Communication and Early Contributions

COMMUNICATION AND EARLY CONTRIBUTIONS

STEFANO BARBIERITulane University

AbstractIn the context of the voluntary provision of a public good,I study the interaction of communication and dynamics incontributions. I modify the cheap talk, joint-project frame-work of Agastya, Menezes, and Sengupta (2007) by consid-ering early contributions that precede communication andby assuming partially funded projects are not completelyworthless. First, I show environments in which, without earlycontributions, parties cannot be trusted to communicatesincerely, so that the joint endeavor always fails. Then, forthe same environments, I demonstrate that early contribu-tions can create the conditions for mutual trust, improvethe outcome of subsequent communication, and eventuallyincrease welfare. This trust-building role of early contribu-tions matches well with the aims that negotiation scholars at-tribute to preconditions for negotiations and to preliminaryconcessions.

1. Introduction and Related Literature

Negotiations are fundamental to the success of joint projects, especially whenthe benefits of the common endeavor are uncertain. Many theorists havestudied joint projects, formalized as public goods that parties may voluntarilyprovide.1 However, most of these studies do not address negotiations. A no-table exception is Agastya, Menezes, and Sengupta (2007), henceforth AMS,

Stefano Barbieri, Department of Economics, 206 Tilton Hall, Tulane University, NewOrleans, LA 70118 ([email protected]).

While retaining responsibility for any errors, I thank David Malueg and Andrew Postle-waite for their comments and suggestions. I am also grateful for the comments of ananonymous referee, of the associate editor, and of the editor, John Conley.1 See for example, Admati and Perry (1991) and Marx and Matthews (2000).

Received January 5, 2010; Accepted September 17, 2010.

C© 2012 Wiley Periodicals, IncJournal of Public Economic Theory, 14 (3), 2012, pp. 391–421.

391

392 Journal of Public Economic Theory

who begin their analysis by identifying the element of communication in ne-gotiations: “Rarely, if ever, do parties contemplating a joint project commitresources without engaging in nonbinding discussions on who does what.”The broad literature on negotiations identifies many features that facilitateagreement, including (1) sincere information transmission, (2) dynamics incontributions (or “fractionated concessions”), and (3) mutual trust.2 AMSformally address the first point, by demonstrating the welfare-improving ef-fects of sincere information transmission, which is obtained through a cheap-talk stage that precedes one-shot contributions to a public good. In this paper,I study the interaction of communication and dynamics in contributions,with a focus on the second and third features identified above. In particular,I ask: Can early contributions create the conditions for mutual trust, thusimproving the outcome of subsequent nonbinding communication?

Negotiation scholars provide various examples of contributions or con-cessions that precede (further) discussions, at different points along the life-cycle of a negotiation. For example, at times parties insist on concessions,“goodwill” gestures, or preconditions even before negotiations start. Surely,sometimes preconditions are set too high, in a ploy to stop negotiations frombeginning (Kennedy, 1997). Nonetheless, even Kennedy concedes that “Asconfidence-building measures, some preconditions and the willingness ofthe parties to accept them create the right conditions for a negotiation tobegin . . . . The release of some hostages—the sick, the young, the aged—is a useful precondition to build confidence between the terrorists and theauthorities for the more difficult negotiations that must follow. In business,similar preconditions can be a useful test of intentions” (Kennedy, 1997, pp.111–112).3 In dealing with (how to avoid) breakoffs—the other end of thenegotiation lifecycle—Pruitt states: “In other words, breakoff occurs whenthe other’s expected demand is below the bargainer’s limit. For this reason,it is important for both bargainers or a third party to foster the impressionthat a favorable agreement is possible. This can be done by making or ar-ranging for occasional concessions or preliminary agreements on secondaryissues, a phenomenon we have called maintaining momentum” (Pruitt 1981,p. 232).

Having accepted this fact that some contributions or concessions canprecede further discussion, I build my formal analysis upon the frameworkof AMS, who augment a static private-value contribution game with a priorcheap-talk stage.4 AMS demonstrate that information can be transmittedthrough payoff-irrelevant messages, thus leading to increases in welfare in

2 See for example, Pruitt (1981), especially chapter 4.3 The best examples of preconditions that fit my framework are irreversible, like an ex-change of prisoners between two nations in conflict before peace negotiations begin.4 In the contribution game, players independently and simultaneously make nonrefund-able contributions to fund a discrete public good, which is provided if and only if contri-butions cover the cost of production.

Communication and Early Contributions 393

two ways: (1) whereas without communication the only equilibrium displaysno contributions and is “strongly” inefficient (for large costs of provision),with sufficiently rich messages at their disposal players can achieve an ex anteincentive efficient equilibrium allocation, regardless of cost; and (2) eventhe coarsest possible communication—two messages—results in an outcomethat Pareto dominates any equilibrium without communication.5 The resultsin AMS imply that costless communication “usurps,” at no cost, one of theestablished roles that early contributions to a public good play when infor-mation is asymmetric: information transmission through costly signaling.6

Indeed, my first result shows that there is no welfare-improving role for earlycontributions in the framework of AMS. Thus, their model is not well suitedto study preconditions for negotiations. Therefore, a weakening of AMS’ re-sults is a necessary first step towards my goals.

This preliminary objective is accomplished through a slight change inthe production function for the public good. In particular, I assume thatincomplete projects are not entirely worthless. (The same departure fromthe purely discrete public good case occurs in Marx and Matthews 2000.)Indeed, even a bridge, the textbook example of a discrete public good, mayprovide some utility when incomplete or even ruined. While an incompletebridge does not serve its intended purpose, it can nonetheless be used as afishing pier, as a building block for artificial reefs, or simply as a more ad-vanced starting point for future construction. For simplicity, I analyze thecase in which the utility of a partially funded project becomes smaller andsmaller, approaching zero. Nonetheless, the limiting equilibrium results di-verge from those of AMS: my second result confirms their conclusions, butonly if the cost of the good is below a threshold level. In contrast, if the costof the good is higher, the only equilibrium of the cheap-talk game is theno-contribution equilibrium.

The reason for this stark departure harkens back to the initial quotefrom Pruitt (1981). Suppose a party realizes it has nothing to gain from acompleted project because its value is too small compared to its anticipatedshare of the cost. Then, this party may be expected not to be sincere andtrustworthy in negotiations: this party can lie in order to extract as manyconcessions as possible and then fail to follow through on its commitments,thus leaving the project only partially complete. If incomplete projects areentirely worthless, then this strategy yields zero payoff. This is the same resultas truthfully revealing one’s low value and inducing breakoff of negotiationswithout contributions from any party. The “simple equilibria” that AMS usefor their welfare-improving results rely on this indifference to sustain sincerereporting. In contrast, the above described deviation from sincere report-ing cannot be deterred when partially funded projects generate some util-ity. Therefore, sincere negotiations are possible only if players harbor some

5 Ex ante incentive efficiency is defined in Holmstrom and Myerson (1983).6 See, e.g., Vesterlund (2003) and Andreoni (2006).


hope of profitable completion of the project, regardless of their private in-formation. It follows that the cost of the project cannot be too high, as mysecond result proves.

With these two results, I turn to the interaction of communication anddynamic contributions. In my model, players can make early contributionsbefore receiving their private information.7 Subsequently, players receivetheir private information, engage in communication, and then they haveone last opportunity to contribute. My third and main result demonstrates aheretofore (formally) unexplored role for early contributions that precedecommunication. Early contributions begin the completion process so that,before negotiations start, the remaining cost to completion can be broughtbelow the critical threshold for informative communication. It turns out thatthis “trust-building” role of early contributions is only possible and benefi-cial if communication is sufficiently rich. In particular, my final result showsthat no welfare gains can be realized if only two messages are available at thecheap-talk stage, in contrast with AMS.

The focus and unique characteristic of this paper is the interaction ofdynamic irreversible contributions and communication. Many other papers,in a variety of contexts other than the contribution game, analyze nonbind-ing communication. The reader is directed to AMS and references thereinfor further discussion of the relation between this literature and the contri-bution game with cheap talk.8 As previously discussed, the information sig-naling role of early contributions in models of private, dynamic provision ofa public good with asymmetric information is analyzed in Vesterlund (2003)and Andreoni (2006).9 Beyond AMS, the basic one-shot contribution gamewith private values is analyzed in Menezes, Monteiro, and Temimi (2001)and Barbieri and Malueg (2008). With the exception of AMS, none of thesepapers address communication. Dynamic contributions to a public goodwhen agents are perfectly informed is discussed by, among others, Admatiand Perry (1991), Varian (1994), Marx and Matthews (2000), Lockwood andThomas (2002), Compte and Jehiel (2004) in a bargaining framework, andby Pitchford and Snyder (2004) in the context of a hold-up problem.

The rest of the paper is organized as follows. Section 2 presents themodel. Section 3 contains the two preliminary results: (1) early contribu-tions do not add anything to communication in the purely binary publicgood framework of AMS; and, (2) the efficiency results of AMS do not go

7 This additional stage distinguishes my timing from the one in AMS. Note that, by con-struction, contributions at this stage cannot signal information. Thus, any role that earlycontributions may play is clearly distinct from the one in Vesterlund (2003) and Andreoni(2006), cf. footnote 6.8 In particular, this literature includes Crawford and Sobel (1982), Farrell and Gibbons(1989), Matthews and Postlewaite (1989), and Baliga and Morris (2002).9 Cf. footnote 6. Other (less closely) related models include Yildirim (2006), Komai, Stege-man, and Hermalin (2007), and Bag and Roy (2008).


Figure 1: The production technology

through, for high cost of provision, if the public good is not purely binary.Section 4 uses these two preliminary results to establish a new role for earlycontributions, building trust, and Section 5 concludes.

2. The Model

A joint project between two partners requires a total investment of k > 0to complete. Denote with ci the contribution of Player i , i = 1, 2. The jointproject is modeled as a public good whose quantity G is produced accordingto the following technology:

G(c1 + c2) =⎧⎨⎩

1 if c1 + c2 ≥ k

λc1 + c2

kif c1 + c2 < k,

where 0 ≤ λ < 1. This production technology is depicted in Figure 1.The value vi of a unit of public good is Player i ’s private information.

Values are independently and uniformly distributed on[v, v

].10 Therefore,

if contributions suffice to complete the project, that is, if c1 + c2 ≥ k, then i ’spayoff is vi − ci . Otherwise, i receives −ci + viλ

c1+c2k . Throughout, I assume

2v < k < 2v, so that the decision whether to complete the project is actuallyinteresting, and 4v > v.11

10 Because the departure point of the analysis is AMS’ model, I focus on the case in whichtheir results are simplest and most cogent, that is, uniformly distributed values. Section 5describes some extensions.11 Two relevant implications follow from 4v > v. First, v > 0, a point whose importance isexplored in AMS, see page 4. Second, (4v + 2v)/3 > v. This bound simplifies the analy-sis, because I can apply AMS’ results for the standard contribution game without furtherdevelopments that would detract from the main focus of this paper. An example of how(4v + 2v)/3 > v helps the analysis follows the statements of Lemma 1– 3.


When λ > 0, the technology and payoffs above differ from those of astandard contribution game, in which incomplete projects are worthless.12

Nonetheless, many interesting situations can be usefully modeled with theabove technology and the case λ > 0 is indeed central in Marx and Matthews(2000): “The benefit functions we consider range from binary ones to con-tinuous ones that rise linearly with the cumulative contribution up to a ‘com-pletion point’. In intermediate cases benefits rise linearly with the cumulation andjump up at the completion point. The building of a road network is an example:benefits increase with the number of roads that are built and put into service,and they rise discontinuously at the completion point when the linking roadis finished. Another example is a charity campaign aimed at famine relief ordisease prevention: benefits increase with the number of victims that are fedor treated, but the big payoff comes when the cause of the famine is elimi-nated or a cure for the disease is discovered.”13 While I assume λ < 1, thusensuring that a discrete, positive benefit jump exists as funds approach thecompletion level k, I am particularly interested in the limiting case of λ closeto zero. This is both to simplify the analysis and to facilitate comparisons withextant results, as further discussed in Section 5.

The innovative feature of AMS is the communication stage that precedescontributions. For tractability and comparability, I only consider their twocommunication environments. In the first, players send one of two possiblemessages. In the second, there is a continuum of messages at players’ dis-posal. These two communication environments are referred to as minimallyrevealing and maximally revealing, respectively.

Beyond considering the possibility that partially funded projects yieldsome utility, my most relevant departure from AMS is the addition of anearly contribution stage. Before messages are exchanged, and actually be-fore private information is realized, players can provide “early” irreversiblecontributions c e

1 and c e2, simultaneously and independently.14 Let c e be the

total amount of early contributions, c e = c e1 + c e

2, and denote with kc the re-maining cost to completion after early contributions, kc = k − c e . Finally, c l

1and c l

2 indicate the “late” contributions of players, so that the total contribu-tion of Player i is ci = c e

i + c li , i = 1, 2.

Therefore, the model proceeds in five steps:

1. Players simultaneously and independently make early contributions c e1

and c e2. (This is referred to as the early contribution stage.)

2. Players receive their private information v1 and v2.3. Conditional on the history of early contributions and his own private

value, each agent sends a message mi , i = 1, 2. Messages are chosen

12 For a standard contribution game, that is, with λ = 0, see Admati and Perry (1991).13 See Marx and Matthews (2000), p. 328, emphasis added.14 The fact that players receive information after their strategic interaction has begun linksmy model with the literature on ex post private information, see e.g., Krahmer and Strausz(2008), and references therein. See also the discussion in Section 5.


simultaneously and independently. (This is referred to as the messagestage.)

4. Conditional on the history of early contributions and messages, anddepending on his own private value, each agent simultaneously andindependently makes a late contribution c l

i , i = 1, 2. (This is referredto as the late contribution stage.)

5. The public good is produced as appropriate and agents receivepayoffs.

The equilibrium concept is Perfect Bayesian Equilibrium in pure strate-gies. In particular, at the message stage I consider only equilibrium strategiesin which types either fully separate or bunch in ordered intervals. More for-mally, for any two types vl

i and vhi with vl

i < vhi , if vl

i and vhi send the same mes-

sage, then all types belonging to [vli , vh

i ] send the same message. For brevity,this kind of equilibrium is referred to as “ordered”.15 Finally, because earlycontributions precede the arrival of private information and because the av-erage value is (v + v)/2, if k ≤ v + v, then agents could profitably completethe project at outset. To rule out this uninteresting outcome, I strengthenk > 2v to k > v + v. This last assumption, using λ < 1, also implies λv/k < 1.Therefore, as in the more traditional version of the contribution game, themarginal benefit of increasing one’s contribution to an incomplete projectnever compensates for the associated marginal cost.

3. Preliminary Analysis

In this section, I separately consider the effects of my two modeling innova-tions. First, I augment the discrete public good framework of AMS with anearly contribution stage. Second, I consider the case in which incompleteprojects are beneficial—albeit minimally—in a static, cheap-talk contribu-tion game, i.e., without an early contribution stage. Beyond their intrinsicinterest, the analysis of both departures is instrumental towards establishingthe trust-building role of early contributions in Section 4.

3.1. Early Contributions to a Purely Discrete Public Good

The main result in this section establishes that, when λ = 0, both for min-imally and maximally revealing communication, early contributions cannotimprove welfare.16 Throughout, I use welfare as shorthand for ex ante jointequilibrium payoffs. The next lemma, with proof in the Appendix, begins the

15 The focus on ordered equilibrium has bite in Proposition 2. In Section 5, I relax this re-quirement for minimally revealing communication and show Proposition 2 is not affected.16 This conclusion is not obvious, despite the ex ante incentive-efficiency results in Section4.2 of AMS, because early contributions may relax the incentive compatibility and individ-ual rationality constraints relevant for ex ante incentive-efficient equilibria.


analysis, establishing an upper bound past which early contributions surelybecome counterproductive.

LEMMA 1 (An upper bound for welfare improving early contributions): If k > v +v, then early contributions that reduce kc below (4v + 2v) /3 always yield a negativeex ante expected joint payoff.

From Lemma 1, it follows that there is no equilibrium with c e > k −(4v + 2v) /3. I now recall—without proof—two very useful results for thestandard, private-information contribution game, which, in my model, isany subgame following early contributions c e and some communication ex-change. Therefore, the amount needed to complete the project is kc , andagents contribute only once.

LEMMA 2 (No-contribution equilibrium): If an agent contributes a strictly posi-tive amount in equilibrium, then the probability of completion this agent perceives isstrictly positive. Moreover, if kc > v, then the no-contribution strategy profile is anequilibrium.

LEMMA 3 (AMS, Proposition 4): Denote with F1 and F2 the distributions of players’values. If completion happens with some probability in equilibrium, then there existstwo types v1 and v2 such that H (v1, v2) ≥ kc , where

H (x, y) ≡ x (1 − F2 (y)) + y (1 − F1 (x)) .

Moreover, if each distribution for values is concave, then H is convex in each ar-gument; thus, H assumes its maximum at the boundary. Therefore, if each player’svalue is distributed over an interval, completion in equilibrium requires either that thesum of the lowest possible values exceeds kc , or that one player’s highest possible valueexceeds kc .

As described in AMS, the function H in the previous lemma is noth-ing but the gross surplus in a one-step-function equilibrium. The usefulnessof Lemma 1, in conjunction with 4v > v, becomes now apparent. Thanksto Lemma 1, early contributions can be positive in equilibrium only ifkc ≥ (4v + 2v) /3. Thus, because 4v > v, one obtains kc > v and the im-plications of Lemma 2 and 3 are strengthened. Lemma 2 ensures theno-contribution strategy is always an equilibrium. (This fact simplifies pun-ishment of deviations.) As for Lemma 3, note first that, in any ordered equi-librium, each player’s value at the late contribution stage is conditionallydistributed over an interval, after any message exchange. Therefore, Lemma3 and kc > v imply that if completion can occur with some probability inequilibrium, then it should occur with probability one to maximize welfare.This is because completion can only occur if it surely beneficial, regardlessof the realization of values.


The following proposition, with proof in the Appendix, uses these impli-cations to establish that early contributions cannot improve welfare.

PROPOSITION 1 (Irrelevance of early contributions): If λ = 0 and k > v + v,then early contributions do not improve welfare in any ordered equilibrium.

Proposition 1 demonstrates that the standard contribution game withcheap talk is not suited to answer our main question, can early contributionscreate the conditions for mutual trust and improve the outcome of commu-nication? Indeed, by Proposition 1, there is no use for early contributions.The next section shows that matters change when λ > 0.

3.2. Incomplete Projects Generate Some Utility

For the rest of the paper, I depart from the purely binary public good as-sumption and consider λ > 0. It is immediate to verify that, for λ sufficientlysmall, Lemma 1 and 2 continue to hold. Moreover, if kc > v, the necessarycondition in Lemma 3 strengthens to

H (v1, v2) > kc . (1)

Indeed, because kc > v, each player must contribute to obtain completion,therefore type vi is strictly better off free riding (because λ > 0) ratherthan contributing exactly its expected value, that is, vi (1 − F3−i (v3−i )), fori = 1, 2.

For the rest of this section, I focus on the static, cheap-talk contribu-tion game. In my model, the static, cheap-talk contribution game is any sub-game following early contributions c e ; therefore, the amount needed to com-plete the project is kc and agent i ’s contribution is simply c l

i . The followinglemma demonstrates the most important consequence of λ > 0 at the mes-sage stage.

LEMMA 4 (Completion must follow, with positive probability, all messages sent inequilibrium): Suppose an equilibrium of the static, cheap-talk contribution game ex-ists in which completion happens with strictly positive probability. If kc > v, then allmessages sent in equilibrium must lead to completion with strictly positive probability.

Proof : By contradiction, assume an equilibrium exists in which type v ofPlayer 1 sends message m, but completion never ensues for any messagePlayer 2 may send. By Lemma 2, the payoff of type v, evaluated at the mes-sage stage, is zero. Therefore, the overall payoff of type v is vλ c e

k − c e1. Under

the hypothesis of this lemma, there is some other message m ′ sent by Player1 such that completion happens with strictly positive probability. Becausekc > v, it must be the case that Player 2 makes a strictly positive expectedcontribution. Therefore, the following deviation is strictly profitable for v,and thus a contradiction to equilibrium: send message m ′ and contribute


nothing after any message sent by Player 2. Indeed, the deviation generates

this expected payoff: vλc e +E [c l

2]k − c e

1. �Lemma 4 impedes the workings of the “simple” equilibria in AMS, be-

cause they depend on some types separating at the message stage and ac-cepting the sure failure of the project, while other types that send differentmessages end up completing the project. Simple equilibria are very impor-tant for the efficiency conclusions of AMS. In fact, the next proposition showsthat, for sufficiently large costs of provision, the result of AMS that comple-tion is always possible with cheap talk does not hold in my set-up.

PROPOSITION 2 (No completion): If kc ≥ v + v and λ > 0, then the only or-dered equilibrium of the static, cheap-talk contribution game is the no-contributionequilibrium.

Proof : Denote with I (v) the interval of types of Player 1 that sends the samemessage sent by v. In any ordered equilibrium, H (v1, v2) remains convex inv1 after the message stage (see Lemma 3). Indeed, conditional value distribu-tions at the late contribution stage are uniform. (Concavity would suffice forthe argument.) Therefore, H achieves its maximum over I (v) either for v1 =v, or for v1 = sup I (v). In the latter case, H (v1, v2) = sup I (v) ≤ v < kc . Inthe former, H (v1, v2) ≤ v + v2 ≤ v + v ≤ kc . Hence, by the necessary con-dition in Equation (1), no completion occurs after v sends his equilibriummessage. Therefore, Lemma 4 implies no completion is the only equilibriumoutcome. �

Proposition 2 identifies a necessary condition for completion: Playersmust be willing to contribute when they are sure that one of them has thelowest possible value and one has the highest.17 The following two propo-sitions construct equilibria that show this condition is also sufficient for λ

sufficiently small.

PROPOSITION 3 (Completion with minimally revealing communication): Inthe game with minimally revealing communication, ∃λ > 0 : ∀λ ∈ (0, λ), if(4v + 2v) /3 < kc < v + v, then an ordered equilibrium of the static, cheap-talk con-tribution game exists in which completion occurs with strictly positive probability.

Proof : At the message stage of the equilibrium I construct, types below acutoff v send message “l”, while types above v send message “h”; v is cho-sen such that v + v > kc . At the (late) contribution stage, c l

1 = c l2 = 0 after

message pair (l, l). After (h, h), types larger than v contribute c l1 = c l

2 = kc/2and any type v < v—necessarily, this type previously deviated from equilib-rium by sending h rather than l—contributes kc/2 if v − kc/2 > vλ

kc /2+c e

k

17 The result is reminiscent of the “strong” inefficiency result in Theorem 2 of Menezeset al. (2001).


and contributes nothing otherwise (early contributions are sunk). After mes-sage pairs (l, h) and (h, l), the sharing rule on the equilibrium path isnonegalitarian: (1) all types of the player who sent h and that are largerthan v contribute a fraction s > 1/2 of the cost kc ; (2) all types of the playerwho sent l and that are smaller than v contribute (1 − s) kc ; and, (3) deviat-ing types best respond, similarly to what happens after message pair (h, h).Note that completion does not occur if both values are below v; otherwise,completion surely happens. This is the difference with the “simple” equilib-ria in AMS, in which completion occurs if and only if both values are abovev. To complete the proof, one needs to choose values for v and s such that,for any λ sufficiently small, the above described strategies constitute an equi-librium. This verification proceeds along similar lines to the one for the up-coming Proposition 4 and is here omitted. Full details are available uponrequest. �

The following is the analogue of Proposition 3 for maximally revealingcommunication.

PROPOSITION 4 (Completion with maximally revealing communication): Inthe game with maximally revealing communication, ∃λ > 0 : ∀λ ∈ (0, λ), if(4v + 2v) /3 < kc < v + v, then an ordered equilibrium of the static, cheap-talk con-tribution game exists in which completion occurs with strictly positive probability.

Proof : The completion region of the symmetric equilibrium I construct isthe striped area in Figure 2 . Two quantities help to describe the equilibrium.The first, g , is the intersection of the boundary of the completion region withthe two axis, where v1 = v or v2 = v. The second quantity, p , is given as

p = 34

kc − 2g + 32

v. (2)

At the message stage, Player 1’s strategy is described in three parts: (1) forv1 ∈ [v, p ) send message “below p ”, (2) for v1 ∈ [p , g] send message m1

equal to v1, and (3) for v1 ∈ (g, v] send message “above g”. The strategy ofPlayer 2 is analogous, but for ease of exposition I label the three possibilitiesa, b , and c , respectively.

I now describe the contribution behavior on the equilibrium path.18 It isconvenient to divide the possible message pairs into regions, as in Figure 2.Beginning with Region 1 where Player 1 has sent message “below p ”, thereare three subcases:

(a) Player 2’s message is “below p ” and no completion occurs. Players donot contribute.

18 Off-equilibrium path deviations are followed by the deviator best responding, eitherwith a contribution that exactly completes the project, or with no contribution at all, as inProposition 3.


g

g

p

p

v

v

v

Reg. 1 Reg. 2 Reg. 3

c

b

a

v1

v2

Figure 2: Illustration of equilibrium.

(b) Player 2’s message is m2 ∈ [p , g] and no completion occurs. Playersdo not contribute.

(c) Player 2’s message is “above g”. If v1 ∈ [v, p ), then c l1 = s , and if

v2 ∈ (g, v], then c l2 = kc − s , where s is an amount later determined.

Completion always occurs on the equilibrium path.

In Region 2, after Player 1 has sent some message m1 ∈ [p , g], equilib-rium play continues with:

(a) m2 is “below p ” and no completion occurs. Players do not contribute.

(b) m2 ∈ [p , g]. For m1 + m2 ≥ g + p , if v1 ∈ [p , g] , then c l1 = kc

2 +13 (m1 − m2) and if v2 ∈ [p , g] , then c l

2 = kc2 + 1

3 (m2 − m1). For m1 +m2 < g + p no payments are made.

(c) m2 is “above g”. If v1 ∈ [p , g] , then c l1 = s + v1 − p . If v2 ∈ (g, v],

then c l2 = kc − (s + v1 − p ).

In Region 3, after Player 1 has sent message “above g”, equilibrium playcontinues with:


(a) m2 is “below p ”. By symmetry to 1(c), v1 ∈ (g, v] pays kc − s and v2 ∈[v, p ) pays s .

(b) m2 ∈ [p , g]. By symmetry to 2(c), v1 ∈ (g, v] pays kc − (s + v2 − p )and v2 ∈ [p , g] pays s + v2 − p .

(c) m2 is “above g”. If vi ∈ (g, v], then c li = kc/2, for i = 1, 2.

The proof shows that ∃ε > 0 : ∀ε ∈ (0, ε) there exists an equilibriumwith g = v − ε, for λ sufficiently small. The value for s and the verificationthat the above is an equilibrium are in the Appendix. �

Finally, I show that the ex ante incentive-efficiency result of AMS is pre-served, as long as Proposition 4 holds.

COROLLARY 1 (Approaching incentive efficiency): Under the hypotheses of Propo-sition 4, there exists an ordered equilibrium yielding a completion region that is ar-bitrarily close to the one deriving from the ex ante incentive efficient allocation thatmaximizes the sum of players’ payoffs.

Proof : It is sufficient to compare the value of p in Equation (2), as g ↑ v,with the value of x0 in Proposition 19 in AMS to see that the completion re-gion described in the proof of Proposition 4 approaches the ex ante incentiveefficient one that maximizes the sum of players’ payoffs, described by AMSin their Proposition 19.19 �

4. Early Contributions

The analysis so far reveals that cheap talk can effectively transmit informa-tion, but only if the cost of completing the project at the message stage, kc , isbelow the critical threshold v + v. If the overall cost, k, is larger, and if playerscan contribute before messages are exchanged, then an intriguing possibil-ity arises. Early contributions can reduce kc below v + v, thus allowing sub-sequent communication to be meaningful. This role for early contributionsmatches well with what Kennedy (1997) argues about preconditions for ne-gotiations, and with the rationale for occasional concessions or preliminaryagreements that Pruitt (1981) puts forth (as reported in the Introduction,see Kennedy, 1997, pp. 111–112 and Pruitt 1981, p. 232). Of course, to estab-lish such a result, one must verify that players are actually willing to sustainthe sure cost of early contributions in exchange for the potential benefitsbrought about by meaningful communication, keeping in mind that suchbenefits depend on the richness of the communication environment.

In this section I establish that in the game with minimally revealing com-munication players are not willing to bear the cost of early contributions.20

19 See also the discussion in the proof of Lemma 1, in particular Equation (A2).20 This is an interesting counterpoint to the result of AMS that even the coarsest possibleform of communication, i.e., two messages, always improves players’ welfare.


On the contrary, in the game with maximally revealing communication, earlycontributions can be worthwhile and they can increase welfare.

PROPOSITION 5 (Minimally revealing communication cannot justify early contri-butions): If k ≥ v + v, then there is no ordered equilibrium in which early contribu-tions are strictly positive in the game with minimally revealing communication.

Proof : A necessary condition for agents to be willing to provide early con-tributions in equilibrium is that the sum of ex ante equilibrium payoffs isnon-negative. Otherwise, at least one agent has a profitable deviation to no-contribution. Therefore, the calculations for ex ante welfare in Section 3.1,especially those in the proof of Proposition 1, are again relevant. Orderedequilibria can be described at the cheap talk stage by cutoffs zi , i = 1, 2:types smaller than zi send message n and types larger than zi send mes-sage y . From the necessary condition (1) and Lemma 4, meaningful com-munication can take place only if zi + v > kc , for i = 1, 2. Under this condi-tion, as demonstrated in the proof of Proposition 1 [see Equation (A3) andthe following discussion], ex ante welfare is bounded above by v + v − k <

0. Therefore, there cannot be an equilibrium with strictly positive earlycontributions. �

Turning now to maximally revealing communication, I focus on thestrategies described in the proof of Proposition 4 as continuation equilib-rium after the early contribution stage, to prove the following positive re-sult.

PROPOSITION 6 (Maximally revealing communication and beneficial early con-tributions): In the game with maximally revealing communication, ∃λ > 0 : ∀λ ∈(0, λ), if v + v < k < (v + v) + ( 3

8

)2(v − v), then there exists an equilibrium in

which early contributions are strictly positive. In this equilibrium players enjoy strictlypositive ex ante expected payoffs. In any other ordered equilibrium with zero early con-tributions, players’ payoffs are zero.

Proof : Define W E as

W E ≡(

38(v − v)

)2

(2v − kc )3 − (k − kc ), (3)

and let η > 0 be such that kc = k − 2η satisfies (4v + 2v) /3 < kc < v + v andW E > 0. (The quantity W E will be shown below to be an upper bound forequilibrium utility.) Note that for the range of k in the proposition suchη exists. I claim the following is an equilibrium: (1) at the early contribu-tion stage, c e

1 = c e2 = η > 0; (2) at the communication stage, after any his-

tory other than (η, η) players babble and then they do not contribute; and(3) at the communication stage, after history (η, η) the equilibrium strategy


described in the proof of Proposition 4 is played, with g sufficiently close to v.To see that the above is indeed an equilibrium, note that Proposition 4 andkc > (4v + 2v) /3 > v, along with Lemma 2 (for the no-contribution strat-egy), imply no profitable deviations are available at the message and at thelate contribution stages. Moving back to the early contribution stage, thereis only one constraint to consider: agents must prefer to contribute η ratherthan zero. As λ → 0 and g ↑ v, p converges to 3kc/4 − v/2, by Equation (2)in the proof of Proposition 4. Therefore, the ex ante utility of contributing η

for Player i converges to

12

[∫ v

(3kc /4−v/2)

∫ v

(3kc /4−v/2)+v−v1

v1 + v2 − kc

(v − v)2 dv2dv1 − (k − kc )]

= W E

2> 0,

where W E is given in Equation (3). (The last inequality follows from kc =k − 2η and the earlier choice of η.) Moreover, the utility of not contribut-ing at the early contribution stage is λη, which converges to zero as λ → 0.

Therefore, the equilibrium verification is complete: by continuity, λ can bechosen sufficiently small so that agents prefer to contribute η rather thanzero. In this equilibrium, each agent’s payoff is arbitrarily close to W E /2 > 0.

To complete the proof, note that in any other ordered equilibrium withc e = 0, one has kc = k > v + v, so that agents’ payoffs are zero by Propo-sition 2 and Lemma 2. �

The following corollary demonstrates that even maximally revealingcommunication is insufficient to justify early contributions, if k is larger thanthe upper bound in Proposition 6.

COROLLARY 2 (An upper bound for k): If k ≥ (v + v) + ( 38

)2(v − v), then

the only ordered equilibrium of the cheap-talk contribution game is the no-contributionequilibrium.

Proof : Proposition 2 establishes that if kc ≥ v + v, then the project is nevercompleted; therefore, by Lemma 2, players never contribute. Lemma 1 showsthat if kc ≤ (2v + 4v)/3, then at least one player’s payoff is nonpositive;therefore, there cannot be an equilibrium where kc ≤ (2v + 4v)/3. (Theweak inequality follows from λ > 0.) Corollary 1 implies that if (2v + 4v)/3 <

kc < v + v, then players’ payoffs never sum up to more than W E in Equa-tion (3). Therefore, necessary conditions for existence of an equilibrium inwhich early contributions are positive are kc < v + v and W E ≥ 0. Note thatW E in Equation (3) is strictly increasing in kc for kc < v + v. Therefore, ifW E

∣∣kc =v+v ≤ 0, then the corollary is proven. By Equation (3), this condition

is ensured by k ≥ (v + v) + ( 38

)2(v − v), the upper bound for k in the state-

ment of the corollary. �


The proof of Corollary 2 incidentally demonstrates that an upper boundfor equilibrium welfare is

W E∣∣kc =v+v = (v + v) +

(38

)2

(v − v) − k. (4)

The following corollary identifies how early contributions should be chosento approach the bound in Equation (4), if it is possible to do so.

COROLLARY 3 (The welfare-maximizing choice of early contributions): Assumev + v < k < (v + v) + ( 3

8

)2(v − v). There exists λ > 0 : ∀λ ∈ (0, λ), as c e ↓

k − (v + v), the equilibrium described in the proof of Proposition 6 yields welfare thatapproaches the upper bound in Equation (4).

Proof : The result follows from W E in Equation (3) increasing in kc , andfrom the proof of Proposition 6, which shows an equilibrium in which, asg ↑ v, joint welfare approaches W E . Together, they imply early contributionshould be as small as possible, but sufficient to leave kc < v + v, to ensureexistence of the equilibrium in the proof of Proposition 6. �

5. Discussion and Conclusion

Within the context of a joint project that two parties can contribute to, myanalysis explores the link between early contributions and information trans-mission in negotiations. In particular, the main question I am interested inis: Can early contributions create the conditions for mutual trust, thus im-proving the outcome of subsequent nonbinding communication? My answer,in the context of a cheap talk, dynamic contribution game in which partiallyfunded projects generate some utility, is Yes. My argument proceeds in threesteps:

1. I show that cheap talk—one way of modeling information transmis-sion in negotiations—can be effective in increasing agents’ equilib-rium payoffs from a joint project, but only if, at the moment in whichnegotiations start, the remaining cost to completion is sufficientlysmall. Proposition 2 identifies this critical threshold cost. This firststep demonstrates a need that communication, by itself, cannot sat-isfy, in contrast with what one obtains in a model in which incompleteprojects are worthless, see Proposition 1.

2. If the overall cost of the project exceeds the value in Proposition 2,early contributions can reduce the cost to completion below this criti-cal level, thus allowing sincere transmission of information in negoti-ations. This second step connects dynamics in contributions with theroles that negotiation scholars identify for preconditions for negotia-tions (Kennedy 1997) or preliminary agreements (Pruitt 1981): to cre-ate the conditions for mutual trust, thus allowing negotiations to begin


(or to continue, in a model with more opportunities to contribute andto receive new information.)

3. I show conditions under which it is worthwhile for agents to makethe sure sacrifice of early contributions in exchange for the potentialbenefits of successful negotiations. In particular, the quality of infor-mation transmission plays a critical role in this cost-benefit analysis.Indeed, Proposition 6 shows that sufficiently rich communication cangenerate benefits that justify the cost of early contributions. On thecontrary, Proposition 5 demonstrates that if information transmissionis very coarse, then the cost of early contributions is always larger thanthe potential benefit stemming from communication. This third stepformally verifies that the role for early contributions prefigured in step2 can be sustained as an equilibrium.

The negative result for minimally revealing communication, Proposition5, is interesting also in relation with the finding in AMS that even very coarsecommunication can improve agents’ payoffs.21 While the contrast arises be-cause of different technologies, one may wonder about the role played bymy restriction to ordered equilibria. In particular, Proposition 2, which isthe basis of the negative result in Proposition 5, makes use of this restriction.To show robustness of this result, I next extend Proposition 2 to all purestrategy equilibria for the game with minimally revealing communication.

PROPOSITION 7: If kc > v + v and λ > 0, then the only pure-strategy equilib-rium of the static, cheap-talk contribution game with minimally revealing communica-tion is the no-contribution equilibrium.

Proof : By Proposition 2, one only needs to consider nonordered equilibriain which some types send message “h” and all other types send message “l”. Ifirst establish two claims, with proof in the Appendix.

CLAIM 1: Completion after message pair (h, h) and completion after message pair(l, l) are incompatible.

CLAIM 2: Completion after message pairs (m1 = l, m2 = h) and (m1 = h,

m2 = l) are incompatible.Claims 1 and 2 imply that there exist some type of some player that is

sending a message fully expecting the ensuing probability of completionto be zero. Therefore, Lemma 4 implies the only equilibrium is the no-contribution equilibrium. �

Extending the negative results in Propositions 2 and 7 to the more gen-eral case considered in AMS of concave distribution functions for values

21 “. . . allowing exchange of binary messages is enough for greater efficiency . . .” see AMS,p. 3.


requires little additional work.22 Similarly, Proposition 2 easily extends tothe n-player case as long as v + (n − 1)v ≤ kc , that is, if every agent is pivotal.It is also possible to extend the positive results in Propositions 3 and 4 tomore general distributions, but, as AMS remark upon, the exact form of theequilibrium payments is highly distribution specific.

The positive results in Propositions 3 and 4 are derived for the limit-ing case in which the parameter λ, which scales the units of public goodproduced in partially funded projects, is converging to zero. This limitingcase is simplest to analyze. Moreover, the fact that the negative result ofPropositions 2 holds for any positive value for λ, including arbitrarily smallones, lends a nice symmetry to the statements of the results.

Most importantly, though, the limiting case λ ↓ 0 allows a clear under-standing of the contributions of this paper in relation to the extant resultsin the literature on the private provision of public goods. First, this paperborrows the technology from Marx and Matthews (2000), who, in a publicly-known-value framework, show that dynamics may allow efficient completionsotherwise impossible in a static setting. This result is similar to what I obtainin Proposition 6. However, the forces they identify are not what generatesmy result. Indeed, as λ ↓ 0, the static game of Marx and Matthews (2000)admits equilibrium outcomes that approach efficiency,23 so the differencebetween their static and dynamic games vanishes with λ. This does not hap-pen in my model, as the comparison of Propositions 2 and 6 shows. Indeed,as described at the beginning of this section, my result has everything to dowith private information and communication, and both aspects are absentfrom Marx and Matthews (2000). Second, the technology and payoffs in mymodel converge to those in AMS, as λ ↓ 0. When λ = 0, they present the veryinteresting result that communication may resolve any inefficiency derivingfrom private information, regardless of the completion cost. Therefore, mynegative result in Proposition 2, which implies that, for an arbitrarily smallbut positive λ, the cost to completion cannot be too high for efficiency toobtain, constitutes an important qualification to a very relevant result.

Beyond λ ↓ 0, it is also worth discussing the assumption that early contri-butions precede the arrival of information. I view this assumption as simpli-fying a model with “ex post private information,” such as the one in Krahmerand Strausz (2008). In these models, agents become progressively better in-formed, and, in particular, additional information arrives after the strategicinteraction among players has begun. As long as the passage of time bringsthe possibility that agents receive further information (i.e., if they reviseor become more confident in their assessment of how much they stand to

22 The point is explicitly made in the proof of Proposition 2.23 Using the equilibrium described in footnote 5 on p. 333 of Marx and Matthews (2000),in a two-player setting completion can occur if the threshold is below (v1 + v2)/(1 + λ).Thus, as λ ↓ 0, projects whose cost approaches v1 + v2 may be completed, implying thatfull efficiency is obtained in the limit.


v

v

vv1

v2

v

v

vv1

v2

x0

x0

y0

y0

a b

Figure 3: Ex ante efficient completion region.

benefit from the public good), then the main force I identify survives: first-period contributions reduce the amount remaining to completion so thatsecond-period information can be credibly communicated through costlessmessages. In other words, it is not necessary that early contributions pre-cede all information, but only that they precede some nontrivial information.Therefore, my timing, while clearly a simplification of the real-world situa-tions I introduce in Section 1, is sufficiently rich to show this effect.

Finally, one interesting open question is whether there is any benefit toambiguity in messages for maximally revealing communication, i.e., whethera nonordered equilibrium can lead to completion when ordered ones fail.This topic is subject of current research.

Appendix

Proof of Lemma 1: I first derive an upper bound for ex ante total welfareassuming that, after the early contribution stage, players obtain the ex anteincentive-efficient outcome that maximizes the sum of payoffs. The proofis completed showing this upper bound for welfare is negative, under thehypothesis of the lemma.

Applying the results of Ledyard and Palfrey (1999, 2007) and Williams(1987), the ex ante incentive-efficient probability of completion p (v1, v2)identifies a linear completion boundary with slope −1 in the (v1, v2)space, as in parts (a) and (b) of Figure 3, where the completion areasare striped. Ledyard and Palfrey (2007) and Williams (1987) show thatthe following constraint, subsuming incentive compatibility and individual


rationality, must be satisfied:

�(p ) ≡∫

v1

∫v2

(v1 − (v − v1) + v2 − (v − v2) − kc )

× p (v1, v2)1

(v − v)2dv1dv2 ≥ 0,

(A1)

where the first term under the integral sign is the sum of virtual valuations,for a uniform distribution, minus the remaining cost of provision. Moreover,Williams points out that this constraint must hold with equality, when onemaximes the sum of players’ payoffs.

Using � in Equation (A1), I now show that kc < (4v + 2v)/3 rules outthe case depicted in part (a) of Figure 3, where the ex ante probability ofcompletion is strictly smaller than 1/2.

Denote the intersection of the completion boundary with the line v1 = vas x0. A direct calculation yields

�(p ) =∫ v

x0

∫ v

x0+v−v1

(2v1 − v + 2v2 − v − kc )(v − v)2

dv2dv1

= 16

(v − x0)2

(v − v)2(2v − 3kc + 4x0). (A2)

The requirement � = 0 then implies either x0 = (3kc − 2v)/4 < v, which isimpossible, or x0 = v, which is clearly not optimal. Therefore, the probabilityof completion must be larger or equal to 1/2, as in part (b) of Figure 3.

Having established the shape of the optimal completion frontier, I nowset up the welfare maximization problem. The choice variables are kc , whichidentifies the amount of early contributions given the total cost k, and y0,

which identifies the ex ante incentive-efficient outcome given kc . Note that, inFigure 3b, y0 denotes the intersection of the completion boundary with theline v1 = v. � in Equation (A1) may be calculated in three separate areas:(1) v1 ∈ [v, y0) and v2 ∈ [y0 + v − v1, y0], (2) v1 ∈ [v, y0] and v2 ∈ [y0, v],and (3) v1 ∈ [y0, v] and v2 ∈ [v, v]. Making explicit the dependence on y0

and kc , one obtains

�(y0, kc ) = (y0 − v)2

(v − v)2

(43

y0 − v − 12

kc + 23

v)

+ (v − y0)(y0 − v)(v − v)2

(2y0 − v − kc + v)

+ (v − y0)(v − v)(v − v)2

(y0 − kc + v).


Similarly, total ex ante expected welfare, including the cost of earlycontributions, is

W (y0, kc , k) = 16

(y0 − v)2

(v − v)2(4y0 − 3kc + 2v)

+ (v − y0)(y0 − v)(v − v)2

(y0 − kc + v + v

2

)

+(v − y0)(v − v)(v − v)2

(y0 + v

2+ v − kc

)− (k − kc ).

Thus, for y0 ∈ [v, v] and kc ∈ [2v, (4v + 2v)/3], the welfare-maximizing out-come solves24

maxy0,kc

W (y0, kc , k) s .t. �(y0, kc ) ≥ 0.

Now, I first demonstrate that there is no interior solution to this prob-lem. Then, I show the objective function is negative at the boundary, thusconcluding the proof. By contradiction, suppose that an interior solutionto this problem exists. Since ∂�

∂y0(v − v)2 = (y0 − v) (kc − 2y0 + 2v − 2v) >

2 (y0 − v) (v − y0) > 0, the constraint qualification is satisfied. At the inte-rior solution, the first-order condition holds, so that for some μ ≥ 0 oneobtains

∂W∂y0

+ μ∂�

∂y0= 0 = ∂W

∂kc+ μ

∂�

∂kc.

The first-order conditions are incompatible with μ = 0, because ∂W∂kc

(v −v)2 = 1

2 (y0 − v)2 > 0. Moreover, at the optimal solution kc <4v+2y0

3 , since∂�∂kc

< 0 and

�

(y0, kc = 4v + 2y0

3

)=

(13

)(v − y0) (y0 − v)

(v − v)2 (y0 − 2v + v) < 0.

Therefore, at the putative interior solution, μ > 0 and kc <4v+2y0

3 .

I now claim that, under these last two inequalities, the second-order con-dition implies any critical point is a strict minimum, and thus a contradic-tion to the existence of an interior maximum. The second-order sufficient

24 Without loss of generality, I extended the possible choices of kc to include the bound-aries to ensure existence of a solution.


condition for an interior minimum is

det

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0∂�

∂kc

∂�

∂y0

∂�

∂kc

∂2W∂k2

c+ μ

∂2�

∂k2c

∂2W∂kc∂y0

+ μ∂2�

∂kc∂y0

∂�

∂y0

∂2W∂kc∂y0

+ μ∂2�

∂kc∂y0

∂2W

∂y 20

+ μ∂2�

∂y 20

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

< 0.

Substituting ∂�∂kc

= − 1μ

∂W∂kc

and ∂�∂y0

= − 1μ

∂W∂y0

from the first-order condition

and noting that ∂2W∂k2

c+ μ∂2�

∂k2c

= 0, straightforward calculations show the above

determinant is (y0−v)4

4μ2(v−v)4 (2y0 − 3(1 + μ)kc + 4v + 2vμ + 4vμ), an increasing

function of kc . Substituting the upper bound of 4v+2y03 for kc , the determi-

nant is therefore strictly smaller than (y0−v)4

4μ2(v−v)4 (2μ(v − y0)) < 0; thus, therecannot be an interior maximum.

I now show that at the boundary of the constraint set the maxi-mand is negative, thus concluding the proof of the lemma. If y0 = v, thenW (y0, kc , k) = 1

6 (4v − 3kc + 2v) − (k − kc ), an increasing function of kc thatat kc = (4v + 2v) /3 equals v + v − k < 0. If y0 = v, then � (y0, kc ) ≥ 0 andkc ≥ 2v jointly require kc = 2v. Also in this case W (y0, kc , k) equals v + v −k < 0. The other possible solutions, that is, those with kc = (4v + 2v) /3 orkc = 2v, yield identical results. �

Proof of Proposition 1: The strategy of proof is the same for both minimallyand maximally revealing communication: (1) identify ordered equilibria thatmaximize joint-welfare conditional on any amount of early contributions c e ,(2) evaluate welfare, including the cost of c e , and (3) show that the welfare-maximizing level of c e is zero.

For minimally revealing communication, in equilibrium all types ofPlayer i above zi send message y , while those below zi send message n, fori = 1, 2. There are two possibilities to consider. The first is kc > v + v. By thediscussion following Lemma 3, one can restrict attention to “simple” equi-libria: z1 + z2 = kc and completion occurs if and only if two y messages aresent, in which case Player i pays zi . No contributions are made in all othercases. In equilibrium, ex ante welfare is

maxz1

∫ v

z1

∫ v

kc −z1

v1 + v2 − kc

(v − v)2 dv2dv1 − (k − kc )

= 1(v − v)2

(v − kc

2

)3

− (k − kc ),


where the maximum is attained at z1 = kc/2. The previously displayed valueis strictly increasing in kc for kc > v + v; therefore, the optimal level of kc isk. Moreover, welfare is strictly positive, because kc > 2v.

The second possibility is (4v + 2v)/3 ≤ kc ≤ v + v, that is, higher levelof early contributions.25 It remains true that after two n messages comple-tion cannot happen in equilibrium, because of Lemma 3 and 4v > v, whichimplies kc > v. However, if zi + v ≥ kc not only is completion possible aftermessage pair (y , n), but, to maximize welfare, completion should happenwith probability one. Therefore, an upper bound for welfare is

W U = T {z1 ≥ kc − v}∫ v

z1

∫ z2

v

v1 + v2 − kc

(v − v)2dv2dv1

+ T {z2 ≥ kc − v}∫ v

z2

∫ z1

v

v1 + v2 − kc

(v − v)2dv1dv2

+∫ v

z1

∫ v

z2

v1 + v2 − kc

(v − v)2dv2dv1 − (k − kc ),

(A3)

where T is an indicator function that takes the value 1 if true and 0 if false.If z1 < kc − v and z2 < kc − v, then one falls back on the “simple” equi-librium previously analyzed. Otherwise, W U in Equation (A3) is decreas-ing in z1 and z2. Setting z1 = z2 = kc − v, the upper bound for welfare inEquation (A3) simplifies as v − k + v < 0. Therefore, calculated over bothpossibilities, the best level of kc is k, implying c e = k − kc = 0: early contribu-tion play no welfare-improving role.

I now turn to maximally revealing communication. Since one can restrictattention to kc ≥ (4v + 2v) /3 by Lemma 1, Proposition 19 in AMS applies.By this result, there exists an ex ante incentive efficient equilibrium that max-imizes the sum of players’ ex ante payoffs if x0 = 3

4 kc − 12 v, where x0 denotes

the intersection of the completion boundary with the line v1 = v as in Fig-ure 3a. In this equilibrium, welfare is∫ v

x0

∫ v

x0+v−v1

v1 + v2 − kc

(v − v)2dv2dv1 − (k − kc ) = 9

64(2v − kc )3

(v − v)2− (k − kc ),

a strictly increasing function of kc , for kc ≥ (2v + 4v) /3. Therefore, the bestlevel of kc is k, implying c e = 0: early contribution play no welfare-improvingrole. �

Equilibrium verification for strategies in the Proof of Proposition 4: I analyze Player1’s incentives; considerations for Player 2 follow by symmetry. First, I collecta set of inequalities ensuring equilibrium. Second, I show a solution to thecollected inequalities, for λ sufficiently small.

25 By Lemma 1, it is pointless to consider kc < (4v + 2v)/3.


Beginning at the (late) contribution stage, recall the value of p givenin Equation (2):

p = 34

kc − 2g + 32

v. (2)

For the rest of this verification, I refer to the inequality ensuring that playersprefer to contribute rather than to free ride as NFR, for brevity. Moreover, Imake reference to the regions in Figure 2. In Region 1(c), NFR for Player 1is v − s ≥ vλ kc −s+c e

k . Note that if the left-hand side of the previous inequalityis strictly positive, then there exists λ sufficiently small so that the inequal-ity holds. To simplify notation then, all NRF constraints are written in suchfashion. Therefore, the first inequality collected is

v > s . (A4)

In addition, Player 1’s payment is admissible if 0 ≤ s ≤ kc . However, be-cause kc > (4v + 2v)/3 > v, if Player 1 prefers to contribute rather than freeride, then his payments are necessarily strictly smaller than kc . Moreover,since payments sum up to kc , if Player 2’s payments are strictly smaller thankc , then Player 1’s payments are positive. Therefore, by symmetry, paymentsare admissible if NFR holds, in all Regions. Note as well that, since kc > v,the no-contribution strategy is always an equilibrium.

In Region 2(c), NFR is satisfied if v1 − (s + v1 − p ) > 0, or simply

p > s, (A5)

while in Region 2(b), NFR is satisfied if, for any v1 and v2, we have v1 − kc2 −

13 (v1 − v2) > 0.

The most taxing situation occurs when v1 and v2 are as small as possibleon the equilibrium path, that is along the line v2 = (p + g) − v1. Thus, usingp in Equation (2), the constraint is satisfied for all v1 ∈ [p , g] if it holds forv1 = p , requiring

g < v. (A6)

Moving to Region 3(c), NFR is satisfied if

g > kc/2, (A7)

while in Region 3(b), NFR is satisfied if v1 − (kc − (s + v2 − p )) > 0, whichis sure to hold when it is true for v1 = g and v2 = p , or

g − (kc − s) > 0. (A8)

Finally, in Region 3(a), NFR is satisfied when g > kc − s , which is again Equa-tion (A8).

I now analyze the message stage. I consider deviations to adjacent mes-sages and I defer the analysis of the other deviations. In Region 1, type v1 = pmust be indifferent between m1 = p or m1 = “below p ”. This is true by con-struction, because the expected probability of completion and the expected


payment are the same after the two messages.26 Moving to Region 2, notethat Q(v1), the expected probability of completion for type v1 at the mes-sage stage, is

Q(v1) =∫ v

g+p −v1

1v − v

dv2,

and that the expected payment of v1, X (v1), is

X (v1) = 1v − v

[∫ g

p +g−v1

(13

(v1 − v2) + kc

2

)dv2 +

∫ v

g(s + v1 − p )dv2

].

The necessary (and locally sufficient) condition for incentive compatibilityX ′(v1) = v1Q ′(v1) is satisfied by the value of p in Equation (2), which is ad-missible when

v < p < g . (A9)

Finally, in Region 3, type v1 = g must be indifferent between “g” or “aboveg”. The utility of “g” is

1v − v

(g(v − p ) −

∫ g

p

(13

(g − v2) + kc

2

)dv2 −

∫ v

g(s + g − p )dv2

),

and the utility of “above g” is

1v − v

(g(v − v)−(p − v)(kc − s)−

∫ g

p(kc − (s + v2 − p ))dv2 −(v − g)

kc

2

).

Equality of these two expressions pins down s :

s = −2g2 + kc (v − 2v) + g(kc + 2v)2(v − v)

. (A10)

In two steps, I now provide a solution to the system (2) and Equations(A4)–(A10). First, I show that if g = v, then the strict inequalities Equations(A4), (A5), (A7), and (A9) hold, using Equations (2) and (A10). Second, Ishow that Equations (A6) and (A8) are satisfied if one reduces g to v − ε.Therefore, by continuity, for ε sufficiently small the system (2) and Equa-tions (A4)—(A10) is satisfied. Thus, no deviation on the equilibrium path isprofitable for λ sufficiently small.

First, if g = v, then p = 3kc/4 − v/2 and s = kc − v, using Equations (2)and (A10). Therefore Equations (A4), (A5), (A7), and (A9) hold, because(4v + 2v)/3 < kc < v + v < 2v. Second, note that Equations (A6) and (A8),that is g < v and g − (kc − s) > 0, fail for g = v, because they would holdwith equality rather than with strict inequality. Reducing g by ε, then g < vis satisfied. All that is left to show is g − (kc − s) > 0 for g = v − ε, which

26 See payments and completion probabilities for Regions 1(c) and 2(c).


is implied by s ′(g)|g=v < −1. Indeed, from Equation (A10), one obtainsdsdg |g=v = kc −4v+2v

2(v−v) < − 32 , where the last inequality follows from kc < v + v.

Therefore, for sufficiently small ε, the choice of g = v − ε satisfies all con-straints. This concludes the main part of the proof. To complete the verifi-cation, I now show that deviations off the equilibrium path at the commu-nication stage are not profitable for the previous choice of g = v − ε and ε

positive, but small.Throughout, the necessary conditions X ′(v1) = v1Q ′(v1) and Q ′(v1) ≥

0, ∀v1 ∈ [v, v], hold. They are also sufficient to discourage deviations tosmaller messages because, after sending the message of type v′

1 < v1, typev1 behaves off the equilibrium path exactly as v′

1 does on the equilibriumpath. On the contrary, there is no guarantee that the utility of type v1 thatpretends to be v′

1 > v1 is v1Q(v′1) − X (v′

1), because this is not a direct mech-anism. In particular, v1 may decide to free ride when v′

1 completes in equi-librium. I present in its entirety the proof that types v1 ∈ (p , g) optimallychoose m1 = v1 rather then any other message m1 ∈ (v1, g). All other proofsare similar and they are available upon request.

Define as U D(m1) the utility of deviating to a messages m1 ∈ [v1, g],

U D(m1) ≡∫ g

p +g−m1

max{

v1 −(

13

(m1 − v2) + kc

2

),

λv1

kc − 13

(m1 − v2) − kc

2+ c e

k

⎫⎪⎬⎪⎭

dv2

v − v

+∫ v

gmax

{v1 − (s + m1 − p ) , λv1

kc − (s + m1 − p ) + c e

k

}dv2

v − v.

I now show that, for v1 ∈ (p , g), the maximum of U D(m1) is attained at m1 =v1. Fix the previous choice of g = v − ε, with ε > 0 but small. Note that thereexists λ sufficiently small such that type v1 prefers to complete the projectrather than free ride, for any m1 ≤ v1 + ε. Indeed, for the relevant ranges,one obtains

v1 −(

13

(m1 − v2) + kc

2

)≥ v1 −

(13

(m1 − (p + g − m1)) + kc

2

)

≥ 13

v1 − 23ε + 1

3p + 1

3g − kc

2≥ 1

3ε > 0,

using p in Equation (2), and also that v1 − (s + m1 − p ) = p − s − ε > 0,

where the last inequality follows by Equation (A5) and ε small. Once theform of U D is established for m1 ≤ v1 + ε, one may easily verify that U D isstrictly concave for p ≤ m1 ≤ v1 + ε, with a strict maximum at m1 = v1. Toanalyze deviations beyond m1 = v1 + ε, denote the difference in utility fortype v1 between sending message v1 and message v1 + ε as � > 0, which doesnot depend on λ, and consider how U D changes as one increases m1 beyond


v1 + ε. There are two possibilities. First, if type v1 prefers to complete theproject even for the smallest possible v2, that is, when v2 is such that m1 +v2 = p + g , then U D is strictly decreasing in m1 for m1 > v1 + ε. Indeed, sincethe second addendum in U D above is decreasing in m1, dU D/dm1 is smallerthan

d(∫ g

p +g−m1

(v1 −

(13

(m1 − v2) + kc

2

))dv2

v − v

)dm1

= v1 − m1 + ε

v − v< 0,

using p in Equation (2) and g = v1 − ε. The second possibility arises whenv1 is indifferent between completion and free riding for some v∗

2 ∈ (p , g).Nonetheless, dU D/dm1 is bounded above by a finite term multiplied by λ.

Indeed, dU D/dm1 is smaller than

d

(∫ v∗2

p +g−m1

λv1kc − 1

3 (m1 − v2) − kc2 + c e

kdv2

v − v

)

dm1

+d

(∫ g

v∗2

(v1 −

(13

(m1 − v2) + kc

2

))dv2

v − v

)

dm1

< λv1

k(v − v)

(c e + 2

3g + 1

2kc + −m1 + 2

3p − 1

3v∗

2

).

Thus, there exists λ sufficiently small such that any potential increase in U D

brought about by a message beyond v1 + ε cannot make up for the sure loss� of moving from m1 = v1 to m1 = v1 + ε, since � is independent of λ. �

Proof of Claim 1: I proceed by contradiction, assuming that completion oc-curs both after (h, h) and after (l, l). The “conditional” version of Lemma 3,and in particular the necessary condition in Equation (1), implies that thereexists four types, vh

1 , vl1, vh

2 , and vl2, such that:

vh1 · Pr

{(v2 > vh

2 )|(m2 = h)} + vh

2 · Pr{(

v1 > vh1 )|(m1 = h

)}> v + v , (A11)

so that completion after (h, h) is possible, and

vl1 · Pr

{(v2 > vl

2)|(m2 = l)} + vl

2 · Pr{(

v1 > vl1)|(m1 = l

)}> v + v , (A12)

so that completion after (l, l) is possible. Since values are uniformly dis-tributed, it follows that, for i = 1, 2 and for j = l, h,

Pr{vi > v j

i ∩ mi = j}(v − v) ≤ v − v j

i . (A13)


Therefore, I now claim the following must hold, for i = 1, 2 and for j = l, h:

v ji > (v − v) Pr

{v3−i > v j

3−i ∩ m3−i = j}

+ (v − v) Pr{v3−i ≤ v j

3−i ∩ m3−i = j}. (A14)

To see why Equation (A14) is true, suppose that, for the sake of simplicity, itis contradicted for vh

1 , that is,

vh1 ≤ (v − v) Pr

{v2 > vh

2 ∩ m2 = h} + (v − v) Pr

{v2 ≤ vh

2 ∩ m2 = h}. (A15)

It then follows that

v + v < vh1 · Pr

{(v2 > vh

2

)|(m2 = h)} + vh

2 · Pr{(

v1 > vh1

)|(m1 = h)}

(by (A11))

≤ vh1 · Pr

{(v2 > vh

2

)|(m2 = h)} + vh

2

≤ vh1 · Pr

{(v2 > vh

2

) ∩ (m2 = h

)}Pr

{(v2 > vh

2

) ∩ (m2 = h

)} + Pr{(

v2 ≤ vh2

) ∩ (m2 = h

)} + vh2

≤ Pr{(

v2 > vh2

) ∩ (m2 = h

)}· v1

Pr{(

v2 > vh2

) ∩ (m2 = h

)} + Pr{(

v2 ≤ vh2

) ∩ (m2 = h

)} + vh2

≤ Pr{(

v2 > vh2

) ∩ (m2 = h

)}(v − v) + vh

2 (by (A15))

≤ v − vh2 + vh

2 , (by (A13))

and the first and last inequalities imply v < 0, which is impossible.Having established Equation (A14), one can rewrite it as

Pr{v3−i >v j

3−i ∩ m3−i = j}(v − v)<v j

i − Pr{v3−i ≤ v j

3−i ∩ m3−i = j}(v − v),

(A16)

and combine it with Equations (A11) and (A12) to obtain

vh1 − Pr

{v2 ≤ vh

2 ∩ m2 = h}(v − v) + vh

2

− Pr{v1 ≤ vh

1 ∩ m1 = h}(v − v) > v + v , (A17)

and

vl1 − Pr

{v2 ≤ vl

2 ∩ m2 = l}(v − v) + vl

2

− Pr{v1 ≤ vl

1 ∩ m2 = l}(v − v) > v + v , (A18)

respectively. To see how Equation (A17) is obtained, rewrite Equation (A11)as

vh1 · Pr

{(v2 > vh

2

) ∩ (m2 = h

)}Pr

{(v2 > vh

2

) ∩ (m2 = h

)} + Pr{(

v2 ≤ vh2

) ∩ (m2 = h

)}+ vh

2 · Pr{(

v1 > vh1

)|(m1 = h)}

> v + v ,


and, since x/ (x + y) is increasing in x for y ≥ 0, one can substitute Pr{(v2 >

vh2 ) ∩ (m2 = h)} from Equation (A16) into the above and obtain

vh1 ·

vh1

v − v− Pr{v2 ≤ vh

2 ∩ m2 = h}vh

1

v − v

+ vh2 · Pr{(v1 > vh

1 )|(m1 = h)} > v + v ,

which simplifies as vh1 − Pr{v2 ≤ vh

2 ∩ m2 = h}(v − v) + vh2 · Pr{(v1 >

vh1 )|(m1 = h)} > v + v. Applying the same procedure to the term

Pr{(v1 > vh1 )|(m1 = h)} above, one obtains Equation (A17). Equation (A18)

is derived similarly. Now, after summing Equations (A17) and (A18), oneobtains ∑

i=1,2

∑j=l,h

[v j

i − (Pr

{vi ≤ v j

i ∩ mi = j})

(v − v)]

> 2 (v + v) . (A19)

However, Equation (A19) can never be satisfied, since for i = 1, 2 the follow-ing holds,∑

j=l,h

Pr{

vi ≤ v ji ∩ mi = j

}(v − v)

≥∑j=l,h

Pr{vi ≤ min{vh

i , vli } ∩ mi = j

}(v − v) ≥ min{vh

i , vli } − v ,

because types are uniformly distributed and Pr{mi = l} + Pr{mi = h} = 1. �

Proof of Claim 2 : By contradiction, using the necessary condition in Equa-tion (1), there exists four types vh

1 , vl1, vh

2 , and vl2, (not necessarily the same

as in the Proof of Claim 1), such that

vl1 · Pr{v2 > vh

2 ∩ m2 = h}Pr{v2 > vh

2 ∩ m2 = h} + Pr{v2 ≤ vh2 ∩ m2 = h}

+ vh2 · Pr{v1 > vl

1 ∩ m1 = l}Pr{v1 > vl

1 ∩ m1 = l} + Pr{v1 ≤ vl1 ∩ m1 = l} > v + v,

if completion after (m1 = l, m2 = h) occurs, and such that

vh1 · Pr{v2 > vl

2 ∩ m2 = l}Pr{v2 > vl

2 ∩ m2 = l} + Pr{v2 ≤ vl2 ∩ m2 = l}

+ vl2 · Pr{v1 > vh

1 ∩ m1 = h}Pr{v1 > vh

1 ∩ m1 = h} + Pr{v1 ≤ vh1 ∩ m1 = h} > v + v,

if completion after (m1 = h, m2 = l) occurs.With steps similar to those in the Proof of Claim 1, using Pr{v2 >

vh2 ∩ m2 = h} ≤ v−vh

2v−v , if the previous two displayed inequality hold, then one


obtains

vl1 − (v − v) Pr{v2 ≤ vh

2 ∩ m2 = h}+ vh

2 − (v − v) Pr{v1 ≤ vl1 ∩ m1 = l} > v + v, and

vh1 − (v − v) Pr{v2 ≤ vl

2 ∩ m2 = l}+ vl

2 − (v − v) Pr{v1 ≤ vh1 ∩ m1 = h} > v + v,

respectively. Summing these two inequalities leads to Equation (A19),and the same impossibility result as in the Proof of Claim 1 applies.

�

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