S Y t t S .
3. Rational Updating: the investors’ belief function ˆ ( | , )t S follows Bayes’s
caps do not bind, each equilibrium price p∗i (q∗ (t) , p∗ (t)) equals the conditional
expected payout fi (t) of asset i: the profile e∗ is fairly priced. It follows from
(100) and (101) that the type-t seller’s revenue and payoff functions are ρ∗ (t) and
the payoff u∗ (t) given by RLP. If the seller chooses a pair (q, p) that never occurs
in equilibrium, investors’ beliefs µ∗ (·|q, p) put positive weight only on types t that
have the lowest opportunity cost of giving up their equilibrium payoff u∗: those in
the set T (q) = argmint ′ [u∗ (t ′)+δq f (t ′)]. Hence, beliefs are intuitive. In particular,
if the set T (q) is a singleton t, then investors believe the seller is of type t. If T (q)
contains both types t = 1,2, investors believe the seller is of that type t for which
the deviation revenue q [p∧ f (t)] under symmetric information is minimized. If this
deviation revenue is the same for the two types, then investors’ beliefs do not matter.
60
We can assume, e.g., that they believe that t = 1, although any other beliefs would
give the same results.
The above specification and argument implies that, in e∗, a type-t seller sells the quantities
q∗ (t) and gets the revenue ρ∗ (t) and payoff u∗ (t) that solve RLP. It remains only to show
that e∗ is the unique intuitive PBE. More precisely, (a) it is an intuitive PBE, and (b) any
other intuitive PBE has identical equilibrium behavior and payoffs to e∗.10
Proposition 1. The profile e∗ is an intuitive PBE.
Proposition 2. If e=(q(·) , p(·) ,µ (·|·, ·) , p(·, ·)) is an intuitive PBE of A2NM, with outcome
u(·) and price function p(·), then the seller makes the same quantity choices and receives
the same payoffs in e as in e∗: for each type t, q(t) = q∗ (t) and u(t) = u∗ (t). Moreover,
if the quantity qi (t) = q∗i (t) of an asset i sold by a type t is positive, the resulting price
of asset i is the same in the two equilibria and equals the fair value of the asset: pi (t) =
p∗i (t) = fi (t).
4.2 Relaxing Monotonicity: Proofs
PROOF OF CLAIM 10. Part 1. First, a f (2) > a f (1) by (93) whence a /∈ Q2 by (98).
Hence either q∗1 (2) or q∗2 (2) is less than one; assume w.l.o.g. that q∗2 (2)< 1. Thus, if (103)
does not hold then for ε in (0,1−q∗2 (2)], the vector q′ = q∗ (2)+ (0,ε) is also in Q2. By
(98), q′ offers type 2 a higher payoff under symmetric information than q∗ (2) does. Hence
q∗ (2) is not optimal for type 2 in Q2, a contradiction.
Part 2. Suppose not: q∗1 (2) < 1 and q∗2 (2) > 0. We will derive a contradiction. Let
ε > 0 be small enough that q∗1 (2)+ ε < 1 and q∗2 (2)− ι > 0 where ι = ε∆1/∆2, whose
denominator ∆2 is positive as noted above. Consider the alternative quantity vector q =
10The other PBE may differ in ways that are not payoff relevant. In particular, it may have different
nonbinding price caps and different beliefs following deviations (as long as these beliefs are intuitive and
serve to deter the deviations). Moreover, if a seller does not sell any shares of a given asset, then the price of
this asset is indeterminate and hence may vary across equilibria.
61
(q1,q2) where q1 = q∗1 (2)+ ε and q2 equals the lesser of q∗2 (2)− ι and one. First, u∗ (1)−
q∆ can be written as the sum of u∗ (1)− q∗ (2)∆, which is nonnegative since q∗ (2) is
in Q2, and [q∗ (2)−q]∆, which by (95) is bounded below by −ε∆1 + ι∆2 which equals
zero by definition of ι . Hence u∗ (1) ≥ q∆, whence q is in Q2. Moreover, the effect on
type 2’s payoff from switching from q∗ (2) to q (under symmetric information) equals the
difference 1−δ in discount factors times the change in revenue q f (2)−q∗ (2) f (2) which,
in turn, equals minε f1 (2)− ι f2 (2) ,ε f1 (2)+1−q∗2 (2). But both elements in the min
are positive: the first by (94)11 and the second by (92) and since ε > 0 and q∗2 (2)≤ 1. We
conclude that q, which is in Q2, is better for 2 than q∗ (2) under symmetric information,
which contradicts the definition of q∗ (2).
Part 3. This holds by part 2 and since, as shown in the proof of part 1, q∗ (2) 6= a.
Part 4. Together with (92), part 3 implies δq∗ (1) f (1) = δa f (1)> δq∗ (2) f (1) which,
substituted into (103), implies q∗ (1) f (1)> q∗ (2) f (2) as claimed. Multiplying each side
by 1− δ , we obtain u∗ (1) > u∗ (2). Clearly, u∗ (2) ≥ 0 since a type 2 seller can always
set q = (0,0). Finally, let ε = u∗(1)a∆
. The denominator is positive by (93). The numerator
is positive by (101) and (102) and less than the denominator by these equations together
with (93). Hence, ε ∈ (0,1) whence the quantity vector (ε,ε) is feasible. Moreover, this
vector is in Q2 since (ε,ε)∆ = εa∆ = u∗ (1) and by (98). It follows that u∗ (2) is not less
than (1−δ )(ε,ε) f (2), which is positive by (92). Hence u∗ (2) and u∗(2)1−δ
= q∗ (2) f (2) are
positive as claimed. Q.E.D.Claim 10
PROOF OF CLAIM 11. Let qi = q∗i (2) for assets i = 1,2. By part 1 of Claim 10, type 1’s
IC constraint binds so
u∗ (1) = q1∆1 +q2∆2. (107)
11By (94), δ f1 (2) f2 (1)< δ f2 (2) f1 (1), whence
f1 (2)∆2 > f2 (2)∆1
which can be rewritten as f1 (2)∆δ2 > f2 (2)∆δ
1 . This, in turn, implies ε f1 (2) > ε f2 (2)∆δ1/∆δ
2 = ι f2 (2) as
claimed.
62
We know from part 3 of Claim 10 that q2 < 1. There are now three cases.
1. Say q1 < 1. Then q2 = 0 by part 3 of Claim 10. Thus, by (107), u∗ (1) = q1∆1 whence
q1 6= 0 and thus q1 > 0 which implies ∆1 > 0. For q1 < 1 we require u∗ (1) < ∆1
(which implies ∆1 > 0).
2. Say q2 > 0. Then q1 = 1 by part 3 of Claim 10. Thus, by (107), q2∆2 = u∗ (1)−∆1
so, since ∆2 > 0 by (95), u∗ (1)> ∆1.
3. Say that neither #1 nor #2 holds: q1 = 1 and q2 = 0. Then by (107), u∗ (1) = ∆1.
As the above three conditions u∗ (1) T ∆1 are mutually exclusive and exhaustive, we can
replace ”if” in cases 1-3 by ”if and only if”. This implies the following three mutually
exclusive possibilities, which corresponds respectively to cases 1-3 above.
1. If ∆1 > u∗ (1) then q∗1 (2) = u∗ (1)/∆1 ∈ (0,1) and q∗2 (2) = 0.
2. If ∆1 < u∗ (1) then q∗1 (2) = 1 and q∗2 (2) =u∗(1)−∆1
∆2∈ (0,1).
3. If ∆1 = u∗ (1) then q∗1 (2) = 1 and q∗2 (2) = 0.
This result is stated more succinctly in equations (104) and (105). Q.E.D.Claim 11
PROOF OF PROPOSITION 1. Beliefs are intuitive by construction and we verify Competitive
Pricing and Rational Updating in the text. To show that e∗ is an intuitive PBE, it thus
remains only to verify Payoff Maximization: that neither type has a (strictly) profitable
deviation. By construction, investors’ beliefs µ∗ following a deviation (q, p) depend on q
but not on p. Hence, if a deviator chooses binding price caps, this can only lower his payoff
from deviating. We can thus assume, w.l.o.g., that a deviator chooses price caps that can
never bind (such as the caps p∗).
We begin with type 1. Let
Γ(q) = q∆−u∗ (1) (108)
63
denote the change in type 1’s payoff from deviating to q and being mistaken for type 2. Let
Λ(q) = [u∗ (2)+δq f (2)]− [u∗ (1)+δq f (1)] (109)
denote the difference between type 2’s and type 1’s opportunity cost of deviating to q. By
construction of µ∗, investors must believe the seller is of type 1 (resp., 2) when Λ(q) is
positive (negative). And when Λ(q) = 0, they think he is of type 1: beliefs are governed by
deviation revenue q f (t) which by (109) and part 3 of Claim 10, is lower for type 1 in this
case. By (97), (101), and (103),
Γ(q∗ (2)) = Λ(q∗ (2)) = 0. (110)
Now suppose type 1 deviates to (q, p). If Λ(q)≥ 0, investors will think the seller’s type
is 1: he gets (1−δ )q f (1) which cannot exceed his equilibrium payoff u∗ (1) as q≤ a. So
such a deviation is not profitable. If instead Λ(q) < 0, investors will think his type is 2
whence his payoff changes by Γ(q), which is nonpositive by the following result.
Claim 12. For any q in [0,1]2, if Λ(q)< 0 then Γ(q)≤ 0.
PROOF OF CLAIM 12. Suppose not: Λ(q)< 0 and Γ(q)> 0. We will derive a contradiction.
For brevity let q denote q∗ (2). Recall that ∆xi is defined in (96). By (92) and (95),
∆x2 > 0 (111)
for all such x, whence
ddx
∆x1
∆x2
∝ f1 (2) f2 (1)− f2 (2) f1 (1)< 0 (112)
by (94). We will rely on the following lemma.
Lemma 5. Let q′ ∈ [0,1]2 be such that Γ(q′)≥ 0 and Λ(q′)< 0. Then q′1 > q1.
Proof. By (110), Γ(q) = 0≤ Γ(q′) whence by (108),
(q1−q′1
) ∆δ1
∆δ2+(q2−q′2
)≤ 0. (113)
64
Also by (110), Λ(q) = 0 > Λ(q′) whence, by (109),(q1−q′1
) ∆11
∆12+(q2−q′2
)> 0. (114)
Subtracting (113) from (114) yields (q1−q′1)(
∆11
∆12− ∆δ
1∆δ
2
)> 0, which by (112) yields q′1 >
q1.
Since Γ(q)> 0, q∆δ > u∗ (1)> 0 by (108) and part 4 of Claim 10. Let α = u∗ (1)/[q∆δ
]∈
(0,1) and q = αq. Since Λ(q)< 0 by assumption, u∗ (1)−u∗ (2) (which is positive by part
4 of Claim 10) exceeds δq∆1 and thus also exceeds δ q∆1 since q = αq and α ∈ (0,1).
Rearranging, we obtain Λ(q)< 0. Since, moreover, Γ(q) = 0 by (108) and (96), Lemma 5
implies that q1 > q1. Hence, by (112),
(q1− q1)
(∆0
1
∆02−
∆δ1
∆δ2
)> 0. (115)
We can also rearrange Γ(q) = Γ(q) = 0 to obtain
(q1− q1)∆δ
1
∆δ2+ q2− q2 = 0. (116)
Adding (115) and (116) yields (q1− q1)∆0
1∆0
2+ q2− q2 > 0. This can be rearranged (with the
result multiplied by 1− δ ) to yield (1−δ ) q∆0 > (1−δ ) q∆0. Moreover, q is in Q2 since
Γ(q) = 0. This contradicts the definition of q as the vector in Q2 that maximizes type 2’s
symmetric-information payoff. Q.E.D.Claim 12
We have shown that a type 1 seller will not deviate. Let us now consider a deviation
(q, p) by a type 2 seller. There are two cases.
1. Λ(q) ≥ 0. Then on seeing q, investors believe the seller’s type is 1. Hence, the
deviation is profitable for the type 2 seller only if
0 < Ω(q) = q [ f (1)−δ f (2)]−u∗ (2) .
If this holds then, by (109), 0 < Λ(q)+Ω(q) = (1−δ )q f (1)−u∗ (1) which is not
possible by (92) and since q≤ a: the deviation is not profitable.
65
2. Λ(q)< 0. Then on seeing q, investors believe the seller’s type is 2. Hence, type 2’s
change in payoffs from this deviation equals Γ(q), which is nonpositive by Claim
12: the deviation is not profitable.
As this verifies Payoff Maximization, e∗ is an intuitive PBE. Q.E.D.Proposition 1
PROOF OF PROPOSITION 2. We first show that in any intuitive PBE, a seller of a given
type is paid the expected value (given her type) of the portfolio that she sells. The precise
property is as follows.
Fair Pricing. A PBE e with quantity function q(·) and price function p(·) is Fairly Priced
if, for each type t, p(t) f (t) = q(t) f (t).
Lemma 6. Any intuitive PBE e is Fairly Priced.
Proof. Let e be intuitive, and suppose type t chooses (q, p) in equilibrium and that investors
respond with price vector p. First suppose the issuance is underpriced: pq < q f (t). Let S
denote the set of types s for which q f (s)< q f (t). There are two cases.
1. S is empty. Then by Competitive Pricing, if the type t seller deviates to (q, f (t)), her
revenue cannot be less than q f (t). As this would be a profitable deviation, this case
is not possible.
2. S is nonempty. Let s′ ∈ S be a type s for which q f (s) is at a maximum among all s
in S: there is no type s in S for which q f (s) exceeds q f (s′). Then q f (s′) < q f (t)
whence, since pq < q f (t), there is a λ ∈ [0,1) such that
q [p−δ f (t)]q [ f (t)−δ f (t)]
< λ <q [p−δ f (s)]
q [ f (t)−δ f (s)]. (117)
Let type t deviate to (λq, f (t)). By (117), λq [ f (t)−δ f (t)] exceeds q [p−δ f (t)],
which equals u(t). And for each s in S, λq [ f (t)−δ f (s)] is less than q [p−δ f (s)]
which, in turn, is not greater than u(s). Accordingly, for each s in S,
u(t)+δ (λq) f (t)< λq f (t)< u(s)+δ (λq) f (s)
66
and hence, since s is intuitive, µ (s|(λq, f (t))) is zero for each s in S. Hence, by
Competitive Pricing, the revenue that results from the deviation (λq, f (t)) is λq f (t)
whence type t’s payoff from the deviation is λq [ f (t)−δ f (t)] which, as noted,
exceeds u(t): the deviation is profitable, which is not possible.
We have shown that in an intuitive PBE e, underpricing cannot occur (except perhaps off the
equilibrium path). Hence, pq≥ q f (t) for any type t who chooses (q, p) in equilibrium. But
by Competitive Pricing, pq = q(
p∧[∑
2t=1 f (t)µ (t|q, p)
])≤ ∑
2t=1 q f (t)µ (t|q, p) . Thus,
pq is bounded above by a convex combination of q f (t) for those types t who choose (q, p)
in equilibrium. But by the prior result, pq is also bounded below by q f (t) for each such
type t, so it cannot be less than the highest such q f (t). Hence all types who choose (q, p)
in equilibrium have the same value of q f (t), and this also equals the revenue pq that they
receive: the equilibrium is Fairly Priced.
Moreover, if a PBE is Fairly Priced, then each asset that is sold in equilibrium must be
priced correctly.
Claim 13. Suppose a PBE e with quantity function q(·) and price function p(·) is Fairly
Priced. Then for each type t, and each asset i for which qi (t)> 0, we have pi (t) = fi (t).
Proof. Suppose not. Then by Competitive Pricing, there is a type t who sells some shares
of an asset i with a binding price cap: qi (t) > 0 and pi (t) < fi (t). Thus, this type’s
revenue pi (t)qi (t) from asset i is less than the value qi (t) fi (t) of shares sold. And by
Competitive Pricing, her revenue p j (t)q j (t) from any other asset j cannot exceed its
true value q j (t) f j (t). Combining these facts, we find that her total issuance revenue
p(t)q(t) must be less than the total value q(t) f (t) of shares she sells, which violates
Fair Pricing.
Now let e be an intuitive PBE with outcome u(·) and price function p(·). By Lemma
6, e is Fairly Priced, whence
u(t) = (1−δ )q(t) f (t) for each type t. (118)
67
Hence, u(1) ≤ (1−δ )a f (1) = u∗ (1). And as e is a PBE, type 1 cannot prefer to imitate
type 2. So since e is Fairly Priced, type 2’s choice q(2) must satisfy q(2)∆ ≤ u(1).
As u(1) ≤ u∗ (1), this implies that q(2) lies in Q2. But u∗ (2) is type 2’s highest payoff
(1−δ )q f (2) among all vectors q in Q2, whence u(2)≤ u∗ (2) as well.
Now suppose that in e, type 1 deviates to (q, p) where q = a and p = f (1)∨ f (2).12 By
Competitive Pricing and part 4 of Claim 10, his resulting revenue is a[∑
2t=1 f (t)µ (t|q, p)
]≥
a f (1) whence his payoff from the deviation is at least u∗ (1). Thus, for 1 not to deviate,
u(1) must equal u∗ (1). By (92) and (118), this implies q(1) = a = q∗ (1) as claimed.
Now consider type 2. If u(2) < u∗ (2), suppose type 2 deviates to (q, p) where q =
λq∗ (2) for some λ ∈(
u(2)u∗(2) ,1
)and p = f (1)∨ f (2). By the prior result, u(1) = u∗ (1) =
q∗ (2)∆> λq∗ (2)∆ and u(2)< λu∗ (2)= (1−δ )λq∗ (2) f (2). Thus, u(1)+δλq∗ (2) f (1)>
λq∗ (2) f (2)> u(2)+δλq∗ (2) f (2) whence, since e is intuitive, µ (2|q, p) = 1 and hence,
by Competitive Pricing, type 2’s payoff from the deviation is λu∗ (2) > u(2). Thus, for
type 2 not to deviate, her payoff u(2) must equal u∗ (2). And since u(1) = u∗ (1) and by
Fair Pricing, q(2) must lie in Q2 (else type 1 will deviate to q(2)). So q = q(2) maximizes
type 2’s symmetric-information payoff (1−δ )q f (2) in Q2 whence, by part 6 of Claim 10,
q(2) must equal q∗ (2) as claimed.
Finally, as q(t) equals q∗ (t) for each type t, Claim 13 implies that the prices pi (t) and
p∗i (t) each equal fi (t) for each asset i for which qi (t) = q∗i (t) is positive. Q.E.D.Proposition 2
5 The Maskin and Tirole (MT) Approach
5.1 Preview
This section will test to what extent the properties our Asset Sale (AS) game carry over
to the more flexible sales procedure of Maskin and Tirole (1992), henceforth ”MT”. We
find that when MT’s approach permits outcomes other than the RLP outcome, these other
12Recall that ∨ denotes the componentwise maximum.
68
outcomes are infinite in number, nonintuitive, and somewhat more likely to involve separation
(as in RLP) than pooling.
We begin, in section 5.2, with the general AS game with T types and n assets. Under
MT’s procedure, the RLP solution (which we will also refer to as the “RSW allocation” for
reasons described below) is the unique intuitive equilibrium. However, other nonintuitive
equilibria cannot be ruled out.
In order to determine when and what sort of other equilibria can exist, we turn to the
case of two types and two assets in section 5.3. The RLP outcome is the unique equilibrium
if the high type seller is not too likely. Otherwise, there also exists a continuum of equilibria
other than RLP. However, many of these nontintuitive equilibria also involve separation
and thus have the same general form as the RLP outcome. Indeed, the set of parameters
for which pooling can occur is a proper subset of the set for which there are non-RLP
separating equilibria. Thus, MT’s approach (without adding the intuitive criterion) does not
yield a unique alternative to RLP, and it actually seems to favor separation over pooling.
In section 5.4 we then apply these results to the Ex Post Security Design Game (DFJ
section 3.3-3.4). While there can be multiple equilibria using MT’s approach, they all
possess the qualitative properties of the RLP outcome identified in DFJ section 3.4: under
the Hazard Rate Ordering property, the issuer sells standard debt with a face value that is
nonincreasing in her type. However, in contrast to the RLP outcome, if the high type is
sufficiently likely then there are outcomes in which the low type issuer is paid more than
the fair value of her security. This loosens her IC constraint, which lets the high type set a
higher face value of his security relative to the RLP outcome. Indeed, if his prior probability
is high enough, this face value can equal that of the low type: the two types can sell the
same standard debt security. Again, none of these non-RLP outcomes are intuitive.
Technically, MT’s key results rely on assumptions do not hold in our AS game. Hence
we must supply proofs that do not use these assumptions. More precisely:
1. MT’s Theorem 1 (which appears below as Proposition 3) assumes the RSW allocation
is interim efficient with respect to some positive beliefs. This assumption lets MT
69
prove Theorem 1 using known properties of sequential equilibria (MT [12, p. 20]).
As we cannot verify the assumption in our setting, we supply a proof that does not
rely on it. We do so by replacing sequential equilibrium with the more general
solution concept of correlated equilibrium (Aumann [2]) - a change that, in our
setting, does not alter the set of equilibrium outcomes.
2. MT’s Theorem 1 relies on their Proposition 5, which assumes two properties that fail
in our setting:
(a) MT [12, p. 5] write ”We will assume that a type i indifference curve is nowhere
tangent to a type j indifference curve (i 6= j).” Our ASSUMPTION A, in contrast,
permits two adjacent types to have the same information about one or more
assets whence their preferences over these assets coincide.
(b) Later MT [12, p. 11] write ”We will also assume that µ ·(µ ·0)
is not on the
boundary of the feasible set.” In our setting, this means that the quantity qi (t)
sold of each asset, by each type t, lies in the interior (0,ai) of the feasible set.
This condition typically fails in our setting as payoffs are linear. For instance,
under IIS, each quantity sold qi (t) is either 0 or ai with the exception of the
hurdle class quantity qc (t).
Hence we prove MT’s Proposition 5 (which appears below as Claim 21) without the
above two assumptions, relying instead on our ASSUMPTION A (monotonicity).
This section is organized as follows. Section 5.2 studies the general AS game with T types
and n assets. Section 5.3 focuses on the 2x2 case. Section 5.4 applies the results of section
5.3 to the Ex Post Security Design game (DFJ, section 3.3). Omitted proofs appear in
section 5.5.
70
5.2 The General Case
We first summarize the approach of MT and how it will be applied to the setting of our Asset
Sale (AS) model. We generalize MT by permitting any correlated equilibrium (Aumann
[2]) in the last stage of the game, rather than restricting to Nash equilibrium with a public
coordinating device. This permits us to prove a version of MT’s Theorem 1 that does
not require that the RSW allocation is interim efficient with respect to positive beliefs - a
condition that we may not hold in our setting. While in principal the switch to correlated
equilibrium might expand the set of equilibrium outcomes, in practice it does not.13
As applied to our Asset Sale game, MT’s procedure is as follows. The players consist
of a principal and an agent. The principal corresponds to our seller. The agent is best
interpreted as a single risk-neutral investor with deep pockets, who replaces our continuum
of investors. The payoffs of principal and agent depend on three variables q, ρ , and t,
defined as follows.
1. q is a vector of observable and verifiable actions, belonging to a compact, convex set
Q ⊂ℜn. In our AS model, q is the vector (q1, ...,qn) of quantities that the principal
sells to the agent, and
Q = [0,a1]×·· ·× [0,an] (119)
is the set of all feasible quantity vectors.
2. ρ ∈ ℜ is a monetary transfer from the agent to the principal. In the AS model it is
the revenue ρ = pq that the investors pay the seller. To ensure compactness, we will
restrict ρ to lie in [−ρ,ρ] for some arbitrarily large (but finite) constant ρ .
3. t ∈ ϒ = 0, ...,T denotes the principal’s private information or type, with prior
13In a given game, there can be correlated equilibria that are not Nash equilibria - even if the latter include
a public coordinating device (Aumann [2, pp. 70-71]). However, in MT, the principal has enough flexibility
in designing the game that the Nash restriction does not shrink the set of outcomes relative to correlated
equilibrium.
71
probability distribution Π· = ΠtTt=0 ∈ ∆T (the standard T simplex). This is just
as in the AS model.
MT define an outcome o as an action-transfer pair (q,ρ):14
Definition 1. An outcome o is a pair (q,ρ) ∈ O where
O = Q×ℜ⊂ℜn+1 (120)
is the set of all possible outcomes. Let qo and ρo denote the action and transfer, respectively,
of the outcome o: that is, o = (qo,ρo).
The principal and agent have Neumann-Morgenstern (vNM) utility functions V t (o) and
U t (o), respectively, over outcomes o. As the notation indicates, both payoff functions also
depend on the principal’s type t. In our AS model, these payoff functions are15
V t (o) = ρo−δqo f (t) and U t (o) = qo f (t)−ρ
o. (121)
Let
o0 =(−→
0 ,0)
(122)
denote the reservation outcome in which no assets or cash are transferred and thus, by
(121), each player gets a realized payoff of zero. Further define:
Definition 2. An allocation is a menu o· = otTt=0 ∈ A of outcomes, one for each type of
principal, where
A = OT+1 ⊂ℜ(n+1)(T+1) (123)
is the set of all possible allocations.
14In MT, an outcome is denoted µ . We use o since, in our setting, µ is used to denote beliefs. Moreover,
MT work with random outcomes, which are probability distributions over pairs (q,ρ). However, as our
players are risk-neutral, they care only about the expected action Eq and transfer Eρ . Hence there is no gain
in generality from working with random outcomes: any result concerning an outcome (q,ρ) will hold also
for any random outcome whose expectation is (q,ρ).
15In MT, the variables q, ρ, and t are called y, t, and i, respectively.
72
Definition 3. An allocation o· is incentive-compatible if, for all types s and t, V t (ot) ≥
V t (os).
Definition 4. An allocation o· Pareto-dominates an allocation o· if V t (ot) ≥ V t (ot) for
each type t, where the inequality holds strictly for some t.
Definition 5. Let S be a set of allocations. An allocation o· is Pareto optimal in S if o· ∈ S
and o· is not Pareto-dominated by any other allocation in S.
On seeing his type, the principal proposes a mechanism m: a set of actions for each
player and a map from action pairs to outcomes. The agent may reject the mechanism, in
which case the no-trade outcome o0 is played. Without loss of generality, we incorporate
this option as a special action in the mechanism that always leads to the no-trade outcome:
Definition 6. A mechanism m consists of a finite set SPm of actions for the principal, a finite
set of actions SAm for the agent, and, for each pair of actions
(sP,sA) ∈ SP
m×SAm, an outcome
osP,sA
m ∈ O, such that the agent’s action set SAm contains a special action sA
0 that prevents
trade: if the agent selects sA0 then, for any principal’s action sP ∈ SP
m, the outcome osP,sA0
m
selected by m is the no-trade outcome o0.
Let M denote the set of mechanisms m. Eliminating an explicit “accept/reject” stage
lets us specify MT’s game simply as follows.
Stage 1. The principal privately observes his type t and announces a mechanism m in the
set M.
Stage 2. The principal and agent simultaneously choose actions sP ∈ SPm, sA ∈ SA
m. Their
realized payoffs are V t(
osP,sA
m
)and U t
(osP,sA
m
), respectively.
Following the principal’s choice of a mechanism m, the agent forms interim beliefs Π· (m)
about the principal’s type, where Πt (m) is the probability that she assigns to type t.
Our solution concept for stage 2 is an adaptation of strategic form correlated equilibrium
(SFCE) (Forges [6]) to our setting. Why must SFCE be altered? An SFCE is defined for
73
a static, one-shot game in which a player’s beliefs about her opponents’ types are given by
an exogenous prior distribution. Hence, if player 1 believes player 2 has probability zero of
being some type t, then type t indeed cannot occur so the SFCE may assign a suboptimal
action to type t. This feature is not suitable in our setting, where the agent’s beliefs are
given by her interim beliefs Π· (m). In particular, if the mechanism m is unexpected, the
agent’s interim beliefs are arbitrary and so may assign zero probability to a type t who has
positive ex-ante probability - even if it was type t who chose m! Hence, a SFCE would let a
type t deviate to some mechanism m and then play suboptimally in the continuation game,
thus violating the principle of sequential rationality.
We fix this by requiring optimality type-by-type for the principal, regardless of the
agent’s beliefs. More precisely, let
Φm = ∆
∣∣∣(SPm)
T+1×SAm
∣∣∣ (124)
be the set of distributions of pure action profiles (where the principal’s action sP· ∈(SP
m)T+1
specifies the action sPt ∈ SP
m to be taken by each type t).
Definition 7 (CEIGΠ·m ). A distribution πm∈Φm is a correlated equilibrium of the incomplete-
information game with mechanism m and interim beliefs Π·, or “CEIGΠ·m ”, if and only if
1. each type of principal is willing to play each action when told to do so:
∑sA∈SA
m
πm
(sP· ,s
A)
V t(
osPt ,s
A
m
)≥ ∑
sA∈SAm
πm
(sP· ,s
A)
V t(
osPt ,s
A
m
)for any t,sP
· , sP· ∈(SP
m)T+1
;
(125)
2. and the agent is willing to play each action when told to do so:
∑sP· ∈(SP
m)T+1
πm
(sP· ,s
A)
uAm,Π·
(sA,sP
·
)≥ ∑
sP· ∈(SP
m)T+1
πm
(sP· ,s
A)
uAm,Π·
(sA,sP
·
)for any sA, sA ∈ SA
m
(126)
where
uAm,Π·
(sP· ,s
A)=
T
∑t=0
ΠtU t(
osPt ,s
A
m
)(127)
74
is the agent’s expected payoff from the pure strategy profile(sP· ,s
A) under the beliefs
Πt .
Definition 8. A profile Σ of the full game is a triplet(
p··,Π· (·) ,π·
)where pt
m is the
probability that the principal will choose the mechanism m ∈ M given his type t; Πt (m)
is the agent’s posterior probability that the principal’s type is t given that he chose m; and
πm(sP· ,s
A) is the probability, conditional on the principal choosing mechanism m, that the
pure action profile(sP· ,s
A) will be played in stage 2.
Definition 9. An equilibrium of the full game is a profile Σ =(
p··,Π· (·) ,π·
)that satisfies
the following three conditions:
1. for any mechanism m (including those not chosen in equilibrium), πm is a CEIGΠ(m)·
m ;
2. Bayes’s Rule is used whenever possible: for any mechanism m that some type of
principal chooses in equilibrium, the agent’s posterior beliefs are determined by
Bayes’s Rule:
Πt (m) =
Πt ptm
∑Ts=0 Πs ps
m; and (128)
3. for each type t, each mechanism m in the support of pt· is an optimal choice for the
type-t principal given the map π· from mechanisms to correlated equilibria.
5.2.1 Results
For any mechanism m and beliefs Π· ∈ ∆T , let
φm
(Π·)⊂Φm (129)
be the set of associated correlated equilibria of stage 2,16 and let
Um (πm) = ∑sP· ∈(SP
m)T+1
∑sA∈SA
m
πm
(sP· ,s
A)
uAm
(sP· ,s
A)
(130)
16The set Φm of distributions over pure action profiles is defined in (124).
75
and
V tm (πm) = ∑
sP· ∈(SP
m)T+1
∑sA∈SA
m
πm
(sP· ,s
A)
uPm,t
(sP· ,s
A)
(131)
be the resulting payoffs of the agent and of the type-t principal, respectively, when they
play πm in the mechanism m. Let
ψm
(Π·)=(
V ·m (πm) ,Um (πm))
: πm ∈ φm
(Π·)⊂ℜ
T+1×ℜ (132)
denote the set of payoff vectors(V ·,U
)that can occur in correlated equilibria πm given the
mechanism m and interim beliefs Π·.
Claim 14. For any mechanism m and interim beliefs Π·, the agent’s expected payoff Um (πm)
(evaluated using Π·) in any correlated equilibrium πm ∈ φm
(Π·)
is nonnegative.
Proof. The agent can get zero by always ”rejecting” in stage 2; hence, her expected stage-2
continuation payoff Um (πm) in any correlated equilibrium πm in φm
(Π·)
must be nonnegative.
A simple but important type of mechanism m is a Direct Revelation Mechanism DRM.
Definition 10. A Direct Revelation Mechanism (DRM) is a mechanism m with the following
properties. The agent’s action set SAm consists of two actions: accept or reject. The principal’s
action set SPm is just her type space ϒ. If the agent rejects then, for any action ”t” of
the principal, the reservation outcome o0 is implemented: each player gets a payoff of
zero. If the agent accepts and the principal chooses ”t”, then the DRM implements some
prespecified outcome ot .
For the principal, choosing the action t ∈ ϒ is interpreted as claiming to be of type t. A
DRM m is fully specified as the allocation o· = otTt=0 that is implemented when the agent
accepts and the principal truthfully reports his type. Thus, we will also refer to a DRM
as the allocation o· that it is designed to implement. We will also say that the principal
chooses the outcome ot in stage 2; this means that the principal reports the type ”t”.
A DRM equilibrium o· is an equilibrium in which the allocation o· is implemented as a
DRM. More precisely:
76
Definition 11. A DRM equilibrium o· is an equilibrium in which, as long as there has been
no prior deviation, each type of principal chooses the same DRM o· in stage 1 while, in
stage 2, the agent chooses ”accept” and the principal of each type t truthfully reports his
type: the allocation o· is implemented.
Fix an equilibrium Σ =(
p··,Π· (·) ,π·
). By risk neutrality, Σ gives the principal (resp.,
agent)a type-contingent expected payoff of V t (otΣ
)(resp., U t (ot
Σ
)) when the principal’s
type is t, where17
otΣ = ∑
m∈Mpt
m ∑sP· ∈(SP
m)T+1
∑sA∈SA
m
πm
(sP· ,s
A)
osPt ,s
A
m (133)
is the expected outcome for this type. We will refer to o·Σ
as the expected allocation of Σ.
Definition 12. The allocation o· is an equilibrium allocation if there exists an equilibrium
Σ whose expected allocation o·Σ
is o·.
We next show that for every equilibrium Σ, there is a DRM equilibrium that implements
the expected allocation o·Σ
of Σ. Since a DRM equilibrium is also an equilibrium, this
means that o· is an equilibrium allocation if and only if there is a DRM equilibrium that
implements it. This will help us characterize the set of equilibrium allocations.
Claim 15. For any equilibrium Σ, there is a DRM equilibrium o·Σ.
Proof. Section 5.5.
The following definition is from MT [12, p. 10].
Definition 13. An allocation o· is weakly interim efficient (WIE) if and only if (a) it is
incentive compatible -
V t (ot)≥V t (os) for all types t and s (134)
17For any outcomes o = (q,ρ) and o′ = (q′,ρ ′), and any probability π , the expected outcome πo +
(1−π)o′ refers to the outcome in which the portfolio πq+(1−π)q′ is transferred in return for the payment
πρ +(1−π)ρ ′ for sure. By risk neutrality, this deterministic outcome is payoff-equivalent to a lottery in
which outcome o occurs with probability π and outcome o′ occurs with probability 1−π .
77
- and there exists no allocation o· that (a) Pareto dominates o·:
V t (ot)≥V t (ot) for all types t, strictly for some t; (135)
(b) is incentive-compatible:
V t (ot)≥V t (os) for all types t and s; (136)
and (c) gives the agent at least as high a payoff as o· does for each type of principal:
U t (ot)≥U t (ot) for all types t. (137)
MT [12, p. 10] use also the following equivalent formulation.
Claim 16. An allocation o· is WIE if and only if it solves the following maximization
problem for some vector of positive weights wtTt=0 ∈ℜ
T+1++ :
Program I. maxo·∑Tt=0 wtV t (ot) subject to (136) and (137).
Proof. Section 5.5.
Recall that there is no trade in the reservation allocation o·0: each player gets zero.
Definition 14. An allocation o·, with payoffs
V t d=V t (ot) for all t, (138)
is an RSW allocation if and only if, for all t, it solves:
Program IIt . V t = maxo·V t (ot) subject to
V s (os)≥V s (or) for all types r,s; (139)
U s (os)≥U s (os0) for all types s. (140)
Claim 17. There exists an RSW allocation o·. Any RSW allocation o· is WIE.
Proof. The proof is the same as in MT [12, p. 11].
78
For convenience, we now restate the Recursive Linear Program (RLP). Recall that Q,
defined in (119), is the set of all quantity vectors q.
RLP Let Q0 = Q and, for each t = 1, ...,T , let
Qt = q ∈ Q : for all s < t, q [ f (t)−δ f (s)]≤ u∗ (s) (141)
be the set of all quantity vectors that can be assigned to type t which, if accompanied
by revenue ρ = q f (t), do not tempt any lower type to imitate type t. For each
t = 0, ...,T , let
q∗ (t) ∈ argmaxq∈Qt
[q f (t)] , (142)
ρ∗ (t) = q∗ (t) f (t) , and (143)
u∗ (t) = ρ∗ (t)−δq∗ (t) f (t) . (144)
RLP is thus a maximization program that produces a seller’s payoff function u∗, a quantity
function q∗, and a revenue function ρ∗, where u∗ and ρ∗ are unique by DFJ’s PROPOSITION
1.18 Any solution to RLP yields an associated allocation o· where ot = (q∗ (t) ,ρ∗ (t)). In
any such allocation, the payoff q∗ (t) f (t)−ρ∗ (t) of investors is zero by (143).
Crucially, RSW and RLP are equivalent in our setting, in the following sense.
Claim 18. There is a one-to-one correspondence between RSW allocations and solutions
to RLP, which is as follows.
1. Let o· be an RSW allocation. Then the agent’s payoff U t (ot) is identically zero and
the functions q∗ (t) = qot, ρ∗ (t) = ρ ot
, and u∗ (t) =V t (ot) solve RLP.
2. Let (u∗,ρ∗,q∗) solve RLP. Then the allocation o· with ot = (qt ,ρ t) = (q∗ (t) ,ρ∗ (t))
is an RSW allocation.
18Multiple quantity functions q∗ may solve RLP. E.g., if the two assets i and j have the same conditional
expected values (if fi (t) = f j (t) for all t) then the players care only about their total holdings of the two
assets combined. Hence, RLP cannot tell us the number of shares a seller of a given type t will sell of each
asset, although it may pin down the sum qi (t)+q j (t).
79
Proof. Section 5.5.
The following definition is from MT [12, p. 10].
Definition 15. An allocation o· is interim efficient (IE) relative to beliefs Π· if and only if
(a) it is incentive compatible -
V t (ot)≥V t (os) for all types t and s (145)
- and there exists no allocation o· that (a) Pareto dominates o·:
V t (ot)≥V t (ot) for all types t, strictly for some t; (146)
(b) is incentive-compatible:
V t (ot)≥V t (os) for all types t and s; (147)
and (c) gives the agent at least as high expected payoff as o· under the beliefs Π·:
T
∑t=0
ΠtU t (ot)≥ T
∑t=0
ΠtU t (ot) . (148)
MT [12, p. 10] use also the following equivalent formulation.
Claim 19. An allocation o· is IE with respect to the beliefs Π· if and only if it solves the
following maximization problem for some vector of positive weights wtTt=0 ∈ℜ
T+1++ :
Program I. maxo·∑Tt=0 wtV t (ot) subject to (147) and (148).
Proof. Section 5.5.
The following is MT’s Proposition 3, specialized to our model.
Claim 20. For any WIE allocation o·, the set of beliefs Π(o·) for which o· is IE is nonempty
and convex.
Proof. Section 5.5.
80
Theorem 1 in MT relies on the following claim (their Proposition 5). The proof in MT
relies on their assumption that the indifferent curves of different types are never tangent.
In our setting, two adjacent types may have identical preferences. In this case, their
indifference surfaces in (q,ρ) space are identical. Hence, we use a different proof, which
relies on our ASSUMPTION A.
Claim 21. In any equilibrium of the contract proposal game, the payoff of the type t
principal is at least her RSW payoff V t (ot).
Proof. Section 5.5.
We now extend the Intuitive Criterion (Cho and Kreps [4]) to our setting. This criterion
restricts the investor’s beliefs when the principal deviates to a mechanism m that no type’s
strategy puts positive weight on: for all t, ptm = 0. Recall that ϒ is the set of all types t
and V tm (πm), defined in (131), is the type-t principal’s payoff in the correlated equilibrium
πm of the mechanism m. For a fixed equilibrium Σ =(
p··,Π· (·) ,π·
), the payoff V t of a
type-t principal is V t (otΣ
)where ot
Σis the expected outcome for this type, defined in (133).
Consider the following restriction on the agent’s interim beliefs Π· (m) in the equilibrium
Σ, following the principal’s choice of mechanism m, where, for any set S⊂ ϒ of types,
φSm =
⋃Π·∈∆T : supp(Π·)⊂S
φm
(Π·)⊂Φm
denotes the set of correlated equilibria for interim beliefs Π· whose support lies in S.19
IntΣm Fix an equilibrium Σ. For any mechanism m, let Jm ⊂ ϒ denote the (possibly empty)
set of types t for whom the payoff of the principal in Σ exceeds his maximum payoff
in all correlated equilibria of m for any interim beliefs of the agent:
Jm =
t ∈ ϒ : V t (ot
Σ
)> max
πm∈φ ϒm
V tm (πm)
. (149)
19The set φm
(Π·)
of correlated equilibria of m for beliefs Π· is defined at equation (129).
81
Suppose there exists a type t in ϒ− Jm of principal whose payoff in Σ is no higher
than his minimum payoff in any correlated equilibrium of m that results from beliefs
that put zero weight on any type in Jm:
V t (otΣ
)≤ min
πm∈φϒ−Jmm
V tm (πm) . (150)
Then on seeing m, the agent’s interim beliefs Π· (m) put zero weight on any type in
Jm: supp[Π· (m)
]⊂ ϒ− Jm.
IntΣm holds automatically if m is offered by some type in equilibrium. Why? No type in
Jm will choose m, and m is chosen with positive probability in Σ, so by Bayes’s Rule the
agent’s interim beliefs on seeing m must put zero weight on Jm: IntΣm holds.
We use IntΣm to define intuitive beliefs, and an intuitive equilibrium.
Definition 16 (The Intuitive Criterion). Let Σ =(
p··,Π· (·) ,π·
)be an equilibrium. The
beliefs Π· (·) and equilibrium Σ are intuitive if and only if IntΣm holds for every mechanism
m ∈M.
Without strengthening our ASSUMPTION A, we cannot prove the following result,
which is needed to apply MT’s Theorem 1 (stated below as Proposition 3) to our model:
Claim 22. Let o· be an RSW allocation. Then o· is IE with respect to some positive beliefs
Π· ∈ℜT+1++ .20
We will instead rely only on the following weaker property:
Claim 23. Let o· be an RSW allocation. Then o· is IE relative to some nonnegative beliefs
Π· ∈ℜT+1+ .
Proof. Follows from Claims 34 and 17.
We now state and prove our analogue to MT’s Theorem 1. The result shows also that
the RSW allocation is intuitive. Recall that V t , defined in (138), is the RSW payoff of a
principal of type t.
20That is, Πt > 0 for each type t.
82
Proposition 3. An allocation o· is the expected allocation of some equilibrium if and only
if (a) it is incentive compatible:
V t (ot)≥V t (os) for all types t and s; (151)
(b) it is ”profitable in expectation” for the agent under the prior beliefs Π·:
T
∑t=0
ΠtU t (ot)≥ T
∑t=0
ΠtU t (ot
0)
; (152)
and (c) it gives each type of principal at least her RSW payoff:
V t (ot)≥ V t for each type t. (153)
Moreover, if o· is an RSW allocation, there is an intuitive equilibrium with expected allocation
o·.
Proof. Section 5.5.
Proposition 3 shows that any RSW allocation o· is intuitive: it is the expected allocation
of an intuitive equilibrium. The converse also holds:
Proposition 4. Any expected allocation of an intuitive equilibrium is an RSW allocation.
Proof. Section 5.5.
Hence, the sets of intuitive and RSW allocations coincide. Together with Claim 18, this
establishes that the set of intuitive allocations of the above game coincides with the set of
solutions to RLP which, in turn, are the set of intuitive outcomes of our AS game.
We have shown that under the intuitive criterion, the results of our AS game are robust
to the choice of mechanism: it does not matter whether the assets are sold using our
procedure or that of MT. However, without the intuitive criterion, MT’s approach may
permit additional outcomes by Proposition 3.
83
5.3 The MT Approach: 2x2 Case
We next apply the results of section 5.2 to the case of two assets i = 1,2 and two types
t = 1,2.21 Recall that fi (t) is the expected payout of asset i = 1,2 conditional on the
seller’s type being t = 1,2. In order to apply our prior results on the A2NM model (section
4.1), we assume generic parameters. Hence, by ASSUMPTION A (monotonicity),
fi (2)> fi (1)> 0 for each asset i. (154)
Swapping asset indices if needed, we can also assume w.l.o.g. that
f2 (2)/ f2 (1)> f1 (2)/ f1 (1) . (155)
That is, an increase in the seller’s type raises the expected value of asset 2 proportionally
more than that of asset 1: IIS holds. Finally, we normalize the number of shares of each
asset to one: the seller’s endowment vector a is (1,1).
We first preview our results graphically. Figure 1 illustrates the case in which case the
type 2 seller retains all of asset 2 in the RSW allocation.22 Figure 2 illustrates the case in
which the type 2 seller sells all of asset 1 in the RSW allocation. We now explain these
figures.
In any equilibrium allocation o·, let x denote the difference between the low type’s
equilibrium payoff V 1 (o1) and her RSW payoff V 1. We show that in any equilibrium
allocation other than the RSW allocation, x is positive (Claims 26 and 27): the low type
must improve on her RSW payoff. The low type’s payoff increment x is depicted on the
vertical axis of Figures 1 and 2, while the prior probability Π2 of the high type appears on
the horizontal axis.
Let Sx denote the set of equilibrium allocations that raise the low type’s payoff by
exactly x> 0 vis-a-vis the RSW allocation. For simplicity, we focus on the set Sx of efficient
21We deviate from the type numbering of section 5.2, which starts at zero, in order to more easily reference
the results of section 4.1.
22The notation ν , η , ζ , and γ will be defined later.
84
0
0.5
1
1.5
2
2.5
3
3.5
4
0.0
0
0.0
2
0.0
5
0.0
7
0.1
0
0.1
2
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4
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0.2
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1
0.9
4
0.9
6
0.9
8
x:
Lo
w T
yp
e's
Pay
off
In
crem
ent
Π2 : probability of high type
Efficient Equilibria under MT Procedure
2x2 Case, ν > 0
Pooling Equilibria: ! = "
Separating Equilibria with !" = 1, "
" > 0Separating
Equilibria with
!" < 1, "
" = 0
η ζ
ν
γ
No Equilibria in This Region
Figure 1: the case ν > 0.
85
0
0.5
1
1.5
2
2.5
3
3.5
4
0.0
0
0.0
2
0.0
5
0.0
7
0.1
0
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2
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0.9
4
0.9
6
0.9
8
x:
Lo
w T
yp
e's
Pay
off
In
crem
ent
Π2 : probability of high type
Efficient Equilibria under MT Procedure
2x2 Case, ν < 0
Pooling Equilibria: ! = "
Separating Equilibria with !" = 1, "
" > 0
η ζ
γ
No Equilibria in This Region
Figure 2: the case ν ≤ 0.
86
elements of Sx: the elements in Sx that give the greatest social welfare, which we define as
the sum of the unconditional expected payoffs of seller and agent. If Sx is nonempty, then
Sx is a singleton.23
This unique efficient allocation has two key properties. First, the low type sells his
whole portfolio but gets more than the fair value of this portfolio: ρ1 > a f (1). This has the
effect of loosening his incentive compatibility constraint, so the high type can sell more of
his portfolio. In this way, more gains from trade are realized than in the RSW allocation so
welfare is higher. Second, as in the RSW allocation, the high type sells all of asset 1 before
he sells any of asset 2.24
These properties allow us to divide the efficient allocations (over all possible type-1
payoff increments x) into three categories, all of which appear in Figure 1.25 For a range
of payoff increments x there are pooling allocations, in which the high type also sells his
entire portfolio: q2 = a. For an intermediate range of x, there are separating allocations in
which the high type sells all of asset 1 and some of asset 2 (q21 = 1 and q2
2 > 0). Finally, in
Figure 1 there is a low range of increments x for which there are separating allocations in
which the high type sells some of asset 1 but none of asset 2 (q21 > 0 and q2
2 = 0).26
This third set of allocations is present in Figure 1 but not in Figure 2. Why? As noted,
any efficient allocation yields higher welfare than the RSW allocation. And as also noted,
the high type must sell all of asset 1 before any of asset 2, as in the RSW allocation. It
follows that he cannot sell fewer shares of asset 1 than in the RSW allocation, as he cannot
sell more sales of asset 2 to offset this. Finally, Figure 2 concerns the case in which the
23See Claim 29 and Proposition 5.
24These are, respectively, properties P1 and P4 in Claim 28.
25These results appear in Claim 29 and Proposition 5.
26As shown, each type of allocation exists only if the prior probability Π2 of the high type exceeds a
threshold that is nondecreasing in x. Intuitively, the low type seller’s gain x equals the agent’s loss when she
encounters this type. In order to accept a larger such loss x, the probability 1−Π2 of encountering the low
type must be lower.
87
high type sells all of asset 1 in the RSW allocation. Hence, in this case, he must also sell
all of asset 1 in any efficient allocation.
These figures show that in the 2x2 model with MT’s sale procedure, pooling is less
likely than separation. More precisely, for any prior probability Π2 of the high type that
permits pooling, there is a continuum of separating equilibria. But in Figure 1, there is also
a positive-measure set of prior probabilities (the interval Π2 ∈ [η ,ζ ]) for which separating
equilibria exist but pooling equilibria do not. These properties are indeed general, as shown
in Claim 29 and Proposition 5 below.
The final result of this section, Claim 30, characterizes the set of globally efficient
equilibrium allocations: those equilibrium allocations that give the highest social welfare
for any payoff increment x. This set consists of the pooling equilibria depicted in Figures
1 and 2, together with the upper, curved boundary of the region of separating equilibria for
Π2 between η and ζ in Figure 1.27
To proceed, let us define the following set
Definition 17. Let S denote the set of allocations o· that are incentive-compatible for the
principal:
V 1 (o1)≥V 1 (o2) and (156)
V 2 (o2)≥V 2 (o1) ; (157)
that give the agent a nonnegative unconditional expected payoff:
U (o·) =2
∑t=1
ΠtU t (ot)≥ 0 (158)
27In stating this result, we do not mean to imply that the globally efficient equilibrium will be selected. The
equilibrium selection literature has not provided any convincing rationale for this “Pareto criterion”; see, e.g.,
the skeptical discussion in Fudenberg and Tirole [7, pp. 20-22]. While we also focus on efficient equilibria
(for a given payoff increment) in Claim 29 and Proposition 5, our purpose there is to not to make predictions
but rather to shrink the equilibrium set to manageable size while still giving a flavor of the wide variety of
outcomes that can occur.
88
where Πt > 0 is the prior probability of type t; and that give each type of principal at least
her RSW payoff:
V 1 (o1)≥ V 1 and (159)
V 2 (o2)≥ V 2. (160)
Claim 24. 1. S is the set of equilibrium allocations o·.
2. S contains the RSW allocation o·.
3. S is compact.
Proof. Parts 1 and 2 follow from Proposition 3. As for part 3, each qt lies in [0,1]2 whence:
1. by (154), (159), and (160),
ρt ≥ δqt f (t)+V t ≥ V t >−∞ for t = 1,2; (161)
2. by (158), ∑2t=1 Πtρ t ≤ ∑
2t=1 Πta f (t) whence, for s = 1,2,
ρs ≤ (Πs)−1
[−Π
3−sρ
3−s +2
∑t=1
Πta f (t)
]
≤ (Πs)−1
[−Π
3−sV 3−s +2
∑t=1
Πta f (t)
]< ∞ by (161).
Hence, S is compact.
In the next claim, we solve analytically for the RSW allocation and show it is unique.
Under (154), the model we study is a special case of A2NM described in section 4.1. Thus,
the solution to RLP in this section is identical to that characterized by Claims 10 and 11.
And by Claim 18, for each type t, the principal’s RSW payoff Vt equals his RLP payoff
u∗ (t) while the reservation payoff U t (ot0)
of the agent is zero by (121) and (121). This
implies the first part of the following claim. Recall the notation ∆i = fi (2)− δ fi (1) for
i = 1,2 and ∆ = (∆1,∆2) = f (1)− δ f (2) (equation (97)). We will also use the notation
89
∆1i = fi (2)− fi (1) and ∆1 =
(∆1
1,∆12)= f (1)− f (2) from (96). The following facts will
be useful.
∆12∆1−∆
11∆2 = (1−δ ) [ f1 (1) f2 (2)− f2 (1) f1 (2)]> 0 by (155); (162)
∆i−δ∆1i = (1−δ ) fi (2)> 0 by (154); (163)
Also define28
γ = δ(a− q2)
∆1 ∈(0,a∆
1) (164)
and29
ν = γ−δ∆12 < γ, (165)
as well as
η = δ∆11/∆1 (166)
and
ζ = γ/a∆1. (167)
Claim 25. There is a unique RSW allocation o· =(
ρ1, q1) ,(ρ2, q2), with the following
properties.
q1 = a = (1,1) ; (168)
ρ1 = a f (1) ; (169)
V 1 = (1−δ )a f (1) =V 1 (o2)= q2∆; (170)
q21 = min
1,
V 1
∆1
∈ (0,1] ; (171)
q22 = max
0,
V 1−∆1
∆2
∈ [0,1) ; (172)
ρ2 = q2 f (2) ; (173)
V 2 = (1−δ ) q2 f (2) . (174)
28The second equality holds by (179) and the set inclusion by (154), (171), and (172).
29The inequality holds by (154).
90
The following properties also hold:
V 2 >V 2 (o1)= a∆; (175)
V 1−V 2 = δ q2∆
1 (176)
= δ
min
1,
V 1
∆1
∆
11 +max
0,
V 1−∆1
∆2
∆
12
; (177)
V 1−V 2 (o1)= δa∆1; (178)
V 2−V 2 (o1)= γ = δ(a− q2)
∆1 = δa∆
1−(
V 1−V 2)
; (179)
ν = δ∆11−(
V 1−V 2)
; (180)
q21 = min
1,
V 1−V 2
δ∆11
; (181)
q22 = max
0,
V 1−V 2−δ∆11
δ∆12
= max
0,−ν
δ∆12
; (182)
V 1 S ∆1 as V 1−V 2 S δ∆11 as ν T 0; (183)
V 1 = a∆−a∆1; (184)
ζ > η . (185)
Proof. Section 5.5.
An immediate consequence of Claim 25 is that in the RSW allocation, the type-2 seller
retains all of asset 2 (resp., sells all of asset 1) if ν is positive (resp., negative):
Corollary 3. If
ν > 0, (186)
then
V 1 < ∆1 (187)
and type 2 retains all of asset 2 in the RSW allocation:
q2 =
(V 1
∆1,0
)=
(V 1−V 2
δ∆11
,0
). (188)
91
If instead
ν ≤ 0, (189)
then
V 1 ≥ ∆1 (190)
and type 2 sells all of asset 1 in the RSW allocation:30
q2 =
(1,|ν |δ∆1
2
). (191)
Proof. Under (186), equation (187) follows from (183), while (188) is implied by (171),
(172), (181), (182), and (187). If instead (189) holds, then (190) follows from (183), while
(191) results from by (181), (182), and (173).
We next consider elements of S in which each type of seller gets her RSW payoff. We
show that any such allocation must coincide with the RSW allocation o·, which is unique
by Claim 25. The nontrivial part of the proof is to show that the agent’s payoff must be
zero not only ex-ante, but also ex-post.
Claim 26. Let o· be an equilibrium allocation in which each type of seller gets her RSW
allocation payoff: V t (ot) = V t for t = 1,2. Then o· is the unique RSW allocation o·.
Proof. Section 5.5.
We now turn to elements of S in which some type of seller gets more than her RSW
payoff, a property that we write formally:
P0 For some s ∈ 1,2, V s (os)> V s.
In any such allocation, the agent strictly prefers (not) to buy from the high (low) type seller,
while the low type seller gets more than her RSW payoff:
Claim 27. In any equilibrium allocation o· with property P0, U1 (o1)< 0, U2 (o2)> 0, and
x =V 1 (o1)−V 1 > 0.
30
92
Proof. Section 5.5.
We now give a necessary and sufficient condition for there to exist an equilibrium
allocation o· in which the type 1 seller’s payoff V(o1) exceeds her RSW payoff V 1 by
exactly some given x > 0. We also characterize the efficient such allocation: the one that
maximizes the joint surplus
J (o·) =2
∑t=1
Πt [V t (ot)+U t (ot)]= (1−δ )
2
∑t=1
Πtqt f (t) (192)
of seller and agent.31 Let
Sx =
o· ∈ S : V 1 (o1)= V 1 + x
denote the set of equilibrium allocations o· in which the type-1 seller gets exactly x more
than her RSW payoff and let
Sx = o· ∈ Sx : ∀o· ∈ Sx,J (o·)≤ J (o·)
be the efficient frontier of Sx: the set of allocations in Sx that maximize the joint surplus J.
We first show that each element of Sx satisfies four key properties in addition to P0:
Claim 28. Fix x > 0. If Sx is nonempty then so is Sx. Moreover, any allocation o· in Sx has
the following four properties:
P1 The low type sells her entire portfolio (q1 = a) in return for the payment ρ1 = a f (1)+x.
P2 The low type’s IC constraint binds: V 1 (o2)=V 1 (o1).31In MT [12], the agent is ignored in the definition of efficiency. In practice, by (158), this would restrict
us to allocations in which her expected payoff is zero. However, such allocations generally do not maximize
joint surplus. Intuitively, if the type 2 seller gets more than her RSW payoff, we can have her transfer
x = ε(a−q2
)more shares to the agent (for some small ε > 0) in return for an additional payment equal
to type 1’s opportunity cost δx f (1). This raises joint surplus without tempting the type 1 seller to imitate;
however, it also raises the agent’s payoff and thus violates MT’s notion of efficiency.
93
P3 The high type either sells her whole portfolio or gets her RSW payoff: either q2 = a or
V 2 (o2)= V 2 .
P4 The high type sells all of asset 1 before selling any units of asset 2: either q21 = 1 or
q22 = 0.
We will now explicitly characterize the set Sx of efficient allocations for any given
increase x > 0 in the type 1 seller’s payoff vis-a-vis the RSW allocation. The notation
(ρ t ,qt) will refer to the outcome in a given allocation o· that corresponds to type t = 1,2.
As property P4 pins down the outcome(ρ1,q1) of the type 1 seller, outcomes in Sx can
differ only with respect to the type 2 seller’s outcome(ρ2,q2).
We now turn to the formal results that underpin Figures 1 and 2. We first consider the
subset
Spoolx =
o· ∈ Sx : q2 = q1 = a
of efficient allocations in which each type sells their entire portfolio. By P2, this implies
equal transfers as well: ρ1 = ρ2. Hence, by P1, Spoolx is the set of pooling outcomes.
Claim 29. Fix x > 0. The set Spoolx is nonempty if and only if32
Π2 ≥ ζ (193)
and x lies in the interval [γ,Π2a∆
1] (194)
(which is nonempty by (193)). In this case, Spoolx contains a single allocation, in which the
agent pays ρ t = a f (1)+ x to each type t of seller in return for his whole portfolio.
Proof. Section 5.5.
We now turn to the set
Ssepx =
o· ∈ Sx : q2 6= q1 = a
32The notation ζ is defined in (167).
94
of efficient allocations for which q2 6= q1: the separating allocations. In such an allocation,
P3 implies that the high type gets her RSW payoff: V 2 (o2)= ρ2−δq2 f (2) = V 2, whence
ρ2 = δq2 f (2)+V 2. (195)
Hence it suffices to solve for q2, with the corresponding transfer ρ2 given by (195). We
do so in the following claim. Cases 1(a) and 1(b) correspond, respectively, to the lower
and upper regions of separating equilibria in Figure 1, while case 2(a) relates to the single
region of separating equilibria in Figure 2.
Proposition 5. With respect to Ssepx , there are two cases.
1. Case 1. Suppose ν > 0: in the RSW allocation, the type-2 seller retains all of asset
2. There are three subcases.
(a) Case 1(a). If x lies in the interval
(0,ν ] (196)
then Ssepx is nonempty if and only if33
Π2 ∈ [η ,1] , (197)
which has positive measure by (154): the probability of the high type cannot be
too low. In this case, Ssepx contains a single allocation, in which the type 2 seller
retains all of asset 2:
q2 =
(V 1−V 2 + x
δ∆11
,0
). (198)
(b) Case 1(b). If
x ∈ [ν ,γ] , (199)
then Ssepx is nonempty if and only if
Π2 ≥ φ (x) d
= x[
∆1−V 1 +(x−ν)∆2
δ∆12
]−1
. (200)
33The notation η is defined in (166).
95
The function φ (x) is continuous and increasing,34 takes values in (0,1) when
x ≥ ν , and satisfies φ (ν) = η , φ (γ) = ζ , and limx→∞ φ (x) = δ∆12/∆2 > ζ . If
(200) holds, Ssepx contains a single allocation, in which the type 2 seller sells all
of asset 1:
q2 =
(1,
x−ν
δ∆12
). (201)
Finally, the region R defined by (199) and (200) is the union of two regions. The
first, region (i), is defined by the two conditions
Π2 ∈ [η ,ζ ] (202)
and
x ∈[ν ,φ−1 (
Π2)] , (203)
where
φ−1 (
Π2)=
[δ∆1
2
(∆1−V 1
)−ν∆2
]Π2
δ∆12−∆2Π2 . (204)
The second, region (ii), is defined by (199) and
Π2 ∈ [ζ ,1] . (205)
(c) Case 1(c). If x exceeds the upper endpoint of (199) then Ssepx is empty.
2. Case 2. Suppose ν ≤ 0: in the RSW allocation, the type 2 seller sells all of asset 1.
There are two subcases.
(a) Case 2(a). If x lies in the interval
[0,γ] , (206)
then Ssepx is nonempty if and only if
Π2 ≥ ζ , (207)
34Intuitively, φ (x) is increasing in xsince, in order for the agent to participate when the type-1 agent’s
payoff increment x over her RSW payoff is higher, the agent must believe that the agent is more likely to be
of type 2.
96
in which case Ssepx contains a single allocation, in which the quantities q2 sold
by type 2 are given by (201).
(b) Case 2(b). If x exceeds the upper endpoint of (206) then Ssepx is empty.
Proof. Section 5.5.
Finally, we characterize the set
S = o· ∈ S : ∀o· ∈ S,J (o·)≤ J (o·)
of globally efficient equilibrium allocations: those equilibrium allocations that give the
highest social welfare for any payoff increment x. This set consists of the pooling equilibria
depicted in Figures 1 and 2, together with the upper, curved boundary of the region of
separating equilibria for Π2 between η and ζ in Figure 1.
Claim 30. There are three cases.
1. If Π2 ≥ ζ , S consists of those pooling allocations in which, for some x in the interval
(194), the agent pays ρ t = a f (1)+ x to each type of seller for her whole portfolio.
2. If ν > 0 and Π2 ∈ [η ,ζ ), S is a singleton consisting of the allocation in which q2 is
given by (201) for x = φ−1 (Π2).3. In all other cases, S is empty.35
Proof. Section 5.5.
5.4 The MT Approach and Security Design
We now apply the results of section 5.3 to the problem of security design. Assume an issuer
owns a stochastic cash flow Y that takes three possible values: y2 > y1 > y0 = 0. Suppose
the issuer’s beliefs about Y , conditional on his type t, satisfy the Hazard Rate Ordering
35As is Spoolx and Ssep
x for all x > 0.
97
(HRO) property (DFJ, section 2.5). Assume the issuer can sell any monotone security S:
any nondecreasing function S : 0,y1,y2 → ℜ such that, for each possible Y , S (Y ) and
Y −S (Y ) are both in [0,Y ].
How do we apply MT’s approach in this setting? In section 5.2 we defined an outcome
as a pair (q,ρ) consisting of a payment ρ from agent to seller in return for the vector
q = (q1, ...,qn) of quantities of the seller’s assets. In the present setting, the vector q is
replaced by a monotone security S; let O be the set of outcomes (S,ρ). The rest is as in
section 5.2, with the set O of outcomes replaced by O. Let us call this the MTSD game (for
MT - security design).
How do we find the set of equilibria of the MTSD game? As shown in DFJ section 3.3,
a monotone security S is equivalent to a portfolio of assets, where the set of assets is given
by the maximal tranching of the cash flow Y into prioritized, monotone securities. In the
present case with three possible outcomes of the cash flow, this maximal tranching consists
of the two securities F1 (Y ) = y11Y∈y1,y2 and F2 (Y ) = (y2− y1)1Y=y2 . Let MTAS refer to
the 2x2 MT asset sale game in which these are the two assets that the seller wishes to sell.
As in DFJ section 3.3, there is an isomorphism between the MTSD and MTAS, which
is as follows. For any security S in the MTSD game, the quantity of asset i = 1,2 in the
MTAS game is given by qSi = [S (yi)−S (yi−1)]/(yi− yi−1). As shown in DFJ section 3.3,
the monotone security S is payoff-equivalent to the portfolio qS =(qS
1,qS2)
of assets F1 and
F2: for any realization Y of the cash flow, the security S and the portfolio qS imply the same
ex-post transfer from issuer to agent. And any portfolio q = (q1,q2) of the assets F1 and F2
in the MTAS game is payoff-equivalent to the monotone security Sq = qF (Y ) in the MTSD
game, where F (Y ) is the vector (F1 (Y ) ,F2 (Y )) of realized payouts of assets F1 and F2.
We now solve the MTSD game by applying the results of section 5.3 to the MTAS
game. For each type t = 1,2 and each asset i = 1,2, let fi (t) = E [Fi (Y ) |t] denote the
expected payout of asset Fi when the issuer’ type is t. Let f (t) = ( f1 (t) , f2 (t)) denote
the vector of these expected payouts for type t. The function f satisfies (154) since
HRO implies FOSD and since the assets are monotone functions of Y . It satisfies (155)
98
by DFJ’s Proposition 5 and by HRO. Consequently, the set of equilibrium allocations((q1,ρ1) ,(q2,ρ2)) identified in section 5.3 in the MTAS game for this function f is
isomorphic to the set of equilibrium allocations((
S1,ρ1) ,(S2,ρ
2)) of the MTSD game:
for each type t = 1,2, the agent pays the issuer ρ t in return for the monotone security
St (Y ) = qtF (Y ).
Using this equivalence, we can now discuss the main implications of section 5.3 for
the MTSD game. As in section 5.3, we restrict to allocations that are efficient contingent
on giving the low type a variable amount x more than her RSW payoff. By Claim 25, the
RSW allocation is always an equilibrium of the MTAS game. What its equivalent in the
MTSD game? By (168), in the RSW allocation, the low type sells her whole portfolio.
This is equivalent, in the MTSD game, to standard debt (secured by Y ) with face value y2.
By (171) and (172), the type 2 issuer uses a hurdle class strategy (DFJ section 3.4) in the
MTAS game: either q22 is zero (whence the hurdle class c is 1) or q2
1 is one (whence c = 2).
In the first case, the corresponding security in the MTSD game is
q2F (Y ) = q21y11Y∈y1,y2
which is just standard debt with face value q21y1. In the second case, the security sold by
the high type in the MTSD game is
q2F (Y ) = y11Y∈y1,y2+ q22 (y2− y1)1Y=y2
which is just standard debt with face value y1 + q22 (y2− y1). Indeed, this is just the result
of DFJ section 3.4 as the RSW and RLP allocations coincide (Claim 18).
We now turn to efficient equilibrium allocations other than RSW.36 By Claim 28, in
each such allocation the low type sells her whole portfolio which, as shown above, is
equivalent to standard debt with face value y2. As for the high type, Claim 28 implies
that he uses a hurdle class strategy which, as shown in DFJ section 3.4, is also equivalent
to standard debt. Since the low type chooses the highest possible face value, the qualitative
36By Proposition 4, these do not satisfy the intuitive criterion.
99
results of DFJ section 3.4 carry over to the MTSD game: the issuer sells standard debt with
a face value that is nonincreasing in his type.37
How do these non-RSW allocations differ from the RSW allocation? By Claim 27, the
low type is paid some x> 0 more than the fair value of her security. (In the RSW allocation,
she is paid exactly this fair value.) This loosens her IC constraint which, in turn, enables
the high type to raise the face value of the security he sells relative to the outcome of DFJ
section 3.4. In the case in which Π2 exceeds the constant ζ defined in (167), the high type
can raise his face value all the way to the face value y2 chosen by the low type: the two
types can pool, selling the same standard debt security (By Claim 29). As in section 5.3,
neither the pooling outcomes nor the non-RSW separating outcomes survive the intuitive
criterion.
5.5 MT Approach: Proofs
PROOF OF CLAIM 15. Let mΣ denote the DRM o·Σ
defined in (133).38 Further, consider
the profile Σ in which the principal always chooses the mechanism mΣ, the agent accepts,
and the principal chooses the outcome otΣ
corresponding to his type; if any player ever
deviates, they revert to the original equilibrium Σ.39 If Σ is an equilibrium, then it is a
DRM equilibrium that implements o·Σ. We next show that no player ever wants to deviate
in Σ, working backwards.
Stage 2. Suppose there has been no prior deviation: the principal offered mΣ in stage 1.
37More precisely, it is decreasing in his type in the separating equilibria and constant in his type in the
pooling outcomes.
38That is, if the agent accepts and the principal reports ”t”, the mechanism mΣ implements the outcome otΣ.
39Formally, Σ =(
p··,Π· (·) , π·
)where, for all t, pt
mΣ= 1 and, for m 6= mΣ, pt
m = 0. If m = mΣ, the agent’s
beliefs are just the prior beliefs(Π· (mΣ) = Π·) and the stage-2 distribution πmΣputs unit weight on the action
profile(sP· ,s
A)= ((0, ...,T ) ,”accept”). (By sP
· = (0, ...,T ) we mean the pure strategy in stage 2 in which
the principal of each type t truthfully reports his type.) If m 6= mΣ, beliefs and stage-2 play are as in Σ:
Π· (m) = Π· (m) and πm = πm.
100
If the agent chooses ”reject”, she gets zero. Let MΣ ⊂M denote the set of mechanisms that
some type t chooses in Σ with probability ptm > 0. If the agent accepts, she gets
T
∑t=0
ΠtU t (ot
Σ
)=
T
∑t=0
Πt
∑m∈M
ptm ∑
sP· ∈(SP
m)T+1
∑sA∈SA
m
πm
(sP· ,s
A)
U t(
osPt ,s
A
m
)by (121) and (133)
= ∑m∈MΣ
T
∑s=0
Πs ps
m
T
∑t=0
Πt ptm
∑Ts=0 Πs ps
m∑
sP· ∈(SP
m)T+1
∑sA∈SA
m
πm
(sP· ,s
A)
U t(
osPt ,s
A
m
)
= ∑m∈MΣ
T
∑s=0
Πs ps
m ∑sP· ∈(SP
m)T+1
∑sA∈SA
m
πm
(sP· ,s
A) T
∑t=0
Πt (m)U t
(osP
t ,sA
m
)
= ∑m∈MΣ
T
∑s=0
Πs ps
m ∑sP· ∈(SP
m)T+1
∑sA∈SA
m
πm
(sP· ,s
A)
uAm
(sP· ,s
A)
by (127)
= ∑m∈MΣ
T
∑s=0
Πs ps
mUm (πm) by (130)
which is nonnegative by Claim 14: she is willing to accept mΣ. As for a type-t principal,
in Σ he has the option of imitating any other type t ′ yet chooses not to. Thus, his payoff
from doing so, V t(
ot ′Σ
), must not exceed the payoff he gets in equilibrium in Σ, which is
V t (otΣ
). Since these are also the principal’s payoffs from choosing t ′ and t, respectively, in
Σ, he will not deviate in Σ either.
Stage 1. If the principal deviates in stage 1 by proposing some m 6= mΣ, then he and
the agent will conform to Σ thereafter. Thus, his payoffs in Σ and Σ from proposing m are
the same. But this payoff in Σ cannot exceed his payoff from sticking to his equilibrium
strategy in Σ, which is V t (otΣ
): the same payoff that he gets in Σ in equilibrium. Accordingly,
the principal is willing not to deviate in stage 1.
Finally, beliefs in an equilibrium are arbitrary after deviations, so the agent’s beliefs
following an offer m 6= mΣ may equal Π· (m) or anything else. We conclude that Σ is an
equilibrium as claimed. Q.E.D.Claim 15
PROOF OF CLAIM 16. The payoff functions V t and U t are linear in the components qt and
ρ t of o·. Hence any convex combination of any two allocations (o·, o·) that each satisfies
(136) and (137) must also satisfy these conditions. Thus, the set of allocations that satisfy
101
(136) and (137) is convex. The result then follows from Proposition 16.E.2 in Mas-Collel,
Whinston, and Greene [11]. Q.E.D.Claim 16
PROOF OF CLAIM 18.
Part 1. Define the following set.
Definition 18. AUp is the set of allocations o· ∈O that satisfy upwards incentive-compatibility
for the principal:
V s (os)≥V s (ot) for all types 0≤ s < t ≤ T ; (208)
and, for all types, are individually rational for the agent:
U t (ot)≥ 0 for all types 0≤ t ≤ T . (209)
The proof of Part 1 consists of showing (a) any RSW allocation is Pareto optimal in
AUp and (b) any solution to RLP is Pareto optimal in AUp and vice-versa. We begin with
two useful properties of AUp.
Claim 31. AUp contains a Pareto optimal allocation.
Proof. For any allocation o·, let SW (o·) denote the sum ∑Tt=0V t (ot) of payoffs of the
different types of principal. The function SW (·) is linear in the arguments qt and ρ t of
o· and thus continuous in o·. Define AUp0 to be the subset of AUp for which each type of
principal gets a nonnegative payoff:
V t (ot)≥ 0 for all types t. (210)
The set AUp0 is nonempty (it includes o·0), closed (it is defined by weak inequalities of linear
functions), and bounded (as the sum of the payoffs U t (ot)+V t (ot) = (1−δ )qotf (t) is
bounded and, by (209) and (210), each payoff is nonnegative). Hence AUp0 is compact. As
SW (·) is linear and thus continuous, it must have a maximand o· on the nonempty, compact
set AUp0 by the extreme value theorem. Clearly, o· is Pareto optimal in AUp
0 . If there is an
allocation o· in AUp−AUp0 that Pareto dominates o· then V t (ot)≥V t (ot)≥ 0 for each type
t whenceo· is in AUp0 - a contradiction.40
40See Mas-Collel, Whinston, and Greene [11, Prop. 16.E.2].
102
Claim 32. If o· ∈ AUp is not Pareto optimal in AUp, then there exists an allocation o· in AUp
that Pareto dominates o· and is Pareto optimal in AUp.
Proof. By definition 5, there is an allocation o· ∈ AUp that Pareto dominates o·. Let
AUpo· =
o· ∈ AUp : V t (o·)≥V t (o·) for each type t
(211)
be the set of allocations in AUp that are at least as good as o· is for each type t of principal.
The set AUpo· is nonempty (as it contains o·); closed (as it is defined by weak inequalities of
linear functions); and bounded (as the sum of the payoffs U t (ot)+V t (ot) = (1−δ )qotf (t)
is bounded, U t (ot)≥ 0, and V t (ot)≥V t (o·)>−∞). Thus, the function SW (o·)=∑Tt=0V t (ot)
on AUpo· has a maximizer o· in AUp
o· , which is Pareto optimal in AUpo· and thus also in AUp.
(An allocation o· that Pareto dominated o· in AUp would, by transitivity, Pareto dominate
o· and thus lie in AUpo· , contradicting the Pareto optimality of o· in AUp
o· .) Finally, as o· is in
AUpo· , it Pareto dominates o· since o· does.
We next show that the solutions to RLP coincide with the Pareto optimal allocations in
AUp. More precisely, for each Pareto optimal allocation o· in AUp and each type t, the payoff
V t (ot) of the principal equals the seller’s unique payoff u∗ (t) in RLP, while the agent’s
payoff U t (ot) is zero as in RLP. Moreover, the issuance function defined by q∗ (t) = qot
solves RLP. Finally, the expected transfer ρotcoincides with the unique transfer function
ρ∗ (t) in RLP and equals the conditional expected value q∗ (t) f (t) of the issuance in RLP.
The Claim also proves the converse: any solution to RLP corresponds to a Pareto-optimal
element of AUp.
Claim 33. 1. Suppose o· is Pareto optimal in the set AUp. Then for each type t, the
agent’s payoff U t (ot) is identically zero and the functions given by u∗ (t) = V t (ot),
ρ∗ (t) = ρot, and q∗ (t) = qot
solve RLP.
2. Let (u∗,ρ∗,q∗) solve RLP. Then any allocation o· that satisfies qot= q∗ (t) and ρot
=
ρ∗ (t) for all types t is Pareto optimal in the set AUp and, for each type t, yields the
payoffs V t (ot) = u∗ (t) and U t (ot) = 0.
103
Proof. Part 1. Suppose o· is Pareto optimal in AUp. Let t be the lowest type for which
U t (ot) is not identically zero:
0 <U t (ot)= qotf (t)−ρ
ot. (212)
If the IC constraint (208) does not bind for any s < t, then we can raise the transfer ρotthat
type t gets by ι =U t (ot)> 0 without violating (208) or (209), which contradicts the Pareto
optimality ofo· in AUp. Now suppose the IC constraint (208) does bind for some type s < t.
Let
w = min
s≤ t : V s (os) =V s (ot) and U s (ot)> 0
(213)
be the lowest type s ≤ t for which (208) binds and U s (ot) > 0. (If there is no such type
s < t, then w equals t.) By construction, Uw (ot) is positive. Let
ε ∈(
0,min
1,Uw (ot)
2(1−δ )qot f (w)
)(214)
and define41
ι (ε) = min
Uw (ot)
2,min
s<w
(V s (os)−V s (ot)+ εδqot
[ f (w)− f (s)])
> 0. (215)
Now consider the alternative allocation o· given by os = os for s 6= w and ow = (q,ρ) where
q = (1− ε)qotand ρ = ρot − εδqot
f (w)+ ι (ε). For s < w,
V s (os)−V s (ow) =V s (os)−V s (q,ρ)
=V s (os)−[ρ
ot− εδqot
f (w)+ ι (ε)−δ (1− ε)qotf (s)
]=V s (os)−V s (ot)+ εδqot
[ f (w)− f (s)]− ι (ε)
which is nonnegative by (215): no type s < w prefers ow to os. Thus, the IC constraint
(208) is satisfied in o·. Moreover,
Uw (ow) = (1− ε)qotf (w)−
[ρ
ot− εδqot
f (w)+ ι (ε)]
=Uw (ot)− ε (1−δ )qotf (w)− ι (ε)
41If w = 0, then ι is simply Uw (ot)/2 > 0. If w > 0, then the inner min in (215) is positive since for each
s < w, either V s (os)>V s (ot) or U s (ot)≤ 0; the latter and Uw (ot)> 0 jointly imply 0 <Uw (ot)−U s (ot) =
qot[ f (w)− f (s)].
104
which is nonnegative by (214) and (215): the IR constraint (209) for type w is also satisfied
in o·. Hence, o· lies in AUp. But the payoff of the type-w principal in o· is
V w (ow) =[ρ
ot− εδqot
f (w)+ ι (ε)]−δ (1− ε)qot
f (w)
=V w (ot)+ ι (ε)>V w (ot)which contradicts the Pareto optimality of o· in AUp. We conclude that, if o· is Pareto
optimal, then U t (ot) = 0 for all t.
Now let t be the lowest type for which the functions u∗ (t) = V t (ot), ρ∗ (t) = ρot,
and q∗ (t) = qotdo not solve RLP. As U t (ot) = 0, it follows that ρot
= qotf (t), whence
V t (ot) = (1−δ )qotf (t). Hence, if q∗ (t) = qot
solves RLP, so do u∗ (t) = V t (ot) and
ρ∗ (t) = qotf (t). Thus, q∗ (t) = qot
must not solve RLP. By assumption, t is the lowest
type for which the conclusion fails: u∗ (s) = V s (os), ρ∗ (s) = ρos, and q∗ (s) = qos
solve
RLP for each s < t. As o· is Pareto optimal in the set AUp, we have, for all 0 ≤ s < t,
u∗ (s) = V s (os) ≥ V s (ot) = qot[ f (t)−δ f (s)] whence qot
is in Qt . Thus, there must be
a q′ in Qt such that (1−δ )q′ f (t) > (1−δ )qotf (t). Let us now replace the allocation o·
with the allocation o· given by ot = (q′,q′ f (t))and, for all s 6= t, os = os. As U t (ot) = 0,
o· satisfies (209) for t = T . Since q′ is in Qt , for each s < t we have V s (os) = u∗ (s) ≥
q′ [ f (t)−δ f (s)] =V s (ot): o· satisfies (208) for t = T . Thus, o· is in AUp, each type s 6= t
of principal is indifferent between o· and o·, and the type t principal is better off under o·
than o·. This contradicts the Pareto optimality of o· in AUp.
Part 2. Let (u∗,ρ∗,q∗) solve RLP. Let o· be an allocation that satisfies qot= q∗ (t) and
ρot= ρ∗ (t) for all types t. This implies that
V t (ot)= ρot−δqot
f (t) = ρ∗ (t)−δq∗ (t) f (t) = u∗ (t) (216)
(which is positive by PROPOSITION 1) and
U t (ot)= qotf (t)−ρ
ot= q∗ (t) f (t)−ρ
∗ (t) = 0
105
as claimed. Moreover, for s < t,
V s (os) = u∗ (s) by (216)
≥ ρ∗ (t)−δq∗ (t) f (s) as q∗ (t) ∈ Qt and ρ
∗ (t) = q∗ (t) f (t)
= ρot−δqot
f (s) =V s (ot) by hypothesis on o·.
It follows that o· lies in AUp. By Claim 31, AUp contains a Pareto optimal element o·.
By part 1, for each type t, the agent’s payoff U t (ot) is zero and the functions given by
u∗ (t) = V t (ot), ρ∗ (t) = ρ ot, and q∗ (t) = qot
solve RLP. But by PROPOSITION 1, the
solutions u∗ (·) and ρ∗ (·) to RLP are unique. Accordingly, V t (ot) = V t (ot) for each t,
whence o· is Pareto optimal in AUp.
AUp has the following relation to WIE:
Claim 34. 1. If o· is Pareto optimal in AUp, then it is WIE.
(a) If o· is WIE and U t (ot)≥ 0 for each type t, then o· is in AUp.
(b) If o· is WIE and U t (ot) = 0 for each type t, then o· is Pareto optimal in AUp.
Proof. Part 1. By part 1 of Claim 33, the functions u∗ (t) = V t (ot), ρ∗ (t) = ρot, and
q∗ (t) = qotsolve RLP. Hence, by PROPOSITION 1, for any types s and t,42
V t (ot)= u∗ (t)≥ ρ∗ (s)−δq∗ (s) f (t) = ρ
os−δqos
f (t) =V s (ot) . (217)
Thus, o· satisfies (134). Now suppose that o· is not WIE. Then there is an allocation o·
that satisfies (135), (136), and (137). By (136), o· satisfies (208). Since o· is in AUp,
V t (ot) ≥ V t (ot) by (135) and U t (ot) ≥U t (ot) = 0 by (137): o· satisfies (209). Thus, o·
is in AUp and, by (135), Pareto dominates o·, which contradicts the assumption that o· is
Pareto optimal in AUp.
42PROPOSITION 1 characterizes a profile e∗, in which a type t seller sells the quantities q∗ (t) in return
for the revenue ρ∗ (t), and gets the payoff u∗ (t). Part 3 shows that e∗ is an equilibrium, in part by showing
the inequality in (217) holds for all types s and t.
106
Part 2(a). Since U t (ot) ≥ 0 for each type t, o· satisfies (209). Since o· is WIE, it also
satisfies (208) by (134): it is in AUp.
Part 2(b). By part 2(a), o· is in AUp. If o· is not Pareto optimal in AUp, then by Claim
32 there exists an allocation o· in AUp that Pareto dominates o· and is Pareto optimal in
AUp. Hence, by part 1 of Claim 33, the functions given by u∗ (t) =V t (ot), ρ∗ (t) = ρ ot, and
q∗ (t) = qotsolve RLP. Thus, by part 1 of PROPOSITION 1, the allocation o· is incentive
compatible: it satisfies (136). As o· is in AUp, it also satisfies (137) by (209). Finally,
by assumption, o· Pareto dominates o·: o· = o· satisfies (135). But this contradicts the
assumption that o· is WIE.
Part 1 of Claim 18 now follows from Claims 33 and the following result.
Claim 35. Any RSW allocation o· is Pareto optimal in AUp.
Proof. Let o· be an RSW allocation. By Claim 31, there is an allocation o· that is Pareto
optimal in AUp and, by part 1 of Claim 33, the functions given by u∗ (t) =V t (ot), ρ∗ (t) =
ρot, and q∗ (t) = qot
solve RLP. Thus, by part 1 of PROPOSITION 1, the agent’s payoff
U t (ot) is identically zero: o· satisfies (140). By part 3 of PROPOSITION 1, o· also satisfies
(139). Thus, by definition 14, V t (ot) is not less than V t (ot) which, we have shown, equals
u∗ (t). Since, moreover, o· satisfies (208) by (134) and (209) by (140), o· is in AUp.Now
suppose o· is not Pareto optimal in AUp. By Claim 32, there exists an allocation o· in AUp
that Pareto dominates o· and is Pareto optimal in AUp. Hence, o· satisfies (139) by part 1 of
Claim 34 and (140) by part 1 of Claim 33, so V t (ot)≥V t (ot) for all t by Definition 14: o·
does not Pareto dominate o· - a contradiction.
Part 2.An RSW allocation o· exists by Claim 17 and, by part 1, satisfies
V t =V t (ot)= u∗ (t) . (218)
Now consider the alternative allocation o· = (q·,ρ ·) defined by qt = q∗ (t) and ρ t = ρ∗ (t).
We claim that it is an RSW allocation as well. We first verify that it satisfies (139) and
107
(140) for all t. Equation (140) holds since, for each s, U s (os) equals q∗ (s) f (s)−ρ∗ (s)
which, by (143), is zero: the agent’s reservation payoff. Moreover,
V s (os) = ρ∗ (s)−δq∗ (s) f (s) = u∗ (s) (219)
for all s by (144) and
V s (or) = ρ∗ (r)−δq∗ (r) f (s) = q∗ (r) [ f (r)−δ f (s)]
by (143). Hence, to show (139), it suffices to show that u∗ (s) ≥ q∗ (r) [ f (r)−δ f (s)] for
all types r,s. But this has already been shown in the proof of part 3 of PROPOSITION 1.
We conclude that o· solves (139) and (140) for all t. Finally, for each t, V t (ot) attains the
maximum payoff V t possible in Program IIt by (218) and (219). Hence, ot solves Program
IIt for all t: o· is an RSW allocation. Q.E.D.Claim 18
PROOF OF CLAIM 19. The payoff functions V t and U t are linear in qt and ρ t . Hence
any convex combination of any two allocations (o·, o·) that each satisfies (147) and (148)
must also satisfy these conditions. Thus, the set of allocations that satisfy (147) and (148)
is convex. The result then follows from Proposition 16.E.2 in Mas-Collel, Whinston, and
Greene [11]. Q.E.D.Claim 19
PROOF OF CLAIM 20. Let C = o· ∈ A : V t (ot)≥V t (os) for all s, t denote the set of
incentive-compatible allocations. Since the function V t (o) is linear in each element of
o = (qo,ρo), C is a convex subset of A.43 Let o· be WIE. By Claim 16 and Bertsekas [3,
Proposition 5.3.1],44
o· ∈ argmaxo·∈C
∑
Tt=0 wtV t (ot)+∑
Tt=0 ν
t [U t (ot)−U t (ot)]43Defined in (123), A is the set of all possible allocations.
44To satisfy condition (1) in Bertsekas [3, Proposition 5.3.1], begin with o· and then lower the transfer ρ t
that the agent pays the principal by some common (type-independent) amount ε > 0. The resulting allocation
is in C and, moreover, strictly satisfies (137): condition (1) holds.
108
for some positive weights wtTt=0 and nonnegative multipliers ν tT
t=0. Let ν = ∑Tt=0 ν t .
For each t, let Πt equal vt/ν if ν > 0 and 1/(T +1) otherwise. Then
o· ∈ argmaxo·∈C
∑
Tt=0 wtV t (ot)+ν ∑
Tt=0 Π
t [U t (ot)−U t (ot)] .Hence, by Claim 19 and Bertsekas [3, Proposition 5.3.1], o· is IE with respect to the beliefs
Π·: Π(o·) is nonempty. The proof of convexity of the set Π(o·) is identical to that of MT
[12, p. 17]. Q.E.D.Claim 20
PROOF OF CLAIM 21. Let o· be an RSW allocation. By Claim 17, o· is Pareto optimal
in AUp whence, by part 1 of Claim 33, the agent’s payoff U t (ot) is identically zero and
the functions given by u∗ (t) = V t (ot), ρ∗ (t) = ρ ot, and q∗ (t) = qot
solve RLP. Now fix
a small ε > 0 and consider the allocation o· given by, for each type t, qt = qotand ρ t =
ρ ot − (t +1)ε . Then for any types s and t,
V s (ot)=V s (ot)− (t +1)ε. (220)
If type s proposes o·, the agent will accept. Why? First, the agent gets a positive payoff if
type s chooses os as
U s (os) = (s+1)ε > 0. (221)
Moreover, type s will not imitate any type t > s since V s (os)−V s (ot) =V s (os)−V s (ot)+
(t− s)ε by (220) which is positive as o· is in AUp. Finally, if type t imitates type s < t then,
by ASSUMPTION A, the agent gets no less than U s (os) which again is positive by (221).
As the agent knows her payoff will be positive,she will accept. As for the principal, if her
type is s then by proposing o· and then choosing os she receives V s (os)−(s+1)ε by (220).
Thus, she will deviate from any putative equilibrium that offers her less than this. As this is
true for all ε > 0,her payoff in any equilibrium cannot be less than V s (os). Q.E.D.Claim 21
PROOF OF PROPOSITION 3.
By Claim 23, the RSW allocation o· is IE relative to some nonnegative beliefs Π· ∈
ℜT+1+
109
Part 1: only if. Condition (151) is necessary since, otherwise, the principal could
strictly gain by imitating a different type. Condition (152) is necessary since, otherwise,
the agent could do better by always playing ”reject”. And (153) is necessary by Claim 21.
We conclude that (151), (152), and (153) are necessary conditions for o· to be the expected
allocation of an equilibrium.
Part 2: if. Consider any allocation o·= o· that satisfies (151), (152), and (153). We will
prove the existence of a DRM equilibrium o·, which is intuitive if o· is an RSW allocation
o·.
In such an equilibrium, the principal always chooses the DRM o·. Thus, if the principal
conforms to the equilibrium in stage 1, the agent’s interim beliefs are simply her prior
beliefs Π·: by (152), she will accept in stage 2. Moreover, the principal is willing to
truthfully report his type in the DRM o· by (151). It remains only to show that there exist
interim beliefs that deter the principal from deviating to a mechanism other than o·, and
that these beliefs are intuitive if o· is an RSW allocation.
For any alternative mechanism m ∈M, define
∆Σm =
Π· ∈ ∆T : Πt = 0 for all t ∈ Jm
if Jm 6= ϒ
∆T if Jm = ϒ
(222)
where Jm, defined in (149), is the set of types t who will never choose m over the RSW
allocation. If some types are not in Jm, then ∆Σm is the set of interim beliefs that assign
probability zero to all types in Jm; else it is the full set of all possible beliefs. As m is
unexpected, any interim beliefs Π· (m) are consistent with Bayes’s Rule. If, in addition, the
interim beliefs Π· (m) lie in ∆Σm, then they satisfy IntΣm:
Claim 36. ∆Σm is compact and the beliefs in ∆Σ
m satisfy IntΣm.
Proof. Compactness is immediate from (222). As for IntΣm, if Jm 6= ϒ then the beliefs in ∆Σm
put zero weight on any type in Jm by (222). And if Jm = ϒ, then no type t in ϒ−Jm satisfies
(150) as ϒ−Jm is empty: IntΣm holds for any beliefs. In both cases, IntΣm holds for all beliefs
in ∆Σm.
110
We will show the following condition holds for each m.
Deter(m). For each mechanism m, there exist interim beliefs Π· (m) in ∆Σm and associated
equilibrium payoffs(V · (m) ,U (m)
)in Ψm
(Π· (m)
)such that V t (m) ≤ V t for all t:
the type-t principal does not prefer m to ot .
Suppose Deter(m) holds for each out-of-equilibrium mechanism m. Consider the strategy
profile in which any unexpected mechanism m leads to the beliefs Π· (m) in ∆Σm and associated
equilibrium payoffs(V · (m) ,U (m)
)described in Deter(m). Then as long as the agent will
surely accept the DRM o·, a type-t principal is willing to choose o· and then, by (151), to
choose the outcome ot . And since each type of principal chooses o·, the agent’s interim
beliefs in response to o· are her prior beliefs Π·, so she is indeed willing to accept by (152).
Hence, the DRM o· is supported by an equilibrium. If, further, o· = o·, then IntΣm holds
since Π· (m) lies in ∆Σm: the given equilibrium is intuitive, so the RSW allocation o· is the
expected allocation of an intuitive equilibrium as claimed.
It remains only to prove Deter(m) for each mechanism m other than the DRM o·.
Suppose, to the contrary, that there is a mechanism m for which Deter(m) fails to hold: for
any interim beliefs Π· (m) in ∆Σm and equilibrium payoffs
(V · (m) ,U (m)
)in Ψm
(Π· (m)
),
there is a type t for whom V t (m)> V t . We will derive a contradiction. A key step is the use
of Kakutani’s [9] fixed point theorem. To verify the assumptions of Kakutani’s theorem,
we must first derive a series of technical results.
Claim 37. Let S be any compact subset of ∆T . The correspondences φm : S⇒ Φm and
ψm : S⇒ ℜT+1×ℜ defined in (129) and (132) are upper hemicontinuous and, for any
interim beliefs Π· ∈ S, the sets φm
(Π·)
and ψm
(Π·)
are nonempty and convex.
Proof. We first show an equivalence between CEIGΠ·m and Aumann’s [2] original notion of
correlated equilibrium in a symmetric information setting.
Lemma 7. A distribution πm∈Φm is a CEIGΠ·m if and only if it is a correlated equilibrium
(Aumann [2]) of the complete information game with pure strategy sets(SP
m)T+1 and SA
m
111
and payoff functions
uPm
(sP· ,s
A)=
T
∑t=0
V t(
osPt ,s
A
m
). (223)
and uAm,Π·
for the principal and agent, respectively. These payoff functions are continuous
in the interim beliefs Π·, and the pure strategy sets(SP)T+1 and SA are finite.
Proof. Equation (125) holds if and only if
∑sA∈SA
m
πm
(sP· ,s
A)
uPm
(sP· ,s
A)≥ ∑
sA∈SAm
πm
(sP· ,s
A)
uPm
(sP· ,s
A)
for any sP· , s
P· ∈
(SP
m)T+1
(224)
Why? If (125) holds then, summing over types t, one obtains (224). Conversely, if (125)
does not hold then, for some type t and actions sP· ,s
P· , ∑sA∈SA
mπm(sP· ,s
A)V t(
osPt ,s
A
m
)<
∑sA∈SAm
πm(sP· ,s
A)V t(
osPt ,s
A
m
). Defining sP
· by sPt = sP
t and, for each t ′ 6= t, sPt ′ = sP
t ′ , we have
∑sA∈SA
m
πm
(sP· ,s
A)
uPm
(sP· ,s
A)< ∑
sA∈SAm
πm
(sP· ,s
A)
uPm
(sP· ,s
A),
whence (224) fails, as claimed. Hence, πm is a CEIGΠ·m if and only the conditions (126)
and (224) hold. By Fudenberg and Tirole [7, p. 57], these two conditions are necessary
and sufficient for πm to be a correlated equilibrium in the sense of Aumann [2]. Finally,
continuity and finiteness are trivial.
By Lemma 7 and Cotter’s [5] result for the degenerate symmetric information case, the
correspondence φm is upper hemicontinuous on ∆T and, for any interim beliefs Π· ∈ ∆T ,
the set φm
(Π·)
is nonempty and convex. By (132), ψm
(Π·)
is nonempty since φm
(Π·)
is. By (130) and (131), for any λ in [0,1],
λ(V ·m (πm) ,Um (πm)
)+(1−λ )
(V ·m(π′m),Um
(π′m))
=(V ·m(λπm +(1−λ )π
′m),Um
(λπm +(1−λ )π
′m))
whence ψm
(Π·)
is convex as well.
It remains to show that φm and ψm are upper hemicontinuous on any compact set
S ⊂ ∆T . Since φm is upper hemicontinuous on ∆T , if(
Π·n,πnm
)→(
Π·,πm
)with Π·n ∈ ∆T
112
and πnm ∈ φm
(Π·n
)for all n, then πm ∈ φm
(Π·)
, so it suffices to show that if Π·n ∈ S for
all n, then Π· = limn→∞ Π·n is also in S, but this must hold as S is compact: φm is upper
hemicontinuous on S. As for upper hemicontinuity of ψm on S, suppose(
Π·n,(V ·n,Un
))→(
Π·,(V ·,U
))with Π·n ∈ S and
(V ·n,Un
)∈ ψm
(Π·n
)for all n. As just shown, Π· ∈ S, so we
must merely show that(V ,U
)∈ ψm
(Π·)
. By (132), for each nthere is an πnm ∈ φm
(Π·)
such that V ·n = V ·m (πnm) and Un = Um (πm). By taking subsequences if needed, we may
assume that πm = limn→∞ πnmexists. Hence,
(Π·n,π
nm
)→(
Π·,πm
)with πn
m ∈ φm
(Π·n
)for
all n. Thus, by the upper hemicontinuity of φm on S, πm ∈ φm
(Π·)
whence, by (132),(V ·m (πm) ,Um (πm)
)∈ ψm
(Π·)
. Finally, since πnm converges to πm, (130) and (131) imply
that(V ·m (πn
m) ,Um (πnm))=(V ·n,Un
)converges to
(V ·m (πm) ,Um (πm)
)but by assumption
limn→∞
(V ·n,Un
)=(V ·,U
)whence
(V ·,U
)=(V ·m (πm) ,Um (πm)
)∈ ψm
(Π·)
as claimed:
ψm is upper hemicontinuous on S. This concludes the proof of Claim 37.
Let
Ψm =⋃
Π·∈∆T
ψm
(Π·)
(225)
be the set of all correlated equilibrium payoff vectors in the mechanism m, for any interim
beliefs Π· in ∆T .
Claim 38. Ψm is compact.
Proof. We first show that it is bounded. Since we assume ρ ∈ [−ρ,ρ] for some arbitrarily
large but finite ρ , the worst that could happen to the principal is to pay the agent ρ and also
give the agent his entire portfolio; his payoff would then be at least−ρ−δa f (T ). The best
that could occur is for him to retain his whole portfolio and receive a transfer of ρ from the
agent, giving him a payoff of ρ . Hence, the principal’s payoff lies in [−ρ−δa f (T ) ,ρ] for
each type. Likewise, the agent’s payoff lies in [−ρ,ρ +a f (T )]. Hence, Ψm is bounded.
We now show that it is closed. Let(V ·n,Un
)→(V ·,U
)with
(V ·n,Un
)∈ Ψmfor all n.
Then by definition of Ψm, for each n there is a Π·n in ∆T such that(V ·n,Un
)∈ ψm
(Π·n
).
By taking subsequences if needed, we can assume that Π·n converges to a limit Π·. That is,(Π·n,(V ·n,Un
))→(
Π·,(V ·,U
))with Π·n ∈ ∆T and
(V ·n,Un
)∈ψm
(Π·n
)for all n. As shown
113
in the proof of Claim 37, this implies that Π· ∈ ∆T and(V ,U
)lies in ψm
(Π·)
which is a
subset of Ψm: Ψm is closed.
By Claim 23, the RSW allocation is IE with respect to some nonnegative prior beliefs
Π·. Let us now modify the game in two ways: (a) the agent’s prior beliefs are Π· rather
than Π· and (b) the principal must choose either the mechanism m or the RSW DRM o·. In
the modified game G, let (V ·,U) ∈Ψm denote the correlated equilibrium payoff vector that
results when the principal chooses m. In this situation,
Pt (V ·,U) = argmaxp∈[0,1]
[pV t +(1− p)V t
](226)
is the type-t principal’s set of optimal probabilities of choosing m. Let P· (V ·,U)=∏Tt=0 Pt (V ·,U)
be the corresponding set of optimal probability vectors45 and let
P(Ψm) = P· (V ·,U) : (V ·,U) ∈Ψm
be the image of Ψm under P·. Clearly, P(Ψm) is a subset of
PΣm =
p· ∈ [0,1]T+1 : pt = 0 for all t ∈ Jm
. (227)
Claim 39. The correspondence P· : Ψm⇒ PΣm is upper hemicontinuous and, for any payoff
vector (V ·,U) ∈Ψm, the set P· (V ·,U) is nonempty and convex. Moreover, PΣm is compact.
Proof. Compactness of PΣm is immediate from (227). For any V · in ℜT+1, Pt (V ·) is
clearly nonempty and convex whence so is P· (V ·). For upper hemicontinuity, suppose
that (V ·n, p·n)→ (V ·, p·)with V ·n ∈ S and p·n ∈ P· (V ·n)for all n. Then V · ∈ S as S is compact.
It remains to show that p· ∈ P· (V ·) - or, equivalently, that pt ∈ Pt (V ·) for each type t.
This is trivial if V t = V t since then Pt (V ·) = [0,1] which must contain pt . If instead
V t < (>)V t , then Pt (V ·) equals 0 (resp., 1) but also, for high enough n, V tn < (>)V t
whence ptn equals 0 (resp., 1) and thus limn→∞ pt
n is also 0 (resp., 1) which lies in Pt (V ·), a
contradiction. Since this is so for each type t, p· = limn p·n lies in P· (V ·) as claimed.
45The symbol ∏ here denotes the Cartesian product.
114
In the modified game G, let m lead to the correlated equilibrium payoff vector (V ·,U).
Let the principal’s strategy be p· where pt is the type t principal’s chance of choosing m.
If the principal chooses m, what is the set of possible interim beliefs Π· of the agent if her
prior is Π·?First, if the principal’s ex-ante probability P(p·) = ∑Tt=0 ptΠt of choosing m is
zero, the choice of m is unexpected: the agent’s beliefs are arbitrary. In this case, we will
restrict beliefs to lie in the set ∆Σm defined in (222). On the other hand, if P(p·) is positive,
the agent’s beliefs are uniquely given by Bayes’s Rule: Πt = ptΠt/P(p·) for all t. Hence,
Πt is positive only if pt > 0 which, by (227), implies t /∈ Jm since p· ∈ PΣm. Thus, Πt lies in
∆Σm in this case as well. Combining the two cases, our assumptions imply that for each p·
in PΣm, the agent’s interim beliefs Π· on seeing m will lie in the set
β (p·) =
(
p0Π0
P(p·), ..., pT ΠT
P(p·)
)⊂ ∆Σ
m if P(p·)> 0
∆Σm if P(p·) = 0
(228)
which, in each case, is a subset of ∆Σm. Let β
(PΣ
m)=
β (p·) : p· ∈ PΣm⊂ ∆Σ
m be the image
of PΣm under β .
Claim 40. The correspondence β : PΣm ⇒ ∆Σ
m is upper hemicontinuous and, for any vector
p· ∈ PΣm of probabilities, the set β (p·) is nonempty and convex.
Proof. By (228), β (p·) is nonempty and convex. For upper hemicontinuity, suppose
(p·n,x·n)→ (p·,x·)with p·n ∈ PΣ
m and x·n ∈ β (p·n)for all n. Then p· ∈ PΣm since PΣ
m is compact
(Claim 39). It remains to show that x· ∈ β (p·). Since x·n ∈ β (p·n)⊂ ∆Σm for all n and ∆Σ
m is
closed, x· = limn→∞ x·n is also in ∆Σm; thus, if P(p·) is zero we are done. If P(p·) is positive,
then, since the function P(p·n) is continuous in p·n, there is a n∗ such that, for all n > n∗,
P(p·n) is positive whence x·n ∈ β (p·n) must equal(
p0nΠ0
P(p·n), ...,
pTn ΠT
P(p·n)
)and thus46
x· = limn→∞
x·n = limn→∞
(p0
nΠ0
P(p·n), ...,
pTn ΠT
P(p·n)
)=
(p0Π0
P(p·), ...,
pT ΠT
P(p·)
)∈ β (p·) .
Hence β is upper hemicontinuous.
46This uses the fact that f (x,y) = g(x)/h(y) is continuous at (x0,y0) if g and h are continuous and h(y0) 6=
0.
115
Let Dm denote the Cartesian product Ψm × PΣm × ∆Σ
m, which is compact by Claims
38, 39, and 36, resp. Consider the correspondence r : Dm ⇒ Dm that maps each d =((V ·,U) , p·,Π·
)in Dm to r (d) = ψm
(Π·)×P· (V ·)× β (p·). For any d in Dm, r (d) is
nonempty and convex since ψm
(Π·)
, P· (V ·), and β (p·) are by Claims 37, 39, and 40,
resp. Moreover, r is upper hemicontinuous since ψm, P·, and β have this property by the
same Claims. Thus, by Kakutani’s [9] fixed point theorem, r has a fixed point((V ·∗,U∗) , p·∗,Π
·∗
)∈ Dm (229)
satisfying
(V ·∗,U∗) ∈ ψm
(Π·∗
), (230)
p·∗ ∈ P· (V ·∗) , and (231)
Π·∗ ∈ β (p·∗) . (232)
By (230), there is a correlated equilibrium π∗m ∈ φm
(Π·)
such that
(V ·∗,U∗) =(V ·m (π∗m) ,Um (π∗m)
). (233)
Moreover, the beliefs Π·∗ are in ∆Σm by (229) and thus satisfy IntΣm by Claim 36.
For the mechanism m, we have now constructed beliefs Πt (m) = Π·∗ (which satisfy
IntΣm) and associated correlated equilibrium payoffs(V · (m) ,U (m)
)=(V ·∗,U∗
)of the stage-
2 game. By hypothesis, there is a type t whose payoff V t (m) exceeds V t .
What is the expected type-contingent outcome in the stage-2 equilibrium of m? For
each pure action profile(sP· ,s
A) ∈ (SPm)T+1×SA
m, the probability is π∗m(sP· ,s
A) that(sP· ,s
A)is played. If, moreover, the principal’s type is t, the resulting outcome is osP
t ,sA
m for sure.
Thus, the expected outcome conditional on the principal choosing m and his type being t is
ot∗ = ∑
sA∈SAm
∑sP· ∈(SP
m)T+1
π∗m
(sP· ,s
A)
osPt ,s
A
m . (234)
Now consider the following scenario of the modified game G.
116
Stage 1: the principal of type t proposes m with probability pt∗ and an RSW allocation
o· with complementary probability.
Stage 2: if the principal chose o·, the agent plays ”accept” and the principal chooses
the outcome ot that corresponds to his type t. If the principal chose m, the players play the
correlated equilibrium π∗m whence the expected outcome, for a principal of type t, is ot∗.
We claim that the agent’s ex ante expected payoff in Σ is nonnegative. This payoff is
T
∑t=0
Πt [pt∗U
t (ot∗)+(1− pt
∗)
U t (ot)]≥ T
∑t=0
Πt pt∗U
t (ot∗)
by (140). (235)
If Πt pt∗ is zero for all t, we are done as the right hand side is zero. Else ∑
Ts=0 Πs ps
∗ is
positive, whence the right hand side of (235) has the same sign as
T
∑t=0
Πt pt∗
∑Ts=0 Πs ps
∗U t (ot
∗)=
T
∑t=0
Πt (m)U t (ot
∗)
by (128) with Π· = Π
·
= ∑sA∈SA
m
∑sP· ∈(SP
m)T+1
π∗m
(sP· ,s
A) T
∑t=0
Πt (m)U t
(osP
t ,sA
m
)by (121) and (234)
= ∑sA∈SA
m
∑sP· ∈(SP
m)T+1
π∗m
(sP· ,s
A)
uAm
(sP· ,s
A)
by (127)
= Um (π∗m) by (130)
which is nonnegative by Claim 14.
Now let us give the principal a third option at the initial stage in G: he can propose
the DRM o· defined, for each t, by ot = pt∗o
t∗+(1− pt
∗) ot . If the agent accepts and the
principal chooses ot , the players get the same type-contingent payoffs as in Σ above. Thus,
as Σ gives the agent a nonnegative payoff, she will accept the DRM. And as the principal
does not want to deviate in Σ, he will choose ot : o·is incentive-compatible. But recall
that m is a mechanism for which, for any beliefs Π· ∈ ∆T and any associated equilibrium
payoffs(V ·,U
), there is a type t who does better than in the RSW allocation: V t > V t .
Letting Π· = Π· (m), it follows that there is a type t for which V t∗ > V t and thus, by (230),
pt∗ = 1, so V t (ot) > V t . Since, by (230), V t (ot) = max
V t∗ ,V
t≥ V t , we have produced
an allocation, o·, that is incentive-compatible, Pareto dominates o·, and gives the agent a
117
nonnegative payoff under the beliefs Π·. Thus, by definition 15, o· is not IE under the
beliefs Π· - a contradiction. Q.E.D.Proposition 3
PROOF OF PROPOSITION 4. Let Σ =(
p··,Π· (·) ,π·
)be an intuitive equilibrium whose
expected allocation is o· = o·Σ= (q·,ρ ·). Assume w.l.o.g. that Σ is a DRM equilibrium.47
When considering deviations by the principal, it will suffice to focus on simple mechanisms
in which the agent has two actions, accept and reject, and the principal has one action: ”do
nothing”. If the agent accepts, a given outcome (q,ρ) is implemented; else the reservation
allocation (in which both get zero) is implemented. We will refer to such a mechanism,
abusing notation slightly, as m = (q,ρ).
We first show that Σ is fairly priced: the amount ρ t that the agent pays to a principal of
each type t equals the conditional expected value qt f (t) of the portfolio that she receives.
For suppose not. First suppose that the portfolio of some type t is underpriced: ρ t < qt f (t).
Let s be the largest type such that qt f (s) < qt f (t). (If there is no such type s, then the
principal can instead select the mechanism m = (qt ,qt f (t)− ε) for any ε ∈ (0,qt f (t)−ρ t)
and the agent will accept so Σ is not an equilibrium.) Choose any λ ∈ [0,1) satisfying
ρ t−δqt f (t)qt [ f (t)−δ f (t)]
< λ <ρ t−δqt f (s)
qt [ f (t)−δ f (s)]. (236)
Such a λ must exist as the ratio on the right is increasing in −qt f (s) since ρ t < qt f (t).
Also choose any
ε ∈(0,λqt [ f (t)−δ f (t)]−ρ
t−δqt f (t))
(237)
where the interval is nonempty by the first inequality in (236). Now suppose type t deviates
to the mechanism m = (λqt ,λqt f (t)− ε). By ASSUMPTION A, for all s′ < s, qt f (s′)≤
qt f (s) < qt f (t) and so, by (236), λqt [ f (t)−δ f (s′)] is less than ρ t − δqt f (s′) which, in
turn, is no greater than V s′(
os′)
since s′ is willing not to imitate t in Σ. Thus, since ε > 0
47If not, use Σ to construct the equivalent DRM equilibrium Σ as in footnote 39. It has the same equilibrium
payoffs of principal and agent for each type of principal. Moreover, beliefs following any deviation m of the
principal are the same in Σ as in Σ. Hence, since πm is intuitive, so is the belief function πm of Σ: We have
thus constructed an intuitive DRM equilibrium that implements the expected allocation o·Σ
of Σ.
118
by (237), in any stage-2 correlated equilibrium following a deviation to m, each type s′ ≤ s
gets less than her equilibrium payoff V s′(
os′)
: each type s′ ≤ s is in the set Jm defined in
(149). Now suppose the principal chooses m and the agent is sure the agent’s type is not in
Jm. Then by ASSUMPTION A, she is sure that the portfolio λqt is worth at least λqt f (t).
As accepting thus gives her at least ε > 0, she will accept in any correlated equilibrium
πm.48 This implies, in turn, that a type-t principal gains by deviating to m for any πm
since,by (236) and (237), V t (ot) = ρ t −δqt f (t) is less than λqt [ f (t)−δ f (t)]. It follows
from IntΣm that on seeing m, the agent’s interim beliefs Π· (m) must put zero weight on any
type in Jm: she will accept. But then t will surely defect: Σ is not an equilibrium.
We have shown that no portfolio is underpriced in Σ: for each t, ρ t ≤ qt f (t). Can
there be overpricing? No: as each agent has positive ex-ante probability Πt , this (with
the absence of underpricing) would imply that the agent’s ex-ante expected payoff from
accepting the DRM Σ is negative - which contradicts Claim 14.
Since Σ is fairly priced, the price paid by the agent equals the value of the assets she
receives for each type t:
ρt = qt f (t) for t = 0, ...,T. (238)
Hence, the payoff U t (ot) = qt f (t)−ρ t of the agent is identically zero. By Claim 21, the
payoff V t (ot) of the type-t principal is not less than V t which, in turn, equals the solution
u∗ (t) to RLP by Corollary 18. We claim that V t (ot)≤ u∗ (t) as well and that qt lies in the
set Qt defined in (141).If not, let t be the lowest type for which V t (ot) > u∗ (t). By fair
pricing,
V t (ot)= ρt−δqt f (t) equals (1−δ )qt f (t) for each t. (239)
But by (142), (143), and (144), u∗ (t) equals maxq∈Qt [(1−δ )q f (t)]. Thus, if V t (ot) >
u∗ (t), then qt must not lie in Qt . By (141), this means that there is a type s < t whose
equilibrium payoff V s (os) = u∗ (s) in Σ is less than qt [ f (t)−δ f (s)] which in turn, by
48Since the principal’s choice in stage 2 is trivial, correlated equilibrium implies merely that the agent will
act optimally given her beliefs.
119
(238), equals his payoff ρ t−δqt f (s) from imitating t in Σ. Thus, Σ is not an equilibrium -
a contradiction.
We have shown that in any intuitive equilibrium Σ, the payoff V t (ot) of the type-t
principal equals his RSW payoff V t which, in turn, equals u∗ (t) and the agent’s corresponding
payoff U t (ot) also equals her RSW payoff of zero. Moreover, qt ∈ Qt . If qt does not
maximize qt f (t) in this set then
V t (ot)= (1−δ )qt f (t) by (239)
< (1−δ )q∗ (t) f (t) by (142)
= u∗ (t) by (143) and (144)
=V t (ot) as shown above.
This is a contradiction. We thus have confirmed that q∗ (t) = qt , ρ∗ (t) = ρ t , and u∗ (t) =
V t (ot) are a solution to RLP. Accordingly, by part 2 of Claim 18, the allocation o· is an
RSW allocation as claimed.
We have shown that any DRM o· that satisfies (151), (152), and (153) is supported by
an equilibrium as claimed. Moreover, for each mechanism m ∈M, the property IntΣm holds
since the agent’s interim beliefs Π· (m) have been constructed to lie in ∆Σm. Thus, if o· is
an RSW allocation o·, the equilibrium is intuitive as well: any RSW allocation o· is the
expected allocation of an intuitive equilibrium, as claimed. Q.E.D.Proposition 4
PROOF OF CLAIM 25. Equations (168) through (174) are proved in the text, and (184)
follows from (96), (97), and (170). Substituting (169) into V 2 (o1) = ρ1− δa f (2) we
obtain the equality in (175). The RSW allocation o· is an equilibrium allocation by Claim
24 and q2 6= q1 = a by Claim 25, whence the inequality in (175) follows from the following
lemma.
Lemma 8. In any equilibrium allocation o· in which q2 6= q1 = a, V 2 (o2)>V 2 (o1).
By (170) and (174), V 1− V 2 equals q2 [∆− (1−δ ) f (2)] which in turn, by (97), equals
δ q2∆1, proving (176). As for (178),
V 1−V 2 (o1)= (1−δ )a f (1)−a [ f (1)−δ f (2)] = δa∆1
as claimed whence, by (176),
V 2−V 2 (o1)= [V 1−V 2 (o1)]−[V 1−V 2]= δ
(a− q2)
∆1
proving (179). Equation (180) follows from (164) and (165). As for (181) and (182), there
are two cases.
1. If V 1 ≤ ∆1, (171) and (172) imply q21 =
V 1
∆1and q2
2 = 0. Substituting these into (174),
clearing denominators, and multiplying by−1, we obtain (δ −1)V 1 f1 (2) =−V 2∆1.
Adding ∆1V 1 to both sides then yields δV 1∆11 = ∆1
(V 1−V 2
), which proves (a) the
< and = parts of (183), (b) that the second elements of the mins in (171) and (181)
are equal, and (c) that the second entry of the max in (181) is zero. This verifies (181)
and (182) for this case.
2. If V 1 > ∆1, (171) and (172) imply q21 = 1 and q2
2 = V 1−∆1∆2
. Substituting these into
(174) and clearing denominators yields
V 2∆2 = (1−δ )
[f1 (2)∆2 +
(V 1−∆1
)f2 (2)
].
Isolating the ∆2 terms on the right hand side yields(V 1−∆1
)[(δ −1) f2 (2)] = ∆2
−V 2− (δ −1) f1 (2)
.
121
Adding(
V 1−∆1
)∆2 to both sides and using (97) to substitute for ∆2 on the left and
for ∆1 on the right we obtain(V 1−∆1
)δ∆
12 = ∆2
V 1−V 2−δ∆
11
,
which (a) proves the > part of (183) and (b) implies that the second elements in the
maxes in (172) and (182) are equal. Hence, by hypothesis that V 1 > ∆1, the second
element of the max in (182) is positive, whence the second element in the min in
(181) exceeds 1. Finally, This verifies (181) and (182) for this case.
As for (177), by (171), (172), and (176), there are two cases.
1. If V 1 ≤ ∆1 then q2 =(
V 1
∆1,0)
so
V 1−V 2 = δV 1
∆1∆
11. (240)
2. If V 1 > ∆1 then q2 =(
1, V 1−∆1∆2
)so
V 1−V 2 = δ
∆
11 +
V 1−∆1
∆2∆
12
. (241)
Combining the two cases we obtain (177). Finally, we verify (184). We compute
ζ =γ
a∆1 >δ∆1
1∆1
= η ⇐⇒ ∆11a∆
1 <∆1γ
δ= ∆1
(a− q2)
∆1 = ∆1a∆
1−∆1q2∆
1
⇐⇒0 <(∆1−∆
11)
a∆1−∆1q2
∆11 (242)
When ν > 0, by (171), (172), and (183), this becomes
0 <(∆1−∆
11)
a∆1−∆1
V 1
∆1∆
11 = (1−δ ) f1 (1)a∆
1−V 1∆
11
= (1−δ ) f1 (1)a [ f (2)− f (1)]− (1−δ )a f (1) [ f1 (2)− f1 (1)]
= (1−δ ) f1 (1)a [ f (2)− f (1)]−a f (1) [ f1 (2)− f1 (1)]
= (1−δ ) f1 (1) [ f1 (2)− f1 (1)]+ f1 (1) [ f2 (2)− f2 (1)]− [ f1 (1)+ f2 (1)] [ f1 (2)− f1 (1)]
= (1−δ ) f1 (1) [ f2 (2)− f2 (1)]− f2 (1) [ f1 (2)− f1 (1)]
= (1−δ ) f1 (1) f2 (2)− f2 (1) f1 (2)
122
which is true by (155). When ν ≤ 0, by (171), (172), and (183), (242) becomes
0 <(∆1−∆
11)
a∆1−∆1
(∆
11 +
V 1−∆1
∆2∆
12
)
=−∆11a∆
1 +∆1
(a∆
1−∆11−
V 1−∆1
∆2∆
12
)
=−∆11a∆
1 +∆1∆1
2∆2
(a∆−V 1
)=−∆
11a∆
1 +∆1∆1
2∆2
a∆1 by (184)
= ∆1
[∆1
2∆2−
∆11
∆1
]a∆
1
which is positive by (154) and (162).
Q.E.D.Claim 25
PROOF OF CLAIM 26. By Claim 25, it suffices to show that o· is an RSW allocation.
Since V 1 = (1−δ )a f (1) is the maximum gains from trade when the seller’s type is 1,
V 1 (o1)+U1 (o1) ≤ V 1 = V 1 (o1) whence U1 (o1) ≤ 0 and thus, by (158), U2 (o2) ≥0. First suppose U1 (o1) = 0. Then o· clearly satisfies (140). As o· is an equilibrium
allocation, it also satisfies (139). Since, moreover, it gives each type of seller her RSW
payoff, it solves Program IIt in Definition 14 and hence is an RSW allocation as claimed.
Now suppose instead that U1 (o1)< 0, whence U2 (o2)> 0; we will prove a contradiction.
Consider the alternative allocation o· defined by o1 = o1 and o2 = o2. We have
U1 (o1)=U1 (o1)= 0 by part 1 of Claim 18;
U2 (o2)=U2 (o2)> 0 by assumption;
V 1 (o2)=V 1 (o2)≤V 1 (o1)=V 1 (o1)=V 1 (o1) by construction, as o· is an equilibrium,
by hypothesis, and by construction, resp.;
V 2 (o2)=V 2 (o2)=V 2 (o2)≥V 2 (o1)=V 2 (o1) by construction, by hypothesis,
by Claim 17, and by construction, resp.
123
Accordingly, o· satisfies (139) and (140) and gives each type of seller her RSW payoff.
Thus,o· solves Program IIt in Definition 14 and so is an RSW allocation. However, U2 (o2)>0, which contradicts part 1 of Claim 18.
Q.E.D.Claim 26
PROOF OF CLAIM 27. By part (c) of Proposition 3, the payoff V 1 (o1) of the low type
seller cannot be less than her RSW payoff V 1 which, by parts 1 of Claims 10 and Claim
18, equals (1−δ )a f (1). But this is the maximum gains from trade V 1 (o1)+U1 (o1) =(1−δ )q1 f (1) when the seller’s type is low. Hence U1 (o1) ≤ 0 and thus, by (158),
U2 (o2) ≥ 0. Accordingly, if U1 (o1) = 0, then o· satisfies (140). As o· is an equilibrium
allocation, it also satisfies (139) and thus, by Definition 14, V t (ot) ≤ V t for each t, which
contradicts P0. We conclude that U1 (o1)< 0 whence, by (158), U2 (o2)> 0.
Now suppose that V 1 (o1)≤ V 1; we will derive a contradiction. By (159), V 1 (o1)= V 1,
and by P0, V 2 (o2)> V 2. By part 2 of Claim 18, the above unique solution to RLP induces
an RSW allocation o· defined by ot = (qt , ρ t) = (q∗ (t) ,ρ∗ (t)) for t = 1,2. Consider the
allocation o· defined by o1 = o1 and o2 = o2. We have
U1 (o1)=U1 (o1)= 0 by part 1 of Claim 18;
U2 (o2)=U2 (o2)> 0 as shown above;
V 1 (o2)=V 1 (o2)≤V 1 (o1)=V 1 (o1)=V 1 (o1) by construction, as o· is an equilibrium,
as shown above, and by construction, resp.;
V 2 (o2)=V 2 (o2)≥V 2 (o2)≥V 2 (o1)=V 2 (o1) by construction, as shown above,
by Claim 17, and by construction, resp.
Accordingly, o· satisfies (139) and (140) yet gives type 2 seller more than her RSW payoff
- contradicting Definition 14.
Q.E.D.Claim 27
PROOF OF CLAIM 28. The set S is compact by Claim 24. Hence, Sx is also compact
124
as it adds an equality to the set of constraints that define S. Thus, if Sx is nonempty, then
the continuous function J has a maximum on Sx by the Extreme Value Theorem: Sx is
nonempty as well.
Now suppose Sx is nonempty and let o· ∈ Sx. We will show that o· satisfies P1-P4 in turn.
Suppose first that o· violates P1. First suppose q1 6= a. Let q1 = a and ρ1 = ρ1+δ(a−q1) f (1),
and o2 = o2. One can see that V t (ot) =V t (ot) for t = 1,2: o· satisfies P0 and (159) since
o· does. Moreover, V 1 (o2)=V 1 (o2) since o2 = o2, and
V 2 (o1)−V 2 (o1)= ρ1−ρ
1−δ(a−q1) f (2) = δ
(a−q1) [ f (1)− f (2)]≤ 0
whence o· satisfies (156) since o· does. Finally,
U1 (o1)−U1 (o1)= (1−δ )(a−q1) f (1)> 0 (243)
by (154) while U2 (o2) = U2 (o2) whence o· satisfies (158) as o· does: o· is in Sx. But
by (243), J (o·) > J (o·): o· is not in Sx, a contradiction. Finally, using q1 = a, we set
V 1 (o1) = ρ1− δa f (1) equal to the required payoff of V 1 + x and use (170) to obtain
ρ1 = a f (1)+ x.
Now say o· violates P2. By (156), this implies V 1 (o2)<V 1 (o1). Hence q2 6= a since
otherwise type 2 would deviate to o1.49 For any ε > 0, consider the alternative allocation o·
defined by o1 = o1, q2 = (1− ε)q2+εa, and ρ2 = ρ2+δ(q2−q2) f (2). By construction,
V 1 (o1)=V 1 (o1)= V 1+x, V 2 (o2)=V 2 (o2), and U2 (o2)>U2 (o2), which imply (157),
(160), and (158). Finally, since V 1 (o2)<V 1 (o1), we can choose ε > 0 small enough that
(156) holds so o· is in Sx whence (as U2 (o2)>U2 (o2)) the original allocation o· is not in
Sx - a contradiction.
Suppose now that o· violates P3: q2 6= a and V 2 (o2) > V 2. For any ε > 0, consider
the alternative allocation o· defined by o1 = o1, q2 = (1− ε)q2 + εa, and ρ2 = ρ2 +
δ(q2−q2) f (1). By construction, V 1 (o1)=V 1 (o1)= V 1+x. By P2, V 1 (o2)=V 1 (o2)=
V 1 (o1)=V 1 (o1) and U2 (o2)>U2 (o2)which imply (156) and (158), respectively. Moreover,
49If q2 = a = q1 then V 2(o1)−V 2
(o2)= ρ1−ρ2 =V 1
(o1)−V 1
(o2)> 0.
125
by P1 and P2,
V 2 (o2)−V 2 (o1)= ρ2−ρ
1 +δ(a−q2) f (2)
=V 1 (o2)−V 1 (o1)+δ(a−q2)
∆1
>V 1 (o2)−V 1 (o1)= 0
where the inequality relies on ASSUMPTION A and genericity. Thus, if ε is small enough,
(157) and (160) hold whence o· is in Sx. But since U2 (o2)−U2 (o2)> 0, o· is not in Sx -
a contradiction.
Finally, suppose o· violates P4: q21 < 1, q2
2 > 0, and hence, by P3, V 2 (o2)= V 2. Let us
now raise q21 slightly while lowering q2
2 and adjusting ρ2 so as to leave the payoffs of each
type of seller from o2 unchanged. We will show that this makes the agent better off when
facing the type-2 seller, so o· is not in Sx - a contradiction. For any small constants ε, ι ,ω ,
consider the alternative allocation o· defined by o1 = o1 = o1, q21 = q2
1+ε , q22 = q2
2− ι , and
ρ2 = ρ2+ω . Given ε > 0, we seek ι and ω such that the change does not alter the payoff of
either seller from choosing the outcome designated for either seller, whence (156), (157),
(159), and (160) still hold. As o1 = o1, V t (o1)=V t (o1) for t = 1,2. Now,
V 1 (o2)−V 1 (o2)= ω−δε f1 (1)+δι f2 (1)
and
V 2 (o2)−V 2 (o2)= ω−δε f1 (2)+δι f2 (2) .
Subtracting and dividing by δ , the two expressions are equal as long as ι = εψ where
ψ = ∆11/∆1
2 is positive by (154). Finally, the second expression (and thus the first) is zero
as long as ω = δε [ f1 (2)−ψ f2 (2)]. With these choices, the agent’s payoff from selling to
type 2 changes by
U2 (o2) − U2 (o2) = ε f1 (2) − ι f2 (2) − ω = ε (1−δ ) [ f1 (2)−ψ f2 (2)]
If we multiply the term in square brackets by ∆12, which is positive by (154), it becomes
∆12 f1 (2) − ∆
11 f2 (2) = f1 (1) f2 (1)
[f2 (2)f2 (1)
− f1 (2)f1 (1)
]
126
which is positive by (155). Accordingly, U2 (o2)>U2 (o2), so o· is not in Sx - a contradiction.
We conclude that o· satisfies P1, P2, P3, and P4, as claimed. Q.E.D.Claim 28
PROOF OF CLAIM 29. By Claim 28, any such allocation satisfies P1-P4. Given P1,
q1 = q2 implies q2 = a. Moreover, by P2, ρ2 = ρ1 = a f (1)+x: a pooling menu is offered.
As the agent’s payoff must be nonnegative, using the notation U (o·) of (158),
0≤U (o·) =2
∑t=1
Πt (qt f (t)−ρ
t)= a2
∑t=1
Πt f (t)− [a f (1)+ x]
= Π2a∆
1− x.
Accordingly, we must have x ≤ Π2a∆1. By Claim 24, for this to be an equilibrium it
suffices that the type 2 seller gets at least his RSW payoff: that V 2 (o2) = ρ1−δa f (2) =
a [ f (1)−δ f (2)]+x is not less than V 2 or, equivalently, x≥ V 2−a [ f (1)−δ f (2)] = V 2−
V 2 (o1). Combining these inequalities and using (179), x must lies in the interval (194) as
claimed. Moreover, for this interval to be nonempty, we must have δ(a− q2)∆1 ≤Π2a∆1,
which is equivalent to (193) as claimed.50 Q.E.D.Claim 29
PROOF OF PROPOSITION 5. Let o· ∈ Sx satisfy q2 6= q2. By P2 we have V 1 (o2) =ρ2−δq2 f (1) =V 1 (o1)= V 1 + x, whence
ρ2 = δq2 f (1)+V 1 + x. (244)
Combining (195) and (244) to eliminate ρ2 yields
x = δq2∆
1−(
V 1−V 2). (245)
By P4, there are now two subcases.
1. Assume (186). Then by (171), (181), and (182), type 2’s RSW outcome satisfies
(188). There are now two intervals in which x may lie.
50The right hand side of (193) is positive by (154) and (174). It is less than one since q2 6= a by (171) and
(172) and thus V 2 = (1−δ ) q2 f (2) (using (174)) is less than (1−δ )a f (2) =V 2(o1)+a∆1 by (154).
127
(a) If x lies in (196) then, by (154) and (245), q2 is given by (198) so, as x→ 0, q2
converges to q2 by (188) and hence, by (195), ρ2 converges to δ q2 f (2)+ V 2
which equals ρ2 by (173) and (174). It follows that the agent’s expected payoff
(158) converges to her RSW payoff of zero as x→ 0. For x > 0, her type-
contingent payoff U t (ot) = qt f (t)−ρ t equals−x when t = 1 and, using (195),
U2 (o2)= q2 f (2)−ρ2 = (1−δ )q2 f (2)−V 2
= (1−δ )V 1−V 2 + x
δ∆11
f1 (2)−V 2
when t = 2. Hence, using Π1 = 1−Π2, her unconditional expected payoff
(158), for a given increment x in type 1’s payoff, is
U (o·) =U (o·|x) =−x+Π2
[(1−δ )
V 1−V 2 + xδ∆1
1f1 (2)+ x−V 2
](246)
which at the top endpoint of (196) equalsV 1−V 2−δ∆11 +Π2
[∆1−V 1
]which
by (187) is nonnegative if and only if51
Π2 ≥
δ∆11−(
V 1−V 2)
∆1−V 1= η , (247)
which confirms (197). Alternatively, one can differentiate (246) with respect to
x:
ddx
U (o·|x) =−1+Π2 (1−δ ) f1 (2)+δ∆1
1δ∆1
1=−1+Π
2 ∆1
δ∆11=−1+
Π2
η.
(248)
As U (o·|x) is zero when x = 0, the constraint (158) holds at x if and only if this
derivative is nonnegative, which is equivalent to (197).
51By (97), (181), (181), and (186),
δ∆11−(
V 1−V 2)
∆1−V 1=
δ∆11
∆1
1− V 1−V 2
δ∆11
1−V 1/∆1=
δ∆11
∆1= η .
128
(b) If
x≥ ν (249)
then, by (245), q2 is given by (201). The requirement that q22 ≤ 1 then imposes
the additional constraint on x that
x≤ δa∆1−(
V 1−V 2)= γ (250)
by (179) and (164); thus, x must lie in (199) as claimed.52 It remains to check
that the agent’s payoff (158) is nonnegative. As in part 1(a), her type-contingent
payoff U t (ot) = qt f (t)−ρ t satisfies
U1 (o1)=−x and (251)
U2 (o2)= (1−δ )q2 f (2)−V 2. (252)
Thus, using (201) and Π1 = 1−Π2, condition (158) can be rewritten as
0≤U (o·|x) =−x+Π2
(1−δ )
[f1 (2)+ f2 (2)
V 1−V 2 + x−δ∆11
δ∆12
]+ x−V 2
.
(253)
The left hand side is linear and continuous in x, so U (o·|x)=U (o·|ν)+(x−ν)∂U (o·|x)/∂x
as ∂U (o·|x)/∂x is constant over x > ν . Thus, using (253) and (165) to compute
U (o·|ν) =−ν +Π2[(1−δ ) f1 (2)+ν−V 2
]=−ν +Π
2(
∆1−V 1)
and using (253) to compute
∂U (o·|x)∂x
=−x+Π2(1−δ ) f2 (2)
1δ∆1
2+1=−1+Π
2 ∆2
δ∆12,
we find that U (o·|x) is nonnegative at x in (199) if and only if
0≤U (o·|x) =−ν +Π2(
∆1−V 1)+(x−ν)
[−1+Π
2 ∆2
δ∆12
]=−x+Π
2[
∆1−V 1 +(x−ν)∆2
δ∆12
](254)
52Intuitively, for o· to be an equilibrium, type 2’s payoff V 2(o1)+ x from type 1’s outcome o1 cannot
exceed her payoff V 2(o2)
from her own outcome which, in turn, equals V 2 by P3.
129
or equivalently if Π2 ≥ φ (x) (defined in (200)) as claimed. We next show that
φ (x) is in (0,1) for x in (199). As x≥ ν > 0, it suffices to show that
x < ∆1−V 1 +(x−ν)∆2
δ∆12
⇐⇒x[
1− ∆2
δ∆12
]< ∆1−V 1−ν
∆2
δ∆12
⇐⇒− (1−δ ) f2 (2)δ∆1
2x < ∆1−V 1−ν
∆2
δ∆12
⇐⇒0 < (1−δ ) f2 (2)x+(
∆1−V 1)
δ∆12−∆2ν
As this expression is increasing in x, it suffices to check the inequality at x = ν
when it becomes
0 <(
∆1−V 1−ν
)δ∆
12
or, equivalently, by (154),
0 < ∆1−V 1−ν = ∆1−V 1−δ∆11 +(
V 1−V 2)
= (1−δ ) f1 (2)−V 2 = (1−δ )[
f1 (2)− q2 f (2)]
by (174)
= (1−δ )[∆1−V 1
] f1 (2)∆1
by (171), (172), and (187)
which is positive by (154) and (187) as claimed. Next, φ (x) is of the form xα+βx
where β = ∆2δ∆1
2> 1 by (163), so
φ′ (x) =
α
(α +βx)2
which is clearly finite (so φ is continuous). It is positive if and only if
0 < α = ∆1−V 1−ν∆2
δ∆12
⇐⇒δ∆12
(∆1−V 1
)> ν∆2 =
[δ∆
11−(
V 1−V 2)]
∆2 =
[δ∆
11−δ
V 1
∆1∆
11
]∆2 by (188)
⇐⇒δ∆12
(∆1−V 1
)∆1 > δ∆
11
(∆1−V 1
)∆2⇐⇒ ∆
12∆1 > ∆
11∆2
130
which holds by (162). We next must evaluate φ (x) at the endpoints of (199).
At the lower endpoint we obtain
φ (ν) =ν
∆1−V 1=
δ∆11−δ q2∆1
∆1−V 1by (180) and (176)
=δ∆1
1−δV 1
∆1∆1
1
∆1−V 1=
δ∆11
∆1= η
as claimed, while at the upper endpoint we obtain
φ (γ) =γ
∆1−V 1 +(γ−ν) ∆2δ∆1
2
=γ
∆1−V 1 +δ∆12
∆2δ∆1
2
=γ
∆1 +∆2−V 1=
γ
a∆− (1−δ )a f (1)=
γ
a∆1 = ζ by (175) and (179)
as claimed. Next, it is trivial from (200) to see that limx→∞ φ (x) equals δ∆12/∆2;
this exceeds ζ as φ is increasing and φ (γ) = ζ . Finally, since φ is continuous
and increasing on [ν ,γ] and maps this interval to [η ,ζ ], it has a continuous
and increasing inverse function φ−1 defined on [η ,ζ ] that maps this interval to
[ν ,γ], and for(x,Π2) ∈ [ν ,γ]× [η ,ζ ] , Π2 ≥ φ (x) if and only if x≤ φ−1 (Π2).
Hence, the region R defined by (199) and (200) is the union of two regions. The
first, region (i), is the subset of R on which Π2 is in [η ,ζ ] . Since, at all points
in R, x lies in [ν ,γ] , Π2 ≥ φ (x) is equivalent to x ≤ φ−1 (Π2). Hence, region
(i) is defined by the two conditions (202) and (203). The second, region (ii),
is the subset of R on which (205) holds. Since φ (x) ≤ ζ when x is in [ν ,γ],
condition (200) holds automatically if, in addition, (199) holds, region (ii) is
defined simply by (205) and (199).
2. Suppose (189) holds. In this case, x> 0 clearly cannot be less than δ∆11−(
V 1−V 2)
.
Hence, reasoning as in case (a)(ii), x must lie in (206) and q2 must be given by (201).
This has two implications. First, since q22≤ 1, x must lie in (206) as claimed. Second,
as x→ 0, q2 converges to q2 by (191) and hence, by (195), ρ2 converges to δ q2 f (2)+
V 2 which equals ρ2 by (173) and (174). So as in case (a)(i), the agent’s expected
payoff (158) converges to her RSW payoff of zero as x→ 0. Since, moreover, the
131
agent’s expected payoff is again given by (253), a necessary and sufficient condition
for there to be a separating equilibrium with x in (206) is that
0≤ ddx
U (o·|x) =−1+Π2(1−δ ) f2 (2)
δ∆12
+1=−1+Π
2 ∆2
δ∆12
or, equivalently, that
Π2 ≥
δ∆12
∆2. (255)
Since (189),
γ∆2
δ= ∆2
(a− q2)
∆1 by (164)
= ∆2
((1,1)−
(1,
V 1−∆1
∆2
))∆
1 by (172) and (181)
= ∆2
(1− V 1−∆1
∆2
)∆
12 =
(∆1 +∆2−V 1
)∆
12 = ∆
12 a [ f (2)−δ f (1)]− (1−δ )a f (1)
= ∆12a∆
1
and thus δ∆12
∆2= γ
a∆1 , whence, (255) is equivalent to (207) when (189) holds. Q.E.D.Proposition 5
PROOF OF CLAIM 30. Clearly, if o· is in S, it must also be in Sx for that x equal to the
(constant) increment V 1 (o1)− V 1 that the type 1 seller gets in o·. Thus, to characterize
S, we can restrict attention to allocations that are in Sx for some payoff increment x. Any
element of S must therefore satisfy P1 and P4: type 1 sells her whole portfolio and type
2 sells all of asset 1 before she sells any of asset 2. Moreover, by (192), any element of
S must maximize (in S) the value ∑2t=1 Πtqt f (t) of assets sold. Hence, every allocation in
Spoolx (for any x) must be in S since, in each such allocation, the type 2 seller sells her entire
portfolio. Moreover, if Spoolx is nonempty for some x, then no element of any Ssep
x can be in
S as any such allocation involves type 2 retaining some units of asset 2. Finally, by Claim
29, Spoolx is nonempty for some x if and only if Spool
γ is nonempty, which is so if Π2 ≥ ζ . It
follows that, if Π2 ≥ ζ , S consists only of pooling allocations in which, for some x in the
interval (194), the agent pays ρ t = a f (1)+x to each type of seller for her whole portfolio.
132
Now suppose Π2 < γ: Spoolx is empty for any x > 0. Hence, any element of S must be in
Ssepx for some x > 0. But by Proposition 5, an increase in x always leads to higher welfare
as it leads type 2 to sell more of one asset and no less of the other. Thus, for any Π2 < γ ,
S consists of the (by Proposition 5) singleton element of Ssepx for the highest x for which
Ssepx is nonempty. As there is no such x when either (a) ν > 0 and Π2 < η or (b) ν ≤ 0 and
Π2 < γ , S is empty in these cases. But if ν > 0 and Π2 ∈ [η ,γ), the highest x for which
Ssepx is nonempty is x = φ−1 (Π2) by case 1(b) of Proposition 5. This proves the result.
Q.E.D.Claim 30
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