Free riders and the optimal prize in public‐good funding lotteries

We prove the existence and uniqueness of an equilibrium in a game where players, whose preferences exhibit constant absolute risk aversion or constant relative risk aversion, contribute to a public good via lottery‐ticket purchases. Contrasting models with risk neutrality, we show that an equilibrium with a strictly positive amount of the public good may not exist without a sufficient number of participants who are not too risk‐averse. We show that players who are more risk‐averse purchase fewer lottery tickets and are more likely to free ride in equilibrium. In fact, it is possible for free riders to place a larger value on the public good than do those who contribute. In a symmetric equilibrium, we show that an upper bound exists for the amount of the public good, even though there are infinitely many participants. Furthermore, we derive a lottery prize that maximizes the amount of the public good in a symmetric equilibrium and find that such a prize always results in an overprovision of the public good.     

Social equilibrium and cycles on compact sets

One proof of existence of general equilibrium assumes convexity and continuity of a preference correspondence on a compact convex feasible set W. Here the existence of a local equilibrium for a preference field which satisfies, not convexity, but the weaker local acyclicity is shown. The theorem is then applied to a voting game, σ, without veto players. It is shown that if the dimension of the policy space is no greater than ν(σ) ? 2, where ν(σ) is the Nakamura number of the game, then no local cycles may occur and a local equilibrium must exist. With convex preferences, then, there will exist a choice of the game from W.     

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.     

Providing public goods in the absence of strong institutions

This paper proposes a simple two-stage mechanism to establish positive contributions to public goods in the absence of powerful institutions to provide the public good and to sanction free-riders. In this mechanism players commit to the public good by paying a deposit prior to the contribution stage. If there is universal commitment, deposits are immediately refunded whenever a player contributes her specified share to the public good. If there is no universal commitment, all deposits are refunded and the standard game is played. For suitable deposits, prior commitment and full ex post contributions are supported as a subgame-perfect Nash equilibrium for the resulting game. As the mechanism obviates the need for any ex post prosecution of free-riders, it is particularly suited for situations where players do not submit to a common authority as in the case of international agreements.     

Co-evolution of Preferences and Information in Simple Games of Trust

In standard rational choice modelling decisions are made according to given information and preferences. In the model presented here the `information technology' of individual decision-makers as well as their preferences evolve in a dynamic process. In this process decisions are made rationally by players who differ in their informational as well as in their preference type. Relative success of alternative decisions feeds back on the type composition of the population which in turn influences rational decision-making. An indirect evolutionary analysis of an elementary yet important basic game of trust shows that under certain parameter constellations the population dynamics of the evolutionary process specify a unique completely mixed rest point. However, as opposed to previous studies of preference formation in the game of trust there is no convergence to but only circumventing the rest point if the informational status of individuals evolves rather than being chosen strategically.     

Evolutionary stability of discrimination under observability

We study the evolution of preferences under perfect and almost perfect observability in symmetric 2-player games. We demonstrate that if nature can choose from a sufficiently general preference space, which includes preferences over outcomes that may depend on the opponent's preference-type, then, in most games, only discriminating preferences (treating different types of opponents differently in the same situation) can be evolutionary stable and some discriminating types are stable in a very strong sense in all games. We use these discriminating types to show that any symmetric outcome which gives players more than their minmax value in material payoffs (fitness) can be seen as equilibrium play of a player population with such strongly stable preferences.     

Evolutionary stability of discrimination under observability

We study the evolution of preferences under perfect and almost perfect observability in symmetric 2-player games. We demonstrate that if nature can choose from a sufficiently general preference space, which includes preferences over outcomes that may depend on the opponent's preference-type, then, in most games, only discriminating preferences (treating different types of opponents differently in the same situation) can be evolutionary stable and some discriminating types are stable in a very strong sense in all games. We use these discriminating types to show that any symmetric outcome which gives players more than their minmax value in material payoffs (fitness) can be seen as equilibrium play of a player population with such strongly stable preferences.     

Bounding preference parameters under different assumptions about beliefs: a partial identification approach

We show how bounds around preferences parameters can be estimated under various levels of assumptions concerning the beliefs of senders in the investment game. We contrast these bounds with point estimates of the preference parameters obtained using non-incentivized subjective belief data. Our point estimates suggest that expected responses and social preferences both play a significant role in determining investment in the game. Moreover, these point estimates fall within our most reasonable bounds. This suggests that credible inferences can be obtained using non-incentivized beliefs.     

Imitation and the Incentive to Contribute Early in a Sequential Public Good Game

Whether motivated by reciprocity or conformity, imitation is common in public good contexts. We consider the incentive for an agent to contribute to a public good if he expects imitation from others. Using a sequential public good game with exogenous ordering, we show that agents early enough in the sequence who believe imitation to be sufficiently likely would want to contribute. By contributing, they expect total contributions to increase significantly. We also show that preferences determine how early an agent need be, that the observed share of imitators in experiments is sufficiently high to warrant contribution and that an increase in group size reduces the incentive to contribute.     

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.     

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.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.     

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.     

Heterogeneity,impatience, and dynamic private provision of a discrete public good

We analyze dynamic private provision of a discrete public good by heterogeneous agents, who differ in terms of their levels of impatience, in a differential game framework. In contrast to the strategic complementarity result for homogeneous individuals, we show that an asymmetric completion Markov perfect equilibrium exists, where the individual contributions and the strategic behaviors depend crucially on an impatience differential: the difference in rates of time preference across groups of individuals. When this differential is insignificant, contributions of both types of individuals are strategic complements. On the other hand, when this impatience differential exceeds a threshold, the contributions of impatient individuals become strategic substitutes, whereas the contributions of patient individuals remain strategic complements. We show that group size has an interesting role to play in the strategic behavior: increasing the number of patient individuals aggravates the incentives to free‐ride by the impatient agents. We also derive a condition under which all the socially beneficial projects get completed in the equilibrium.     

Stated beliefs versus inferred beliefs: A methodological inquiry and experimental test

Belief elicitation in game experiments may be problematic if it changes game play. We experimentally verify that belief elicitation can alter paths of play in a two-player repeated asymmetric matching pennies game. Importantly, this effect occurs only during early periods and only for players with strongly asymmetric payoffs, consistent with a cognitive/affective effect on priors that may serve as a substitute for experience. These effects occur with a common scoring rule elicitation procedure, but not with simpler (unmotivated) statements of expected choices of opponents. Scoring rule belief elicitation improves the goodness of fit of structural models of belief learning, and prior beliefs implied by such models are both stronger and more realistic when beliefs are elicited than when they are not. We also find that “inferred beliefs” (beliefs estimated from past observed actions of opponents) can predict observed actions better than the “stated beliefs” from scoring rule belief elicitation.     

Robustness of equilibria in anonymous local games

This paper studies the robustness of symmetric equilibria in anonymous local games to perturbations of prior beliefs. Two priors are strategically close on a class of games if players receive similar expected payoffs in equilibrium under the priors, for any game in that class. I show that if the structure of payoff interdependencies is sparse in a well-defined sense, the conditions for strategic proximity in anonymous local games are strictly weaker than the conditions for general Bayesian games of Kajii and Morris (1998) [11] when attention is restricted to symmetric equilibria. Hence, by exploiting the properties of anonymous local games, it is possible to obtain stronger robustness results for this class.     

The effects of (incentivized) belief elicitation in public goods experiments

Belief elicitation is an important methodological issue for experimental economists. There are two generic questions: 1) Do incentives increase belief accuracy? 2) Are there interaction effects of beliefs and decisions? We investigate these questions in the case of finitely repeated public goods experiments. We find that belief accuracy is significantly higher when beliefs are incentivized. The relationship between contributions and beliefs is slightly steeper under incentives. However, we find that incentivized beliefs tend to lead to higher contribution levels than either non-incentivized beliefs or no beliefs at all. We discuss the implications of our results for the design of public good experiments.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

Rational conjectures and evolutionary beliefs in public goods games

This paper applies conjectural variations (CVs) to a model of public good provision and shows that CVs are superior to Nash beliefs. In addition to imposing consistency, as Bresnahan, I show that consistent conjectures (CCs) are obtained from individual payoff maximization. CCs emerge as the unique subgame perfect Nash equilibrium (NE) in a two-stage game in which beliefs are chosen in Stage 1 and quantities in Stage 2. There is an individual payoff advantage to non-Nash behavior, generating a Prisoner's Dilemma in conjectures in addition to the usual free-rider problem associated with public goods. The correct and payoff maximizing conjecture is the unique equilibrium in an evolutionary framework against a player with Nash conjectures. The consistent conjecture equilibrium is the unique evolutionary equilibrium when both players conjectures evolve. Hence, the NE prediction is too optimistic when players have rational conjectures.     

Strategic budgets in sequential elimination contests

We study a sequential elimination contest where contestants (e.g., campaigns) spend resources that are provided by strategic players called backers (e.g., donors). In the unique symmetric equilibrium, backers initially provide small budgets, increasing their contributions only if their contestant wins the preliminary round. If backers are only allowed to provide budgets at the start of the game, as opposed to before each round, spending is higher. When unspent resources are refunded to the backer, total spending is higher than when all resources are sunk costs. We also study an extension of the model with asymmetric players.