Discrete public goods under threshold uncertainty

A discrete public good is provided when total contributions exceed the contribution threshold, yet the threshold is often not known with certainty. I show that the relationship between the degree of threshold uncertainty and equilibrium contributions and welfare is not monotonic. For a large class of threshold probability distributions, equilibrium contributions will be higher under increased uncertainty (e.g., a mean-preserving spread) if the public good's value is sufficiently high. Otherwise, and if another condition on the distribution's mode is met, contributions will be lower. The same result also obtains if a single-crossing condition of the pdfs is met.     

SEQUENTIAL TWO‐PLAYER GAMES WITH AMBIGUITY*

If players' beliefs are strictly nonadditive, the Dempster–Shafer updating rule can be used to define beliefs off the equilibrium path. We define an equilibrium concept in sequential two‐person games where players update their beliefs with the Dempster–Shafer updating rule. We show that in the limit as uncertainty tends to zero, our equilibrium approximates Bayesian Nash equilibrium. We argue that our equilibrium can be used to define a refinement of Bayesian Nash equilibrium by imposing context‐dependent constraints on beliefs under uncertainty.     

Contests with Alternative Public‐Good Prizes

I study contests in which a society of players compete, by expending irreversible effort, over which one of alternative prizes should be awarded to them by the decision maker. The prizes are public goods and/or public bads for the players. The players choose their effort levels simultaneously and independently. I define each player's valuation spread as the difference between his valuations for the two public‐good/public‐bad prizes. I establish that the players' equilibrium effort levels depend solely on their valuation spreads, and that the players never expend positive effort for both prizes in equilibrium. Further, I establish that in equilibrium only players with the widest positive valuation spread and players with the widest negative valuation spread expend positive effort. Finally, I establish that the equilibrium effort level expended for each prize and the equilibrium total effort level are determined only by the widest positive valuation spread and the widest negative valuation spread.     

Private Information in the BBV Model of Public Goods

To analyze the private provision of a public good in the presence of private information, we explore the connections between two frameworks: the binary public good model with threshold uncertainty and the standard continuous model à la Bergstrom et al. Linearity of best responses in others' contributions is key to matching the two frameworks. We identify all utility functions that display this linearity, and we provide conditions ensuring that the minimal properties that Bergstrom et al. require for utilities are satisfied. Using techniques developed in the threshold uncertainty framework, we show existence and uniqueness of the Bayes‐Nash equilibrium—thus generalizing existing results—and we analyze its comparative statics properties. In particular, under the reasonable assumption that agents' income is stochastic and private information, we complement the full‐information crowding‐out and redistribution results of Bergstrom et al. If the government taxes agents' income proportionally and redistributes (expected) revenues lump sum, equilibrium public good provision can increase or decrease, even if the set of contributors is unchanged. Similarly, we show that crowding‐out can be one‐for‐one, less than one‐for‐one, or more than one‐for‐one. Finally, we extend our results to a multidimensional framework in which agents' unit costs of contributions are also private information.     

Round‐Robin Tournaments with a Dominant Player

We analyze the subgame perfect equilibrium of the round‐robin tournament with one strong (dominant) and two weak players, and we compare this tournament and the one‐stage contest with respect to the players' expected payoffs, expected total effort, and their probabilities of winning. We find that if the contest designer's goal is to maximize the players' expected total effort, then – if the asymmetry between the players is relatively low – the one‐stage contest should be used. However, if the asymmetry is relatively high, then the round‐robin tournament should be used.     

Contribution to a public good under subjective uncertainty

This paper examines how voluntary contributions to a public good are affected by the contributors' heterogeneity in beliefs about the uncertain impact of their contributions. It assumes that contributors have Savagian preferences that are represented by a two‐state‐dependent expected utility function and different beliefs about the benefit that will result from the sum of their contributions. We establish general comparative statics results regarding the effect of specific changes in the distribution of beliefs on the (unique) Nash equilibrium provision of the public good, under certain conditions imposed on the preferences. We specifically show that the equilibrium public good provision is increasing with respect to both first‐ and second‐order stochastic dominance changes in the distribution of beliefs. Hence, increasing the contributors' optimism about the uncertain benefit of their contributions increases aggregate public good provision, as does any homogenization of these beliefs around their mean.     

Speculative Attacks with Multiple Sources of Public Information*

We propose a speculative attack model in which agents receive multiple public signals. Diverse pieces of public information can be taken into account differently by different players and are likely to lead to different appreciations ex post. This process defines players' expected private values of a successful attack. The main result shows that equilibrium uniqueness depends on two conditions: (i) signals are sufficiently dispersed and (ii) private beliefs about signals' relative precision differ sufficiently. We derive some implications for information dissemination policy. Transparency in this context is multidimensional: it concerns the publicity of announcements, the number of signals disclosed and their precision.     

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.     

Asymptotic Efficiency for Discriminatory Private Value Auctions

We consider discriminatory auctions for multiple identical units of a good. Players have private values, possibly for multiple units. None of the usual assumptions about symmetry of players' distributions over values or symmetry of equilibrium play are made. Because of this, equilibria will typically involve inefficient allocations. Equilibria also become very difficult to solve for. Using an approach which does not depend on explicit equilibrium calculations we show that such auctions become arbitrarily close to efficient as the number of players, and possibly the number of objects, grows large, and provide a simple characterization of limit equilibria.     

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.     

Bargaining foundation for ratio equilibrium in public‐good economies

We provide a bargaining foundation for the concept of ratio equilibrium in public‐good economies. We define a bargaining game of alternating offers, in which players bargain to determine their cost shares of public‐good production and a level of public good. We study the stationary subgame perfect equilibrium (SSPE) without delay of the bargaining game. We demonstrate that when the players are perfectly patient, they are indifferent between the equilibrium offers of all players. We also show that every SSPE without delay in which the ratios offered by all players are the same leads to a ratio equilibrium. In addition, we demonstrate that all equilibrium ratios are offered by the players at some SSPE without delay. We use these results to discuss the case when the assumption of perfectly patient players is relaxed and the cost of delay vanishes.     

Reciprocal relationships and mechanism design

We study an incomplete information game in which players can coordinate their actions by contracting among themselves. We model this relationship as a reciprocal contracting procedure where each player has the ability to make commitments contingent on the other players' commitments. We differ from the rest of the literature on reciprocal contracting by assuming that punishments cannot be enforced in the event that cooperation breaks down. We fully characterize the outcomes that can be supported as perfect Bayesian equilibrium outcomes in such an environment. We use our characterization to show that the set of supportable outcomes with reciprocal contracting is larger than the set of outcomes available in a centralized mechanism design environment in which the mechanism designer is constrained by his inability to enforce punishments against non‐participants. The difference stems from the players' ability in our contracting game to convey partial information about their types at the time they offer contracts. We discuss the implications of our analysis for modelling collusion between multiple agents interacting with the same principal.     

Optimal contracting with altruism and reciprocity

Motivated by the recent experimental evidence on altruistic behavior, we study a simple principal–agent model where each player cares about other players' utility, and may reciprocate their attitude towards him. We show that, relative to the selfish benchmark, efficiency improves when players are altruistic. Nevertheless, in contrast to what may be expected, an increase in the degree of the agent's altruism as well as a more reciprocal behavior by players has ambiguous effects on efficiency. We also consider the effects of the presence of spiteful players and discuss how monetary transfers between players depend on their degrees of altruism and spitefulness.     

Beliefs and Public Good Provision with Anonymous Contributors

We analyze a static game of public good contributions where finitely many anonymous players have heterogeneous preferences about the public good and heterogeneous beliefs about the distribution of preferences. In the unique symmetric equilibrium, the only individuals who make positive contributions are those who most value the public good and who are also the most pessimistic; that is, according to their beliefs, the proportion of players who most like the public good is smaller than it would be according to any other possible belief. We predict whether the aggregate contribution is larger or smaller than it would be in an analogous game with complete information and heterogeneous preferences, by comparing the beliefs of contributors with the true distribution of preferences. A trade‐off between preferences and beliefs arises if there is no individual who simultaneously has the highest preference type and the most pessimistic belief. In this case, there is a symmetric equilibrium, and multiple symmetric equilibria occur only if there are more than two preference types.     

The Private Provision of Public Goods under Uncertainty: A Symmetric-Equilibrium Approach

Various studies have examined whether increased uncertainty about the non‐Nash response of others to an individual's voluntary contribution to a public good affects that individual's contribution so as to mitigate the free‐rider problem. We extend this single‐agent approach to the analysis of a symmetric equilibrium. We provide conditions on group size and endogenous relative risk aversion that imply increased equilibrium contributions in response to greater uncertainty about the productivity of each individual's contribution to the actual level of the public good. These results enable us to broaden the circumstances in which the theory predicts that increased uncertainty reduces free riding.     

INTIMIDATION: LINKING NEGOTIATION AND CONFLICT

A challenger wants a resource initially held by a defender, who can negotiate a settlement by offering to share the resource. If Challenger rejects, conflict ensues. During conflict, each player could be a tough type for whom fighting is costless. Therefore, nonconcession intimidates the opponent into conceding. Unlike in models where negotiations happen in the shadow of exogenously specified conflicts, offers made during negotiations determine how conflict unfolds if negotiations fail. In turn, how conflict is expected to unfold determines the players' negotiating positions. In equilibrium, negotiations always fail with positive probability, even if players face a high cost of conflict. Allowing multiple offers leads to brinkmanship—the only acceptable offer is the one made when conflict is imminent. If negotiations fail, conflict is prolonged and not duration dependent.     

Interlinkage and Generous Tit‐for‐Tat Strategy

We investigate an infinitely‐repeated prisoners' dilemma with imperfect monitoring and consider the possibility that the interlinkage of the players' distinct activities enhances implicit collusion. We show a necessary and sufficient condition for the existence of a generous tit‐for‐tat Nash equilibrium. Such an equilibrium, if it exists, is unique. This equilibrium achieves approximate efficiency when monitoring is almost perfect, where the discount factors are fixed.     

Continuous time limits of repeated games with imperfect public monitoring

In a repeated game with imperfect public information, the set of equilibria depends on the way that the distribution of public signals varies with the players' actions. Recent research has focused on the case of “frequent monitoring,” where the time interval between periods becomes small. Here we study a simple example of a commitment game with a long-run and short-run player in order to examine different specifications of how the signal distribution depends upon period length. We give a simple criterion for the existence of efficient equilibrium, and show that the efficiency of the equilibria that can be supported depends in an important way on the effect of the player's actions on the variance of the signals, and whether extreme values of the signals are “bad news” of “cheating” behavior, or “good news” of “cooperative” behavior.     

Teaching Nash Equilibrium and Dominance: A Classroom Experiment on the Beauty Contest

Abstract: The authors' aim in this article was to show how the use of classroom experiments may be a good pedagogical tool to teach the Nash equilibrium (NE) concept. The basic game is a version of the beauty contest game (BCG), a simple guessing game in which repetition lets students react to other players' choices and converge iteratively to the equilibrium solution. The authors perform this experiment with undergraduate students with no previous training in game theory. After four rounds, they observe a clear decreasing tendency in the average submitted number in all groups. Thus, the findings show that by playing a repeated BCG, students quickly learn how to reach the NE solution.     

On Bargaining for an Indivisible Good

This note studies contracts involving lotteries for the purpose of trading an indivisible good. Lottery contracts allow players to reach agreements for trading the good even if the reservation price of the potential buyer is below that of the potential seller. When certainty exchanges are possible, lottery contracts often dominate them. An interesting feature of the contracts is that the probability of trading the good depends on the players' bargaining powers, but the payments do not.Journal of Economic LiteratureClassification Numbers: C78.