The efficacy of cheap talk in collective action problems

Abstract

Incomplete information exacerbates the problems inherent in collective action. Participants cannot efficiently coordinate their actions if they do not know each other’s preferences. I investigate when ordinary communication, or cheap talk, may resolve mutual uncertainty in collective action problems. I find that the efficacy of communication depends critically on the relationship between contributions and the value of the joint project. The incentive barriers to honesty are highest when every contribution increases the project’s value. Participants then have a strict incentive to say whatever would induce others to contribute the most, so cheap talk lacks credibility. By contrast, when contributions may be marginally worthless, such as when the project has no value unless contributions hit a certain threshold, communication may help participants avoid wasted effort. Using these findings, I identify which collective action problems in politics might benefit from communication and which require more expensive solutions to overcome uncertainty.

Keywords

Cheap talk collective action communication public good provision

1. Introduction

Collective action problems—and how to solve them—are a longstanding concern in political science and political economy. Because of the incentive to free-ride, voluntary contributions to joint projects are likely to be insufficient even in the best of circumstances (Olson, 1965). But many collective action problems face an additional hurdle to cooperation that the classical analysis ignores: mutual uncertainty among potential contributors to the common good. For example, a citizen who opposes an autocratic government may be unsure of her fellow citizens’ willingness to risk their lives in a protest. Similarly, countries that have a shared security goal may not know how much their allies are willing to mobilize to meet that goal.

Uncertainty makes the hard problem of collective action even harder. A potential contributor cannot calibrate her own actions when she does not fully understand the incentives of other participants. Incomplete information raises two questions that do not arise in the classic collective action problem. First, is it worth contributing at all? In a project that requires everyone’s participation to succeed, one player’s unwillingness to contribute makes everyone else’s contributions worthless. With incomplete information, potential participants may refrain for fear that their partners are insufficiently committed. Second, how should the project’s costs be divided among participants? When the participants are fully informed, the equilibrium solution to a standard voluntary collective action problem entails the contributor who values the project most highly taking on a disproportionate share of the effort (Olson and Zeckhauser, 1966). With incomplete information, however, players may not know who values the project most and thus may not be able to coordinate on an optimal division of labor (Palfrey et al., 2017).

In this paper, I examine the simplest possible mechanism by which participants in collective action could resolve mutual uncertainty without outside involvement—ordinary communication, which formal theorists usually model as cheap talk (Crawford and Sobel, 1982; Farrell and Rabin, 1996). When, if at all, can participants credibly reveal information and thus coordinate their actions through cheap talk? I investigate this question using a simple but general model of collective action under incomplete information. I find that the possibility of meaningful communication depends critically on the relationship between individual contributions and the outcome of collective action. The incentive barriers to honest communication are strongest in continuous collective action problems—those in which all contributions have some marginal benefit, though possibly a small one (e.g. carbon emission reduction). But in threshold problems, in which one participant’s contribution may be worthless if her partners do not contribute enough (e.g. building a bridge), at least a limited form of communication is possible in equilibrium.

A simple logic drives the main findings. In any collective action problem, each participant at least weakly prefers greater contributions by her partners. In continuous problems, in which every contribution has positive marginal value, this preference is strict—a player is always better off if others give more rather than less. This strict preference undermines honest communication through cheap talk. In order for cheap talk to work, a player’s private information (here, her marginal cost of contributing to the joint project) must affect what she wants the other players to do; otherwise, all ‘types’ of a player will prefer to say the same thing (Aumann, 1990). Specifically, in a continuous collective action problem, a player would always want to say whatever would induce her partner to give the most, whether this be by overstating her own costs of contribution (if the players’ efforts are substitutes) or by understating them (if they are complements).

The incentives are similar, yet different in a critical way, in collective action problems where a fixed threshold determines whether the project is achieved. Below the threshold, the marginal benefit of a contribution—one’s own or one’s partner’s—is zero. At the margin, then, a potential contributor may be indifferent whether her partner gives more or less. More to the point, if a player is sure not to contribute enough for the threshold to be met (e.g. because she values the project very little), then she will be indifferent about her partner’s contribution. This indifference turns out to be critical for the possibility of communication. It means a participant will be willing to reveal that she will not give enough to meet the threshold, even though this will encourage others to contribute less than they might have otherwise. By contrast, in a collective action problem where all contributions make at least a small difference, a participant would be strictly worse off if she said something that made her partners contribute less.

The analysis speaks to a broad cross-section of political science research, given the ubiquity of collective action problems with incomplete information in politics. One area of application is global public goods problems, such as the ongoing refugee crisis and the effort to reverse climate change. These initiatives are complicated not only by countries’ incentives to free-ride, but also by their uncertainty of where each other’s breaking points lie. Another application in the international arena is the provision of collective security, a classic collective action problem (Olson and Zeckhauser, 1966) that takes place in an environment of high uncertainty (Fearon, 1995; Jervis, 1976). In domestic politics, perhaps the most prominent collective action problem is revolution or other means of overthrowing the government (Tilly, 1978). Potential participants may not know how likely their fellow citizens are to participate, especially in regimes without free media, making coordination difficult without some means of sharing information. Some facets of democratic politics, such as campaign fundraising, also have the features of collective action problems.

The main upshot of my findings is that we should expect uncertainty to impede cooperation in these areas, above and beyond the difficulties inherent in any collective action problem, even if the participants can freely communicate with each other. Uncertainty can only be resolved through mechanisms more ‘expensive’ than mere talk, such as costly signaling of one’s intent to participate, direct monitoring by external actors, or binding commitment to a transfer scheme. At best, political actors may reveal through cheap talk when they are so unwilling to contribute, that the project is doomed to be worthless. But even this can only take place when there is a fixed, commonly known threshold for contributions below which the project has literally no value, a rarity in political collective action problems.

This analysis contributes to the political economy literature on collective action and public good provision. In a seminal study of the finance of public goods, Samuelson (1954) identifies how individual incentives to misrepresent demand for the collective good might undermine efficient taxation and provision. Subsequently, various papers in the mechanism design literature identified transfer schemes by which efficient provision might be achieved (Clarke, 1971; d’Aspremont and Gérard-Varet, 1979; Groves, 1973). My analysis shows that the consequences of incomplete information for efficient collective action are considerably more severe when we move from a publicly financed public good to the voluntary contribution setting. In fact, in the class of continuous problems I consider, in which all contributions have positive marginal value, communication cannot recover even part of the efficiency loss due to incomplete information. Without a central authority that can commit participants to a transfer scheme, as assumed in the mechanism design literature, it will be difficult to achieve coordination through voluntary information revelation. The difference between the centralized and decentralized approaches is particularly important for political scientists who study collective action problems in which there is no central authority (e.g. in international politics) or where the object of collective action is to overthrow that authority (e.g. revolutions or coups).

This paper is most closely related to a recent set of work in political economy that considers the efficacy of cheap talk in various public goods games (Agastya et al., 2007; Costa and Moreira, 2012; Palfrey et al., 2017). These analyses consider discrete problems, in which the public good is provided if contributions exceed a known threshold and has zero value otherwise. I show that the findings of effective cheap talk in these models do not extend to problems in which the value of the public good increases continuously with contributions, as in environments like that of Bergstrom et al. (1986). My analysis is most similar to that of Barbieri (2012), who shows that the possibility of communication in threshold models is sensitive to assumptions about the public goods technology. He extends the baseline model of Agastya et al. (2007) to allow contributions below the threshold to have a positive but small marginal value, finding that it is more difficult to sustain influential communication when small contributions are not totally wasted. In this paper, I show that this intuition extends to a significantly broader class of collective action models, and I obtain entirely new results on the possibility and nature of communication in threshold models.

The other major literature this paper contributes to concerns the role of communication in coordinating political action. The question of what makes cheap talk effective is important to the study of international diplomacy (Kydd, 2003; Sartori, 2002; Trager, 2010) and domestic policy debates (Austen-Smith, 1990). These literatures, drawing from the seminal study of cheap talk by Crawford and Sobel (1982), conclude that the efficacy of communication is a function of interest alignment among the actors involved. My results qualify this conclusion. Even partial communication is ineffective in the baseline case, though the model, like those aforementioned, is a mixed-motive game. I show that what matters is not just some alignment of preferences over outcomes, which indeed is a necessary condition for communication, but also that a player’s private information affects what she wants the other players to do. Otherwise, all types will prefer to send the same message—namely, whichever one yields the most-preferred outcome from the other player—so cheap talk will lack credibility.² In my analysis, this condition is the key difference between the baseline model in which communication is ineffective and the extensions in which cheap talk works.

Section 2 presents the model of collective action and communication. I derive results for continuous problems in Section 3 and for threshold problems in Section 4. Section 5 discusses the theory’s implications for a variety of applications in political science. Section 6 concludes.

2. The model

I model a two-player collective action problem with incomplete information and pre-play communication. I look for conditions under which communication is influential—i.e. it reveals some information and affects the selection of contributions.

2.1. Contribution subgame

In the contribution subgame, two players, labeled 1 and 2, individually and simultaneously choose how much effort to contribute to a joint project or goal. Throughout the analysis, i denotes a generic player and j her partner. A player’s contribution is denoted $x_{i}$ ; the feasible set of contributions for each player is a compact interval, $X_{i} = [0, {\bar{x}}_{i}]$ .

Each player’s effort affects the value of the joint project or its likelihood of success. Given the contributions $x_{1}$ and $x_{2}$ , the project’s value is $p (x_{1}, x_{2}) \geq 0$ .³ If we interpret the project as a public good, p is its production function. I assume throughout the analysis that p is non-decreasing in both arguments. As a regularity condition to assure the existence of a pure strategy equilibrium in every contribution subgame, I also assume p is upper semicontinuous. Further conditions on the production and cost functions are enumerated in numbered assumptions. In particular, I highlight the difference between when p is strictly increasing in its arguments, so all contributions are valuable, and when p has flat spots, so some contributions may be worthless.

Each player has private information about her ability or willingness to contribute to the joint project. Let $t_{i}$ denote a player’s type, and let her cost of contributing $x_{i}$ as a function of her type be $c_{i} (x_{i}, t_{i})$ . I assume each cost function is strictly increasing in $x_{i}$ and that a player may always avoid costs by contributing nothing: $c_{i} (0, \cdot) = 0$ . Again to ensure equilibrium existence, I assume each $c_{i}$ is lower semicontinuous in $x_{i}$ .

The information structure is as follows. Each player’s type is drawn from a finite, though possibly large, set $T_{i}$ with $N_{i}$ elements, ordered $t_{i 1} < \dots < t_{i N_{i}}$ . The prior distribution of $t_{i}$ is $π_{i} \in Δ T_{i}$ , which is common knowledge. To avoid trivialities, I assume each $π_{i} >> 0$ .⁴ To obtain stronger results, I sometimes consider the special case of the game with one-sided incomplete information. When I do so, I assume without (further) loss of generality that player 2 is the one whose type is common knowledge; i.e. $N_{2} = 1$ and $π_{21} = 1$ .

Notice that players’ types affect their individual cost functions,⁵ but not the joint production function. The assumption that one’s private information only affects one’s own payoff is standard in analyses of incomplete information in public goods problems (Green and Laffont, 1977). This rules out, for example, private signals about the nature of the production technology. Therefore, each player only cares about her partner’s type insofar as it affects how much her partner will contribute.

Given these assumptions about the production and cost technologies, each player’s utility function is

u_{i} (x_{i}, x_{j}, t_{i}) = p (x_{i}, x_{j}) - c_{i} (x_{i}, t_{i})

(1)

With this payoff structure, the contribution subgame is a Bayesian potential game (Monderer and Shapley, 1996; van Heumen et al., 1996) with potential function

P (x_{1}, x_{2}, t_{1}, t_{2}) = p (x_{1}, x_{2}) - c_{1} (x_{1}, t_{1}) - c_{2} (x_{2}, t_{2})

(2)

(see also Myatt and Wallace, 2009). Next, I extend the game to include a prior communication stage.

2.2. Cheap talk communication

I model communication with a messaging subgame that precedes the contribution subgame. After learning their types, each player simultaneously sends a message $m_{i}$ from the message space $M_{i}$ . To avoid trivialities, I assume each $| M_{i} | \geq 2$ . The messages are publicly observed before players choose their contributions. Messages are cheap talk: every type of each player may send any message, and the chosen messages have no direct effect on either player’s payoff (Crawford and Sobel, 1982; Farrell and Rabin, 1996).

Communication can only affect outcomes through its effect on the players’ beliefs. Each player’s message shapes her partner’s beliefs about her type, which in turn may affect the equilibrium of the contribution subgame. For example, a player may contribute more if she believes her partner is relatively unwilling to contribute than if she expects her partner to take up most of the burden. For every $m_{i} \in M_{i}$ , let ${\hat{π}}_{i} (m_{i}) = ({\hat{π}}_{i 1} (m_{i}), \dots, {\hat{π}}_{i N_{i}} (m_{i})) \in Δ T_{i}$ represent j’s updated beliefs about i’s type after receiving $m_{i}$ . The crucial question is when it is in a player’s interest to reveal her type—or even just some information about her type—given how doing so will affect her partner’s beliefs.

To summarize, the sequence of play is as follows:

Nature privately informs each player of her type, $t_{i} \in T_{i}$ .

Each player simultaneously sends a message, $m_{i} \in M_{i}$ .

Each player observes her partner’s message and updates her beliefs about $t_{j}$ to ${\hat{π}}_{j} (m_{j})$ .

Each player simultaneously chooses a contribution, $x_{i} \in X_{i}$ .

The game ends and payoffs are realized.

A messaging strategy is a function $μ_{i} : T_{i} \to M_{i}$ that prescribes a message for each type of each player. A contribution strategy is a function $ω_{i} : T_{i} \times M_{i} \times M_{j} \to X_{i}$ that prescribes how much each type contributes after each possible pair of messages. An assessment $(μ_{i}, ω_{i}, ({\hat{π}}_{i} (m_{i}))_{m_{i} \in M_{i}})_{i = 1, 2}$ is a tuple containing both players’ strategies and belief systems.

At times, to keep the analysis tractable, I restrict attention to interval messaging strategies, in which each player’s messaging strategy reveals an interval of the type space in which her type lies. Formally, a messaging strategy $μ_{i}$ is interval messaging if, for all $t_{i}, t'_{i}, t''_{i} \in T_{i}$ such that $t_{i} < t'_{i} < t''_{i}$ , $μ_{i} (t_{i}) = μ_{i} (t''_{i}) = m_{i}$ implies $μ_{i} (t'_{i}) = m_{i}$ . This type of strategy is common in models of cheap talk communication (Crawford and Sobel, 1982). Figure 1 illustrates the distinction between interval and non-interval messaging. Note that a fully separating messaging strategy, in which $μ_{i}$ is one-to-one (every type sends a distinct message), is always an interval strategy. So is a ‘babbling’ strategy, in which $μ_{i}$ is constant (every type sends the same message). Consequently, in the special case where $| T_{i} | \leq 2$ , every messaging strategy for player i is trivially an interval strategy.

Figure 1.

Illustration of messaging strategy types with $N_{1} = 7$ .

2.3. Influential communication

As this is a multistage game of incomplete information with observed actions, the appropriate solution concept is perfect Bayesian equilibrium (Fudenberg and Tirole, 1991). An assessment is an equilibrium if each player’s strategy is sequentially rational given her beliefs and the other player’s strategy, and beliefs are updated in accordance with Bayes’ rule whenever possible. Throughout the analysis, I restrict attention to pure strategy equilibria. The potential function (2) is upper semicontinuous and thereby attains its maximum, which in turn corresponds to a pure strategy equilibrium (van Heumen et al., 1996, Corollary 5.4).⁶

The interesting question is not just when players might reveal their private information, but when that information sharing can help them coordinate their actions. In the language of cheap talk models, an equilibrium in which a player reveals information and that revelation affects the outcome is an influential equilibrium. An equilibrium of this model is influential if, on the path of play, at least one type of one player contributes different amounts depending on the message she receives. A formal definition follows.

Definition 1. An equilibrium is influential if there is a type $t_{i} \in T_{i}$ and types $t_{j}, t'_{j} \in T_{j}$ such that

ω_{i} (t_{i}, μ_{i} (t_{i}), μ_{j} (t_{j})) \neq ω_{i} (t_{i}, μ_{i} (t_{i}), μ_{j} (t_{j}'))

□

An immediate consequence of this definition is that a babbling equilibrium, in which every type of each player sends the same message, cannot be influential. However, not all non-babbling equilibria are influential. For example, an equilibrium in which player 1 fully reveals her type but player 2 always contributes $x_{2} = 0$ on the path of play is non-babbling and non-influential.

In cheap talk models like this one, influential equilibria are least likely to exist, though not necessarily impossible (Baliga and Morris, 2002; Seidmann, 1990), when every type of a player has the same preferences over the other player’s actions (see Aumann, 1990). In an influential equilibrium, one player’s message effectively dictates what the other will do. So for different types to choose different messages, they must prefer different actions by their partner—or at least be indifferent. Therefore, a key task of the present analysis will be to separate those instances of collective action where this uniformity of preference exists from those where it does not.

3. Continuous problems

I first consider collective action problems in which all contributions are at least somewhat valuable. These include public goods problems in which the amount of the good produced is a continuous and strictly increasing function of the contributions. All-or-nothing projects whose success is probabilistic also fit the bill (Nitzan and Romano, 1990), provided that the probability of success is continuous in the contributions. One example is reducing carbon emissions in order to slow global warming—every reduction has some effect, though perhaps a small one. Military cooperation is another example. Even if a military operation has a discrete goal, such as the removal of a particular regime, the exact level of force necessary to achieve it probably is not known in advance. Greater deployments simply increase the chance of success.

The main result is that in a wide class of continuous collective action problems, communication either does not occur or has no effect on contribution behavior. The intuition behind the result is simple. If all contributions are valuable, then each player strictly prefers that her partner give as much as possible. Ideally, then, a player would send whichever message induced the greatest contribution by her partner. And since talk is cheap, there is nothing to prevent one from doing so. Therefore, influential communication cannot be sustained as an equilibrium. In order for different types to be willing to send different messages, they must want different things from their partner. But that is not true here—a player always wants the same thing, namely the greatest possible contribution, regardless of her own willingness to give.

I consider a natural set of collective action problems in which there are diminishing marginal returns to contribution and the two players’ contributions are substitutes.⁷ Concavity and substitutability are common assumptions in Cournot contribution games, a popular class of models of public goods and collective action (see Myatt, 2007). The following assumption formalizes the class of games under consideration.

Assumption 1.

p is strictly increasing, twice continuously differentiable, and strictly concave.

$x_{1}$ and $x_{2}$ are substitutes: $\partial^{2} p / \partial x_{1} \partial x_{2} \leq 0$ .

Each $c_{i}$ is twice continuously differentiable and convex in $x_{i}$ .

Each $\partial c_{i} / \partial x_{i}$ is strictly increasing in $t_{i}$ .

Two relevant properties follow from this assumption. First, because higher types have higher marginal costs of contribution, they are less willing to give to the joint project. Second, because the players’ contributions are substitutes, the more a player gives, the less her partner prefers to give. In combination, these imply that the way to induce one’s partner to give most is to mimic a high type—in other words, to understate one’s willingness to contribute. This incentive to feign unwillingness is what precludes influential communication in equilibrium.

In order to prove that there is no influential equilibrium in interval messaging, I show that every type of a player contributes weakly more after receiving a ‘high’ message—i.e. one that comes from types that have high costs and thus are relatively unwilling to contribute—than after receiving a ‘low’ message. The proof uses the method of monotone comparative statics (Ashworth and Bueno de Mesquita, 2006; Milgrom and Shannon, 1994). Without explicitly solving for equilibria, which would require placing specific functional forms or distributional assumptions limiting the generality of the results, I show that a player’s equilibrium contributions increase monotonically as she moves from believing her partner’s type is in a ‘low’ set to a ‘high’ one.

Monotone comparative statics usually arise in games of strategic complementarities, in which a player’s best response is an increasing function of the other players’ actions. But in the type of collective action problem considered here, we have the opposite—the more a player thinks her partner will give, the less she prefers to give herself. So this is a game of strategic substitutes. To identify monotone comparative statics nonetheless, I transform this into a game of strategic complements by redefining one player’s action as the additive inverse of her contribution (see Amir, 1996).

Let $Γ ({\hat{π}}_{1}, {\hat{π}}_{2})$ denote a contribution subgame in which player 1 believes $t_{2} ~ {\hat{π}}_{2}$ , player 2 believes $t_{1} ~ {\hat{π}}_{1}$ , and these beliefs are common knowledge.⁸ Then let the transformed contribution subgame, denoted $\tilde{Γ} ({\hat{π}}_{1}, {\hat{π}}_{2})$ , be a two-player strategic form game with strategy spaces

\begin{matrix} {\tilde{X}}_{1} = {{\tilde{x}}_{1} \in X_{1}^{N_{1}} | {\tilde{x}}_{11} \leq \dots \leq {\tilde{x}}_{1 N_{1}}}, \\ {\tilde{X}}_{2} = {{\tilde{x}}_{2} \in X_{2}^{N_{2}} | {\tilde{x}}_{21} \geq \dots \geq {\tilde{x}}_{2 N_{2}}} \end{matrix}

and utility functions

{\tilde{u}}_{1} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) = \sum_{n = 1}^{N_{1}} [\sum_{m = 1}^{N_{2}} {\hat{π}}_{2 m} p ({\bar{x}}_{1} - {\tilde{x}}_{1 n}, {\tilde{x}}_{2 m}) - c_{1} ({\bar{x}}_{1} - {\tilde{x}}_{1 n}, t_{1 n})],

(3)

{\tilde{u}}_{2} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) = \sum_{n = 1}^{N_{2}} [\sum_{m = 1}^{N_{1}} {\hat{π}}_{1 m} p ({\bar{x}}_{1} - {\tilde{x}}_{1 m}, {\tilde{x}}_{2 n}) - c_{2} ({\tilde{x}}_{2 n}, t_{2 n})]

(4)

The transformed contribution subgame differs from the original in a few respects that make it more amenable to monotone comparative statics analysis. First, player 1’s action represents the inverse of her contribution, so $\tilde{Γ} ({\hat{π}}_{1}, {\hat{π}}_{2})$ is a game of strategic complements. The greater ${\tilde{x}}_{i}$ is, the greater the optimal choice of ${\tilde{x}}_{j}$ . Substantively, this means each actor prefers to contribute less the more she expects her partner to contribute (since ${\tilde{x}}_{1}$ represents the inverse of player 1’s contribution). The following lemma states this result formally.⁹

Lemma 1. Let Assumption 1 hold. $\tilde{Γ} ({\hat{π}}_{1}, {\hat{π}}_{2})$ is supermodular.

Second, the transformed subgame represents the Bayesian game in ex ante form, with each player choosing a vector of contributions corresponding to each of her own types so as to maximize total payoffs across types. Third, players are restricted to strategies in which contributions weakly decrease by type. Nonetheless, for the purposes of equilibrium analysis, the transformed contribution subgame is isomorphic to the original, as the following lemma states.

Lemma 2. Let Assumption 1 hold. $\tilde{x} \in {\tilde{X}}_{1} \times {\tilde{X}}_{2}$ is a Nash equilibrium of $\tilde{Γ} ({\hat{π}}_{1}, {\hat{π}}_{2})$ if and only if the strategy profile defined by

\begin{matrix} x_{1} (t_{1 n}) = {\bar{x}}_{1} - {\tilde{x}}_{1 n}, n = 1, \dots, N_{1}, \\ x_{2} (t_{2 m}) = {\tilde{x}}_{2 m}, m = 1, \dots, N_{2} \end{matrix}

is a Bayesian Nash equilibrium of $Γ ({\hat{π}}_{1}, {\hat{π}}_{2})$ .

I have already noted that the contribution subgame is a potential game (Monderer and Shapley, 1996), which guarantees the existence of an equilibrium in pure strategies. Under the differentiability and concavity conditions of Assumption 1, the potential function is smooth and strictly concave, so the equilibrium is unique (Neyman, 1997), as stated in the following result.

Lemma 3. Let Assumption 1 hold. $\tilde{Γ} ({\hat{π}}_{1}, {\hat{π}}_{2})$ has a unique equilibrium.

In cheap talk models like this one, equilibrium uniqueness in the final stage is important for the question of whether influential communication is possible. For example, models in which cheap talk may affect bargaining (Farrell and Gibbons, 1989; Ramsay, 2011) rely on multiplicity of equilibria. These models admit both ‘no trade’ equilibria in which bargaining is sure to fail and ‘trade’ equilibria in which serious offers are exchanged. Because of this multiplicity, a player may rationally expect the message ‘I will stand firm’ to lead to more aggressive behavior by some types of her partner (those with whom the no-trade equilibrium is played) and more conciliatory behavior by others (those who proceed to bargain seriously). In the environment I analyze here, by contrast, cheap talk cannot generate a diversity of responses simply by sorting players into different equilibria.

With uniqueness pinned down, the only remaining question for the possibility of influential communication is how each player responds to her partner’s messages. In order for influential cheap talk to be incentive-compatible, a message that causes some types to increase their contribution must cause other types to decrease theirs. To see why, consider an influential strategy profile, in which different types of player i send distinct messages $m_{i}$ and $m'_{i}$ such that at least one type of player j contributes more after receiving $m'_{i}$ . Now suppose no type of player j contributes more after receiving $m_{i}$ . Then this strategy profile cannot be an equilibrium, since every type of player i strictly prefers to send the message $m'_{i}$ . Sending $m'_{i}$ guarantees her partner will contribute at least as much if she had sent $m_{i}$ , and with positive probability her partner will contribute even more.

Unfortunately for the prospects of influential communication, this uniformity of responses is exactly what happens under interval messaging in the broad class of models characterized by Assumption 1. Specifically, in equilibrium, every type of player j contributes weakly less after receiving a ‘low type’ (high willingness) message from player i than after receiving a ‘high type’ (low willingness) message. Figure 2 illustrates this finding,¹⁰ and the following result formalizes it. It considers a range of contribution subgames, holding fixed the distribution of one player’s type while varying the relative chance of a lower versus higher type of the other player.¹¹ As a player’s partner becomes more likely to have a high cost of contribution, all types of that player (weakly) increase their own contribution in equilibrium.

Figure 2.

Equilibrium responses by player 2 along the path of play in the contribution subgames when player 1’s messaging strategy separates herself into ‘low,’ ‘medium’, and ‘high’ types. In the left panel, player 2’s messaging strategy is uninformative; in the right panel, she employs the same messaging strategy as player 1. In either case, as stated in Lemma 4, every type of player 2 contributes (weakly) more after receiving the ‘medium’ message than after the ‘low’ message and still more after receiving the ‘high’ message.

Lemma 4. Let Assumption 1 hold. Let ${\hat{π}}_{1}^{L}, {\hat{π}}_{1}^{H} \in Δ T_{1}$ , ${\hat{π}}_{2} \in Δ T_{2}$ , and $α, β \in [0, 1]$ . Assume the support of ${\hat{π}}_{1}^{H}$ consists of higher types than that of ${\hat{π}}_{1}^{L}$ ; i.e. $min {n | {\hat{π}}_{1 n}^{H} > 0} > max {n | {\hat{π}}_{1 n}^{L} > 0}$ . Let

{\hat{π}}_{1}^{α} = α {\hat{π}}_{1}^{H} + (1 - α) {\hat{π}}_{1}^{L}

and define ${\hat{π}}_{1}^{β}$ analogously. Let ${\tilde{x}}^{α}$ and ${\tilde{x}}^{β}$ be the equilibria of $\tilde{Γ} ({\hat{π}}_{1}^{α}, {\hat{π}}_{2})$ and $\tilde{Γ} ({\hat{π}}_{1}^{β}, {\hat{π}}_{2})$ , respectively. If $α \leq β$ , then ${\tilde{x}}^{α} \leq {\tilde{x}}^{β}$ .

It is almost immediate from this result that there cannot be influential communication when players use interval messaging strategies and Assumption 1 holds. For example, imagine an equilibrium in which player 1’s messaging strategy partitions her type into two intervals (‘low cost’ and ‘high cost’). According to Lemma 4, along the path of play, every type of player 2 must contribute weakly more following the ‘high cost’ message than after the ‘low cost’ message. If the equilibrium is influential, then at least one type of player 2 must be contributing strictly more. But then even the low types of player 1 would strictly prefer to send the ‘high cost’ message so as to induce the greatest possible contribution by player 2, contradicting the assumption of equilibrium. Therefore, even if there is an equilibrium in which one player wholly or partly reveals her type via an interval messaging strategy, this revelation does not affect the ultimate choice of contributions. The following proposition summarizes this finding.

Proposition 1. Let Assumption 1 hold. There is no influential equilibrium in interval messaging.

This finding, though negative on the face of it, has important implications for the study of collective action problems. Even without uncertainty, collective action problems are difficult to overcome, particularly when contributions are voluntary and non-refundable, as in the setting I study here. Private information may exacerbate the problem, as players who do not know how much each other is willing to contribute cannot even coordinate on a second-best division of labor. What Proposition 1 shows is that there is no cheap solution to the additional problems created by uncertainty, at least in the class of games covered by Assumption 1. In order to have information exchange that improves collective action outcomes in these continuous problems, the players must engage in costly signaling, participate in institutions that can independently solicit information about members’ willingness to contribute, or use some other costly mechanism.

Before moving on to threshold problems, I briefly state two additional results for the continuous case. First, if the players’ contributions were complements instead of substitutes, influential communication would remain impossible in equilibrium. Suppose Assumption 1(b) was reversed to say $\partial^{2} p / \partial x_{1} \partial x_{2} \geq 0$ . Then the (non-transformed) contribution subgame has strategic complements: the more a player expects her partner to give, the more she wants to give. From there all the subsequent results go through. In fact, they would go through if the game were generalized to have an arbitrary number of players. There is still a pervasive incentive to misrepresent, which precludes any influential communication. What differs is the nature of the misrepresentation. Whereas before it was profitable to understate one’s willingness to chip in, here it would be best to overstate.

Second, the assumptions on the production function can be relaxed even further if only one player has private information about her willingness to contribute. The only condition that is required is that the value of the project be strictly increasing in the contribution of the player whose type is common knowledge.¹² The formal statement of this condition, which is weaker than Assumption 1, is as follows.

Assumption 1W. p is strictly increasing in $x_{2}$ .

With this assumption in hand, it is simple to prove that there cannot be influential messaging with one-sided incomplete information. Imagine an influential strategy profile, in which distinct types of player 1 send distinct messages, leading to distinct responses by player 2. Since there is only one type of player 2, it must be the case that one of these responses is greater—and thereby strictly better for all types of player 1, under Assumption 1W—than the other. But then, by the same logic as in the proof of Proposition 1, every type of player 1 would strictly prefer to send the message that yielded the greater contribution by her partner. Therefore, an influential strategy profile cannot be incentive-compatible. The following result follows immediately from this line of reasoning.

Proposition 2. If there is one-sided incomplete information and Assumption 1W holds, there is no influential equilibrium.

I have identified a broad set of conditions in continuous collective action problems under which communication does nothing to coordinate behavior or improve the efficiency of outcomes. The basic problem is that in the continuous setting, each player strictly prefers for her partner to contribute as much as possible. There is accordingly a strict incentive to deviate from an influential messaging strategy, namely to say whatever would get one’s partner to contribute the most. In the next section, I consider discrete problems in which these strict preferences do not hold, at least not globally. Although the difference might seem minor at a glance, it turns out to be critically important for the possibility of influential communication.

4. Threshold problems

The negative findings in the previous section seemingly contradict a recent spate of analyses claiming that cheap talk may help contributors coordinate their actions in the presence of incomplete information (Agastya et al., 2007; Costa and Moreira, 2012; Palfrey et al., 2017). In contrast to the collective action problems I considered above, the models in these articles consider discrete, or threshold, public goods. The value of such a good is zero unless contributions meet a fixed, commonly known threshold. For example, a public works project whose cost is known in advance and that provides no value if not completed (e.g. a bridge) is a threshold good.

In this section, I identify the strategic differences between continuous and threshold problems, and I explain why the possibility of influential communication in the threshold setting does not carry over to the continuous one. In a continuous collective action problem, every contributor strictly prefers greater contributions by her partner. Even if the marginal benefit of your partner’s contributions is fairly low, it comes at no cost to you. But in threshold problems, the preference for one’s partner to give more becomes weak, at least sometimes. If a player is unwilling to contribute enough for the threshold to be met, then she is indifferent between her partner giving nothing and her partner making a wasted contribution. This indifference is crucial for the possibility of communication, as I illustrate below.

The key feature of the collective action problems I consider in this section is that some contributions are worthless, at least at the margin. For example, in a simple threshold model where the good is provided only if $x_{1} + x_{2} \geq θ$ , then a contribution $x_{i} \in (0, θ)$ is wasted if $x_{j} < θ - x_{i}$ . More generally, as stated in the following assumption, we can allow for a set of contribution profiles (not necessarily defined by a summation) for which the project fails and has no value.

Assumption 2. There exists a set $Z \subseteq X_{1} \times X_{2}$ , with $| Z | > 1$ , such that $p (x_{1}, x_{2}) = 0$ if and only if $(x_{1}, x_{2}) \in Z$ .

This assumption encapsulates a wide variety of collective action problems with (marginally) worthless contributions.¹³ For example, in the kind of additive threshold problem discussed above, $Z = {(x_{1}, x_{2}) | x_{1} + x_{2} < θ}$ . Another example would be a public good with a Cobb–Douglas production function, $p (x_{1}, x_{2}) = {Kx}_{1}^{α} x_{2}^{β}$ , for which $Z = {(x_{1}, x_{2}) | min {x_{1}, x_{2}} = 0}$ .

4.1. Numerical example

I begin with a parameterized example that demonstrates the possibilities—and limitations—of communication in threshold problems. Consider the game in which the action spaces are $X_{1} = X_{2} = [0, \frac{1}{2}]$ , the production function takes the thresholded quadratic form

p (x_{1}, x_{2}) = {\begin{matrix} 0 & x_{1} + x_{2} < \frac{1}{2}, \\ 1 - {(1 - x_{1} - x_{2})}^{2} & x_{1} + x_{2} \geq \frac{1}{2} \end{matrix}

(5)

and the cost functions take the linear form $c_{i} (x_{i}, t_{i}) = x_{i} t_{i}$ . Figure 3 illustrates the difference between the thresholded production function here and its continuous analogue. The production function here is the kind of thresholded function that Myatt and Wallace (2009) consider.

Figure 3.

On the left, the function $p (x_{1}, x_{2}) = 1 - (1 - x_{1} - x_{2})^{2}$ . On the right, the thresholded version given in equation (5).

I assume one-sided incomplete information, both to keep the example simple and to best highlight the contrast with the continuous case (where the impossibility of influential communication does not depend on concavity or other functional form restrictions; see Proposition 2). Specifically, let $t_{2} = 3$ and let $T_{1} = {t_{1}^{L}, t_{1}^{M}, t_{1}^{H}}$ where $\frac{3}{2} < t_{1}^{L} < t_{1}^{M} \leq 3 < t_{1}^{H}$ .

In this example, the value of the project at the threshold is $\frac{3}{4}$ , and consequently the most a player of type $t_{i}$ is willing to spend in order to make the difference between meeting the threshold or not is

\bar{x} (t_{i}) = {\begin{matrix} \frac{1}{2} & t \leq \frac{3}{2}, \\ \frac{3}{4 t_{i}} & t > \frac{3}{2} \end{matrix}

Past the threshold, the marginal benefit of a contribution is 1 or less, so only a player of type $t_{i} < 1$ would spend beyond that point.

This game has an influential equilibrium¹⁴ in which the two players coordinate to meet the threshold if and only if it is individually rational to do so. In the messaging stage, player 1 sends the ‘low’ message $m_{1}^{L}$ if $t_{1} \in {t_{1}^{L}, t_{1}^{M}}$ and the ‘high’ message $m_{1}^{H} \neq m_{1}^{L}$ if $t_{1} = t_{1}^{H}$ . In the contribution subgame, following the ‘low’ message, either type of player 1 contributes $x_{1} = \bar{x} (t_{1}^{M}) \geq \frac{1}{4}$ and player 2 contributes $x_{2} = \frac{1}{2} - \bar{x} (t_{1}^{M})$ , so the threshold is met exactly.¹⁵ Following the ‘high’ message, both players contribute $x_{1} = x_{2} = 0$ , so the project fails but no effort is wasted.

It is trivial to confirm that the proposed contribution strategies form Bayesian Nash equilibria in their respective subgames. It is also easy to see why $t_{1}^{L}$ and $t_{1}^{M}$ prefer not to deviate from sending $m_{1}^{L}$ , as doing so would result in a lower contribution by their partner. What is more surprising is that it is not profitable for $t_{1}^{H}$ to deviate to sending the low message. Doing so would yield a greater contribution by player 2, but this contribution has no value to the high type of player 1. The most this type is willing to spend to meet the threshold is

\bar{x} (t_{1}^{H}) = \frac{3}{4 t_{1}^{H}} < \frac{1}{4}

which, when combined with player 2’s contribution of $x_{2} \leq \frac{1}{4}$ , is not enough. Therefore, the high type is indifferent between her own messaging strategy and a deviation. This indifference, in contrast to the strict preference for a greater contribution by the other player in the continuous case, enables influential communication.

Not only is influential communication possible, it may also provide an efficiency gain over the no-communication equilibrium. In the game without communication, as the prior probability of $t_{1}^{H}$ approaches 1, the most player 2 is willing to contribute approaches 0. So when this prior probability is sufficiently high, even the low type of player 1 will be unwilling to make up the difference. (Recall that $t_{1}^{L} > \frac{3}{2}$ , so the low type is not willing to meet the threshold on her own.) Consequently, the project will sometimes fail even though it would have been feasible had player 2 known $t_{1} \leq t_{1}^{M}$ .

Nonetheless, in this equilibrium, some potential social welfare gains are being left on the table. If the players’ types were common knowledge, the contribution scheme that would maximize social welfare is

(x_{1}, x_{2}) = {\begin{matrix} (\bar{x} (t_{1}^{L}), \frac{1}{2} - \bar{x} (t_{1}^{L})) & t_{1} = t_{1}^{L}, \\ (\bar{x} (t_{1}^{M}), \frac{1}{2} - \bar{x} (t_{1}^{M})) & t_{1} = t_{1}^{M}, \\ (0, 0) & t_{1} = t_{1}^{H} \end{matrix}

But consider a strategy profile in which player 1’s message reveals her type exactly, and then the two players coordinate on this contribution scheme. Such a strategy cannot be an equilibrium, as $t_{1}^{L}$ has a strict incentive to mimic $t_{1}^{M}$ in the messaging stage. By doing so, $t_{1}^{L}$ could lower her own contribution to $\bar{x} (t_{1}^{M})$ yet still have the project succeed—the same benefit at less cost. Even when communication recovers some of the efficiency loss due to incomplete information in a threshold collective action problem, it may still not allow players to achieve what they could have under full information.

The upshot of this example is twofold. First, in threshold collective action problems, cheap talk communication may help players reveal information so as to better coordinate their contributions. This possibility arises because, unlike in the class of continuous problems studied in the previous section, a player may be indifferent at the margin about whether her partner gives more or less. More to the point, a player who is unwilling to give enough to meet the threshold does not lose anything by revealing this. This brings us to the second takeaway, which is that even when influential cheap talk is possible in threshold problems, its efficacy may still be limited (Costa and Moreira, 2012; Palfrey et al., 2017). Once a player expects the threshold to be met, she once again strictly benefits from greater contributions by her partner. A player may honestly reveal whether she is willing to contribute at all, but there are incentive barriers to more detailed revelations.

4.2. Formal results

I now turn to some more general findings on communication and contributions in collective action problems characterized by Assumption 2. I focus on results that reinforce the lessons of the numerical example above. This is a wide class of games, so it is difficult to pin down more than broad patterns. The previous literature on cheap talk in threshold games contains more detailed findings on specific models within this class (Agastya et al., 2007; Barbieri, 2012; Costa and Moreira, 2012; Palfrey et al., 2017).

First, I establish that it is incentive-compatible for a player to reveal that she is unwilling to contribute enough to meet the threshold. What complicates this is that how much a player is willing to contribute, and whether that is enough to meet the threshold, may depend at least partly on what she expects her partner to do. To accommodate this, I use an iterated strict dominance criterion to define unwillingness to meet the threshold. Let each $X_{i}^{0} (t_{i}) = X_{i}$ . Then for each $k \in N_{+}$ , let $X_{i}^{k} (t_{i}) \subseteq X_{i}$ be the set of contributions that are not dominated for $t_{i}$ given that player j makes a contribution that is not dominated for all types in the $k - 1$ ’th iteration: $x_{i} \in X_{i}^{k} (t_{i})$ if and only if there is no $x'_{i} \in X_{i}$ such that $u_{i} (x_{i}', x_{j}, t_{i}) > u_{i} (x_{i}, x_{j}, t_{i})$ for all $x_{j} \in ⋃_{t_{j} \in T_{j}} X_{j}^{k - 1} (t_{j})$ . Let $X_{i}^{\infty} (t_{i}) = ⋂_{k = 0}^{\infty} X_{i}^{k} (t_{i})$ denote the set of contributions that survive iterated strict dominance for $t_{i}$ .

I call type $t_{i}$ of player iunwilling to meet the threshold if $(x_{i}, x_{j}) \in Z$ for all $x_{i} \in X_{i}^{\infty} (t_{i})$ and all $x_{j} \in ⋃_{t_{j} \in T_{j}} X_{j}^{\infty} (t_{j})$ . In other words, $t_{i}$ is unwilling to meet the threshold if it is strictly dominated for her to make up the difference between the threshold and any rationalizable contribution by her partner. In the numerical example above, $t_{1}^{H}$ is unwilling to meet the threshold. To see why, notice that $X_{2}^{1} (t_{2}) = [0, \bar{x} (t_{2})] = [0, \frac{1}{4}]$ . We then have $u_{1} (x_{1}, x_{2}, t_{1}^{H}) < u_{1} (0, x_{2}, t_{1}^{H})$ for all $x_{1} \in X_{1} \ {0}$ and $x_{2} \in X_{2}^{1} (t_{2})$ , so $X_{1}^{2} (t_{1}^{H}) = {0}$ and thus $X_{1}^{\infty} (t_{1}^{H}) = {0}$ . This implies $x_{1} + x_{2} < \frac{1}{2}$ for all $x_{1} \in X_{1}^{\infty} (t_{1}^{H})$ and $x_{2} \in X_{2}^{\infty} (t_{2}) \subseteq [0, \frac{1}{4}]$ , so $t_{1}^{H}$ is unwilling to meet the threshold.

If a player is unwilling to meet the threshold, there is no downside to revealing this through cheap talk. In the end, the project will have zero value either way; she is indifferent between whether this occurs with her partner giving nothing or making a wasted contribution. At the same time, if a player is willing to meet the threshold, it is obviously beneficial to reveal this, allaying the other player’s fear of making a contribution that will go to waste. Therefore, as the following proposition states, threshold collective action games have equilibria in which players’ messages reveal whether they are willing to meet the threshold.

Proposition 3. Let Assumption 2 hold. For each player i, let $T_{i}^{U} \subseteq T_{i}$ be the set of types that are unwilling to meet the threshold. For any distinct $m_{i}^{U}, m_{i}^{W} \in M_{i}$ , there exists an equilibrium in which each player’s message reveals whether she is unwilling to meet the threshold

μ_{i} (t_{i}) = {\begin{matrix} m_{i}^{U} & t_{i} \in T_{i}^{U}, \\ m_{i}^{W} & t_{i} \notin T_{i}^{U} \end{matrix}

Along the path of play, in any contribution subgame in which either $m_{i} = m_{i}^{U}$ , both players contribute $x_{i} = 0$ .

A couple of caveats are in order. First, unwillingness to meet the threshold, as I have defined it here, is a strong condition. In order for a type to be unwilling to meet the threshold, any rationalizable contribution under any set of beliefs by that type must result in failure. There may be other equilibria in which high-cost types that do not quite meet this stringent condition separate themselves, but Proposition 3 does not speak to them. Second, the equilibrium described in the proposition need not be influential. Depending on the parameters of the game, including the prior distribution of types, the equilibrium in the subgame that follows both players sending $m_{i}^{W}$ may still entail zero contributions.

Despite these caveats, Proposition 3 captures a substantive difference between continuous and threshold collective action problems. In continuous problems, each player strictly prefers greater contributions by her partner, regardless of her own type. There is consequently a major incentive barrier to information revelation through cheap talk, as each player simply prefers to say whatever would induce her partner to give as much as possible. In threshold problems, by contrast, some types—namely, those who value the project so little that they are unwilling to give enough for it to have a chance of completion—are indifferent about how much their partner gives. This indifference enables meaningful communication, which may allow players to coordinate to avoid wasted contributions.

There is more room for communication in collective action problems with known thresholds than in the continuous case. But there are still limits (Costa and Moreira, 2012; Palfrey et al., 2017). In the example above, cheap talk could help avert wasted contributions in equilibrium, but it could not lead to coordination on a socially efficient contribution scheme. Once a player expects the threshold to be met, she once again strictly prefers for her partner to give more, raising the same incentive problems as in the continuous case.

Proposition 3 shows that cheap talk may in equilibrium reveal who is willing or unwilling to meet the threshold. I now characterize a class of threshold problems in which this is essentially all cheap talk can do—there is no coordination beyond whether to contribute at all. To make this analysis tractable, I focus on one-sided incomplete information (player 2’s type is common knowledge). I also must impose the condition that the value of the project strictly increases with contributions once the threshold is met. The following assumption, which is stronger than Assumption 2, formalizes this condition.

Assumption 2S. There exists a set $Z \subseteq X_{1} \times X_{2}$ , with $| Z | > 1$ , such that $p (x_{1}, x_{2}) = 0$ if and only if $(x_{1}, x_{2}) \in Z$ . If $(x_{1}, x_{2}) \notin Z$ , $x'_{1} \geq x_{1}$ , and $x'_{2} > x_{2}$ , then $p (x'_{1}, x'_{2}) > p (x_{1}, x_{2})$ .

Under these conditions, any influential equilibrium takes a very specific form. The player whose type is private information sends a message of ‘unwilling’ or ‘willing’, revealing nothing further about her type. After the unwilling message, both players give nothing. After the willing message, the player whose type is common knowledge makes a contribution that is insufficient to meet the threshold on its own. This is closely related to the finding by Costa and Moreira (2012), who show in a particular threshold game that extending the message space beyond a yes–no binary does not lead to efficiency gains in equilibrium.

Proposition 4. If there is one-sided incomplete information and Assumption 2S holds, then in any influential equilibrium:

Player 2 always contributes $x_{2} = 0$ or $x_{2} = {\tilde{x}}_{2}$ on the path of play, where ${\tilde{x}}_{2} > 0$ and $p (0, {\tilde{x}}_{2}) = 0$ .

On the path of play, if $t_{1}$ ’s message induces player 2 to contribute $x_{2} = 0$ , then $t_{1}$ spends $x_{1} = 0$ .

Communication is more likely to have some effect in a threshold problem than when all contributions have positive marginal value, but it is not a silver bullet. A player will only pass up the chance for her partner to give more if she knows the greater contribution would be worthless anyway.

5. Discussion

The main finding of the analysis is that the possibility for communication to affect collective action outcomes depends critically on the structure of the collective action problem. In a wide class of continuous problems, in which all contributions are valuable, strong incentives to misrepresent hinder effective communication. There is a greater chance of influential communication in threshold problems, in which contributions may have zero marginal value, though even here the scope of coordination brought on by communication may be limited. I now consider the implication of these findings for collective action problems in various arenas of politics.

5.1. International cooperation

An enduring question in the study of international politics is how to facilitate the provision of global public goods. In the anarchical international system, there is no central authority to enforce contracts or mandate cooperation (Waltz, 1979). States’ uncertainty about each other’s willingness to contribute to the common good only exacerbates the free-rider problem (Jervis, 1976; Keohane, 1984). The most prominent global collective action problems, such as reducing carbon emissions and resettling refugees, are continuous rather than threshold problems; every contribution has some value, though perhaps a small one. Therefore, we should expect cheap talk to do relatively little to coordinate states’ efforts in these areas. A possible exception would be international vaccination campaigns, if the threshold number of vaccinations for herd immunity is known in advance.

International organization theorists have focused on the role of international institutions in promoting cooperation (Keohane, 1984; Martin, 1992; Martin and Simmons, 1998). The inefficacy of cheap talk is itself an argument for institutional involvement—if states could resolve uncertainty through talk alone, then they need only use traditional diplomatic channels. By the same token, the role of institutions is not merely to bring states together; the ‘forum effects’ of institutional deliberation on global public goods provision should be minimal.¹⁶ To be most effective, an institution must have a degree of autonomy from its member states (see Abbott and Snidal, 1998), enough to prioritize overall provision over individual members’ distributive concerns. Though it is unlikely any international institution would have the power to implement a binding transfer scheme like those discussed in the mechanism design literature (d’Aspremont and Gérard-Varet, 1979), one could at least enhance provision by independently collecting information and publicizing its findings to its members.

5.2. International conflict

There is a long tradition of modeling military coalitions as collective action problems (Olson and Zeckhauser, 1966). When nations share security goals, the military effort of one works to the benefit of all. Once again, these are typically continuous problems—even if there is a discrete goal, such as destroying a particular weapons program, the exact amount of military effort required to accomplish it usually is unknown in advance. Success is a probabilistic function of contributions, with additional effort increasing the likelihood of victory at the margin.

A common assumption in the alliance literature is that partners freely share information with one another (Bearce et al., 2006; Konrad, 2012). My findings cast doubt on this assumption, or at least they qualify the mechanism by which it operates. Shared interest in military success is not a sufficient condition for allies (or coalition partners more generally) to reveal information that will aid the effort. The Libyan crisis of 2011, in which confusion prevailed over how much NATO members were willing to contribute and who would lead the joint effort, is one example (Michaels, 2013). If military alliances facilitate information sharing, they must do so through some means separate from ordinary diplomatic communication.

5.3. Revolt and regime change

Revolt is a classic example of a collective action problem (Tilly, 1978). The model results show that private information about willingness to partake in revolt may be a major impediment to revolutionary activity, even if citizens are allowed to communicate freely with each other. If individuals’ efforts in revolt are complementary—i.e. one’s own incentive to participate increases with the number of others involved—then there is an incentive to overstate one’s own likelihood of participation (Barberà and Jackson, 2016). Credible signaling might require costly early action by revolutionary entrepreneurs (Bueno de Mesquita, 2010).

However, two factors might enable credible communication among revolutionaries. The first is if potential participants have prosocial preferences, meaning they at least partially internalize the costs of others. In the model here, where players only internalize their own costs, there is no downside to fooling one’s partner into expending effort that just barely increases the chance of success. But if a player prefers that her partner not expend costs for little benefit, the incentive to overstate disappears. The second exception is when a certain player’s participation is strictly necessary for the movement to have any chance of success, in which case the threshold model applies. Here a coup might be a better example than a mass revolution.¹⁷ If the effort cannot succeed without the involvement of a particular official, and that official does not intend to get involved, it is (weakly) in her interest to say so.

5.4. Lobbying and campaign finance

Lobbying entails spending time and money in hopes of having favorable policies enacted (Tullock, 1980). When multiple interest groups share a policy goal, their lobbying efforts are a kind of collective action problem. Typically this is a continuous problem—lobbying increases the chance of a favorable outcome, but there is no exact threshold.¹⁸ Therefore, according to the results of the model, even if special interest groups have identical policy goals, we should not expect coordination among them to be seamless in the presence of incomplete information. To be able to coordinate effectively, groups must have some means besides cheap talk of signaling their priorities and abilities, or they must be able to make binding commitments to each other. We should expect informational impediments to lobbying coordination to be most severe when legislation of interest arises suddenly, preventing signaling through sequential contributions (Barbieri, 2012), or when issues create temporary alliances between groups, in which case reputational concerns (Sartori, 2002) cannot support honest communication.

Like lobbying, campaign spending entails exerting costly effort in the service of a particular political goal (Meirowitz, 2008). In contemporary American politics, multiple independent actors are involved in the finance of major campaigns—the candidates themselves, the parties that support them, and outside groups like super political action committees (PACs). An important question is whether rules preventing candidates and super (PACs) from explicit coordination have any electoral effect. This analysis suggests one way that they may. If candidates and outside groups cannot engage in the kinds of activities that reduce informational asymmetries within an organization, cheap talk will not help make up the gap. Despite their joint electoral interest with a super PAC, candidates have an incentive to downplay or exaggerate their own fundraising ability, so as to encourage the outside group to take up more of the burden. Fundraising disclosure rules, by providing a verifiable source of information, might alleviate the coordination problem between candidates and outside groups by creating common knowledge of ability and willingness to raise money. The transparency benefits of such rules may thus come at the cost of undermining the independence of candidates’ and outside groups’ expenditures.

6. Conclusion

I have shown that the possibility of influential cheap talk in collective action problems has an important relationship with the shape of the social production function. When all contributions have positive marginal value, each player always wants her partner to give more, creating an incentive problem for cheap talk communication. By contrast, in threshold problems, some players may be indifferent at the margin about others’ contributions. This indifference critically supports the existence of equilibria with influential communication.

In a sense, my analysis of the continuous case mirrors a longstanding conventional wisdom on the efficacy of communication in collective action. In the classic collective action setting, with complete information among participants, it is well known that communication has no strategic effect (Ostrom, 1998).¹⁹ Communication cannot eliminate the incentive to free-ride, which is the core problem of collective action. My analysis of collective action with incomplete information shows that we can take the conventional wisdom further: in a broad set of circumstances, communication also fails to eliminate the auxiliary failures that arise due to uncertainty. If potential contributors do not fully understand their partners’ incentives—how highly they value the project, how costly it is for them to contribute—then they cannot even coordinate on the contribution scheme that would prevail under complete information. In the class of continuous problems I study, cheap talk communication cannot create the common knowledge of preferences that would be necessary for this kind of coordination. In threshold problems, communication may help contributors avoid wasting their effort on hopeless projects, but beyond that it is unlikely to lead to a socially optimal division of labor.

Footnotes

Appendix

Acknowledgements

I thank Brett Benson, Allison Carnegie, Rob Carroll, Josh Clinton, Georgy Egorov, Mark Fey, Hein Goemans, Shawn Ramirez, Kris Ramsay, Scott Tyson, Alan Wiseman, and participants at the Formal Models of International Relations conference (University of Southern California, 2016) for helpful comments.

Author’s Note:

This paper is based on my dissertation chapter “Communication between Allies”.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by a Research Scholar Grant from Vanderbilt University and was completed in part while I was in residence at the Wallis Institute of Political Economy at the University of Rochester.

Notes

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