On the Voluntary Provision of “Weakest-Link” Public Goods:

the case of private information

Stefano Barbieri∗ David A. Malueg†

November 18, 2014

Abstract

We characterize equilibria in a private-provision public-good game where individuals are allowedarbitrary contribution levels and the level of the public good equals the least contribution made by anindividual. Equilibrium comparative statics are derived for the interim Pareto-dominant equilibrium.First, improvements in the cost distribution of even only one player benefit all. Second, even with such“weakest-link” public goods, where greater similarity of preferences would seem to facilitate coordination,increased heterogeneity can increase payoffs. Indeed, increasing the riskiness of cost distributions has anambiguous effect on welfare. Two mechanisms are provided for improving equilibrium payoffs: technologytransfer and cheap-talk communication. While substantial welfare gains are possible, examples show thati) technology transfer may be futile if a “regularity” condition is not satisfied and ii) cheap talk may beuseless if the language for communication is not sufficiently rich.

JEL Codes: D61, D82, H41Keywords: weakest link, voluntary provision, public good, interim efficiency, cheap talk

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∗Department of Economics, 206 Tilton Hall, Tulane University, New Orleans, LA 70118; email: sbarbier@tulane.edu.†Department of Economics, 3136 Sproul Hall, University of California, Riverside, CA 92521; email: david.malueg@ucr.edu.

1 Introduction

We study groups for which performance is determined by the least effort of the members in the group.

Hirshleifer (1983) coined such a relationship a “weakest link” production function.1 This contrasts with the

classic model of private public good provision of Bergstrom, Blume, and Varian (1986), in which the level

of the public good depends on the sum of individuals’ contributions.2 The importance of the weakest-link

scenario has been underscored by its many applications in the literature, including providing levees (Hirsh-

leifer, 1983) or securing alliance perimeters (Murdoch, 1995), dredging successive stretches of a navigation

channel (Harrison and Hirshleifer, 1989), eradicating disease (Arce and Sandler, 2002), producing goods via

an assembly line (Brandts and Cooper, 2006), coauthoring reports (Brandts et al., 2013), and combatting

terrorism (Sandler, 2003).

While many authors have studied such “weakest-link” models of public goods, we suppose players differ

in their abilities to contribute toward the public good, and we model this through private cost parameters.

For three reasons, analyzing the effects of private information in the weakest-link framework is important.

First, in contrast to standard public good settings where free-riding is common, with a weakest-link

technology one agent’s additional effort is effective only if other agents exert as much effort; this lack

of substitutability militates against free riding, but improving efficiency requires coordination. Private

information hampers coordination by introducing uncertainty about others’ efforts. So, fearing others will

provide lower effort, an agent may reduce its own effort below its ideal level, since any effort exceeding

another’s is “wasted.” Therefore, private information may have important consequences pushing toward

inefficiency.

Second, including private information is often a more realistic assumption. Assuming all agents are per-

fectly informed of everyone’s cost of effort is a useful modeling assumption, but it is certainly worthwhile

exploring what happens should this assumption not be met. Moreover, a careful modeling of private in-

formation may offer insights on a sizable, successful, and very interesting experimental literature on the

weakest-link game and the closely related “stag hunt” and “corporate turnaround” games.3 Within this lit-

erature, one concept often proposed to interpret and match theory and experimental results is the “Quantal

Response Equilibrium” (QRE) of McKelvey and Palfrey (1995). McKelvey and Palfrey (2008, p. 541) write:

1See also Harrison and Hirshleifer (1989) for expanded analysis and experimental evaluations. In addition to Hirshleifer’s(1983), another seminal paper on the relation among different public good production functions is Cornes and Sandler (1984),with further developments and applications appearing in Cornes (1993).

2For full-information environments, Cornes and Hartley (2007) present an exhaustive treatment of the relation amongdifferent public-good production functions.

3Because our focus is theoretical rather than experimental, we cannot do justice to this whole strand of the literature;however, important papers include Van Huyck et al. (1990), Palfrey and Rosenthal (1991), Battalio et al. (2001), Brandts andCoopers (2006, 2007), and the more recent Riedl et al. (2011) and Brandts et al. (2013), which also contain comprehensiveliterature reviews.

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The QRE can be viewed as a generalization of statistical models of discrete choice behavior of

individuals to a game theoretic setting. Just like in discrete choice models of individual behavior,

individuals are assumed to have an observed an unobserved component of their utility function.

However, unlike in models of individual choice, the unobserved part of an individual’s utility is

unobserved not just to the econometrician, but also to the other players in the game. Thus, the

game becomes a game of incomplete information. A QRE is simply a Bayesian equilibrium to

this game.

Therefore, our analysis of Bayesian equilibrium in the continuous-action private-information weakest-link

game may be helpful in interpreting these experimental results. For example, it may offer insights about

whether to expect that greater homogeneity of teams is beneficial.

Third, the analysis of private information is a prerequisite to understanding whether information revela-

tion through cheap-talk communication is possible. Several of the previously mentioned experimental papers

investigated communication as a way to improve equilibrium outcomes in the weakest-link game, even with-

out introducing private information. From the point of view of the theory, communication may help in a

full-information framework by allowing players to achieve a correlated equilibrium, rather than just a Nash

equilibrium. This role for communication is well surveyed in Forges (2009), but it is not what we explore

here. We are interested in agents voluntarily revealing their private information to others, so our approach

is inspired by the seminal work of Crawford and Sobel (1982) for private-information environments.

We believe ours is the first formal analysis of the effects of private information within the weakest-link

framework allowing players arbitrary contribution possibilities. Our analysis complements that of Palfrey

and Rosenthal (1991), which, in contrast, considers only binary actions, i.e., agents can either contribute

or not.4 Allowing continuous choice of actions is obviously important for applications; the height of a levee

is crucially important, for instance, not just whether the levee is built or not. Moreover, the binary-action

weakest-link game builds in strong rewards for coordination on the same action: efforts are not wasted and

it is possible to complete the project. It is harder to achieve equilibrium coordination on the same action if

efforts are possibly continuous; moreover, we will demonstrate that this coordination may not be desirable

from an efficiency standpoint. Finally, Palfrey and Rosenthal (1991) show that great benefits can be obtained

in equilibrium via the use of two-message pre-play communication. An interesting question is whether having

the same dimensionality of action and message spaces is important for this result; a natural framework to

explore this issue is when possible actions are continuous, but possible messages are finite.

Our analysis begins with the characterization of equilibria in the symmetric one-shot game without in-

4A companion paper, Barbieri and Malueg (2013), is entirely devoted to the problem of securing nuclear stockpiles fromterrorists. The symmetric theoretical model there is a special case of Palfrey and Rosenthal’s.

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formation sharing. Although many equilibria exist, we are able to identify the “best” one from the players’

perspective, viz., the equilibrium that maximizes the interim expected payoff of each type. Our characteriza-

tion identifies a “regularity” property of the distribution of the private marginal cost of effort, and we show

that regularity is necessary and sufficient for the best equilibrium strategy to be strictly decreasing in cost.

When regularity fails everywhere, uncertainty is pernicious and contributions to the public good are entirely

determined by the choice of a player with the largest possible cost, no matter the actual cost realizations.

More generally, when regularity fails, equilibrium strategies must have “flat spot” (i.e., a contribution level

chosen by an open interval of types). Then with positive probability all players will choose the same level

of effort, an outcome that might be misinterpreted by an outside observer as efficient coordination. On the

contrary, such “flat spots” are a symptom of inefficiency.

Next, focusing on the best equilibrium, we derive comparative statics. A decrease in the number of players

increases individual contributions and increases public good provision, thus benefitting all. We also show

that lower costs benefit all agents, and we extend this analysis to two-player situations where contributors

are ex ante asymmetric. In particular, if even one player’s cost distribution improves, then both players’

contributions and interim utilities (weakly) increase.5 Given the importance of coordination, one may expect

a decrease in riskiness of cost distributions to increase payoffs. But this is not necessarily the case. And

since a “more risky” distribution of cost can be thought of as increasing (ex post) heterogeneity in team

composition, our results demonstrate that the effects of heterogeneity are complex.6

Many of the Introduction’s applications of weakest-link technology can be considered examples of pro-

duction in teams. There is an extensive literature analyzing the efficiency consequences of heterogeneity

and knowledge transfers among peers, with mixed findings. For example, Jackson and Bruegmann (2009,

pp. 85–86)7 write:

Economists have long been concerned with human capital spillovers. . . . When workers and their

colleagues are complementary inputs in production, improvements in coworker quality may in-

crease a worker’s own productivity. There is evidence of such spillovers. . . . However, Jonathan

E. Guryan, Kory Kroft, and Matthew J. Notowidigdo (2009) find no evidence of peer effects

between randomly assigned golf partners in professional tournaments, suggesting the importance

of context.

Therefore, we conclude our analysis by considering “positive” avenues of cooperation among agents, with

particular focus on technology transfer (or peer learning) and communication. Benefits of technology transfer

5To ensure improvement for every possible realization, we use first-order stochastic dominance shifts in cost distributions.6A general discussion of this issue, including a brief literature review, appears in Barbieri and Malueg (2010).7For theoretical analyses of technology transfer and learning in full information environments, using different production

functions than ours, also see Jovanovic and Nyarko (1995), Antonetti and Rufini (2008), and references therein.

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were foreshadowed by our comparative statics results. We model technology transfer between agents as

occurring when an agent with a less favorable distribution of cost is able, for example, to “adopt” the more

favorable cost distribution of another player. As a result of such a transfer, all agents (weakly) increase their

efforts, benefitting all. Nevertheless, we show by example that such efforts may be be completely wasted in

the sense that players’ contribution strategies remain unchanged after the transfer.

Lastly, we show communication has great potential for increasing efficiency, even where technology trans-

fer alone is ineffective. We introduce a pre-play exchange of cheap-talk messages; full revelation of one’s

own cost is an equilibrium outcome, and agents are as well-off ex post as in the best equilibrium of the

full-information game. Thus, considerable welfare gains are possible through communication. Moreover,

with communication, technology transfers are never wasted. However, it is crucial that communication

be unimpeded. In a setting where full revelation of information is not feasible, we show limited informa-

tion transmission is entirely futile: any equilibrium outcome with communication can also be implemented

without communication.

Beyond the previously mentioned papers, our work is related to a final strand of literature, namely,

the one studying the voluntary provision of discrete public goods with private information via “positive”

mechanisms, rather than using an uninterested mediator with full commitment power as in the standard

mechanism design approach (see among others Alboth et al., 2001; Barbieri and Malueg, 2008a and 2008b;

Laussel and Palfrey, 2003; Martimort and Moreira, 2010; and Menezes et al., 2001). This literature has

analyzed variations of two-player contribution games in which the success of the collective action depends

on the sum of individual efforts. Our use of the weakest-link technology, rather than summation, is key to

extending the analysis to more than two players. With the summation technology and private information,

players face the daunting problem of computing the distribution of the sum of others’ contributions. With

the weakest-link technology one need only concern oneself with the distribution of the least contribution in

the group, a much easier task.8

The rest of the paper proceeds as follows. Section 2 describes the n-player symmetric model and Section 3

characterizes equilibria, identifying the unique Pareto dominant equilibrium. A two-player asymmetric model

is analyzed in Section 4, and these results are used to understand the comparative statics with respect to

distributions of costs. Section 5 analyzes the scope for improving efficiency through technology transfer and

information sharing. The Appendix contains proofs and a numerical example illustrating our results.

8The weakest-link and the closely related best-shot technologies have recently been explored as alternatives to summa-tion in full-information group contests, e.g. Chowdhury et al. (2013a,b), Chowdhury and Topolyan (2013), and Kolmar andRommeswinkel (2013). Therefore, our results may be of help in extending this literature to accommodate private information.

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2 The Model

We study the problem of n (≥ 2) symmetric players who simultaneously contribute to the funding of a

public good. We assume players have quasilinear payoffs, a standard assumption in the literature on private

provision of public goods when there is private information. Each player’s benefit from amount G of the

public good is given by v(G). We assume there exists a nonnegative number G (≤ ∞) such that v′ > 0 and

v′′ < 0 on (0, G); additionally, limG↑G v′(G) = 0. Thus, over the relevant range, the marginal utility of the

public good is positive and strictly decreasing. We let ci denote player i’s marginal cost of contributing to

the public good. Each player’s marginal cost is independently drawn from the atomless (except possibly at

c) cumulative distribution function (cdf) F having support [c, c] and continuous probability density function

(pdf) f on (c, c]. A player knows his own marginal cost of provision but no one else’s. If player i contributes

gi and the realized value of the public good is G, then his net payoff is v(G) − cigi.9 If the vector of

contributions is (g1, . . . , gn), then through the the “Weakest-Link” production function the realized level of

the public good is GWL ≡ min{g1, . . . , gn}.

3 Equilibria

3.1 Characterization

We now derive Bayesian equilibria. Taking as given the strategies of the others, a player with cost c chooses

his own contribution g to maximize −cg + E[ v(GWL) ]. We denote the equilibrium strategy of player i as

gi : [c, c] → <+. Usual incentive compatibility arguments imply that every gi will be weakly decreasing in

cost. The following proposition simplifies the search for Bayesian equilibria.

Proposition 1 (Continuity and symmetry). 1) All equilibrium strategies are continuous. 2) All equilibria

are symmetric.

Because the level of the public good is determined by the player with the highest cost, it is helpful to

define c[k]1 as the maximum realization of k independent draws from the distribution F . Let H and h denote

the cdf and pdf of c[n−1]1 . The next proposition characterizes all equilibria, which must, by Proposition 1,

exhibit symmetric continuous strategies.

Proposition 2 (Characterization). Consider a continuous and non-increasing function g : [c, c]→ <+. The

function g is a (symmetric) equilibrium strategy if and only if, for almost all c ∈ [c, c], all of the following

9As an alternative approach, one could introduce private information through the income elasticity for the public good. Forexample, if we were to use the utility function log(w − gi )+αG, where α is private information to each player, nothing essentialwould change in the analysis below.

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conditions are satisfied:

1. If g(c) > 0, then

−c+ v′(g(c)) ≥ 0, (1)

−c+ v′(g(c))H(min{s : g(s) = g(c)}) ≤ 0, (2)

−c+ v′(g(c))H(max{s : g(s) = g(c)}) ≥ 0; (3)

and, ∀c such that g is strictly decreasing at c, H(c)v′(g(c)) = c. (4)

2. If g(c) = 0, then only (2) applies.

While a formal proof of Proposition 2 is in the Appendix, here we sketch the intuition behind conditions

(1) and (4). A player with cost c is sure his contribution will determine the equilibrium level of GWL. So

condition (1) simply says that such a player contributes an amount for which the marginal benefit of effort is

at least as large as its marginal cost. Next, suppose that at c the equilibrium strategy g is strictly decreasing.

If player 1 has marginal cost c but acts as if his cost were c′, his payoff would be

U(c′ | c) = −cg(c′) +H(c′) v(g(c′)) +

∫ c

c′v(g(c))h(c) dc. (5)

Consequently, where g is differentiable (which, by monotonicity of g, is almost everywhere), we find

∂U(c′ | c)∂c′

= g′(c′) [−c+H(c′)v′(g(c′))] ; (6)

in equilibrium c′ = c, so, where g′(c) < 0, the first-order condition 0 = ∂U(c′ | c)∂c′

∣∣∣c′=c

implies v′(g(c)) = c/H(c),

which is condition (4). Conditions (2) and (3) handle the possibility of flat spots in the equilibrium strategy.

3.2 Regularity

Because v′ is strictly decreasing, (4) implies that an equilibrium strategy can be everywhere strictly decreasing

only if c/H(c) is strictly increasing. We say that the game is regular if c/H(c) is strictly increasing over

[c, c]. Letting

ψ(x) =

(v′)−1(x) if x ≤ v′(0)

0 if x > v′(0),

we see in the regular case that an equilibrium strategy is given by g(c) = ψ(c/H(c)). We emphasize that when

the game is regular, there exists an equilibrium without flat spots. However, when the game is not regular,

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every equilibrium strategy must have at least one flat spot. Example 1 illustrates the best equilibrium in

the regular case.

Example 1 (Power function cost distributions).

Suppose F (c) = ca on [0, 1], where a > 0. As c/H(c) = c1−(n−1)a, the game is regular if and only if

a < 1/(n− 1). Suppose v(G) = G− 12G

2. In the regular case, the strictly decreasing equilibrium strategy is

g(c) = 1− c1−(n−1)a, which is positive for all c < 1. If this game is not regular, the only equilibrium is the

zero-contribution equilibrium.

While the regular case is fairly easy to analyze, frequently regularity fails to hold. Because H(c) =

(F (c))n−1

, we see that

0 <d(c/H(c))

dc=F (c)− (n− 1)f(c)c

(F (c))n;

so regularity holds if and only if

f(c)

F (c)<

1

(n− 1)c. (7)

Thus, regularity depends on the distribution of costs and the number of players. Integrating (7) and using

the boundary condition F (c) = 1, we find a necessary condition for regularity is that F (c) > (c/c)1

n−1 for

any c ∈ (c, c). Consequently, if F is atomless and c > 0, then the provision game is not regular. Finally, note

that if c/H(c) is decreasing (a situation guaranteed by convexity of F ), then all equilibrium strategies are

constant.

3.3 The “best” equilibrium

Of all the equilibria in the weakest-link provision game, we focus our analysis on the pointwise largest

contribution function. We adopt this selection because the equilibrium is interim Pareto dominant; indeed,

over all equilibria it maximizes the interim expected payoff of every type. Therefore, we refer to this as

the “best” equilibrium. This follows from the next lemma, which restates a familiar incentive-compatibility

result to fit our framework.

Lemma 1 (Equilibrium interim utility). Fix an (symmetric) equilibrium strategy g, and for this equilibrium

let the interim equilibrium payoff be denoted by U∗(c) ≡ U(c′ = c | c), where U(c′ | c) is defined in (5). Then

U∗(c) = U∗(c) +

∫ c

c

g(s) ds. (8)

By Lemma 1, interim utility is maximized by choosing the largest possible g(c) and the largest U∗(c). For

regular provision games, this will coincide with ψ(c/H(c)). More generally, in both regular and non-regular

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cases, the largest equilibrium contribution function g(c) coincides with the lowest weakly decreasing function

lying above ψ(c/H(c)). The corresponding contribution function is depicted in Figure 1.

The key to understanding the observations above is condition (1), which may be rewritten as g(c) ≤

ψ(c/H(c)) = ψ(c): the equilibrium function g must end at or below ψ(c). Coupled with continuity of g and

condition (4), which may be restated as g(c) = ψ(c/H(c)) when g is strictly decreasing, it implies that any

strictly decreasing segment of the graph of ψ(c/H(c)) is an upper bound for g, unless the same range of

contributions appears in another strictly decreasing segment of ψ(c/H(c)) further to the right (i.e., for larger

costs). So, in Figure 1 the segment of ψ(c/H(c)) between c1 and c2 is an upper bound for g, but the one

between c2 and c3 is not, since the same range (ψ(c3/H(c3)), ψ(c2/H(c2))) appears also further to the right,

i.e. between c4 and c5. To conclude the reasoning, note that in equilibrium type c is sure the quantity of

public good provided is g(c) for a payoff of v(g(c)) − cg(c). Therefore, and as prescribed by the previous

procedure, U∗(c) is maximized when g(c) = ψ(c).

c cc1 c2 c3 c4 c5

ψ(c1/H(c1))

ψ(c2/H(c2))

ψ(c3/H(c3))

ψ(c)

g(c)

ψ(c/H(c))

Figure 1: The largest g in a symmetric irregular case

For regular cases, Figure 1 has a very clear implication about “flat spots.” With regularity, for those

equilibria with flat spots, the flat spots are necessarily at the top; for any c ∈ (c, c), the function g(c) ≡

min{ψ(c/H(c)), ψ(c/H(c)} is also an equilibrium strategy. Also, for any γ ∈ [0, ψ(c)], the strategy g(c) ≡ γ

also constitutes an equilibrium. Though strategies with a flat spots imply that, with positive probability,

all players choose the same level of effort, this should not be misinterpreted by an outside observer as

efficient coordination. On the contrary, given regularity, such coincidence of contributions is a symptom of

inefficiency.

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3.4 Comparative statics

We next examine how the best equilibrium in the weakest-link game changes with the number of players and

the distribution of costs. In the symmetric full-information game, the (best) equilibrium is characterized

by v′(g) = c, and thus it is independent of the number of players. In contrast, with private information

the best equilibrium strategy is pointwise decreasing in the number of players. This follows because H(c)

is decreasing in n, implying that ψ(c/H(c)) is decreasing in n. As described in the previous section, this

implies the largest equilibrium strategy decreases pointwise as n increases.10 And since the best payoff to a

player with cost c does not change with n, Lemma 1 implies interim payoffs weakly decrease with n.

While we leave to Section 5 a full investigation of the efficiency properties of equilibrium, Example 1

clearly shows how private information can create severe inefficiencies. Indeed, if in Example 1 the game is not

regular, cooperation collapses—equilibrium has contributions of zero with probability 1. Also, inefficiencies

arise even if agents are ex ante symmetric. This contrasts with the symmetric, full-information setting, where

the first-order condition for individual optimality, i.e., v′(g) = c, and the first-order condition for efficiency,

i.e., nv′(g) = nc, coincide—indeed, the predominant expectation in Hirshleifer (1983) is that inefficiencies

will be limited, if present at all.11 Intuitively, inefficiencies arise because the weakest-link private provision

game rewards coordination but asymmetric information makes coordination more difficult.

One naturally expects that a shift to lower costs (in the sense of first-order stochastic dominance, FOSD,

indicating an unambiguous improvement in costs) would lead to contribution strategies that are greater, as,

for each realization of his own cost, each player believes the others’ costs are now more likely to be low and

therefore each is more inclined toward greater contributions. This can be verified in Example 1, and, indeed,

this comparison holds generally generally. If F FOSD F , then, for any equilibrium strategy of the symmetric

game with distribution of costs given by F , there is in the game with distribution F an equilibrium strategy

that is everywhere at least as large as when the distribution is F . This follows from the simple fact that for all

c, c/(F (c))n−1 ≤ c/(F (c))n−1, so (because ψ is decreasing) the least weakly decreasing function lying above

ψ(c/(F (c))n−1) is everywhere as large as the least weakly decreasing function lying above ψ(c/(F (c))n−1).

This discussion establishes the following.

Proposition 3 (First-order stochastic dominance). Suppose there are n players. Suppose F and F are two

distributions of cost, and suppose that F (c) ≤ F (c) for all c, with strict inequality for some c. For all types,

the best equilibrium under F is at least as good as the best equilibrium under F .

While the effects of a FOSD shift in costs are intuitive, we next ask, What are the effects of second -order

10As n increases, the distribution of the realized public good is, therefore, sure to shift leftward in the sense of first-orderstochastic dominance.

11Therefore, both for efficiency and number-of-players considerations, our results are closer to the findings of the Cornes andHartley’s (2007) “weaker-link” than to Hirshleifer’s (1983) weakest-link.

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stochastic dominance (SOSD) shifts in cost distributions? An answer would also address the question of

whether homogeneity or diversity in a team is preferable efforts because, while maintaining ex ante symmetry,

a “more risky” distribution can be interpreted as yielding greater diversity ex post. As discussed earlier,

uncertainty about others’ contributions leads a player to contribute less than his individually preferred level.

Therefore, in the weakest-link game, one might expect that decreasing uncertainty about others’ costs would

ease coordination of efforts and thereby increase payoffs. However, this intuition is incomplete.

Consider the simplest case of two distributions related by (mean-preserving) SOSD and satisfying the

single-crossing property. Suppose regularity holds for both distributions. Equation (4) implies that the same

pattern of crossings of the cdfs carries over to equilibrium strategies. Thus, if F1 and F2 are related by single

crossing and F1 SOSD F2, then for large c efforts under the less risky cdf are larger than those under the

more risky cdf; and for small c efforts are greater under the more risky cdf. Therefore, for costs near c, (8)

implies that interim payoffs decrease as riskiness increases by moving from cdf F1 to F2. However, under

the more risky distribution players with low costs increase their efforts, and for them it is unclear whether

their interim payoffs will fall or rise. Consequently, it is also unclear whether increasing riskiness of the cost

distribution will decrease or increase ex ante payoffs. The following example shows either possibility may

arise.

Example 2 (Second-order stochastic dominance).

Increasing riskiness can increase payoffs. Let there be two players and consider the following family

of cdfs:

F 1(c, t) = t+ (4c− 1)(1− t)2 ∀c ∈[1/4, h1(t)

], where h1(t) =

2− t4(1− t)

, ∀t ∈ [0.4, 1).

For any t, expected cost is 3/8 and F 1(1/4, t) = t; for any two levels for t, the resulting cdfs cross only once

where both are strictly increasing. The situation is depicted in the left panel of Figure 2, for some t < 0.5,

illustrating that larger values of t make F 1[c, t] “more risky” in the sense of SOSD.

For n = 2, regularity is guaranteed for t > (3 −√

5)/2 ≈ 0.38. Suppose now that v(G) = 2√G.

Therefore, by (4), g(c, t) =(F 1(c, t)/c

)2in the best equilibrium. Substitution of this g into (8) shows

that increasing riskiness of the cdf raises interim utility for cost realizations near 1/4, but lowers interim

utility for larger costs. And we find ex ante expected utility strictly increases, with E[U | t = .4 ] ≈ 2.18 and

limt↑1 E[U | t ] = 4.

10

0.5

t

1/4

1

h1(t) 3/4

c

F 1(c, t)

F 1(c, .5)

c0

1

1 h2(t)

t

0.5

F 2(c, .5)

Figure 2: Representatives of a family of cdfs related by mean-preserving spreads

Increasing riskiness can decrease payoffs. Let there be three players and consider the following

family of cdfs:

F 2(c, t) = t+ 2(1− t)2c ∀c ∈[0, h2(t)

], where h2(t) =

1

2(1− t), ∀t ∈ [1/2, 3/4].

The expected cost is 1/4 for any t, F 2(0, t) = t; for any two levels for t, the resulting cdfs cross only once where

both are strictly increasing. The situation is depicted in the right panel of Figure 2. Larger values of t make

F 2[c, t] “more risky.” Suppose now that v(G) = 2G− 12G

2. Regularity is guaranteed for t ≥ 1/2. Now by (4),

g(c, t) = 2−c/(F 2(c, t)

)2in the best equilibrium. From (8) and the boundary condition U(c) = 1

2 (2−h2(t))2,

we find that increasing riskiness reduces interim utility for all cost realizations. Moreover, increasing riskiness

decreases expected utility, with E[U | t = .5 ] ≈ 1.39 and E[U | t = .75 ] ≈ 1.31.

Our symmetric model is sufficient to highlight the inefficiencies in the weakest-link provision game.

However, to understand the implications of our framework for peer learning, we next consider agents that

are ex ante different, so that they can learn from one another.

4 The case of two asymmetric players

We now consider a two-player asymmetric provision game. For simplicity, we assume the support of marginal

cost is [c, c] for each player, but we now allow for distinct cdfs of cost, F1 and F2. While the equilibrium

characterization becomes more complex, several results we derived for the symmetric game remain valid.

Indeed, part 1 of Proposition 1 does not make use of symmetry, and, thus, holds for asymmetric settings too:

any equilibrium strategies g1 and g2 must be continuous. Part 2 of Proposition 1 has the following analogue.

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Proposition 4 (Common range and flat spots). The range of g1 coincides with the range of g2. Moreover,

g1 is strictly decreasing when taking values in any interval (γl, γh) if and only if g2 is strictly decreasing when

taking values in (γl, γh).

We now characterize equilibria, with particular focus on the best equilibrium, which is identified using

the following analogue of Lemma 1 (proven as was Lemma 1).

Lemma 2 (Equilibrium interim utility). Fix an equilibrium (g1, g2), and for this equilibrium let the interim

equilibrium payoff of agent i be denoted by U∗i , for i = 1, 2. We then have

U∗i (ci) = U∗i (c) +

∫ c

ci

gi(s) ds.

By Lemma 2, interim utility is maximized by choosing the largest feasible U∗i (c) and gi(c). The largest

U∗i (c) is found by choosing g1(c) = g2(c) such that v′(g1(c)) = v′(g2(c)) = c. To find the largest possible

gi(c), as we did for the symmetric case, we first focus on costs where equilibrium strategies are strictly

decreasing. Fixing player 2’s strategy g2(c2), type c1’s utility of contributing γ is

−c1 γ + v(γ) Pr(g2 ≥ γ) +

∫{c2: g2(c2)<γ}

v(g2(c2)) dF2(c2),

or, letting φ2 ≡ g−12 ,

−c1γ + v(γ)F2(φ2(γ)) +

∫ c

φ2(γ)

v(g2(c2)) dF2(c2);

differentiating with respect to γ, we obtain the first-order condition

−c1 + v′(γ)F2(φ2(γ)) = 0. (9)

Letting φ1 ≡ g−11 and replicating the same steps for player 1 and noticing that, in equilibrium, c1 =

φ1(g1(c1)), we obtain the following system of functional equations that characterizes equilibrium, for any

equilibrium contribution level γ at which φ1 and φ2 are strictly decreasing:

φ1(γ) = v′(γ)F2(φ2(γ))

φ2(γ) = v′(γ)F1(φ1(γ)).

(10)

The application of the implicit function theorem to (10) yields the following sufficient condition for regularity,

12

i.e., for strategies to be strictly decreasing in cost:12

c1f1(c1)

F1(c1)· c2f2(c2)

F2(c2)< 1 for all c1, c2. (11)

Finally, at any flat spot the following conditions, analogous to (2) and (3), must hold for i, j = 1, 2, i 6= j:

v′(gi(c))Fj(

min{s : gi(s) = gj(c)})≤ ci ≤ v′(gi(c))Fj

(max{s : gi(s) = gj(c)}

).

We see from (10) that the derivation of equilibrium strategies reduces to finding the appropriate solution

to two equations in two unknowns. Because of the technicalities involved for non-regular environments, we

present the procedure to find the best equilibrium in the Appendix. Example 3 illustrates equilibrium for

the regular asymmetric case.

Example 3 (Asymmetric power function cost distributions).

Suppose F1(c) = ca and F2(c) = cb on [0, 1], where a < b and ab < 1. Also, suppose v(G) = G − 12G

2.

Using the conditions of (10) we find

φ2(γ) = (1− γ)(φ1(γ))a = (1− γ)[(1− γ)(φ2(γ))b

]a= (1− γ)1+a(φ2(γ))ab;

rearranging the extremes now yields (φ2(γ))1−ab = (1 − γ)1+a, so that g2(c) = γ = 1 − c1−ab1+a . Similarly,

g1(c) = 1− c1−ab1+b . As is readily checked, we have g2(c) > g1(c). Thus, the “less able” player follows a more

generous strategy. This reflects the complementarity of efforts in the weakest-link framework: being matched

with a worse partner leads a player to contribute less. This interpretation is reinforced by the comparison

to the symmetric case in Example 1, for which we take F1(c) = F2(c) = ca and obtain g0(c) = 1 − c1−a as

the best equilibrium contribution function.13 Note that a < b also implies g2(c) < g0(c). Thus, the overall

ranking of contribution functions is g0(c) > g2(c) > g1(c). This change can be understood with (9). The

first term in (9) gives player 1’s marginal cost while the second term is the (expected) marginal benefit of

an increase in γ. Even if, with the change in distribution, player 2 did not alter her strategy, player 1 would

see a decreased marginal benefit of increasing γ, compared to when player 2’s value has cdf F1. This direct

effect causes player 1 to reduce his contribution for each c1. Indirect effects are present as well: because

of player 1’s induced reduced contribution function, for each c2 player 2 sees a lower marginal benefit of

contribution, which causes player 2 to reduce her contribution, causing player 1 to reduce his contribution

12The necessary condition for equilibrium strategies g1 and g2 to be strictly decreasing is that the inequality in (11) besatisfied weakly for all (c1, c2) such that g1(c1) = g2(c2).

13The assumptions a < b and ab < 1 ensure a < 1, as required in Example 1.

13

further. Thus, in the asymmetric equilibrium, both players contribute less than under g0. And the change is

larger for player 1, who experiences both direct and indirect effects, than for player 2, who is only indirectly

affected.

The next proposition shows the comparative statics highlighted in Example 3 are general.

Proposition 5 (Asymmetric first-order stochastic dominance shifts). Consider the game where players’

distributions are given by the pair (F1, F2) and denote the best equilibrium contribution functions as g1 and

g2. Now consider the game where players’ distributions are given by the pair (F1, F2) and denote the best

equilibrium contribution functions as g1 and g2. Let F2 FOSD F2. Then

g1(c1) ≤ g1(c1) and g2(c2) ≤ g2(c2).

Furthermore, if F2 = F1, then

g1(c) ≤ g2(c) ≤ g1(c) = g2(c).

The inequalities are strict at costs c at which g1 and g2 are strictly decreasing and F2(c) < F2(c).

Proposition 5 also relates to the single game where players’ distributions of cost are ordered. If F2

FOSD F1, then in the game with distributions (F1, F2), player 1’s strategy specifies lower effort for each

cost: g1(c) ≤ g2(c). Nevertheless, player 1 exerts greater effort in the sense of FOSD: Pr(g1(c) ≤ γ) ≤

Pr(g2(c) ≤ γ) , ∀γ ≥ 0.14 Thus, the more able player uses a less generous strategy but expects to provide

greater effort.

The analysis of this section has thus far considered two-sided private information, with special focus on

the effects of cost advantages. It is natural to ask what can be said in the asymmetric setting regarding

SOSD changes in one player’s distribution of cost. We illustrate the possibilities with an example where one

party is well-established but the other is less known.

Example 4 (One-sided asymmetric information).

Assume the marginal cost of player 1 is known and equal to c1, with c1 ∈ (c, c). Meanwhile, player 2’s

marginal cost is privately known—but ex ante it is known to be distributed according to F2 on [c, c].

14Where strategies are strictly decreasing, this comparison follows from the following chain of inequalities:

Pr(g1(c) ≤ γ) = Pr(c ≥ φ1(γ)) = 1− F1(φ1(γ)) = 1−φ2(γ)

v′(γ)(by (10))

≤ 1−φ1(γ)

v′(γ)(because g1 ≤ g2 by Proposition 5)

= 1− F2(φ2(γ)) = Pr(c ≥ φ2(γ)) = Pr(g2(c) ≤ γ) . (by (10))

14

Consider g1 ∈ <+ and a continuous and non-increasing function g2 : [c, c] → <+. Then the pair (g1, g2)

is an equilibrium if and only if

g1 = a and g2(c2) = min{a, ψ(c2)}, where a satisfies v′(a)F (v′(a)) ≥ c1. (12)

Note that ψ(c) is simply a player’s optimal effort in a one-player game. The proof of equilibrium is a special

case of the one for the upcoming Proposition 7. The inequality in (12) is nothing but the analogue of

condition (3); it ensures that player 1 does not want to reduce his contribution below a. Also, condition (12)

establishes an upper bound for a. Among the continuum of equilibria associated with (12), in the best one

the inequality in (12) holds with equality.

v′(a)

1

0

cL/v′(a)

cH/v′(a)

F

F

Figure 3: Effects of reducing the riskiness of player 2’s distribution of cost, from F to F .

We can diagrammatically represent the effects of reducing the riskiness of player 2’s distribution of cost,

first rewriting the equilibrium condition at the best equilibrium as F (v′(a)) = c/v′(a). In Figure 3, F SOSD

F . The effect of reducing riskiness from F to F depends on player 1’s cost. If c1 = cL, then, as shown,

this reduction in riskiness means that, in the best equilibrium, player 1’s marginal utility v′(a) will increase,

which corresponds to a reduction in player 1’s effort. Intuitively, player 1 reduces his effort because, given

his relatively low marginal cost, a reduction in the riskiness of player 2’s cost distribution increases the

likelihood that player 1 is not the weakest link, meaning at the initial effort level he more often makes

excessive contributions and could therefore benefit from a reduction in effort. This reduction in player 1’s

effort also causes player 2 to use a strategy that is lower, and for some costs is strictly so. These two effects

imply player 1’s utility and the interim utilities for all types of player 2 fall. Alternatively, if player 1’s cost

is high, c1 = cH , the effect is reversed, with the reduction in risk causing player 1 to increase his effort in

the best equilibrium. Correspondingly, player 1’s utility and the interim utilities for all types of player 2

rise.

15

5 Cooperation

Now we consider two “positive” avenues for cooperation: technology transfer as an instance of peer learning

and communication. To take advantage of Proposition 5, we view technology transfers as occurring prior to

the realization of costs. Accordingly, this is modeled as a change in a player’s ex ante capacity for providing

the public good—in particular, one player receives the transfer if his distribution of cost moves toward the

more favorable distribution associated with another player (in the sense of FOSD).

Proposition 5 suggests this technology transfer will raise the level of the public good and thereby benefit

all players. Indeed, substantial gains may be possible. In Example 5, found in the Appendix, total expected

welfare increases by 100% after technology transfer, rising from 38% to 68% of the expected full-information

efficient levels. Moreover, the payoff to the transferring party rises by 75%. Thus, the transferring party may

willingly bear some or all of the cost associated with the technology transfer, e.g., volunteering to teach peers

in her group. However, if the best equilibrium after the transfer is the constant one where all types contribute

the most preferred amount for the type c player, then such a transfer does not change the equilibrium (as

the upper end of the support of cost remains unchanged).15 Therefore, our results can be interpreted as

suggesting that, while one should expect peer knowledge transmission to have positive effects, it remains

possible that no advantages are derived from it, in accord with the reading of the empirical literature by

Jackson and Bruegmann (2009) quoted in the Introduction.

The other avenue for cooperation we envision is interim cheap-talk communication occurring after players

observe their costs but before they make their contributions, a la Palfrey and Rosenthal (1991). We augment

our basic game with a round of communication, with timing as follows:

1. Agents receive their private information;

2. Agents exchange non-binding messages;

3. Taking into account their private information, messages sent, and updating upon the messages received,

agents make their contributions; and then

4. Payoffs are realized.

In this framework, the analysis in the previous sections corresponds to subgames starting at point 3. In

particular, while we have shown that multiple equilibria exist, we focus on the best equilibrium, for simplicity

and to avoid creating spurious multiplicity; let ueqi (c1, . . . , cn) denote the ex post payoff to player i in the

best equilibrium.

15Such a situation arises where v(G) = G− (1/2)G2, F1(c) = c on [0, 1], and F2(c) = c2 on [0, 1]. After technology transfer,both players’ costs are uniformly distributed on [0, 1] and regularity fails throughout.

16

As for the consequences of communication, we begin by noting that in a one-player contribution game,

a player with cost c < v′(0) would contribute the “stand-alone” amount gsa(c) ≡ ψ(c); thus, v′(gsa(c)) = c.

Now suppose each agent i makes a report c′i and then contributes Galt ≡ min{gsa(c′1), . . . , gsa(c′n)}. Let

ualti (c1, . . . , cn) denote the ex post payoff to player i in the game with communication, under the contribution

strategy just described and truthful reporting. We then obtain the following result.

Proposition 6 (Communication with a rich message space). Suppose c ≤ v′(0). Truthful revelation

followed by each agent contributing Galt is an equilibrium of the game with communication. Moreover,

ualti (c1, . . . , cn) > ueqi (c1, . . . , cn) for almost all (c1, . . . , cn).

Proposition 6 shows cheap-talk communication can resolve all inefficiencies deriving from asymmetric

information—players coordinate by contributing the stand-alone level for the largest announced cost.16 In-

deed, ex post equilibrium contribution levels coincide with those of the best equilibrium in the full-information

game with the realized costs. Here, too, Example 5, shows gains in ex ante total expected welfare can be

substantial. Moreover, technology transfer in combination with communication can be effective even where

technology transfer alone would have no effect (as in the situation described in footnote 15).

Proposition 6 assumes agents can exactly communicate their realized costs. A natural question is then:

does cheap talk mitigate inefficiency even if messages cannot fully reveal information? The answer is No, as

the next proposition shows. For simplicity, we conduct the proof of our negative result in the context of the

simplest version of our model, namely, the one-sided asymmetric information model of Section 4. Player 1’s

cost c1 is commonly known. Player 2’s cost c2 is private information, and c2 is distributed according to the

continuous cdf F2 over [c, c]. While Proposition 6 continues to hold if a continuum of messages is available,

it fails if the number of possible messages is finite, implying full revelation of information is impossible. We

then obtain the following stark result showing that communication brings no welfare improvement.

Proposition 7 (Communication with a finite message space). Consider the one-sided asymmetric informa-

tion model of Example 4 and augment it with a round of communication. If the number of available messages

is finite, then the outcome of the best equilibrium of the game with communication can also be implemented

in the game without communication.

The intuition behind Proposition 7 is as follows. Suppose player 1’s cost is publicly known and player 2’s is

private information. Suppose two messages, M1 and M2, are each sent by player 2 with positive probability

in the equilibrium with communication. Conditional on the message, player 1’s best response is a pure

strategy, by concavity of the weakest-link technology and of v(·). Let ai denote player 1’s equilibrium action

16And since our augmented model is nothing but another mechanism, the payoff dominance result of Proposition 6 establishesinefficiency of any equilibrium of the standard game described in Section 2, by any of the definitions of Holmstrom and Myerson(1983) for private-information environments.

17

following message Mi, i = 1, 2. The key step is to show that a1 = a2. Let g2(c2 |Mi) denote player 2’s

equilibrium strategy following message Mi. In equilibrium it must be that g2(c2 |Mi) = min{ai, gsa(c2)}

for any c2 that sends message Mi. Beyond that, however, it must be that player 1 is the weakest link with

positive probability (conditional on Mi); otherwise he could increase his expected payoff by slightly reducing

his effort. Consequently, in equilibrium, for each message sent with positive probability, there is a positive-

mass set of c2 types that are constrained to enjoy less than their stand-alone levels of the public good. Now,

if ai < aj , then the constrained types sending message ai would do better to switch their messages to Mj

and slightly increase their effort, invalidating the equilibrium. Thus, it must be the case that a1 = a2. This

argument can be repeated for all pairs of messages sent with positive probability, implying that messages

have no effect on player 1’s choice of action, in stark contrast with what happens in Proposition 6. Hence,

all types of player 2 could as well send a single message, or none at all.17

6 Conclusion

We have analyzed the effects of private information in a model of voluntary provision of a public good

produced with a weakest-link technology allowing players arbitrary contribution levels. Private informa-

tion introduces uncertainty about others’ efforts, so the fear that others will provide lower effort leads an

agent to reduce its own efforts below its ideal, stand-alone level because any effort exceeding another’s is

“wasted.” Therefore, the most immediate consequence of private information is that, in contrast with the

full-information framework, coordination is hampered and inefficiencies ensue. In addition, the introduction

of private information changes the appearance of utility-maximizing strategies. With full information, agents

should coordinate on the same level of effort. The same is true with private information and binary actions.

But with continuous actions and private information (and a “regularity” property of the distribution of

private information), the best equilibrium strategy is strictly decreasing, i.e., it has no flat spots. In other

words, given regularity, in the best equilibrium agents choose the same effort level with probability zero.

For simplicity, comparative statics were examined focussing on the interim Pareto dominant equilibrium.

We found that, quite naturally, if in the symmetric game all player experience an improvement in costs (lower

in the sense of FOSD), then equilibrium strategies increase, as do interim and ex ante payoffs. Moreover, if

one player experiences an improvement in costs, all players benefit. In contrast with FOSD, the effects of

reducing the riskiness of the cost distribution are less clear. A mean-preserving reduction in the riskiness of

individual costs could increase or decrease equilibrium payoffs. The ambiguity arises because a reduction in

riskiness makes a high-cost type more likely to be the weakest link, so he may be more willing to increase

17The remainder of the proof establishes that player 1 exerting effort level a and player 2 using the strategy g2(c2) =min{a, gsa(c2)} indeed constitutes an equilibrium in the game without communication.

18

his effort toward his stand-alone level; but a low-cost type is less likely to be the weakest link, so he is likely

to reduce his effort. The overall effect on payoffs accounts for the mix of these effects.

We considered two routes for improving efficiency. Where agents’ costs arise from different probability

distributions, efficiency might improve if an agent with a high-cost distribution is able to adopt the stochas-

tically lower distribution of another agent—we view this as a transfer of technology or as an instance of peer

learning (teaching). Such learning, however, does not guarantee better outcomes. For example, when the

low-cost distribution is not “regular” the technology transfer might have no effect on equilibrium.

Arguably, communication is more central to improving efficiency. Through a voluntary information

exchange, uncertainty about costs can be removed, allowing players to use the stand-alone provision level of

the least-willing player as a focal point. This information exchange can also render effective a previously futile

technology transfer. However, the positive results about information exchanges on a purely voluntary and

non-verifiable basis are fragile, despite players’ interests being closely aligned. Indeed, we have shown that the

positive effects of information transmission take place only if the uncertainty about most desired contribution

levels is completely resolved: in our continuous-action setup, communication is entirely ineffective if it relies

on a finite message space. This is a surprising contrast with the setup in which actions are binary.

We conclude with some consideration of robustness of our model. Instead of the pure weakest-link

aggregation function, one might consider a weaker-link production function according to which, say, the

least and second-least efforts determine the level of the public good, with the least effort playing a greater

role (see, e.g., Arce and Sandler, 2001; and Cornes and Hartley, 2007). No discontinuous changes appear

in moving from a weakest-link toward weaker-link technologies. This is because the uncertainty generated

by private information makes the isoquants of the expected public good production function (taking as

arguments contribution strategies, not single amounts) “smooth,” even for the weakest-link technology.

However, it remains true that the importance of coordination is diminished as one moves away from weakest

link and progressively increases the marginal product of non-minimal contributions. This effect manifests

itself in an expanded range of parameter values for which regularity is satisfied.

Finally, an important area for additional research is to model further cooperation through in-kind transfers

that supplement another agent’s efforts.18 With the introduction of private information, two additional issues

arise. The first is the possibility of using supplements as signals. The second is the possibility of further

hampering cheap-talk communication, because if reporting high costs promotes transfers from others, then

agents’ incentives become less well aligned. With these complications, a full analysis of transfers is beyond

the scope of this paper.

18For a theoretical perspective in full-information environments, see Vicary (1990), Sandler and Vicary (2001), and Vicaryand Sandler (2002). For a closely related notion, see the experimental treatment of “help” in Brandts et al. (2013) and referencestherein.

19

Appendix

The following lemma is used in the proof of Proposition 1.

Lemma 3 (Necessary conditions). Consider a Bayesian equilibrium {gj}nj=1. The following two inequalities

must hold for gi(ci) > 0:

−ci + v′(gi(ci))∏j 6=i

limγ↑gi(ci)

(1− Pr(gj ≤ γ)) ≥ 0, (A-1)

and

−ci + v′(gi(ci))∏j 6=i

limγ↓gi(ci)

(1− Pr(gj ≤ γ)) ≤ 0; (A-2)

for gi(ci) = 0 only (A-2) applies.

Proof. Consider a Bayesian equilibrium {gj}nj=1 and let K−i denote the induced cdf of the minimum effort

of agents {1, ..., i− 1, i+ 1, ..., n}. Type ci’s utility of contributing γ is then

(1−K−i(γ))v(γ) +

∫ γ

0

v(s) dK−i(s)− c1γ = v(γ)− v(0)K−i(0)−∫ γ

0

K−i(s)v′(s) ds− c1γ, (A-3)

after integration by parts (see Billingsley, 1995, Theorem 18.4, for the case in which K−i need not be

differentiable). This shows type ci’s utility is continuous in γ and differentiable almost everywhere with

respect to γ, with derivative equal to

v′(γ)(1−K−i(γ))− ci, (A-4)

except for a set (of γ) having Lebesgue measure zero (Billingsley, 1995, Theorem 31.3). Therefore, a necessary

condition for gi(ci) to maximize utility is that the left derivative in (A-4) is positive and that the right

derivative is negative. Equations (A-1) and (A-2) simply impose these requirements. Clearly, for a corner

solution gi(ci) = 0, only the condition on the right-hand derivative applies.

Proof of Proposition 1. Part 1). We prove this by showing that the best-response of type ci to the equilib-

rium strategies {gj}j 6=i contains only one value. This, in addition to the continuity of ci’s utility in (A-3)

established in the Proof of Lemma 3, allows us to invoke the Theorem of the Maximum to conclude that

gi(ci) is continuous in ci. To see that ci’s best response is a unique effort level, it’s sufficient to notice that,

in the derivative of ci’s utility calculated in (A-4), 1−K−i is weakly decreasing and v′ is strictly decreasing.

Part 2). A Bayesian equilibrium (gj)j must satisfy g1(c) = · · · = gn(c), for otherwise the player with

the largest contribution at c could reduce his contribution (when his cost parameter is c) to mink gk(c) and

thereby raise his expected payoff, as the realized level of the public good would not change.

20

Now, without loss of generality, assume it is the strategies of players 1 and 2 that are not the same.

There are two cases to consider. In the first, at the moment c ≥ c in which g1 and g2 become different, both

strategies are strictly decreasing.19 This implies that there exists an interval of contribution levels such as g

in Figure 4 in which the same contribution is made by two different types.

g2(c)

g

g1(c)

c2c1c c

c

Figure 4: Both strategies are decreasing

Because the number of players is finite, we can choose g so that no strategy of any player has a flat spot

at g. Note how Lemma 3 leads to

v′(g)F (c2)∏s>2

(1− Pr(gs ≤ g)) = c1 (A-5)

because the limits in (A-1) and (A-2) are identical as soon as there is no gj that has a flat spot at g. Similarly,

v′(g)F (c1)∏s>2

(1− Pr(gs ≤ g)) = c2. (A-6)

Equations (A-5) and (A-6) imply F (c1) c1 = F (c2) c2, which is impossible because F (c) c is strictly increasing

and c1 6= c2. The previous argument can be extended to any segment of equilibrium strategies to conclude

that there cannot exist a range of contributions such that equilibrium strategies are strictly decreasing and

different. Therefore, in our quest for asymmetric equilibria, we can restrict attention to situations such that,

at the moment in which the first divergence between strategies occurs, at least one strategy remains flat,

as depicted in Figure 4 below, where g1 and g2 are flat at the beginning, but, for costs larger than c1, g2

becomes strictly decreasing at least up to c2 while g1 remains flat at least up to c2.

Note that, by the previous discussion, there cannot be any contributions of player 1 between g and g, so

by Proposition 1, this implies that player 1’s strategy must be constant all the way up to c1. Now, applying

19Formally, c ≡ inf{c ≥ c | g1(c) 6= g2(c)}.

21

g2(c)

g

g

g1(c)

c2c1c c

Figure 5: One strategy is locally constant and the other is decreasing.

equation (A-2) in Lemma 3 for type c2 of player 2, we obtain

−c2 + v′(g)∏j 6=2

limγ↓g

(1− Pr(gj ≤ γ)) ≤ 0. (A-7)

As for type c2 of player 1, applying equation (A-1) in Lemma 3, we obtain the necessary condition

−c2 + v′(g)∏j 6=1

limγ↑g

(1− Pr(gj ≤ γ)) ≥ 0. (A-8)

The conditions (A-7) and (A-8) are incompatible. To see this, observe that equations (A-7) and (A-8) imply

v′(g)∏j 6=2

limγ↓g

(1− Pr(gj ≤ γ)) ≤ v′(g)∏j 6=1

limγ↑g

(1− Pr(gj ≤ γ)),

which, because v′(g) > v′(g), in turn implies

∏j 6=1

limγ↑g

(1− Pr(gj ≤ γ)) >∏j 6=2

limγ↓g

(1− Pr(gj ≤ γ)).

And this last inequality is contradicted by

∏j 6=1

limγ↑g

(1− Pr(gj ≤ γ)) ≤∏j 6=2

limγ↑g

(1− Pr(gj ≤ γ)) ≤∏j 6=2

limγ↓g

(1− Pr(gj ≤ γ)),

where the first inequality follows because g1 is uniformly larger than g2, and the second inequality follows

because eventually all elements of the subsequence converging to g are larger than all elements of the

subsequence converging to g, since g > g. Therefore we can exclude the case depicted in Figure 5 as well,

thus concluding the proof, because any other possible configuration would involve jumps, which are ruled

out by part 1 of this proposition.

Proof of Proposition 2. Necessity. This follows from Lemma 3. In particular, when g is strictly decreasing

at c, the limits in (A-1) and (A-2) are the same, which yields (4).

22

Sufficiency. The derivative of U(c′ | c) (see (5)) with respect to c′ is, almost always,

∂U

∂c′= g′(c′) [−c+H(c′)v′(g(c′))] .

This, along with condition (4), implies that any deviation from g(c) to a different contribution γ such that

γ = g(c′) for some c′ ∈ [c, c] is not profitable. Indeed, consider for instance c′ > c. We obtain

U(c′ | c) = U(c′ = c | c) +

∫ c′

c

g′(c) [−c+H(c)v′(g(c))] dc.

By condition (4), the integrand in the above expression is either zero (when g′(c) = 0), or it is strictly

negative for any c > c:

g′(c) [−c+H(c)v′(g(c))] = g′(c) [−c+ c] < 0.

Therefore, since g is continuous, it follows that U(c′ | c) < U(c | c). The case c′ < c is similar. To conclude

the proof, we now rule out deviations to contributions γ outside the range of g. Because of the weakest-link

technology, it is obvious that if γ > g(c), then a deviation to γ is dominated by one to g(c). Consider now

γ < g(c). The utility of such deviation is −cγ + v(γ), with derivative

−c+ v′(γ) > −c+ v′(c) ≥ 0,

where the last inequality follows from (1). Therefore, a deviation to γ < g(c) is dominated by one to g(c),

thus concluding the proof of sufficiency.

Proof of Lemma 1. Using (5), we obtain

dU∗(c)/dc = −g(c)− cg′(c) + h(c)v(g(c)) +H(c)v′(g(c))g′(c)− v(g(c))h(c)

= −g(c)− g′(c)(c−H(c)v′(g(c)))

= −g(c),

where the last equality follows from Proposition 2, since almost everywhere either g′(c) = 0 or g′(c) < 0, in

which case v′(g(c))H(c) = c. The lemma follows by integration of dU∗(c)/dc = −g(c).

Proof of Proposition 4. As for the symmetric case, g1(c) must equal g2(c), otherwise the type making the

larger contribution could reduce it without repercussions on the public good produced and saving on the cost.

Also, note that g1(c) = g2(c). To see this, suppose to the contrary that g1(c) > g2(c), which, by continuity—

23

established in Proposition 1—implies that there exists c2 < c such that g1(c) = g2(c2) = γ > 0. Now,

Lemma 3 implies v′(g2(c2)) = c2 and v′(g1(c))F2(c2) ≥ c, leading to the contradiction v′(γ)F2(c2) ≥ v′(γ).

After having so established that the range of g1 and g2 is the same, note that if g1 is strictly decreasing when

taking values in (γl, γh), then (A-1) and (A-2) imply that the following must hold for g2 = γ:

−c2 + v′(γ)F1(g−11 (γ)) = 0, ∀γ ∈ (γl, γh).

Note that the above uniquely identifies c2 as a function of γ, implying that g2 cannot have a flat spot for a

contribution level equal to γ.

Derivation of the best equilibrium for the asymmetric model

Figure 6 shows how to use the conditions in (10) to find the largest possible contribution functions and

thus the best equilibrium for the asymmetric model. This figure assumes marginal costs distributed over

[0, 1] and a value function v with v′(0) = 1. Additionally, it is assumed that c2 is uniformly distributed. The

nonlinear curves correspond to φ2 = v′(γ)F1(φ1) for different values of γ. Increasing the value of γ scales

downward the curve v′(γ)F1(φ1). It is convenient to refer to this curve as φ2 = ρ1(φ1|γ). The straight lines

correspond to the graph of φ1 = v′(γ)F2(φ2), and increases in γ rotate this line counter-clockwise. Similarly,

we refer to this curve as a second function that identifies φ2 as a function of φ1 for given γ: φ2 = ρ2(φ1|γ).

In other words, ρ2 is defined as F−12 (φ1/v

′(γ)). Curves of equal thickness correspond to a common value

of γ. Thus, when γ = 0 the relevant lines are the thinnest ones, and the only solutions to (10) are at

(c1, c2) = (1, 1) and (c1, c2) = (p, p). Therefore, in our quest for the best equilibrium, we choose g’s to

satisfy two requirements. First, we set g1(1) = g2(1) = 0, since 1 = v′(0) characterizes the contribution

that maximizes U∗1 (1) and U∗2 (1). The alternative would have contribution functions g1 and g2 such that

g1(c1) = 0 and g2(c2) = 0 for c1 > p and c2 > p. Second, in looking for the best (largest) equilibrium

strategies, we hope to increase gi whenever possible as cost is reduced from c. While the pair (g1, g2) is not

the best equilibrium, because it leads to contribution functions that cannot be strictly larger than g1 and

g2, we keep following (g1, g2) to illustrate equilibrium more generally.

To check whether strictly decreasing segments can indeed be part of the proposed equilibrium strategies,

note that marginally increasing γ to γ rotates curves to the ones of medium thickness, where the only

solutions to (10) are at (c1, c2) = (s, s′) and (c1, c2) = (q, q′). Thus, our hope that g′1(1) < 0 and g′2(1) < 0 is

validated, since s < 1 and s′ < 1. Therefore, we continue our process to find the best equilibrium by setting

g1(s) = g2(s′) = γ, provisionally setting g′1(s) < 0 and g′2(s′) < 0, and further increasing γ to once more

apply (10). In contrast, note that the hope that g′1(p) < 0 and g′2(p) < 0 is not confirmed in equilibrium,

24

0 1

1

ρ2(φ1 | γ)

r

r′

q′

s′

p q sr

ρ1(φ1 | 0) ≡ v′(0)F1(φ1)

ρ1(φ1 | γ) ≡ v′(γ)F1(φ1)

v′(γ)F2(φ2)

v′(γmax)F1(φ1)

c1, φ1

c2, φ2

Figure 6: Graphs determining φi(γ) in the asymmetric case.

25

since q < p and q′ < p. In other words, the condition for regularity (11) is violated at c1 = c2 = p. This

reasoning allows us to state a graphical interpretation of condition (11): ρ1 (the curved lines in Figure 6)

must intersect ρ2 (the straight lines in Figure 6) from above.

The maximum value of γ achievable in equilibrium occurs for the highest level of γ that just leaves the

corresponding curves for players tangent. In Figure 6 this occurs when γ = γmax; the intersection then

occurs at (c1, c2) = (r, r′). In Figure 7 the maximum equilibrium contribution functions are shown as g1 (in

green) and g2 (in red).

c

g1(c)

g2(c)

0 q′ r r′ s s′ 1

γmax

γ

Figure 7: Graphs determining contribution functions in the asymmetric case.

Summing up, the above described procedure to construct the best equilibrium (g1, g2) for two generic

distributions F1 and F2 consists of these three steps:

1. Best equilibrium strategies end at the contribution level γ that maximizes type c’s utility, v(γ) − cγ.

We label this level γ. Since we assumed v′(0) ≥ c, γ is characterized by v′(γ) = c.

2. For any γ > γ, we identify g1 and g2 through the largest intersection between ρ1(φ1|γ) and ρ2(φ1|γ), if

such φ1 exists in [c, c]. Note that at this largest intersection regularity is satisfied, i.e., ρ1 intersects ρ2

from above.20 This is true because ρ1 and ρ2 are continuous and because ρ1 ends below ρ2. Indeed, ρ1

ends below the 45◦ line, since ρ1(c) = v′(γ)F1(c) = v′(γ) < v′(γ) = c, and a similar chain of inequalities

shows that ρ2(φ1|γ) ends above the 45◦ line.

3. Flat spots in best equilibrium strategies occur when the largest intersection between ρ1(φ1|γ) and

ρ2(φ1|γ) jumps discontinuously, say from φα1 to φβ1 with φα1 > φβ1 , with a marginal increase in γ. Such

a jump may occur if we start from a level γ that makes ρ1 and ρ2 (locally) tangent. Therefore, we

20Strictly speaking, ρ1 does not intersect ρ2 from below.

26

set g1(c1) = γ for any c1 ∈ [φβ1 , φα1 ] and we construct g2 similarly. Finally, if there is no intersection

between ρ1 and ρ2 for γ > γ, then we set φβ1 = c.

0

c

cc1, φ1

c2, φ2

c

c

φ1 = v′(γ)F2(φ2)

φ1 = v′(γ)F2(φ2)

ρ2(φ1 | γ)

φ2 = v′(γ)F1(φ1)︸ ︷︷ ︸ρ1(φ1 | γ)

φ1(γ) φ1(γ)

φ2(γ)

φ2(γ)

45◦

Figure 8: Equilibrium effects of a FOSD shift in player 2’s cost.

Proof of Proposition 5. The following proof makes extensive use of the previous derivation of the best equi-

librium for the asymmetric model. Recall first that best equilibrium strategies are decreasing and continuous,

and they end at the same contribution level for both pairs of distributions. If g1(·) is constant at this level,

then g2 is too constant at this level by Proposition 4, and this proposition is trivially proven. If not, then

we now show that, for any γ in the range of gi,

max{ci : gi(ci) ≥ γ} ≤ max{ci : gi(ci) ≥ γ},

with strict inequality under the appropriate condition on F2, thus concluding the proof of first part of the

27

proposition.

For any γ in the range of gi, we define φi(γ) ≡ max{ci : gi(ci) = γ}, i = 1, 2. Thus, φi(γ) is simply the

inverse of gi when {ci : gi(ci) = γ} is single-valued. Similarly, φi is defined from gi. Because (g1, g2) is the

best equilibrium, condition (10) holds at (φ1, φ1), by steps 2 and 3 in the previous derivation of the best

equilibrium for the asymmetric model. As in the previous derivation, we use ρ1(φ1|γ) and ρ2(φ1|γ) to refer

to the curves defined by the two equations in (10).

The next step of our argument is best understood using Figure 8, where the blue curve represents ρ1

and the red curve represents ρ2, calculated through F2(c2). Now, because (g1, g2) is the best equilibrium,

(φ1, φ1), is the largest intersection between ρ1(φ1|γ) and ρ2(φ1|γ) and ρ1 intersects ρ2 from above, by step

2 in the previous derivation of the best equilibrium. Moreover, again by step 2, ρ1 must end below ρ2.

Now, moving from F2 to F2 lowers ρ2 to the black curve, and the change is strict at φ1(γ) if F2(φ2(γ)) <

F2(φ2(γ)). However, it is still true that ρ1 must end strictly below ρ2. Hence, the largest intersection at

which ρ1 cuts ρ2 from above moves to the north-east. Thus, as shown in the figure, φ1(γ) ≤ φ1(γ) and

φ2(γ) ≤ φ2(γ), with strict inequality if F2(φ2(γ)) < F2(φ2(γ)). Therefore, we have demonstrated that, for

any γ in the range of gi,

max{ci : gi(ci) ≥ γ} ≤ max{ci : gi(ci) ≥ γ},

with strict inequality under the appropriate condition on F2, thus proving the first part of the proposition.

Now suppose further that F2 = F1. Then, as illustrated in Figure 8, ρ1 and ρ2 are symmetric about

the 45◦ line. Increasing player 2’s cost to cdf F2 moves ρ2 as depicted by the red curve. Therefore, any

intersection of this red curve and the unchanged (blue) ρ1 curve must therefore lie above the 45◦ line,

implying φ1(γ) ≤ φ2(γ), from which it now follows that

g2(φ2(γ)) = γ = g1(φ1(γ)) ≥ g1(φ2(γ)),

where the inequality follows because g1 is weakly decreasing. The inequality is strict under the appropriate

condition on F2.

Proof of Proposition 6. The Pareto improvement brought about by the communication can be understood

as follows. In any equilibrium of the basic game, each type c ∈ (c, c) contributes less than gsa(c), this type’s

most desirable level (because an increase in contribution is enjoyed with probability less than 1). For any

realized vector of costs (c1, . . . , cn) ∈ (c, c)n, the communication game therefore results in a strictly higher

level of the public good, namely, the stand-alone level of the highest-cost player. And in the communication

game all players are better off than in any equilibrium to the basic game. Those for whom g(c) < Galt end up

28

contributing more than in the basic game, but their payoffs are higher because they (exactly) finance a level

of the public good closer to their preferred levels. And any player for whom g(c) > Galt has a two-fold benefit:

under the communication game he enjoys a greater level of the public good and he contributes less than in

the basic-game equilibrium. Finally, truth-telling is a weakly-dominant strategy in the communication game.

To see this, observe that in equilibrium all players contribute a common amount Galt. Player i with cost ci

would most prefer Galt to be gsa(ci); and, by concavity of v, i’s utility decreases as the distance between

Galt and gsa(ci) increases. Now, reporting a cost c′i > ci can only lower Galt, weakly moving the outcome

further from the player’s ideal. In particular, for c′i > ci, we have

min{gsa(c′1), . . . , gsa(c′i), . . . , gsa(c′n)} ≤ min{gsa(c′1), . . . , gsa(ci), . . . , g

sa(c′n)} ≤ gsa(ci),

for any possible combination of the other players’ announcements. Similarly, reporting a cost less than ci

either has no effect on the outcome or moves it further from the player’s ideal. Consequently, untruthful

reporting of cost cannot improve a player’s payoff, regardless of the strategies the others are using.

Example 5 (Technology transfer and communication).

Building on Examples 1 and 3, we assume n = 2, v(G) = G − 12G

2, costs are distributed on [0, 1],

and either (i) both players have costs distributed according to F (c) =√c or (ii) one has cost distributed

according to F1(c) =√c and the other has cost distributed according to F2(c) = c.

Both players face F (c) =√c. The best equilibrium contribution function is g(c) = 1−

√c. Ex ante total

expected utility is then

W eq = 2

∫ 1

0

∫ c1

0

(2v(g(c1))− c1g(c1)− c2g(c2)) dF (c2) dF (c1) =1

3.

We now turn our attention to welfare in the cheap-talk communication game. Using v′(G) = 1−G, we have

Galt = min{1− c1, 1− c2} = 1−max {c1, c2}; thus, ex post joint utility evaluates to

2(1−max {c1, c2})− (1−max {c1, c2})2 − (c1 + c2)(1−max {c1, c2}).

Therefore, ex ante utility is

W alt = 2

∫ 1

0

∫ c1

0

(2 (1− c1)− (1− c1)

2 − (c1 + c2) (1− c1))dF (c2)dF (c1) =

4

9.

Finally, we consider the classical socially efficient allocation. Efficiency requires that both players contribute

29

the same amount, say, g; total welfare at these contributions is therefore 2v(g)−(c1+c2)g, which is maximized

at Geff = 1− (c1 + c2)/2. Thus, ex post utility at the efficient solution evaluates to

2v(Geff

)− c1Geff − c2Geff =

1

4(2− c1 − c2)2.

Therefore, ex ante utility is

W eff =

∫ 1

0

∫ 1

0

1

4(2− c1 − c2)

2dF (c2) dF (c1) =

22

45.

Player 1 faces F1(c) =√c and player 2 faces F2(c) = c. The best equilibrium contribution functions

have player 1 use g1(c) = 1 − c 14 and player 2 use g2(c) = 1 − c 1

3 . Because g2(c2) ≥ g1(c1) if and only if

c2 ≤ (c1)3/4, ex ante utility is now calculated as

W eq =

∫ 1

0

∫ (c1)3/4

0

[2v(g1(c1))− c1g1(c1)− c2g2(c2)] dF2(c2) dF1(c1)

+

∫ 1

0

∫ 1

(c1)3/4[2v(g2(c2))− c1g1(c1)− c2g2(c2)] dF2(c2) dF1(c1)

=1

6.

Again, in the cheap-talk communication game, individual contributions are Galt = 1−max {c1, c2}. There-

fore, ex ante utility is

W alt =

∫ 1

0

∫ c1

0

[2v(1− c1)− (c1 + c2)(1− c1)] dF2(c2) dF1(c1)

+

∫ 1

0

∫ 1

c1

[2v(1− c2)− (c1 + c2)(1− c2)] dF2(c2) dF1(c1)

=1

3.

Finally, the efficiency calculation is exactly as before, but for the change in distributions. Thus, ex post utility

at the efficient solution evaluates to

W eff =

∫ 1

0

∫ 1

0

1

4(2− c1 − c2)

2dF2(c2) dF1(c1) =

23

60.

Table 1 summarizes the welfare calculations in the two scenarios above. These calculations highlight the

welfare-improving effects of communication and technology transfer. Technology transfer—resulting in both

players having cdf√c—raises equilibrium welfare by 100%. Interestingly, without technology transfer, the

30

Table 1: Welfare calculations Example 5. (Numbers in parentheses denote percentages of the efficient welfarelevel.)

F1(c) =√c and F2(c) = c F1(c) = F2(c) =

√c

W eq 30180 (43.5%) 60

180 (68.2%)

W alt 60180 (87.0%) 80

180 (90.9%)

W eff 69180

88180

same increase in welfare is achieved in the equilibrium of the contribution game with initial cheap talk. The

combination of technology transfer and cheap talk raises welfare by another 33%, for an overall increase of

167%.

Proof of Proposition 7. Consider a perfect Bayesian equilibrium Enm to the game with communication. Let

{M1, ...,Mnm} denote the set of messages sent with strictly positive probability in Enm .

For each k, let Hk(s) denote the induced cdf of the effort by player 2, conditional on 2 having sent

message Mk. We now proceed to characterize necessary conditions for equilibrium strategies, in 5 points.

1. Given Mk, 1’s best-response is a pure strategy. For a given effort γ by player 1, we can write player 1’s

expected payoff, given message Mk was received, as

V (γ |Mk) = (1−Hk(γ))v(γ) +

∫ γ

0

v(s) dHk(s)− c1γ

= v(γ)− v(0)Hk(0)−∫ γ

0

Hk(s)v′(s) ds− c1γ,

where the second line follows using integration by parts (see Billingsley, 1995, Theorem 18.4, for the case

in which Hk need not be differentiable). Now, the above shows V is continuous in γ and differentiable

almost everywhere with respect to γ, with

V ′(γ |Mk) = v′(γ)(1−Hk(γ))− c1, (A-9)

except for a set (of γ) having Lebesgue measure zero (Billingsley, 1995, Theorem 31.3). Because 1−Hk

is weakly decreasing and v′ is strictly decreasing, we see that player 1’s best response is a unique effort

level; thus, for each Mk, player 1’s best response is some effort level ak.

2. Player 2’s best response when contributing. If player 2 sends Mk, his best response to ak is

g2(c2 |Mk) = min {ak, gsa(c2)} . (A-10)

31

3. g2(· |Mk) has a flat spot that is taken on with strictly positive (conditional) probability. If ak = 0,

then g2(c2 |Mk) is identically equal to 0. If ak > 0, from equation (A-9) we see it must be that

0 ≤ limγ↑ak V′(γ |Mk), which implies limγ↑ak(1−Hk(γ)) > 0. Therefore,

Pr(g2(c2 |Mk) = ak) = 1− limγ↑ak

Pr(g2(c2 |Mk) < γ)

= limγ↑ak

(1−Hk(γ))

> 0. (A-11)

The set {c2 ∈Mk | c2 ≤ v′(ak)} is just the “flat spot” of player 2’s types that choose effort ak and the

foregoing shows it has strictly positive conditional probability (given Mk was sent).

4. Equilibrium entails a1 = a2 · · · = anm. Otherwise, a type in the flat spot of a message with corre-

spondingly “low” effort by 1 will want to switch messages in order to achieve an outcome closer to his

stand-alone level gsa(c2).

5. An upper bound on a, independent of nm. Let a denote the effort taken by Player 1 with probability 1.

By (A-10), we have g2(c2 |Mk) = min {a, gsa(c2)} for all c2 who announce Mk and all k. Moreover, by

(A-11)

Pr(Mk ∩ {c2| c2 ≤ v′(a)}) > 0, for k = 1...n.

Now, by (A-9), we have

v′(a)Pr(Mk ∩ {c2| c2 ≤ v′(a)})

Pr(Mk)≥ c1, (A-12)

and since∑nm

k=1 Pr(Mk ∩ {c2| c2 ≤ v′(a)}) Pr(Mk) = F (v′(a)) and∑nm

k=1 Pr(Mk) = 1 (by points 2

and 3), condition (A-12) implies

v′(a)F (v′(a)) ≥ c1. (A-13)

Since the game with no communication is equivalent to the game with nm = 1, the above also establish

that the conditions in Example 4 are necessary. The following will establish that the conditions in Example 4

are sufficient, that the upper bound on a is achieved when nm = 1, and therefore communication is not welfare

improving. In other words, we claim that for any a that solves (A-13), the following is an equilibrium, outcome

equivalent to Enm :

a. All types of player 2 send message M1, so no meaningful communication takes place;

b. player 1 contributes the amount a; and

32

c. player 2 contributes min {a, gsa(c2)} .

To see the validity of this claim, it suffices to show that the strategy in part b is optimal for player 1. Note

that any increase in contribution beyond a is counterproductive. Consider now a contribution γ < a. Using

H1(γ) = 1−F ((gsa)−1(γ)) = 1−F (v′(γ)), equation (A-9) implies that the derivative of player 1’s payoff at

γ < a is

v′(γ)F (v′(γ))− c1 > v′(a)F (a)− c1 ≥ 0,

where the first inequality follows by strict convexity of v while the second follows by (A-13). Thus, a is

indeed the optimal choice of player 1. Therefore, communication with a finite number of messages does not

help.

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