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.     

A decomposition algorithm for N-player games

An N-player game can be decomposed by adding a coordinator who interacts bilaterally with each player. The coordinator proposes profiles of strategies to the players, and his payoff is maximized when players’ optimal replies agree with his proposal. When the feasible set of proposals is finite, a solution of an associated linear complementarity problem yields an equilibrium of the approximate game and thus an approximate equilibrium of the original game. Computational efficiency is improved by using vertices of a triangulation of the players’ strategy space for the coordinator’s pure strategies. Computational experience is reported.     

A bargaining model in general equilibrium

The goal of the paper is to present a simple model of rational endogenous household formation in a general equilibrium framework in which Pareto optimality at the economy level is not necessarily obtained. The simplest example of household formation is the case in which pairs of individuals engage themselves in a bargaining process on the division of some wealth: if an agreement on the distribution is (not) reached, we can say that the household is (not) formed. The vast majority of existing bargaining models predicts agreements on an efficient outcome. A seminal paper by Crawford (Econometrica 50:607–637, 1982) describes a very simple game with incomplete information in which, even with rational agents, disagreement causes welfare losses. We embed that model in a general equilibrium framework and present some results on equilibria both in the bargaining game and the associated exchange economy. Crawford’s results support Schelling’s intuition on the reasons of disagreement: it may arise if players’ commitments are reversible. Crawford shows that high probabilities of reversibility tend to favor the bargaining impasse, in fact with low probability. We prove that even if those probabilities are arbitrarily close to zero, disagreement is an equilibrium outcome, with high probability. That conclusion seems to be an even stronger support to Schelling’s original viewpoint. In the exchange economy model with that noncooperative bargaining game as a first stage, we present significant examples of economies for which equilibria exist. Because of disagreement, Pareto suboptimal exchange economy equilibria exist for all elements in the utility function and endowment spaces and they may coexist with Pareto optimal equilibria even at the same competitive prices.     

Competition and coordination in experimental minority games

This work presents experimental results on a coordination game in which agents must repeatedly choose between two sides, and a positive fixed payoff is assigned only to agents who pick the minoritarian side. We conduct laboratory experiments in which stationary groups of five players play the game for 100 periods, and manipulate two treatment variables: the amount of information about other players’ past choices and the salience of information regarding the game history (i.e., the length of the string of past outcomes that players can see on the screen while choosing). Our main findings can be summarized as follows: aggregate efficiency in the game is in most cases significantly higher than the level corresponding to the symmetric mixed strategy Nash equilibrium. In addition, providing players with information about individual choices in the group does not improve aggregate efficiency with respect to when such information is absent. Displaying information about more rounds than just the previous one, on the other hand, seems to have a positive effect on aggregate efficiency. At the individual level, we find a stronger statistical relation between players’ current choices and their own past choices than between players’ choices and previous aggregate outcomes. In addition, the depth of the relation between present and past choices seems to be affected by the prompt availability of information about the game history. Finally, we detect evidence of a mutual co-adaptation between players’ choices over time that is partly responsible for the high level of efficiency observed.      

Citizen-candidates,lobbies, and strategic campaigning

We study a spatial model of political competition in which potential candidates need a fixed amount of money from lobbies to enter an election. We show that the set of pure strategy Nash equilibria in which lobbies finance candidates whose policies they prefer among the set of entrants coincides with the set of Nash equilibria with weakly less than two entering candidates. Fixing lobbies’ preferences, if the total amount of money held by lobbies is finite, there exists some minimal distance between the two candidates’ positions. This minimal distance is a bound for all such Nash equilibria and is independent of the distribution of voters’ preferences. I would like to thank John Duggan, Al Slivinski, and William Thomson for useful comments and suggestions. Dan Kovenock and two anonymous referees also provided detailed comments and pointed out several errors. All errors are my own.     

From ultimatum to Nash bargaining: Theory and experimental evidence

We consider a sequential two-party bargaining game with uncertain information transmission. When the first mover states her demand she does only know the probability with which the second mover will be informed about it. The informed second mover can either accept or reject the offer and payoffs are determined as in the ultimatum game. Otherwise the uninformed second mover states his own demand and payoffs are determined as in the Nash demand game. In the experiment we vary the commonly known probability of information transmission. Our main finding is that first movers’ and uninformed second movers’ demands adjust to this probability as qualitatively predicted, that is, first movers’ (uninformed second movers’) demands are lower (higher) the lower the probability of information transmission. JEL Classification C72 · C78 · C92     

Coordination of strategic responses to security threats: Laboratory evidence

Laboratory experiments are used to evaluate the extent to which players in games can coordinate investments that diminish the probability of losses due to security breaches or terrorist attacks. In this environment, economically sensible investments may be foregone if their potential benefits are negated by failures to invest in security at other sites. The result is a coordination game with a desirable high-payoff, high-security equilibrium and an undesirable low-security equilibrium that may result if players do not expect others to invest in security. One unique feature of this coordination situation is that investment in security by one player generates a positive externality such that all other players’ expected payoffs are increased, regardless of those other players’ investment decisions. Coordination failures are pervasive in a baseline experiment with simultaneous decisions, but coordination is improved if players are allowed to move in an endogenously determined sequence. In addition, coordinated security investments are observed more often when the largest single security threat to individuals is preventable by their own decisions to invest in security. The security coordination game is a “potential game,” and the success of coordination on the more secure equilibrium is related to the notion of potential function maximization and basin of attraction.      

Supply function equilibria of pay-as-bid auctions

This paper characterizes the Nash equilibrium in a pay-as-bid (discriminatory), divisible-good, procurement auction. Demand by the auctioneer is uncertain as in the supply function equilibrium model. A closed form expression is derived for a one shot game. Existence of an equilibrium is ensured if the hazard rate of the demand distribution is monotonically decreasing with respect to the shock outcome and sellers have non-decreasing marginal costs. Multiple equilibria can be ruled out for markets, for which the auctioneer’s demand exceeds suppliers’ capacity with a positive probability. The derived equilibrium can be used to model strategic bidding behavior in pay-as-bid electricity auctions, such as the balancing mechanism of United Kingdom. Offer curves and mark-ups of the derived equilibrium are compared to results for the SFE of a uniform-price auction.      

A “Super” Folk Theorem for dynastic repeated games

We analyze dynastic repeated games. These are repeated games in which the stage game is played by successive generations of finitely-lived players with dynastic preferences. Each individual has preferences that replicate those of the infinitely-lived players of a standard discounted infinitely-repeated game. Individuals live one period and do not observe the history of play that takes place before their birth, but instead create social memory through private messages received from their immediate predecessors. Under mild conditions, when players are sufficiently patient, all feasible payoff vectors (including those below the minmax of the stage game) can be sustained by sequential equilibria of the dynastic repeated game with private communication. In particular, the result applies to any stage game with n  ≥  4 players for which the standard Folk Theorem yields a payoff set with a non-empty interior. We are also able to characterize fully the conditions under which a sequential equilibrium of the dynastic repeated game can yield a payoff vector not sustainable as a subgame perfect equilibrium of the standard repeated game. For this to be the case it must be that the players’ equilibrium beliefs violate a condition that we term “inter-generational agreement.” A previous version of this paper was circulated as Anderlini et al. (2005). We are grateful to Jeff Ely, Leonardo Felli, Navin Kartik, David Levine, Stephen Morris, Michele Piccione, Andrew Postlewaite, Lones Smith and to seminar audiences at Bocconi, Cambridge, CEPR-Guerzensee, Chicago, Columbia, Edinburgh, Essex, Georgetown, Leicester, LSE, Northwestern, Oxford, Rome (La Sapienza), Rutgers, SAET-Vigo, Stanford, SUNY-Albany, UCL, UC-San Diego, Venice and Yale for helpful feedback.     

Robust rational turnout

We show that equilibria of a class of participation games (Palfrey and Rosenthal in Public Choice 41(1):7–53, 1983; Journal of Public Economics 24(2):171–193, 1984) exhibit minimal heterogeneity of behavior so that players’ mixed strategies are summarized by at most two probabilities. We then establish that, except for a finite set of common costs of participation, these games are regular. Thus, equilibria of these voting games are robust to general payoff perturbations and survive in nearby games of incomplete information. Thanks to participants of the 2006 MPSA conference for comments on an early version.     

Keynesian unemployment as a nash equilibrium with endogenous wage/price setting: An example

A simple 3 good, 1 consumer, 1 firm model of fixed price, quantity constrained equilibrium is developed. A game is then defined on the set of (globally unique) equilibria. The consumer sets the money wage, the firm sets the money price of output (money is numeraire). Nash solutions of the game exist and may involve Keynesian unemployment but never involve Classical unemployment or Repressed inflation.     

Dictator game giving: altruism or artefact?

Experimental dictator games have been used to explore unselfish behaviour. Evidence is presented here, however, that subjects’ generosity can be reversed by allowing them to take a partner’s money. Dictator game giving therefore does not reveal concern for consequences to others existing independently of the environment, as posited in rational choice theory. It may instead be an artefact of experimentation. Alternatively, evaluations of options depend on the composition of the choice set. Implications of these possibilities are explored for experimental methodology and charitable donations respectively. The data favour the artefact interpretation, suggesting that demand characteristics of experimental protocols merit investigation, and that economic analysis should not exclude context-specific social norms.

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Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.      

Bargaining with revoking costs

A simple two stage bilateral bargaining game is analyzed. The players simultaneously demand shares of a unit size pie. If the demands add up to more than one, the players simultaneously choose whether to stick to their demand or accept the other?s offer. While both parties sticking to their offers leads to an impasse, accepting a lower share than the original demand is costly for each party. The set of pure strategy subgame perfect equilibria of the game is characterized for continuously differentiable payoff and cost functions, strictly increasing in the pie share and the amount conceded, respectively. Higher cost functions are shown to improve bargaining power. The limit equilibrium prediction of the model, as the cost functions are made arbitrarily high, selects a unique equilibrium in the Nash Demand Game that corresponds to a Proportional Bargaining Solution of Kalai (1977).     

A multi-agent bilateral bargaining model with endogenous protocol

People do bargain over how to bargain. We examine the role of individuals’ ability to pursue certain bargaining protocols in a multi-agent bilateral bargaining model. Bargaining protocols are not completely settled, but will emerge endogenously in equilibrium. We show that players’ ability to partially influence bargaining protocols plays a crucial role in determining equilibrium outcomes. When discounting is not too high, there are multiple subgame perfect equilibria, including inefficient ones. As the number of players increases, both the set of discount factors that support multiple equilibrium outcomes and the set of the first proposing player’s equilibrium payoffs expand. The maximum loss of efficiency increases with respect to the discount factor. We would like to thank Hongbin Cai, John Conlon, Andrew Daughety, Taiji Furusawa, Byoung Heon Jun, Akira Okada, Ping Wang, and two anonymous referees for their comments and suggestions. Both authors gratefully acknowledge financial support from the Social Sciences and Humanities Research Council of Canada.     

A K% MONEY GROWTH LEADERSHIP RULE IN AN INTERNATIONAL MONETARY POLICY GAME

This paper examines a version of the Friedman k% money growth rule in an open economy monetary policy game. Using the two-country model proposed by Canzoneri and Henderson (1991), we show that, in response to asymmetric aggregate demand shocks, the Pareto-efficient outcome can be achieved by a policy that we call a k% money growth leadership rule. Following that rule, one country, the leader, sets her money supply growth rate, and the follower sets her money supply growth rate so as to keep the sum of nominal money supply growth at k%. We show that this policy yields the same outcome as does cooperative equilibrium. We also show that alternative policy rules, such as keeping exchange-rate adjusted money supply growth at k%, or forming a currency union, will not lead to the Pareto-efficient outcome in response to these demand shocks. ( JEL E5, F3)     

The Colonel Blotto game

In the Colonel Blotto game, two players simultaneously distribute forces across n battlefields. Within each battlefield, the player that allocates the higher level of force wins. The payoff of the game is the proportion of wins on the individual battlefields. An equilibrium of the Colonel Blotto game is a pair of n-variate distributions. This paper characterizes the unique equilibrium payoffs for all (symmetric and asymmetric) configurations of the players’ aggregate levels of force, characterizes the complete set of equilibrium univariate marginal distributions for most of these configurations, and constructs entirely new and novel equilibrium n-variate distributions.I am grateful to Jason Abrevaya, Dan Kovenock, James C. Moore, Roger B. Nelsen, and three anonymous referees for very helpful comments. A version of this paper was presented at the 2005 Midwest Economic Theory Meetings. This paper is based on the first chapter of my Ph.D. dissertation     

Behavior in one-shot traveler’s dilemma games: model and experiments with advice

We report results of one-shot traveler’s dilemma game experiments to test the predictions of a model of introspection. The model describes a noisy out-of-equilibrium process by which players reach a decision of what to do in one-shot games. To test the robustness of the model and to compare it to other models of introspection without noise, we introduce non-binding advice. Advice has the effect of coordinating all players’ beliefs onto a common strategy. Experimentally, advice is implemented by asking subjects who participated in a repeated traveler’s dilemma game to recommend an action to subjects playing one-shot games with identical parameters. In contrast to observations, models based on best-response dynamics would predict lower claims than the advised. We show that our model’s predictions with and without advice are consistent with the data.      

Decisiveness of contributors’ perceptions in elections

We consider a model of two-candidate elections with a one-dimensional policy space. Spending on campaign advertisements can directly influence voters’ preferences, and contributors give the money for campaign spending in exchange for promised services if the candidate wins. We find that the winner of the election depends crucially on the contributors’ beliefs about who is likely to win and the contribution market tends toward nonsymmetric equilibria in which one of the two candidates has no chance of winning. If the voters are only weakly influenced by advertising or if permissible campaign spending is small, then the candidates choose policies close to the median voter’s ideal point, but the contributors still determine the winner. Uncertainty about the Condorcet winning point (or its nonexistence) can change these results and generate equilibria in which both candidates have substantial probabilities of winning.     

Strategic approximations of discontinuous games

An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximationof an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.     

Discrete time dynamics in a random matching monetary model

Summary. Under take-it-or-leave-it offers, dynamic equilibria in the discrete time random matching model of money are a “translation” of dynamic equilibria in the standard overlapping generations model. This formalizes earlier conjectures about the equivalence of dynamic behavior in the two models and implies the indeterminacy of dynamic equilibria in the random matching model. As in the overlapping generations model, the indeterminacy disappears if an arbitrarily small utility to holding money is introduced. We introduce a different pricing mechanism, one that puts into sharp focus that agents are forward-looking when they interact. Received: January 18, 2001; revised version: May 25, 2001