Efficient Equilibria in the Voluntary Contributions Mechanism with Private Information

We characterize generally the Bayesian Nash equilibria of a voluntary contributions public goods game for two consumers with private information.The two consumers simultaneously make voluntary contributions to the public good, and the contributions are refunded if the total falls short of the cost of the public good. Several families of equilibria (step‐function, regular and semi‐regular) are studied. Necessary and sufficient conditions for regular and semi‐regular equilibrium allocations to be interim incentive efficient are derived. In the uniform distribution case we prove (i) the existence of an open set of incentive efficient regular equilibria when the cost of production is large enough and (ii) the existence of an open set of incentive efficient semi‐regular equilibria when the cost of production is low enough. Step‐function equilibra are proved to be interim incentive inefficient.     

Private provision of a discrete public good: efficient equilibria in the private-information contribution game

We study the voluntary provision of a discrete public good via the contribution game. Players independently and simultaneously make nonrefundable contributions to fund a discrete public good, which is provided if and only if contributions cover the cost of production. We characterize nonconstant continuous symmetric equilibria, giving sufficient conditions for their existence. We show the common normalization by which players’ values are distributed over [0, 1] is not without loss of generality: if the distribution over this interval has continuous density f with f(0) >  0, then no (nonconstant) continuous symmetric equilibrium exists. We study in detail the case in which players’ private values are uniformly distributed, showing that, generically, when one continuous equilibrium exists, a continuum of continuous equilibria exists. For any given cost of the good, multiple continuous equilibria cannot be Pareto ranked. Nevertheless, not all continuous equilibria are interim incentive efficient. The set of interim incentive efficient equilibria is exactly determined. The authors thank Manfred Dix, George Mailath, Andrew Postlewaite, and an anonymous referee for their comments.     

An Experimental Investigation of Excludable Public Goods

This paper extends the research on incentive compatible institutions for the provision of public goods by imposing a minimum contribution that must be met in order for an individual to enjoy the benefits of the public good. Excluding individuals who do not contribute at least the minimum transforms the linear n-player pure public goods game to an n-player coordination game with multiple, Pareto-ranked Nash equilibria. The experimental results show that exclusion increases contributions to the public good in most cases. However, an increase in contributions may not be sufficient to increase social welfare because there is a welfare cost to excluding individuals when the good is non-rival. Furthermore, exclusion can decrease both contributions and welfare in environments in which individuals fail to coordinate their contributions. The results are sensitive to the minimum contribution requirement and to the relative returns from the public and private alternatives.     

Participation constraints of matching mechanisms

Matching mechanisms have been proposed to improve public good provision in voluntary contributions. However, such decentralized subsidizing mechanisms may not be Pareto‐improving and may suffer from incomplete information and incredible commitment. This paper examines participation constraints of matching mechanisms with small matching rates in two cases of equilibria. At interior equilibria, there always exist small Pareto‐improving matching schemes regardless of preferences. This universal existence is useful for cooperation among heterogeneous players in the context without global information of preferences or at the international level without central governments. At corner equilibria, matching schemes work in different ways and have distinct welfare effects in certain cases, and the existence of Pareto‐improving matching schemes is not universal but is possible under certain conditions. The paper further characterizes Pareto‐improving matching schemes, and shows that it is easier to reach Pareto‐improving matching outcomes if players value more on public goods and have stronger substitution between private and public goods.     

Weak links, good shots and other public good games: Building on BBV

We suggest an alternative way of analyzing the canonical Bergstrom-Blume-Varian model of non-cooperative voluntary contributions to a public good that avoids the proliferation of dimensions as the number of players is increased. We exploit this approach to analyze models in which the aggregate level of public good is determined as a more general social composition function of individual gifts — specifically, as a CES form — rather than as an unweighted sum. We also analyze Hirshleifer's weakest-link and best-shot models. In each case, we characterize the set of equilibria, in some cases establishing existence of a unique equilibrium as well as briefly pointing out some interesting comparative static properties. We also study the weakest-link and best-shot limits of the CES composition function and show how the former can be used for equilibrium selection and the latter to establish that equilibria of some better-shot games are identical to those of the much simpler best-shot game.     

Commitment Robust Equilibria and Endogenous Timing

This paper investigates which equilibria of a game are still viable when players have the opportunity to commit themselves. To that end we study a model of endogenous timing in which players face the trade-off between committing early and moving late. It is shown that mixed (resp. pure) equilibria of the original game are subgame perfect (resp. persistent) in the timing game only when no player has an incentive to move first. Consequently, mixed equilibria are viable only if no player has an incentive to move first. One needs strong evolutionary solution concepts to draw that conclusion for pure equilibria.Journal of Economic LiteratureClassification Number: C72.     

When Does Everyone Contribute in the Private Provision of Local Public Goods?

When does everyone genuinely contribute in the private provision of a local public good? We first introduce a monotonic condition to characterize the relationship between the structure of the network that underlie the noncooperative game of private provision of local public goods on the one hand, and the preferences of the agents on the other, showing that the monotonic condition is a sufficient and necessary condition of existence of a distributed Nash equilibrium (DNE) in which each agent exerts a positive amount of effort to provision of the public good (Theorem 1). We then study the number of equilibria, and, by using the monotonic condition, characterize the condition under which the DNE set is a singleton, a continuum, or null (Theorem 2). As it turns out, the structure of the network and the agents’ preferences jointly shape the effort profile in the provision of local public goods.     

Participation and the provision of discrete public goods: a strategic analysis

This paper considers the Nash equilibria to a game where a discrete public good is to be provided. Each individual may participate by making a fixed contribution. If a sufficient number of contributions are made, the good is provided. Otherwise, the good is not provided. One variant of the rules allows for contributions to be refunded when the good is not provided. For pure strategies, we find that the Nash equilibria with a refund are a superset of those without a refund. For both rules, the efficient number of players contributing is an equilibrium. For mixed strategies, to every equilibrium without a refund there is a corresponding equilibrium with a refund with a higher number of expected contributors. Mixed strategy equilibria ‘disappear’ as the number of players grows large. Some results reported in the experimental literature are discussed in light of these theoretical results.     

Subgame Perfect Equilibria in Stage Games

It is well known that a stage game with infinite choice-sets, unless it contains a public coordination-device in each stage, may have no subgame perfect equilibria. We show that if a game with public coordination-devices has a subgame perfect equilibrium in which two players in each stage use non-atomic strategies, then the game without coordination devices also has a subgame perfect equilibrium. Journal of Economic Literature Classification Numbers: C6, C7, D8.     

Mixed-strategy equilibria in the Nash Demand Game

In the Nash Demand Game, each of the two players announces the share he demands of an amount of money that may be split between them. If the demands can be satisfied, they are; otherwise, neither player receives any money. This game has many pure-strategy equilibria. This paper characterizes mixed-strategy equilibria. The condition critical for an equilibrium is that players’ sets of possible demands be balanced. Two sets of demands are balanced if each demand in one set can be matched with a demand in the other set such that they sum to one. For Nash’s original game, a complete characterization is given of the equilibria in which both players’ expected payoffs are strictly positive. The findings are applied to the private provision of a discrete public good.     

Group efforts when performance is determined by the “best shot”

We investigate the private provision of a public good whose level is determined by the maximum effort made by a group member. Costs of effort are either commonly known or privately known. For symmetric perfect-information games, any number of players may be active and we characterize the unique (mixed-strategy) equilibrium in which active contributors use the same strategy. Increasing the number of active players leads to stochastically lower individual efforts and level of the public good. When information is private, the symmetric equilibrium is in pure strategies. Increasing the number of players yields a pointwise reduction in the equilibrium contribution strategy but an increase in equilibrium payoffs. Comparative statics with respect to costs and levels of risk aversion are derived. Finally, whether information is public or private, equilibria are inefficient—we provide mechanisms that improve efficiency.     

Cost Uncertainty and Unilateral Abatement

Unilateral abatement is sometimes advocated in order to set a good example that will make other countries follow. The aim of this paper is to investigate whether existence of correlated cost uncertainty provides an incentive for a country to undertake unilateral abatement. The theoretical model is driven by two main mechanisms; first, a learning effect, as the follower country might reduce its risk premium as it can observe the cost level in the leader country. Second, there is the public good effect, i.e., the marginal benefit of abatement declines when abatement is a public good and other countries contribute to pollution reductions. Results shows that unilateral abatement would be efficient in reducing uncertainty about the unit costs of abatement if a country with low cost uncertainty would undertake abatement first, while a country with initially high cost uncertainty would follow. However, countries may prefer to act simultaneously because of the larger uncertainties that are inherent in a sequential game.      

Threshold uncertainty in the private-information subscription game

We introduce threshold uncertainty, à la Nitzan and Romano (1990), into a private-values model of voluntary provision of a discrete public good. Players are allowed to make any level of contribution toward funding the good, which is provided if the cost threshold is reached. Otherwise, contributions are refunded. Conditions ensuring existence and uniqueness of a Bayesian equilibrium are established. Further restricting the threshold uncertainty to a uniform distribution, we show the equilibrium strategies are very simple, even allowing for any number of players with asymmetric distributions of values. Comparative statics with respect to changes in players' distributions are derived, allowing changes in both the intensity and the dispersion of values. For example, increased uncertainty, in the sense of mean-preserving spreads of players' distributions of values, increases equilibrium contributions. Finally, we show the equilibrium is interim incentive inefficient. The sharpness of our results greatly contrasts with the more qualified insights of earlier private-values models with known cost threshold, which relied on there being two symmetric players and generally exhibited multiple equilibria.     

Voluntary Participation and Provision of Public Goods in Large Finite Economies

We consider a public good provision game with voluntary participation. Agents participating in the game provide a public good and pay the fees according to a mechanism (allocation rule), while nonparticipants can free‐ride on the participants. We examine how the equilibrium public good provision level is affected by enlarging the population of an economy. We introduce a condition for an allocation rule, the asymptotic uniform continuity in replication (AUCR), which requires that small changes in the population must yield only small changes in the public good provision and which is satisfied by many mechanisms. We show that under AUCR, the equilibrium level of the public good converges to zero as the economy is replicated in the sense of Milleron.     

After you—endogenous sequencing in voluntary contribution games

We examine contributions to a public good when some donors do not know the true value of the good. If donors in such an environment determine the sequence of moves, two contribution orders may arise as equilibria. Either the uninformed and informed donors contribute simultaneously or the informed contribute prior to the uninformed. Sequential moves result in a larger provision of the public good, because the follower mimics the action of the leader, and in accounting for this response the leader chooses to contribute when it is efficient to do so. An experimental investigation of the game shows that the donors predominantly choose to contribute sequentially, and that the resulting contributions are larger than those of the simultaneous-move game. Although the gain from sequential moves is smaller when the sequence is set exogenously, our results suggest that the involved parties would benefit from having sequential moves imposed upon them.     

When Does One Bad Apple Spoil the Barrel? An Evolutionary Analysis of Collective Action

This paper studies collective-action games in which the production of a public good requires teamwork. A leading example is a threshold game in which provision requires the voluntary participation of m out of n players. Quantal-response strategy revisions allow play to move between equilibria in which a team successfully provides, and an equilibrium in which the collective action fails. A full characterization of long-run play reveals the determinants of success; these include the correlation between players' costs of provision and their valuations for the good. The addition of an extra "bad apple" player can "spoil the barrel" by destabilizing successful teams and so offers a rationale for limiting the pool of possible contributors.     

Endogenous Move Structure and Voluntary Provision of Public Goods: Theory and Experiment

In this paper, we examine voluntary contributions to a public good, embedding Varian's (1994) voluntary contribution game in extended games that allow players to choose the timing of their contributions. We show that predicted outcomes are sensitive to the structure of the extended game, and also to the extent to which players care about payoff inequalities. We then report a laboratory experiment based on these extended games. We find that behavior is similar in the two extended games: subjects avoid the detrimental move order of Varian's model, where a person with a high value of the public good commits to a low contribution, and instead players tend to delay contributions. These results suggest that commitment opportunities may be less damaging to public good provision than previously thought.     

The effects of costless pre-play communication: Experimental evidence from games with Pareto-ranked equilibria

Cheap talk is shown to facilitate coordination on the unique efficient equilibrium in experimental order-statistic games. This result is roughly consistent with theoretical predictions according to which cheap talk promotes efficient Nash play. The evidence concerning the mechanisms that theory appeals to is mixed: Frequent agreement of messages and actions is consistent with messages being viewed as self-committing. Risk in the underlying game and the absence of self-signaling messages may explain why message profiles are not unanimous. Time-varying message profiles can be interpreted as evidence for players trying to negotiate equilibria and/or trying to rely on secret handshakes.     

On the formation of interaction networks in social coordination games

There are many situations where two interacting individuals can benefit from coordinating their actions. We examine the endogenous choice of partners in such social coordination games and the implications for resulting play. We model the interaction pattern as a network where individuals periodically have the discretion to add or sever links to other players. With such endogenous interaction patterns we see multiple stochastically stable states of play, including some that involve play of equilibria in the coordination game that are neither efficient nor risk-dominant. Thus the endogenous network structure not only has implications for the interaction pattern that emerges, but it also has a significant impact on the play in the coordination game relative to what would arise if the same interaction network were exogenous.