Structural Rationality in Dynamic Games:

ONLINE APPENDIX

Marciano Siniscalchi

May 2, 2020

A Introduction

This Online Appendix contains supplemental material and elaborations upon the main results

in the paper.

Section B is devoted to the extensive form, and to results that depend upon its specifics.

Subsection B.1 provides a the formal definition of game trees . Subsection B.2 proves Theorem

3. Subsection B.3 defines the extensive form of the elicitation game, of which Definition 8 is a

reduced representation.

Section C proves Proposition 4 and discusses structural assumptions that are sufficient to

ensure that the nested-supports condition holds.

Economics Department, Northwestern University, Evanston, IL 60208; marciano@northwestern.edu.

Earlier drafts were circulated with the titles ‘Behavioral counterfactuals,’ ‘A revealed-preference theory of strate-

gic counterfactuals,’ ‘A revealed-preference theory of sequential rationality,’ and ‘Sequential preferences and se-

quential rationality.’ I thank Amanda Friedenberg, as well as Pierpaolo Battigalli, Gabriel Carroll, Drew Fuden-

berg, Alessandro Pavan, Phil Reny, and participants at RUD 2011, D-TEA 2013, and many seminar presentations

for helpful comments on earlier drafts.

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Section D analyzes the elicitation example in Figure 6 of the paper.

Section E explores alternative characterizations of structural preferences that use lexico-

graphic preferences (§E.1) or different representations of conditional beliefs (§E.2).

Finally, Section F reviews alternative definitions of structural preferences, including one

that was proposed in previous versions of this paper, and explains how the present version

improves upon them. This section also collects a number of unsatisfactory definitions of

preferences over strategies that, while apparently capturing certain intuitions about (weak)

sequential rationality, they actually fail to formally imply it.

B Game Trees and Generic Equivalence Theorem

I first provide a full, but concise description of game trees and extensive-form games (without

chance nodes), using the notation in Osborne and Rubinstein (1994, Def. 200.1, pp-200-201).

Additional notation and results can be found in Online Appendix B.1.

A game tree is a tuple Γ = (N , A, H , P, (Ii )i∈N ); N is the set of players, A is a set of actions, and

H is a finite collection of histories, i.e., finite sequences (a1, . . . , an ) of actions, which contains

the empty sequenceφ. For every history h = (a1, . . . , aL ) ∈H , A(h )≡ {a ∈ A : (a1, . . . , aL , a ) ∈H }

is the set of actions available at h . A history h ∈ H is terminal if A(h ) = ;; denote the set of

terminal histories by Z .

P : H \ Z → N is the player function, which associates with each non-terminal history

h ∈ H \ Z the player on the move at h . Each Ii consists of a partition of P −1(i ), plus the

symbol φ, which corresponds to the beginning of the game (as explained in Section 2, this

ensures that every player’s CCPS includes his prior beliefs). The elements of Ii are player i ’s

information sets. For every i ∈ N , I ∈ Ii \ {φ}, and h , h ′ ∈ I , player i must have the same

moves available at both h and h ′: that is, A(h ) = A(h ′).

The game form is assumed to have perfect recall, as per Def. 203.3 in OR. Briefly, for every

h ∈ P −1(i ), let X i (h ) denote i ’s experience along the history h : that is, the ordered list of all

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information sets owned by i that i encountered along the history h , and the actions she played

there.1 Perfect recall is the requirement that, if h , h ′ ∈ I ∈Ii \ {φ}, then X i (h ) = X i (h ′).

An extensive-form game is an game tree together with payoff assignments ui : Z →R for

every player i ∈N .

The strategic-form objects in Section 2 can be derived from the game form and the payoff

assignments, as follows. For every player i ∈ N , a strategy is a map si : H \Z → A such that

si (h ) ∈ A(h ) for all h ∈ H \ Z , and si (h ) = si (h ′) for all h , h ′ ∈ I ∈ Ii \ {φ}. Si is the set of

strategies for player i ∈ N , and as in the main text, the usual conventions for product sets

apply. For every s ∈ S , ζ(s ) is the terminal history induced by s .2 The set of strategy profiles

reaching I ∈Ii \ {φ} is S (I ) = {s ∈ S : ζ(s ) = (a1, . . . , aL ), ∃` < L : (a1, . . . , a`) ∈ I }; that is, s ∈ S (I )

if some initial segment of ζ(s ) belongs to I . By convention, S (φ) = S . Finally, for every i ∈N ,

the strategic-form payoff function Ui is defined by letting Ui (s ) = ui (ζ(s )) for every s ∈ S .

Under the above assumptions, S (·) and Ui (·) satisfy the properties in Section 2: since this

requires notation that is also used in the proof of Theorem 3, I prove it in the next subsection.

B.1 Further details and properties of extensive-form games

This subsection contains more detailed definitions and properties of game trees. All results

are essentially known and included here only for ease of reference and notational consistency.

Fix a game form Γ = (N , A, H , P, (Ii )i∈N ).

It is convenient to define the concatenation of histories, and of histories and actions. If

h = (a1, . . . , aL ) ∈ AL and (b1, . . . , bM ) ∈ AM , then (h , h ′) = (h , b1, . . . , bM ) = (a1, . . . , aL , h ′) =

(a1, . . . , aL , b1, . . . , bM ).

Histories are ordered by the “initial segment” relation: h < h ′means that h ′ = (h , b1, . . . , bM )

1Formally, if h = (a1, . . . , aL ), let `1, . . . ,`K be the set of indices ` ∈ {1, . . . , L −1} such that P ((a1, . . . , a`−1)) = i ; let

I1, . . . , IK be such that (a1, . . . , a`k−1) ∈ Ik for k = 1, . . . , K . Then X i (h ) = (I1, a`1, . . . , Ik , a`k

).2Formally, ζ(s ) = (a1, . . . , aL ), where a1 = sP (φ)(φ) and, inductively, a`+1 = sP ((a1,...,a`))((a1, . . . , a`)).

3

for some b1, . . . , bM ∈ A; h = φ is a subhistory of all histories, and h ≤ h ′ means that either h

and h ′ are the same sequence, or h < h ′.

Information sets are also ordered by precedence: I < I ′ iff for every h ′ ∈ I ′ there is h ∈ I

with h < h ′. The notation I ≤ I ′ means that either I = I ′ or I < I ′. For players i for which

φ = {φ} is not a partition cell,φ < I for all I ∈Ii .

Fix I ∈ Ii \ {φ}. Since A(h ) = A(h ′) for all h ∈ I , we can abuse notation slightly and write

A(I ) to indicate A(h ) for any h ∈ I . Similarly, write P (I ) to indicate P (h ) for any h ∈ I .

Since a strategy si : H \Z → A for i ∈N must satisfy si (h ) = si (h ′) for all h , h ′ ∈ I ∈Ii \ {φ},

si can also be viewed as a map from Ii \ {φ} to A.

It is convenient to define the set of strategy profiles reaching a history. For every h ∈ H

(terminal or non-terminal), S (h ) = {s ∈ S : h ≤ ζ(s )}. In particular, if z is terminal, then S (z ) =

{s ∈ S : z = ζ(s )}, because by definition a terminal history is not a subhistory of any other

history. Notice that s ∈ S (h ) if there exists z ∈ Z such that h < z and s ∈ S (h ); furthermore, for

every player i ∈N and I ∈Ii \ {φ}, S (I ) =⋃

h∈I S (h ).

It is also useful to define player i ’s information sets Ii (si ) allowed by strategy si : that is, for

every I ∈Ii , I ∈Ii (si ) if and only if si ∈ Si (I ).

The following properties are immediate consequences of the definitions.

Remark 1 (i) For every z , z ′ ∈ Z there is h ∈H \Z such that (a) h < z and h < z ′, and (b) for

all h ′ ∈H with h ′ < z and h ′ < z ′, h ′ ≤ h .

(ii) For all I , J ∈Ii , if I < J then S (I )⊇ S (J ).

Proof: (i) Write z = (a1, . . . , aL ) and z ′ = (b1, . . . , bM ). If z = z ′, then take h = (a1, . . . , aL−1) =

(b1, . . . , bM−1); this may be the empty history if L =M = 1. Otherwise, there is m ∈ {1, . . . , min(L , M )}

such that a` = b` for 1 ≤ ` ≤m − 1, and am 6= bm . Then take h = (a1, . . . , am−1); again, this may

be the empty history if m = 1.

(ii) Fix s ∈ S (J ). By definition, there is h ∈ J with h <ζ(s ). But J < J implies that h ′ < h for

some h ′ ∈ I . Then h ′ <ζ(s ), so s ∈ S (I ).

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I now verify that the properties of S (·) assumed in Section 2 do hold under perfect recall.

In addition, Properties (ii) and (iv) are used in the proof of Theorem 3.

Remark 2 If Γ = (N , A, H , P, (Ii )i∈N ) has perfect recall, then

(i) for every I , J ∈Ii , S (I )∩S (J ) 6= ; implies that S (I ) and S (J ) are nested;

(ii) for every I , J ∈Ii \ {φ} and si , ti ∈ Si (I ), if J < I then si (J ) = ti (J );

(iii) for every I ∈Ii , S (I ) = Si (I )×S−i (I ).

(iv) for all z ∈ Z , S (z ) =∏

j∈N Sj (z ).

Proof: (i) Suppose there are r ∈ S (I )∩S (J ), s ∈ S (I )\S (J ), and t ∈ S (J )\S (I ). In particular, this

implies that I 6= J . By definition, there are hr ∈ I and h ′r ∈ J such that hr < ζ(r ) and h ′r < ζ(r ).

Since I 6= J , hr 6= h ′r , so either hr < h ′r or h ′r < hr . Suppose hr < h ′r ; then, by definition, X i (h ′r )

contains I . Now let h ′ be such that h ′ < ζ(t ) and h ′ ∈ J , which exists because t ∈ S (J ). Since

t 6∈ S (I ), X i (h ′) does not contain I . But then X i (h ′r ) 6= X i (h ′), which contradicts perfect recall.

Suppose instead h ′r < hr ; then X i (hr ) contains J . Let h be such that h < ζ(s ) and h ∈ I , which

exists because s ∈ S (I ). Since s 6∈ S (J ), X i (h )does not contain J . But then X i (hr ) 6= X i (h ), which

again contradicts perfect recall.

(ii) Let s−i , t−i ∈ S−i be such that s ≡ (si , s−i ), t ≡ (ti , t−i ) ∈ S (I ). By definition, there are

h , h ′ ∈ I such that h < ζ(s ) and h ′ < ζ(t ). Suppose that J < I , so by definition there are

h , h ′ ∈ J with h < h and h ′ < h ′. This implies that J and si (J ), and J and ti (J ) respectively, are

elements of X i (h ) and X i (h ′) respectively. But then, by perfect recall, si (J ) = ti (J ).

(iii) Clearly, S (I ) ⊆ Si (I )× S−i (I ). For the converse inclusion, fix s−i ∈ S−i (I ) and ti ∈ Si (I ).

Let si ∈ Si be such that s = (si , s−i ) ∈ S (I ). Let h = (a1, . . . , aL ) ∈ I be such that h < ζ(s ), and let

`1, . . . ,`K be such that P ((a1, . . . , a`−1)) = i if and only if `= `k for some k ; also let Ik be such that

hk ≡ (a1, . . . , a`k−1) ∈ Ik .

I claim that Ik < I for all k . By contradiction, assume that there is h ′ ∈ I such that h ′ 6∈ Ik

for every h ′ < h ′. This implies that Ik is not an element of X i (h ′); however, since hk ∈ Ik and

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hk < h , Ik is an element of X i (h ), so X i (h ) 6= X i (h ′), which contradicts perfect recall.

Since Ik < I for every k , part (ii) implies that a`k= si (Ik ) = ti (Ik ) for every k . Therefore,

h <ζ(ti , s−i ), and so (ti , s−i ) ∈ S (I ).

(iv) As in (iii), it is enough to show that∏

j Sj (z ) ⊆ S (z ). Write z = (a1, . . . , aL ) and h` =

(a1, . . . , a`−1) for ` = 2, . . . , L ; let h1 =φ. For every j ∈N , fix s j ∈ S (z ) arbitrarily. Then, by defi-

nition, z = ζ(s j ) for all j , so s jP (h`)(h`) = a` for all ` = 1, . . . , L and all j . Now define s = (s j

j ) j∈N .

Then sP (h`)(h`) = s P (h`)P (h`)(h`) = a` for all `= 1, . . . , L . Therefore, ζ(s ) = z , i.e., s ∈ S (z ).

Finally, recall that the strategic-form payoff function Ui is defined by Ui (s ) = ui (ζ(s )) for all

s ∈ S , where ui : Z →R. I verify the strategic independence property in Section 2 of the paper.

Remark 3 If Γ = (N , A, H , P, (Ii )i∈N ) has perfect recall, then for all i ∈ N , I ∈ Ii , and si , ti ∈

Si (I ), there is ri ∈ Si (I ) such that Ui (ri , s−i ) =Ui (ti , s−i ) for all s−i ∈ S−i (I ) = S−i (I ), and Ui (ri , s−i ) =

Ui (si , s−i ) for all s−i 6∈ S−i (I ).

[This argument is due to Mailath, Samuelson, and Swinkels (1993).]

Proof: Let ri ∈ Si be a strategy that agrees with si everywhere except at information sets that

weakly follow I , where it agrees with ti . Formally, for every J ∈ Ii , ri (J ) = ti (J ) if I ≤ J , and

ri (J ) = si (J ) otherwise. By Remark 2, since si , ti ∈ Si (I ), si (J ) = ti (J ) for all J ∈ Ii with J < I ;

by construction, r j (J ) = si (J ) for such J . Therefore, ri ∈ Si (I ), and in addition, for every s−i ∈

S−i (I ), there is a unique h ∈ I such that (si , s−i ), (ti , s−i ), (ri , s−i ) ∈ S (h ). At all J ∈ Ii with I ≤ J ,

by construction ri (J ) = ti (J ), so ζ(ri , s−i ) = (h , a1, . . . , aM ) = ζ(ti , s−i ) for suitable a1, . . . , aM ∈ A.

Hence Ui (ri , s−i ) = ui (ζ(ri , s−i )) = ui (ζ(ti , s−i )) =Ui (ti , s−i ).

On the other hand, for s−i 6∈ S−i (I ), by perfect recall (again, see Remark 2) (si , s−i ) 6∈ S (I ),

and hence also (si , s−i ) 6∈ S (J ) for any J ∈ Ii with I ≤ J . Then (si , s−i ) ∈ S (J ) implies that not

I ≤ J , and therefore ri (J ) = si (J ) at all such J . Hence ζ(ri , s−i ) = ζ(si , s−i ), and so Ui (ri , s−i ) =

ui (ζ(ri , s−i )) = ui (ζ(si , s−i )) =Ui (si , s−i ).

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B.2 Proof of Theorem 3

Assume that si is weakly sequentially rational givenµ, but there is ti ∈ Si such that ti �µ si . For

every I ∈Ii , let Bµ(I ) =∪{S−i (J ) : J =µ I }.

I first analyze the case in which Ui (si , s−i ) ≤Ui (ti , s−i ) for all s−i . In this case, Ui (ti , Pµ(I )) ≥

Ui (si , Pµ(I )) for all I ∈ Ii , and Theorem 1 implies that Ui (ti , Pµ(I ∗)) > Ui (si , Pµ(I ∗)) for some

I ∗ ∈ Ii . Hence, there must be t−i ∈ Bµ(I ∗) such that Ui (ti , t−i ) >Ui (si , t−i ). In particular, this

means that z ≡ ζ(si , t−i ) 6= ζ(ti , t−i ) ≡ z ′. Now let h be the longest non-terminal history such

that h < z and h < z ′, per Remark 1 part (i). Then P (h ) = i : otherwise, the move at h is

determined by t−i , so h < (h , tP (h )(h )) < z , z ′, contradiction. Let J ∗ ∈ Ii be such that h ∈ J ∗.

Then si (J ∗) 6= ti (J ∗): otherwise, h < (h , a )< z , z ′, where a = si (J ∗) = ti (J ∗), contradiction.

By weak sequential rationality, Ui (si ,µ(·|J ∗))≥Ui (ti ,µ(·|J ∗)). Since Ui (si , s−i )≤Ui (ti , s−i ) for

all s−i , Ui (si ,µ(·|J ∗)) =Ui (ti ,µ(·|J ∗)). Hence there is t ∗−i ∈ S−i (J ∗) such that Ui (si , t ∗−i ) =Ui (ti , t ∗−i ).

But since si (J ∗) 6= ti (J ∗), ζ(si , t ∗−i ) 6= ζ(ti , t ∗−i ). Therefore, there is a relevant tie for i at J ∗.

Now suppose that there is s ∗−i ∈ S−i such that Ui (si , s ∗−i )>Ui (ti , s ∗−i ). Since ti �µ si , by Theo-

rem 1 there are I , J ∈Ii with I ≥µ J , s ∗−i ∈ S−i ( J ), and Ui (ti , Pµ(I ))>Ui (si , Pµ(I )).

Let I ∈Ii be a ≥µ-maximal element of

Ii = {I ′ ∈Ii : I ′ ≥µ J ,Ui (ti , Pµ(I′))>Ui (si , Pµ(I

′))}. (1)

Such an element exists because I ∈ Ii , ≥µ is transitive, and Ii is finite.

Also, define the set

D = {s−i ∈ S−i : Pµ(I )(s−i )> 0, ζ(si , s−i ) 6= ζ(ti , s−i )}.

Since Ui (ti , Pµ(I ))>Ui (si , Pµ(I )), D 6= ;. Also, for every s−i 6∈D , [Ui (si , s−i )−Ui (ti , s−i )]·Pµ(I )(s−i ) =

0, because either ζ(si , s−i ) = ζ(ti , s−i ), so that the term in square brackets is zero, or the proba-

bility of s−i is zero (or both).

For every s−i ∈ D , let Ji (s−i ) = {J ′ ∈ Ii : (si , s−i ), (ti , s−i ) ∈ S (J ′)}, which is non-empty be-

cause it containsφ. Then let J (s−i ) a <-maximal element ofJi (s−i ).

7

I claim that, for any two s−i , s ′−i ∈ D , the sets S−i (J (s−i )) and S−i (J (s ′−i )) are either disjoint

or nested (in particular, the two sets may coincide). To see this, suppose that there is t−i ∈

S−i (J (s−i ))∩S−i (J (s ′−i )). Then (si , t−i ) ∈ S (J (s−i ))∩S (J (s ′−i )) by perfect recall. By Remark 2 part

(i), S (J (s−i )) and S (J (s ′−i )) are nested. Suppose for definiteness that S (J (s−i )) ⊇ S (J (s ′−i )), and

pick an arbitrary r−i ∈ S−i (J (s ′−i )); by perfect recall, (si , r−i ) ∈ S (J (s ′−i )), so (si , r−i ) ∈ S (J (s−i )) as

well, which implies that r−i ∈ S−i (J (s−i )), as claimed.

Now suppose that, for every s−i ∈ D , S−i (J (s−i )) ⊆ Bµ(I ). Since the sets S−i (J (s−i )), s−i ∈

D , are either disjoint or nested, there is a subset {s 1−i , . . . , s M

−i } ⊆ D such that (1) for every

s−i ∈ D , there is m = 1, . . . , M with S−i (J (s−i )) ⊆ S−i (J (s m−i )); and (2) for distinct `, m = 1, . . . , M ,

S−i (J (s `−i ))∩S−i (J (s m−i )) = ;. Furthermore, for each m = 1, . . . , M , Pµ(I )(S−i (J (s m

−i )))≥ Pµ(I )({s−i })>

0. In particular, by Corollary 4 in the text, J (s m−i )≥

µ I ; since S−i (J (s m−i ))⊆ Bµ(I ), alsoµ(Bµ(I ))|J (s m

−i )>

0, so Pµ(J (s m−i ))(Bµ(I )) > 0 and therefore, by the same Corollary, I ≥µ J (s m

−i ): thus, I =µ J (s m−i ).

Finally, D ⊆⋃

s−i∈D S−i (J (s−i )) ⊆⋃

m S−i (J (s m−i )) ⊆ Bµ(I ), so s−i ∈ Bµ(I ) \

⋃

m S−i (J (s m−i )) implies

that s−i 6∈D and so [Ui (si , s−i )−Ui (ti , s−i )] ·Pµ(I )({s−i }) = 0. Therefore,

∑

s−i

[Ui (si , s−i )−Ui (ti , s−i )] ·Pµ(I )({s−i }) =

=∑

m

∑

s−i∈S−i (J (s m−i ))

[Ui (si , s−i )−Ui (ti , s−i )] ·Pm (I )({s−i }) =

=∑

m

Pµ(I )(S−i (J (sm−i ))

�

Ui

�

si ,µ(·|J (s m−i ))

�

−Ui

�

ti ,µ(·|J (s m−i ))

��

≥ 0.

The last equality follows from the properties of Pµ(I ) and the fact that I =µ J (s m−i ). The inequal-

ity follows from the assumption that si is sequentially rational for µ given u . But this conclu-

sion contradicts the assumption that Ui (ti , Pµ(I )) > Ui (si , Pµ(I )). Therefore, there is s−i ∈ D

such that S−i (J (s−i )) 6⊆ Bµ(I ).

Since s−i ∈ S−i (J (s−i )) and s−i ∈D , Pµ(I )(S−i (J (s−i ))) > 0, so by Corollary 4, J (s−i ) ≥µ I . Sup-

pose that also I ≥µ J (s−i ), so J (s−i ) =µ I : then Bµ(J (s−i )) = Bµ(I ) and so S−i (J (s−i )) ⊆ Bµ(I ),

contradiction: thus, not I ≥µ J (s−i ), and so J (s−i )>µ I .

I claim that si (J (s−i )) 6= ti (J (s−i )). By contradiction, suppose that si (J (s−i )) = ti (J (s−i )). Write

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ζ(si , s−i ) = (a1, . . . , aL ) and ζ(ti , s−i ) = (b1, . . . , bM ). Let h0 = φ. Then h0 < ζ(si , s−i ) and h0 <

ζ(ti , s−i ). If P (h0) 6= i , then a1 = sP (h0)(h0) = b1. If instead P (h0) = i , then {h0} ∈ Ii satisfies {h0} ≤

J (s−i ) and so, by Remark 2 or (in case J (s−i ) =φ) the assumption that si (J (s−i )) = ti (J (s−i )), a1 =

si (h0) = ti (h0) = b1. Inductively, assume that, for some ` <min(L , M ), ak = bk for all k = 1, . . . ,`,

and consider `+1. Let h` = (a1, . . . , a`) = (b1, . . . , b`), so h` <ζ(si , s−i ) and h` <ζ(ti , s−i ). Again, if

P (h`) 6= i , then a`+1 = sP (h`)(h`) = b`+1. If instead P (h`) = i , then h` ∈ J for some J ∈ Ii . I claim

that, in this case, J ≤ J (s−i ), so that Remark 2 or the assumption that si (J (s−i )) = ti (J (s−i )) imply

that a`+1 = si (h`) = ti (h`) = b`+1. To see this, observe that, since (si , s−i ) ∈ S (J (s−i )), by definition

there is h < ζ(si , s−i ) such that h ∈ J (s−i ). Since both h and h` are subhistories of ζ(si , s−i ),

either h = h`, or h` < h , or h < h`. If h = h`, then h` ∈ J (s−i ) and so J = J (s−i ). If h` < h ,

then X i (h ) contains J , and hence so does X i (h ′) for every h ′ ∈ J (s−i ): thus, J < J (s−i ). Finally,

h < h` cannot actually hold: if h < h`, then X i (h`) contains J (s−i ); by perfect recall, every other

h ′ ∈ J must be such that X i (h ′) contains J (s−i ), so h ′ must have a subhistory in J (s−i ): that is,

J (s−i ) < J . Since h` < ζ(si , s−i ) and h` < ζ(ti , s−i ), (si , s−i ), (ti , s−i ) ∈ S (J ): but then, J (s−i ) is not

the<-maximal element ofJi (s−i ), contradiction. It follows that L =M and ζ(si , s−i ) = ζ(ti , s−i ),

which contradicts the fact that s−i ∈D .

To complete the proof, sequential rationality implies thatUi (si ,µ(·|J (s−i ))≥Ui (ti ,µ(·|J (s−i )),

so there is t−i ∈ supp µ(·|J (s−i )) such that Ui (si , t−i ) ≥ Ui (ti , t−i ). By contradiction, suppose

that Ui (si , t−i ) >Ui (ti , t−i ). Then ti �µ si implies that there must be I , J ∈ Ii with t−i ∈ S−i ( J ),

I ≥µ J , and Ui (ti , Pµ(I )) >Ui (si , Pν(I )). Then µ(S−i ( J )|J (s−i )) ≥ µ({t−i }|J (s−i )) > 0, so J ≥µ J (s−i )

by Corollary 4. By transitivity of ≥µ, I ≥µ J (s−i ). Furthermore, as shown above, J (s−i ) >µ I , so

transitivity also implies that I >µ I ≥µ J . But this contradicts the choice of I as a ≥µ-maximal

element of the set Ii in Eq. (1). Therefore, Ui (si , t−i ) =Ui (ti , t−i ). But since si (J (s−i )) 6= ti (J (s−i )),

ζ(si , t−i ) 6= ζ(ti , t−i ). Thus, there is a relevant tie at J (s−i ).

9

B.3 Extensive form of the elicitation game

Fix the game tree and payoffs of the original game, namely Γ = (N , A, H , P, (Ii )i∈N ) and (ui )i∈N ,

and a questionnaire Q = (Qi )i∈N . I now describe the game tree and payoff assignments of the

elicitation game. The objective is to ensure that the corresponding strategy sets and other

derived objects satisfy the properties in Definition 8.

Begin with a description of the elicitation game tree. The player set is N ∗ = N ∪ {c }; the

action set is A∗ = A∪{h , t }. It is useful to distinguish between first-stage and second-stage his-

tories. In the first stage, Chance moves first, at the empty historyφ∗, and chooses an element

of A1c ≡ {h , t }. Then, players move according to their index; player i chooses from A1

i ≡ Si ×Wi .

Hence, stage-1 histories are of the form

φ∗ or�

ac , (s1, w1), . . . , (si−1, wi−1)�

: a 1c ∈ Ac , (s j , w j ) ∈ A1

j j = 1, . . . , i −1. (2)

Second-stage histories reflect the play of the strategies players have committed to in the first

stage. Hence, they take the form

�

ac , (s1, w1), . . . , (sN , wn ), h�

: (s1, . . . , sN ) ∈ S (h ). (3)

It will be convenient to represent these histories by emphasizing strategy profiles, as in

(ac , s , w , h ) or (ac , si , wi , s−i , w−i , h ).

For h =φ, write (ac , s , w ,φ) simply as (ac , s , w ). The set of all histories will be denoted by H ∗.

A history (ac , s , w , z ) is terminal if and only if z is terminal in the original game.

Turn now to information sets. The Chance player has a single one, the root {φ∗}; with some

notational abuse, denote this asφ∗. In the first stage, each player i ∈N has an information set

I 1i =

¦

(ac ,�

s1, w1), . . . , (si−1, wi−1)�

∈H ∗ : ac ∈ A1c , (s j , w j ) ∈ A1

j , j = 1, . . . , i −1©

. (4)

This formalizes the assumption that players do not observe each other’s choices (nor Chance’s

move) in the first stage.

10

In the second stage, for each i ∈ N , (si , wi ) ∈ Si ×Wi , and I ∈ Ii such that si ∈ Si ), keeping

the notation of Definition 8,

(si , wi , I ) =��

ac , (si , wi ), (s−i , w−i ), h�

∈H ∗ : si = si , wi =wi , s−i ∈ S−i (h ), h ∈ I

. (5)

Thus, player i does not observe Chance’s move ac and other players’ choice of bet w−i ; how-

ever, she does recall her own first-stage choices, and does learn that her opponents chose a

strategy that allows I in the original game.

Notice that, consistently with Definition 8, I do not assume that I ∗i includes the symbol

φ∗. This is because I 1i serves the same purpose—it ensures that S ∗(I 1

i ) = S ∗ is a conditioning

event, and hence that a CCPS for i includes i ’s unconditional beliefs.

Turn now to the payoff assignments u ∗j , for j ∈ N ∗. For Chance, u ∗c ≡ 0. For each player

i ∈N , we let

u ∗i�

(ac , s , w , z )�

=

ui (z ) ac = h

1 ac = t ,Qi = (I , (E , p )), wi = E , s−i ∈ E

0 ac = t , wi = E , s−i 6∈ E

p ac = t ,Qi = (I , (E , p )), wi = p , s−i ∈ S−i (I ).

(6)

I now verify that the induced strategy sets S ∗i , strategy correspondence S ∗(·), and payoff

functions U ∗i (·), satisfy the properties in Definition 8.

Chance has a unique information setφ∗, with action set A1c , so S ∗c = A1

c = {h , t }.

Now consider player i ∈ N . Eq. (2) and Eq. (3) for h = φ show that, for any first-period

history h ∗ ∈ I 1i and action (si , wi ) ∈ Si ×Wi , (h ∗, (si , wi )) ∈ H ∗. Therefore, A∗(I 1

i ) = Si ×Wi .

Given a second-period information set (si , wi , I ), Eq. (5) implies that, if h ∗ ∈ (si , wi , I ), then

h ∗ = (ac , s , w , h ) for some ac ∈ Ac , s ∈ S , w ∈W and h ∈ I ; Eq. (3) then implies that (h ∗, a ) =

(ac , s , w , (h , a )) ∈ H ∗ iff s ∈ S ((h , a )); and since P (h ) = i and h ∈ I , a = si (I ). Therefore,

A∗((si , wi , I )) = {si (I )}. This formalizes the statement that player i is committed to action si (I )

at (si , wi , I ).

11

It follows that, for every player i ∈ N , there is a bijection between S ∗i and A∗(I 1i ) = Si ×Wi .

Definition 8 abuses notation and sets S ∗i = Si ×Wi .

Turn now to the strategy map S ∗(·). First, every strategy profile reaches the initial historyφ∗,

so S ∗(φ∗) = S ∗. For every other first-stage information set I 1i , Eq. 2 implies that, for any profile

s ∗ ∈ S ∗, the induced partial history (ac , (s1, w1), . . . , (si−1, w−i )) lies in I 1i . Thus, S ∗(I 1

i ) = S ∗.

Now consider a second-stage information set (si , wi , I ) and a strategy s ∗. Eq. (5) implies

that, first of all, there is no restriction on Chance’s move, but s ∗i (I1

i ) = (si , wi ). Additionally,

let s ∗j (I1j ) = (s j , w j ) for all j 6= i : there is no restriction on w−i , but s−i ∈ S−i (I ). Therefore,

S ∗((si , wi , I )) = {(si , wi )}×S−i (I )×W−i ×S ∗c .

Finally, turn to strategic-form payoffs. The definition of u ∗c implies that U ∗c ≡ 0. For players

i 6= c , fix a profile s ∗ = ((si , wi ), (s−i , w−i ), s ∗c ). The induced terminal history is then (s ∗c , s , w ,ζ(s ))

[at (s ∗c , s , w ) = (s ∗c , s , w ), the player on the move is P (φ); by Eq. (3), there is only one his-

tory featuring a single additional action, namely (s ∗c , s , w , (sP (φ)(φ))); inductively, if s ∗ induces

(s ∗c , s , w , h ), the only continuation history featuring a single additional action is (s ∗c , s , w , (h , sP (h )(h )))].

Eq. (6) implies that

U ∗i (s

∗) = u ∗i (ζ∗(s ∗)) = u ∗i ((s

∗c , s , w ,ζ(s )) =

ui (ζ(s )) =Ui (s ) s ∗c = h

1 s ∗c = t ,Qi = (I , (E , p )), wi = E , s−i ∈ E

0 s ∗c = t ,Qi = (I , (E , p )), wi = E , s−i 6∈ E

p s ∗c = t ,Qi = (I , (E , p )), wi = p , s−i ∈ S−i (I ).

This completes the proof.

12

C Computational considerations

C.1 Proof of Proposition 4

Claim 1: for all µ-sequences I1, . . . , IL such that IL , . . . , I1 is also a µ-sequence, there is ` ∈

{1, . . . , L} such that suppµ(·|I`)⊇ suppµ(·|Im ) for all m ∈ {1, . . . , L}.

If L = 1, the claim is trivially true. Thus, suppose it is true for some L ≥ 1, and con-

sider a µ-sequence I1, . . . , IL , IL+1. Then the inductive hypothesis yields ` ∈ {1, . . . , L} such that

suppµ(·|I`)⊇ suppµ(·|Im ) for all m ∈ {1, . . . , L}. In particular, suppµ(·|I`)⊇ suppµ(·|IL ).

I claim that µ(S−i (IL+1)|I`) > 0. By assumption µ(S−i (IL+1|IL ) > 0, so there is s−i ∈ S−i (IL+1)∩

suppµ(·|IL ). By the inductive hypothesis, s−i ∈ suppµ(·|I`), so µ(S−i (IL+1)|I`)≥µ({s−i }|I`)> 0.

Next, I claim that µ(S−i (I`)|IL+1) > 0. Again, by assumption µ(S−i (IL L )|IL+1) > 0, and since

µ(S−i (IL+1)|IL )> 0 as well, by nested supports either suppµ(·|IL )⊇ suppµ(·|IL+1)or suppµ(·|IL+1)⊇

suppµ(·|IL ). In the first case,µ(S−i (I`)|IL+1)≥µ(suppµ(·|I`)|IL+1)≥µ(suppµ(·|IL )|IL+1)≥µ(suppµ(·|IL+1)|IL+1) =

1. In the second case, since suppµ(·|I`)⊇µ(·|IL ), there is s−i ∈ suppµ(·|IL )with s−i ∈ suppµ(·|I`)⊆

S−i (I`); since suppµ(·|IL+1) ⊇ suppµ(·|IL ), also s−i ∈ suppµ(·|IL+1); but then µ(S−i (I`)|IL+1) ≥

µ({s−i }|IL+1)> 0, as claimed.

Then, by nested supports, either suppµ(·|I`)⊇ suppµ(·|IL+1)or suppµ(·|IL+1)⊇ suppµ(·|I`).

In the first case, ` has the required property for theµ-sequence I1, . . . , IL+1; in the second, L+1

does. In either case, the inductive step is complete.

By Lemma 1 in the paper, the elements of the≥µ-equiv class of any I ∈Ii can be arranged

(potentially with duplicates) in aµ-sequence that begins and ends with I . Let I1, . . . , IL be such

a µ-sequence; by Corollary 3, since I1 = IL = I , IL , . . . , I1 is also a µ-sequence. Hence, Claim 1

applies; denote by ` the index with the properties therein.

Since I` =µ I , Pµ(I )(S−i (I`)) > 0 and µ(·|I`) = Pµ(I )(·|S−i (I`)) by Proposition 3. Moreover,

consider s−i ∈ supp Pµ(I ). Since Pµ(I )(∪{S−i (Im ) : m = 1, . . . , L}) = 1 by the same Proposi-

tion and the definition of the µ-sequence I1, . . . , IL , s−i ∈ S−i (Im ) for some m . Furthermore,

13

since Pµ(I )(S−i (Im )) > 0 and µ(·|Im ) = Pµ(I )(·|S−i (Im )), µ({s−i }|Im ) > 0. Then s−i ∈ suppµ(·|Im ) ⊆

suppµ(·|I`). Therefore, supp Pµ(I )⊆ suppµ(·|I`), so Pµ(I )(S−i (I`))≥ Pµ(I )(suppµ(·|I`))≥ Pµ(I )(supp Pµ(I )) =

1, and so Pµ(I ) =µ(·|I`), as claimed.

C.2 A sufficient condition for nested supports

The following condition can be checked directly on the game form:

Definition 1 A dynamic game has nested information if, for all i ∈ N and I , J ∈ Ii , either

S−i (I )∩S−i (J ) = ; or S−i (I )⊆ S−i (J ) or S−i (J )⊆ S−i (I )

It is immediate to see that nested information implies nested supports for every player

i and CCPS µ ∈ ∆(S−i ,Ii ): if µ(S−i (I )|J ) > 0 and µ(S−i (J )|I ) > 0, then S−i (I ) ∩ S−i (J ) 6= ;; if

S−i (I )⊆ S−i (J ), then for all E ⊆ S−i (I ), µ(E |I ) =µ(E |J )

µ(S−i (I )|J ) by Eq. (1) in the paper, so suppµ(·|I )⊆

suppµ(·|J ); similarly, if S−i (J )⊆ S−i (I ), then suppµ(·|J )⊆ suppµ(·|I ).

Any game in which every player moves at most once on every path has nested information.

Fix one such game, a player i , and I , J ∈ Ii . If (wlog) I =φ, then trivially S−i (I ) = S−i ⊇ S−i (J ).

Otherwise, I and J belong to distinct paths: that is, if h ∈ I , there is no h ′ < h such that

h ′ ∈ J , and conversely. Now fix s−i ∈ S−i (I ). By contradiction, assume s−i ∈ S−i (J ) as well. Let

si ∈ Si (I ) and ti ∈ Si (J ). By perfect recall, (si , s−i ) ∈ S (I ) and (si , s−i ) ∈ S (J ): that is, there are

h , h ′ ∈ H \ Z such that h < ζ(si , s−I ), h ′ < ζ(ti , s−i ), h ∈ I , and h ′ ∈ J . Let h ′′ be the longest

history such that h ′′ ≤ h and h ′′ ≤ h ′. Since i moves only once on each path, P (h ′′) 6= i ; in

particular, h ′′ 6∈ {h , h ′}. Moreover, by definition, there must be actions a , a ′ ∈ A such that

(h ′′, a ) ≤ h and (h ′′, a ′) ≤ h ′, with a 6= a ′. But since P (h ′′) 6= i , sP (h ′′)(h ′′) = a 6= a ′ = sP (h ′′)(h ′′),

contradiction. Thus, s−i 6∈ S−i (J ). Similarly, if s−i ∈ S−i (J ), then s−i 6∈ S−i (I ), so the game has

nested information.

One can model a signalling game as a three-player game, in which the first player is Nature,

the second is the sender, and the third is the receiver. Each player moves only once on each

14

path, so the game has nested information, and hence every CCPS for each of the personal

players has nested supports. Selten’s Horse is also a game in which every player moves once

on each path.

Finally, the Battle of the Sexes with an outside option, Burning Money, and centipede

games all feature nested information. On the other hand, the game in Figure 4 does not have

nested information, and the CCPS in Example 2 does not have nested supports; however, the

CCPS ν ∈∆(S−i ,Ia ) defined by ν({d }|φ) = ν({a }|I ) = ν({c }|J ) does have nested supports.

D Calculations for the game in Figure 6 (Section 6.2)

I first analyze Bob’s preferences. We have (collapsing realization-equivalent strategies, as in

the paper) Sa = {(Out, b ), (Out, p ), (InB , b ), (InB , p ), (InS , b ), (InS , p )}, Sb = {B B , SS},Ia = {φ, K }

with Sa (φ) = Sa (K ) = Sa , andIb = {φ, I , I ′}with Sa (I ) = Sa (I ′) = {(InB , b ), (InB , p ), (InS , b ), (InS , p )}.

Assume that Bob’s beliefsµ satisfyµ({(Out, b ), (Out, p )}|φ) = 1 andµ({(InS , b ), (InS , p )}|I ) =

µ({(InS , b ), (InS , p )}|I ′) = π. One readily verifies that Pµ(I ) = Pµ(I ′) = µ(·|I ) = µ(·|I ′). Table I

summarizes Bob’s payoffs as a function of Ann’s strategy, as well as his conditional expected

payoffs.

sb (Out, b ), (Out, p ) (InB, b ), (InB, p ) (InS, b ), (InS, p ) φ I , I ′

B B 2 1 0 2 1−π

SS 2 0 3 2 3π

Table I: Payoffs and expected payoffs for Bob in Figure 6.

By Theorem 1, since Bob’s payoff is 2 if Ann plays (Out, p ) or (Out, b ), and Sb (I ) = Sb (I ′) =

Sa \ {(Out, b ), (Out, p )}, the ranking of B B vs. SS is pinned down by expected payoffs given

µ(·|I ) = µ(·|I ′). For instance, SS is structurally rational iff π ≥ 14 . This is, of course, exactly the

condition under which S is structurally and weakly sequentially rational in the original game

15

of Figure 1, if he expects Ann to choose S with probability π conditional upon having played

In. Hence, as claimed, Bob’s strategic incentives are preserved.

Now turn to Ann. Since the only conditioning event for her is Sb , her structural preferences

are actually ex-ante EU. Hence, she will choose b at K if and only if she assigns probability at

least p to Bob choosing S (and hence committing to S ) atφ.

E Equivalent definitions of structural preferences

E.1 LPSs and Proof of Theorem 5

Following Blume, Brandenburger, and Dekel (1991b), for an LPS λ = (p1, . . . , pn ) and a vector

r = (r1, . . . , rn−1) ∈ (0, 1)n−1, let r�λ= (1− r1)p1+ r1((1− r2)p2+ r2 . . .+ rn−1pn ) . . .).

(Necessity): Assume that ti �µ si and consider an LPS λ= (p1, . . . , pn ) that generates µ.

Claim: for any sequence (r k )k≥1 ⊂ (0, 1)n−1 with r k → 0, (r k�λ)k≥1 is a perturbation of µ.

Proof: since λ generates µ, for every I ∈Ii there is m with pm (S−i (I ))> 0 and p`(S−i (I )) = 0

for `= 1, . . . , m −1. Thus (r k�λ)(S−i (I ))> 0 for all k . Moreover, for every E ⊆ S−i (I ),

(r k�λ)(E |S−i (I )) =

∏m−1m ′=1 r k

m ′ · (1− r km )pm (E ) +

∑n`=m+1

∏`−1m ′=1 r k

m ′ · (1− r k` )p`(E )

∏m−1m ′=1 r k

m ′ · (1− r km )pm (S−i (I ))+

∑n`=m+1

∏`−1m ′=1 r k

m ′ · (1− r k` )p`(S−i (I ))

=

=(1− r k

m )pm (E ) +∑n`=m+1

∏`−1m ′=m r k

m ′ · (1− r k` )p`(E )

(1− r km )pm (S−i (I ))+

∑n`=m+1

∏`−1m ′=m r k

m ′ · (1− r k` )p`(S−i (I ))

=

=pm (E |S−i (I ))+

∑n`=m+1

∏`−1m ′=m r k

m ′ ·(1−r k

` )p`(E )

(1−r km )pm (S−i (I ))

1+∑n`=m+1

∏`−1m ′=m r k

m ′ ·(1−r k

` )p`(S−i (I ))

(1−r km )pm (S−i (I ))

→µ(E |I ) :

the second equality follows by dividing the numerator and the denominator by the common

factor∏m−1

m ′=1 r km ′ , the third by dividing the numerator and denominator by (1− r k

m )pm (S−i (I ))>

0; and the limit follows because (p k )k≥1 is a perturbation of µ, and each summation is either

empty (if m = n), or contains terms multiplied by at least one factor r km ′ → 0. This proves the

claim.

16

Since (r k�λ)k≥1 is a perturbation ofµ, by assumptionUi (ti , r k�λ)>Ui (si , r k�λ) eventually.

By Proposition 1 in Blume et al. (1991b) (see the Remark at the end of the proof in the Appendix

of that paper), ti �λ si . This completes the proof of necessity.

(Sufficiency): Suppose that ti �σ si for all LPSs σ that generate µ. To show that ti �µ si , I

leverage Theorem 1 in the paper.

As in Section B, for I ∈ Ii , let Bµ(I ) = ∪{S−i (J ) : J =µ I }. Also write I ≥µ s−i for I ∈ Ii and

s−i ∈ S−i (I ) to mean that there is J ∈Ii with I ≥µ J and s−i ∈ S−i (J ). Thus in particular I ≥µ s−i

for all s−i ∈ S−i (I ).

I invoke definitions and results from Appendix A in the paper. Let I1, . . . , IM be a represen-

tative collection for µ, and f the canonical map for I1, . . . , IM .

Claim: Consider a one-to-one function g : {1, . . . , M } →R that agrees with >µ, and define

the LPS λ = (Pµ(In1), . . . , Pµ(InM

)), where n1, . . . , nM is the unique permutation of 1, . . . , M such

that ` <m iff g (In`)< g (Inm). Then λ generates µ.

Proof: for every I there is m such that I =µ Inm, and thus Pµ(Inm

)(S−i (I ))> 0 and Pµ(Inm)(·|S−i (I )) =

µ(·|I )by the properties of Pµ(·). Furthermore, suppose that Pµ(In`)(S−i (I ))> 0, so that Pµ(In`)(Bµ(Inm))>

0. By Corollary 4, Inm≥µ In` . In particular, if m 6= `, then Inm

>µ In` , so g (Inm)< g (In`) and there-

fore m < `. Thus, for ` <m , Pµ(In`)(S−i (I )) = 0. Thus λ generates µ.

To invoke Theorem 1, it is sufficient to show that (i) there is at least one I ∈ Ii such that

Ui (ti , Pµ(I )) >Ui (si , Pµ(I )), and (ii) for all s−i ∈ S−i , if Ui (ti , s−i ) <Ui (si , s−i ) then there is I ∈ Ii

such that Ui (ti , Pµ(I ))>Ui (si , Pµ(I )) and I ≥µ s−i . [This is because then one can take the infor-

mation sets in the statement of Theorem 1 to be all I ∈Ii for which Ui (ti , Pµ(I ))>Ui (si , Pµ(I )).]

For (i), consider the LPS λ obtained by taking g = f in the above Claim. Then λ generates

µ and, by assumption, ti �λ si , so (in the notation of the Claim) there must be some m such

that Ui (ti , Pµ(Inm))>Ui (si , Pµ(Inm

)).

Now consider (ii) and let s−i ∈ S−i be such that Ui (si , s−i )<Ui (ti , s−i ). LetI +i = {I ∈Ii : I ≥µ

s−i }. By contradiction, assume that Ui (ti , Pµ(I ))≤Ui (si , Pµ(I )) for all I ∈I +i .

17

Define g : {1, . . . , M } → R by letting g (m ) = f (m ) if Im ∈ I +i , fmax =maxm :Im∈I +i f (m ), and

g (m ) = f (m ) + fmax + 1 otherwise. Then g is one-to-one. Furthermore, suppose I` >µ Im . If

Im ≥µ s−i , then also I` ≥µ s−i , and so I`, Im ∈ I +i , which implies that g (`) = f (`)< f (m ) = g (m ).

If not Im ≥µ s−i but I` ≥µ s−i , then g (`) = f (`) < fmax + 1 < f (m ) + fmax + 1 = g (m ). Finally, if

neither Im ≥µ s−i nor I` ≥µ s−i , then g (`) = f (`) + fmax + 1) < f (m ) + fmax + 1 = g (m ). Thus,

g agrees with >µ, so by the Claim, the LPS λ = (Pµ(In1), . . . , Pµ(InM

)) induced by g generates µ

(again, n1, . . . , nM are as in the Claim).

By construction, there is m such that In` ∈I+

i iff `≤m . Let

ρ = (Pµ(In1), . . . , Pµ(Inm

),δs−i, Pµ(Inm+1

), . . . Pµ(InM)),

where as usual δs−iis the Dirac measure concentrated on s−i . Then ρ also generates µ: if

δs−i(S−i (K ))> 0 for some K ∈Ii , then s−i ∈ S−i (K ), so S−i (K )≥µ {s−i } and therefore K =µ In` for

some n` with In` ∈ I+

i ; but then, ` ≤m and Pµ(In`))(S−i (K )) = Pµ(K )(S−i (K )) > 0. Thus, δs−iis

never the first measure to assign positive probability to any information set K . On the other

hand, λ does generate µ, and therefore so does ρ. By construction, for all ` ≤ m , In` ∈ I+

i ,

and by assumption this implies Ui (si , Pµ(In`))≤Ui (ti , Pµ(In`)). Furtherore, again by assumption

Ui (si ,δs−i)<Ui (ti ,δs−i

). Hence ti ≺ρ si : contradiction.Thus, (ii) must hold.

E.2 Partially ordered probability systems and structural preferences

Theorem 1 employs the belief Pµ(I ) associated with each I ∈ Ii via Definition 6. The CCPS µ

only appears indirectly, via the definition of the preorder ≥µ per Definition 5.

One can restate the definition of structural preferences in terms of an alternative represen-

tation of beliefs that, in a sense, involves only the objects of interest—the beliefs Pµ(I ), I ∈Ii .

Consider the following definition.

Definition 2 A partially ordered probability system (POPS) for player i ∈ N is a collection

(pI )I∈Ii∈∆(S−i )Ii that satisfies

18

1. for every I , J ∈ Ii , pI = pJ if and only if there exist M > 1 and I1, . . . , IM ∈ Ii such that

I1 = IM = I , IL = J for some L ∈ {1, . . . , M }, and pI`(Si (I`)∩S−i (I`+1))> 0 for `= 1, . . . , M −1;

2. for every I ∈Ii , pI (∪{S−i (J ) : J ∈Ii , pJ = pI }) = 1.

Notice that, in part 2 of the definition, one can take I = J , M = 2, and I1 = I2 (so e.g. J = 1) to

obtain pI (S−i (I ))> 0.

Given a POPS p, one can define a preorder ≥p on Ii by letting I ≥I0 J iff pJ (S−i (I )) > 0,

and letting ≥p be the transitive closure of ≥p0 , as in Definition 5 in the paper. Then part 2 of

the definition states that pI = pJ iff I =p J . Furthermore, when restricted to its equivalence

classes, ≥p is a partial order, and hence induces a partial order over {pI : I ∈Ii }.

For any CCPS µ, the collection�

Pµ(I )�

I∈Iiis a POPS. Conversely, if p = (pI )I∈Ii

is a POPS,

then one can define an array µ = (µ(·|I ))I∈Iiby letting µ(E |I ) = pI (E ∩S−i (I ))/pI (S−i (I )) for all

I ∈ Ii and E ⊆ S−i (I ); the properties in Definition 2 imply that µ is a CCPS. Furthermore, in

either case ≥p=≥µ. (Proofs are available upon request.)

Remark 4 Fix strategies si , ti ∈ Si . Let p = (pI )I∈Iibe a POPS for i ∈ N , and µ the CCPS it

generates. Then ti �µ si iff there exist M > 1 and I1, . . . , IM ∈Ii such that Ui (ti , pIm)>Ui (si , pIm

)

for m = 1, . . . , M , and Ui (ti , s−i )≥Ui (ti , s−i ) for all s−i 6∈⋃

m

⋃

J :Im≥p J S−i (J ).

Thus, as claimed above, the definition of structural preferences can be given entirely in terms

of a player’s POPS.

F Unsatisfactory definitions of structural preferences

This subsection first briefly comments on the definition of structural preferences proposed

in previous versions of this paper, and on an alternative definition based on perturbations. It

emphasizes in what respects the definition in the present version is preferable—in particular,

as regards minimality.

19

It then examines alternatives to Definition 4 (or, more precisely, the characterization thereof

in Theorem 1) that, while apparently sensible, do not imply weak sequential rationality.

F.1 Non-minimal definitions of structural preferences

In previous versions of this paper, structural preferences were defined as a weak, rather than

a strict order.

Definition 3 For si , ti ∈ Si , ti ¼µO LD si if, for every J ∈ Ii with Ui (ti , Pµ(J )) <Ui (si , Pµ(J )), there

is I ∈Ii with I >µ J and Ui (ti , Pµ(I ))>Ui (si , Pµ(I )).

This definition emphasizes the similarity with lexicographic weak preference. One can also

provide a characterization closer to that in Theorem 1: The starting point is the observation

that, if one defines ti �µO LD si as ti ¼

µO LD si and not si ¼

µO LD ti , then ti �

µO LD si is equivalent to:

ti ¼µO LD si and Ui (ti , Pµ(I ))>Ui (si , Pµ(I )) for some I ∈Ii .

Remark 5 ti �µO LD si iff there are M ≥ 1 and I1, . . . , IM ∈Ii such thatUi (ti , Pµ(Im ))>Ui (si , Pµ(Im ))

for all m , and Ui (ti , Pµ(J ))≥Ui (si , Pµ(J )) for all J 6∈⋃

m{K ∈Ii : Im >µ K }.

The key difference between this definition and the one now adopted in the paper (or its

characterization in Theorem 1) arises if there is a profile s−i such that Ui (ti , s−i ) < Ui (si , s−i )

at an information set J such that Ui (ti , Pµ(J )) = Ui (si , Pµ(J )). The characterization of �µO LD

just given implies that the existence of such a profile does not rule out the possibility that

ti �µO LD si . By way of contrast, Theorem 1 shows that, if such a profile exists, there must be

m and J with s−i ∈ S−i (J ) and Im ≥µ J . Intuitively, the definition in the present paper adds

a robustness requirement with respect to the exact spcification of the probabilities Pµ(J ), or,

more broadly, of the CCPS µ. Equalities in expected payoffs are regarded as knife-edge cases

that cannot support a strict preference for ti over si .

Example 1 in the paper illustrates this point. In this game, Pµ(K ) = µ(·|K ) for all infor-

mation sets K of Ann. Recall that µ({t }|φ) = µ({t }|φ) = 12 , so Ua (H L ,µ(·|φ)) = Ua (T ,µ(·|φ)).

20

Moreover, µ({o}|I ) = 1, and Ua (H L , o ) = 2 > 0 = Ua (T , o ). Thus, according to Definition 3,

H L �µO LD T . On the other hand, according to the definition in the present version, H L and

T are incomparable. As explained in the paper, this accounts for the possibility of vanishing

perturbations of Ann’s prior beliefs that, in particular, place slighlty more weight on t than on

h .

In this particular example, ruling out T as a rational strategy has a further unpleasant con-

sequence. The game under consideration is a modification of “Matching Pennies,” and has a

unique Nash equilibrium in which Ann plays H L and T with equal probability. If one rules

out T as a rational strategy, then one must conclude that this game has no Nash equilibrium

in structurally rational strategies (i.e., a Nash equilibrium in which, additionally, each strategy

is a structural best reply to a CCPS for which the prior coincides with the equilibrium conjec-

ture). The definition adopted in this paper does not suffer from this limitation.

Incidentally, the unique Nash equilibrium of this game is, of course, also the unique per-

fect equilibrium. To justify the play of T , one perturbs the Nash equilibrium more towards t

than h—which is exactly the sort of perturbations that are used to argue that T is structurally

rational in the paper.

An alternative to Definition 4 in the present version of the paper uses weak rather than

strict preferences:

Definition 4 si ¼µ∗ ti if, for all perturbations (p k )k≥1 of µ, Ui (si , p k )≥Ui (ti , p k ) eventually. si is

structurally rational given µ if there is no ti ∈ Si with ti �µ∗ si (i.e., ti ¼µ∗ si and not si ¼µ∗ ti .)

This definition of strutural rationality is stronger than the one in the paper. If si is not

structurally rational in the sense of this paper, there is ti with Ui (ti , p k )>Ui (si , p k ) eventually

for all perturbations (p k ) of µ. Thus, a fortiori ti ¼µ∗ si , and it is not the case that si ¼µ∗ ti . But

then ti �µ∗ si , so si is not structurally rational according to Definition 4.

In addition, Definition 4 rules out strategies that are weakly dominated by other pure strate-

gies. If Ui (ti , s−i ) ≥Ui (si , s−i ) for all s−i ∈ S−i , with a strict inequality for at least one t−i ∈ S−i ,

21

then Ui (ti , p k )≥Ui (si , p k ) for all k and all perturbations (p k ), and Ui (ti , q k )>Ui (si , q k ) for any

perturbation ∗q k )k≥1 such that q k ({t−i }) > 0 for all k . Therefore, ti ¼µ∗ si and not si ¼µ∗ ti , so

ti �µ∗ si and si is not structurally rational according to this definition.

One may wish to rule out strategies weakly dominated by pure strategies for reasons in-

dependent from elicitation. However, weak sequential rationality does not eliminate such

strategies in general, and elicitation does not require that they be eliminated. One aspect of

the minimality that characterizes the preferences defined in this paper is that it, too, does not

rule out such strategies.

F.2 Requiring greater payoffs at every information set

The following “eventwise dominance” definition may seem particularly close in spirit to weak

sequential rationality:

Unsatisfactory definition DOM: si ¼µD O M ti iff, for every I ∈Ii , Eµ(·|I )Ui (si , ·)≥ Eµ(·|I )Ui (ti , ·).

Somewhat surprisingly, this definition actually fails to imply weak sequential rationality,

even in simple, perfect-information games with nested strategic information.

Ann

φ

D1

2,1

A1 Bob

d

0, 4

a Ann

ID2

4,3

A23,6

Figure 1: A centipede game. Ann’s CCPS: µ({d }|φ) =µ({a }|I ) = 1

Consider for instance the Centipede game of Figure 1, and assume that Ann’s CCPS µ is

consistent with backward-induction reasoning. Ann’s strategy A1D2 does strictly better than

D1D2 given µ(·|I ) = Pµ(I )—that is, in case Bob chooses a at the second node. Even though

D1D2 does strictly better than A1D2 given Ann’s prior beliefs, Unsatisfactory Definition DOM

22

still deems D1D2 and A1A2 incomparable. As a result, A1D2 is maximal in the order¼µD O M , even

though it is not even optimal ex-ante—let alone weakly sequentially rational.

This example demonstrates that, in order to deliver weak sequential rationality, it is crucial

to take into account the likelihood ordering of information sets. Structural rationality recog-

nizes that Ann’s prior beliefs should take priority over µ(·|I ) = Pµ(I ), and thus discards A1D2.

F.3 A definition that considers all conditional beliefs

The characterization in Theorem 1 restricts attention to the probabilities Pµ(I ). Even in games

in which this coincides with an element of the player’s CCPS, this still implies that not all el-

ements of µ play a direct role. For instance, if S−i (I ) ⊃ S−i (J ) and µ(S−i (J )|I ) > 0, then µ(·|J ) is

not used directly. Weak sequential rationality instead requires optimality at every information

set. One might then be led to consider a notion that takes the original elements of information

sets into account, but still ranks them in terms of likelihood:

Unsatisfactory definition ACB: si ¼µACB ti iff, for every I ∈Ii with Eµ(·|I )Ui (si , ·)< Eµ(·|I )Ui (ti , ·),

there is J ∈Ii such that J >µ I and Eµ(·|J )Ui (si , ·)> Eµ(·|J )Ui (ti , ·).

To see why this definition is inadequate, consider the game in Figure 2.

AnnL R

D

43 +ε

a1

b2

c

1

a

b

c

−1

T−1B

−1T−1B

−1

Ann I

Bob

Figure 2: Ann’s CCPS: µ(a |) =µ(b |) =µ(c |) = 13 ; 0<ε< 1

6 .

Strategy L is strictly dominated for Ann. In addition, if Ann’s CCPS µ assigns equal prob-

ability ex-ante to a , b and c , strategy D yields strictly higher unconditional expected payoff

23

than R , because ε > 0. Thus, D is the unique weakly sequentially rational strategy given µ.

Furthermore, the same payoff inequality implies that it is not the case that R ¼µACB D . How-

ever, consider information set I . Given the associated belief µ(·|Sb (I )), R yields an expected

payoff of 32 ; since ε < 1

6 , D yields a strictly lower expected payoff. As was just noted, D does

do strictly better than R given the prior belief µ(·|); however, it is not the case that Sb >µ Sb (I ),

because µ(Sb (I )|) = 23 . Hence, it is not the case that D ¼µACB R . So, R and D are incomparable

according to Unsatisfactory Definition ACB; in particular, R is maximal, even though it is not

weakly sequentially rational.

Theorem 1 avoids this issue because it employs only the probabilities Pµ(·)—in this exam-

ple, Pµ(φ) =µ(·|φ).

F.4 Comparing payoffs conditional on events allowed by both strategies

The definition of structural preferences compares the expected payoff of strategies si , ti given

beliefs conditional upon events that may not be allowed by si , ti , or even both. As noted in

the main text, this is motivated by the ex-ante nature of structural preferences. However, one

may consider the following alternative, which restricts attention to “common conditioning

events.” These are events F ∈ S−i (Ii ) for which there exist I , I ′ ∈ Ii with si ∈ Si (I ), ti ∈ Si (I ′),

and S−i (I ) = S−i (I ′) = F . (Of course, a special case is I = I ′).

Unsatisfactory definition COM: si ¼µC O M ti iff, for all I , I ′ ∈ Ii such that si ∈ Si (I ) , ti ∈

Si (I ′), S−i (I ) = S−i (I ′), and Eµ(·|I )Ui (si , ·) < Eµ(·|I )Ui (ti , ·), there are J , J ′ ∈ Ii such that si ∈ Si (J ),

ti ∈ Si (J ′), S−i (J ) = S−i (J ′)⊃ S−i (I ) = S−i (I ′), and Eµ(·|J )Ui (si , ·)> Eµ(·|J )Ui (ti , ·).

Note: one may also consider further modifications whereby Pµ(·) is used in lieu ofµ, and/or

set inclusion is replaced with >µ. However, I am going to provide a counterexample in which

the game satisfies nested information (cf. Section C), and in addition Pµ(I ) =µ(·|I ) for all I .

One can show that, if a strategy si is optimal with respect to ¼µC O M (that is, si ¼µC O M ti for

24

all ti ∈ Si ), then si is weakly sequentially rational given µ. However, since¼µC O M is incomplete,

optimal strategies may fail to exist. I have been unable to show that, if si is maximal with re-

spect to¼µC O M (that is, ti �µC O M si for no ti ∈ Si ), then si is weakly sequentially rational (whereas

Theorem 2 establishes this implication for structural preferences). However, even if such a re-

sult were true, it would only hold vacuously in some games. The relation ¼µC O M is not acyclic,

and consequently even¼µC O M –maximal strategies may fail to exist. (Structural preferences are

transitive, so that maximal strategies exist for all finite games.)

To illustrate, consider the game in Figure 3. Notice that this game has nested strategic

information, and a relatively simple multistage structure: Ann and Bob first move simultane-

ously, and then Ann makes a further choice after observing Bob’s action.

Assume that Ann’s CCPS satisfiesµ({o}|) = 1. As asserted, Pµ(K ) =µ(·|K ) for all K ∈Ia , and

the game has nested information. To simplify the presentation, I denote Ann’s strategies by

indicating only the actions specified at information sets not precluded by Ann’s initial choices.

Thus, I write U T T , without specifying whether Ann chooses T ′ or B ′ at I ′, etc.

First, note that U T T �µC O M U T B . The common conditioning events for these strategies

are Sb , Sb (I ) = {t } and Sb (I ) = {m}, and D T B does strictly worse than U T T conditional on

Sb (I )—indeed, it makes a sequentially irrational choice at I .

Second, D T ′′T ′′ �µC O M U T T . The reason is that the only common conditioning events

are Sb and Sb (I ) = Sb (I ′′) = {m}, and D T ′′T ′′ yields 5 given µ(·|I ′′) =µ(·|I ), whereas U T T only

yields 3 given µ(·|I ) =µ(·|I ′′).

Third, M T ′T ′ �µC O M D T ′′T ′′. The common conditioning events are now Sb and Sb (I ′) =

Sb (I ′′) = {b }, and given µ(·|I ′) =µ(·|I ′′, M T ′T ′ does strictly better.

Finally, U T B �µC O M M T ′T ′. The reason is that the only common conditioning events are

Sb and Sb (I ) = Sb (I ′) = {t }, and U T B yields 5, rather than 3, given µ(·|I ) =µ(·|I ′). In particular,

the fact that U T B makes the wrong choice at I is not relevant to the comparison, because

Sb (I ) = {m} is not a commn conditioning event for U T B and M T ”T ′.

This example demonstrates three points. First, the relation¼µC O M admits a strict cycle. Sec-

25

ond, there is no maximal strategy for the relation¼µC O M . In particular, the three strategies that

are weakly sequentially rational given µ, namely U T T , M T ′T ′ and D T ′′T ′′, are all deemed

strictly worse than some other strategy by ¼µC O M . Finally, a cycle can include strategies that

are not weakly sequentially rational.

All difficulties in this example arise because the relation¼µC O M is not transitive. In turn, this

is a consequence of the fact that the set of conditioning events that determine the ranking

of two given strategies depends upon the strategies themselves.3 Structural preferences are

instead defined via all measures Pµ(I ), I ∈Ii ; this delivers transitivity.

3The fact that ri ¼µC O M si and si ¼

µC O M ti does not necessarily yield restrictions on the payoffs conditional

upon reaching information sets that are allowed by boht ri and ti .

26

Ann

U

Bob

t

m

b

0, 0

o

1, 1

Ann

I

T 5, 0

B 0, 0

Ann

I

T 3, 0

B0, 0

M Bob

t

m0, 0

bo

1, 1

Ann

I ′T ′ 3, 0

B ′0, 0

Ann

I ′

T ′ 5, 0

B ′0, 0

D

Bob

t0, 0

m

bo

1, 1

Ann

I ′′T ′′ 5, 0

B ′′0, 0

Ann

I ′′

T ′′ 3, 0

B ′′0, 0

Figure 3: A strict cycle including a sequentially irrational strategy. Ann’s CCPS: µ({o}|) = 1.

27