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.     

Asset price bubbles in Arrow-Debreu and sequential equilibrium

Summary. Price bubbles in an Arrow-Debreu equilibrium in an infinite-time economy are a manifestation of lack of countable additivity of valuation of assets. In contrast, the known examples of price bubbles in a sequential equilibrium in infinite time cannot be attributed to the lack of countable additivity of valuation. In this paper we develop a theory of valuation of assets in sequential markets (with no uncertainty) and study the nature of price bubbles in light of this theory. We define a payoff pricing operator that maps a sequence of payoffs to the minimum cost of an asset holding strategy that generates it. We show that the payoff pricing functional is linear and countably additive on the set of positive payoffs if and only if there is no Ponzi scheme, provided that there is no restriction on long positions in the assets. In the known examples of equilibrium price bubbles in sequential markets valuation is linear and countably additive. The presence of a price bubble means that the dividends of an asset can be purchased in sequential markets at a cost lower than the asset's price. We present further examples of equilibrium price bubbles in which valuation is nonlinear, or linear but not countably additive.     

Maximum efforts in contests with asymmetric valuations

Efforts may be reduced when players with different valuations participate in a contest. This paper considers the problem of designing a contest to elicit maximum aggregate effort from players with asymmetric valuations. Optimal designs for different classes of contest technologies are computed and characterized. A value weighted contest is optimal in the concave case. In the unconstrained case, the optimal contest is equivalent to a first price all-pay auction with a reserve price. The optimal design discounts the effort of the high valuation player in order to induce him to compete vigorously.     

Learning to play games in extensive form by valuation

Game theoretic models of learning which are based on the strategic form of the game cannot explain learning in games with large extensive form. We study learning in such games by using valuation of moves. A valuation for a player is a numeric assessment of her moves that purports to reflect their desirability. We consider a myopic player, who chooses moves with the highest valuation. Each time the game is played, the player revises her valuation by assigning the payoff obtained in the play to each of the moves she has made. We show for a repeated win-lose game that if the player has a winning strategy in the stage game, there is almost surely a time after which she always wins. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1 there is a time after which strategies that are close to subgame perfect equilibrium are played. A single player who adopts this procedure can guarantee only her individually rational payoff.     

Governing Interest Groups and Rent Dissipation

In a contest of group‐specific public goods we consider the effect that managing an interest group has on the rent dissipation and the total expected payoffs of the contest. While in the first group there is a central planner determining its members’ expenditure in the contest, in the second group there are two different possibilities: either all the members are governed by a central planner or they are not. We consider both types of contests: an all‐pay auction and a Logit contest success function. We show that while governing an interest group decreases free‐riding, it may as well decrease the rent dissipation; at the same time the expected payoffs of the groups may also decrease.     

DISCRIMINATION IN A MODEL OF CONTESTS WITH INCOMPLETE INFORMATION ABOUT ABILITY

We study contests with private information and identical contestants, where contestants' efforts and innate abilities generate output of varying qualities. The designer's revenue depends on the quality of the output, and she offers a reward to the contestant achieving the highest quality. We characterize the equilibrium behavior, outcomes, and payoffs for both nondiscriminatory and discriminatory (where the reward is contestant‐dependent) contests. We derive conditions under which the designer obtains a larger payoff when using a discriminatory contest and describe settings, where these conditions are satisfied.     

Core and periphery in networks

We study a model of network formation where the benefits from connections exhibit decreasing returns and decay with network distance. We show that the unique equilibrium network is a periphery-sponsored star, where one player, the center, maintains no links and earns a high payoff, while all other players maintain a single link to the center and earn lower payoffs. Both the star architecture and payoff inequality are preserved in an extension of the model where agents can make transfers and bargain over the formation of links, under the condition that the surplus of connections increases in the size of agents’ neighborhoods. Our model thus generates two common features of social and economic networks: (1) a core-periphery structure; (2) positive correlation between network centrality and payoffs.     

The law of supply in games, markets and matching models

Summary In large games with transferable utility, core payoffs satisfy a comparative statics property: If the proportion of one type of player increases, then the core payoff to that type of player decreases (does not increase). Markets with transferable utility satisfy a similar property: if the aggregate supply of a commodity increases, its value relative to the value of all commodities decreases. In market games, if one type of agent becomes more plentiful, his competitive payoff falls, and its decrease is engineered by a decrease in the relative value of his endowment.We thank Bob Anderson, Joe Farrell, Steve Goldman, Chris Shannon, seminar participants of the Mathematical Economics Seminar at Berkeley (August 1994), the University of Pittsburg (November 1994), Tel Aviv University and the Institute on Rationality at the Hebrew University (January 1995), and especially Vince Crawford for useful discussion.     

Selecting equilibria in common agency games

We characterize equilibrium payoffs of a delegated common agency game in a public good context where principals use smooth contribution schedules. We prove that under complete information, payoff vectors of equilibria with truthful schedules coincide with the set of smooth equilibrium payoffs, including non-truthful schedules. We next consider whether the presence of arbitrarily small amounts of asymmetric information is enough to refine this payoff set. Providing that the extensions of the equilibrium schedules beyond the equilibrium point are flatter than truthful schedules, the set of equilibrium payoffs is strictly smaller than the set of smooth (equivalently, truthful) equilibrium payoffs. Interestingly, some forms of asymmetric information do not sufficiently constrain the slopes of the extensions and fail to refine the payoff set. In the case of a uniform distribution of types and arbitrary out-of-equilibrium contributions, the refinement has no bite. If, however, one restricts out-of-equilibrium behavior in a natural way, the refinement is effective. Alternatively, we may consider an exponential distribution with unbounded support (and hence no out-of-equilibrium choices) and we find that the refinement selects a unique equilibrium payoff vector equal to Lindahl prices.As a separate contribution, equilibria with forcing contracts are also considered both under complete and asymmetric information.     

Individual Differences in EWA Learning with Partial Payoff Information

We extend experience-weighted attraction (EWA) learning to games in which only the set of possible foregone payoffs from unchosen strategies are known, and estimate parameters separately for each player to study heterogeneity. We assume players estimate unknown foregone payoffs from a strategy, by substituting the last payoff actually received from that strategy, by clairvoyantly guessing the actual foregone payoff, or by averaging the set of possible foregone payoffs conditional on the actual outcomes. All three assumptions improve predictive accuracy of EWA. Individual parameter estimates suggest that players cluster into two separate subgroups (which differ from traditional reinforcement and belief learning).     

Group competition for rents

This paper examines rent dissipation in a two-stage group rent-seeking contest without a predetermined distribution rule. the rent in this setting exhibits both public and private good characteristics depending on the stage of the contest. Focusing on the relationship between group size and aggregate rent seeking we find that social waste depends not only on total numbers but also on the distribution of population across groups. We show that group size asymmetry acts to reduce rent dissipation.     

Uniform payoff security and Nash equilibrium in compact games

We introduce a condition, uniform payoff security, for games with compact Hausdorff strategy spaces and payoffs bounded and measurable in players’ strategies. We show that if any such compact game G is uniformly payoff secure, then its mixed extension is payoff secure. We also establish that if a uniformly payoff secure compact game G has a mixed extension with reciprocally upper semicontinuous payoffs, then G has a Nash equilibrium in mixed strategies. We provide several economic examples of compact games satisfying uniform payoff security.     

Existence and Uniqueness of Nash Equilibrium in Electoral Competition Games: The Hybrid Case

This paper analyzes the traditional unidimensional, two‐party electoral competition game when parties have mixed motivations, in the sense that they are interested in winning the election, but also in the policy implemented after the contest. In spite of having discontinuous payoffs, this game, referred to as the hybrid election game, is shown to be payoff secure and reciprocally upper semi‐continuous. Conditional payoffs, however, are not quasi‐concave. Hence, the existence of a pure strategy Nash equilibrium (psne ) is ensured only if parties have homogenous interests in power. In that case, an equilibrium not only exists, but it is also unique. Instead, if parties have heterogeneous motivations, depending upon the relationship between the electoral uncertainty, the aggregate opportunism, and its distribution across parties, a psne may or may not exist. The mixed extension, however, is always better reply secure. Therefore, a mixed strategy Nash equilibrium does indeed exist.     

Evolution of Interdependent Preferences in Aggregative Games

We study the evolution of preference interdependence in aggregative games which are symmetric with respect to material payoffs but asymmetric with respect to player objective functions. We identify a class of aggregative games whose equilibria have the property that the players with interdependent preferences (who care not only about their own material payoffs but also about their payoffs relative to others) earn strictly higher material payoffs than do the material payoff maximizers. Implications of this finding for the theory of preference evolution are discussed. Journal of Economic Literature Classification Numbers: C72, D62.     

On the folk theorem with one-dimensional payoffs and different discount factors

Proving the folk theorem in a game with three or more players usually requires imposing restrictions on the dimensionality of the stage-game payoffs. Fudenberg and Maskin (1986) assume full dimensionality of payoffs, while Abreu et al. (1994) assume the weaker NEU condition (“nonequivalent utilities”). In this note, we consider a class of n-player games where each player receives the same stage-game payoff, either zero or one. The stage-game payoffs therefore constitute a one-dimensional set, violating NEU. We show that if all players have different discount factors, then for discount factors sufficiently close to one, any strictly individually rational payoff profile can be obtained as the outcome of a subgame-perfect equilibrium with public correlation.     

Valuation in infinite-horizon sequential markets with portfolio constraints

Summary. We develop a theory of valuation of assets in sequential markets over an infinite horizon and discuss implications of this theory for equilibrium under various portfolio constraints. We characterize a class of constraints under which sublinear valuation and a modified present value rule hold on the set of non-negative payoff streams in the absence of feasible arbitrage. We provide an example in which valuation is non-linear and the standard present value rule fails in incomplete markets. We show that linearity and countable additivity of valuation hold when markets are complete. We present a transversality constraint under which valuation is linear and countably additive on the set of all payoff streams regardless of whether markets are complete or incomplete. Received: March 9, 2000; revised version: February 13, 2001     

Social norms and trust among strangers

We study the development of a social norm of trust and reciprocity among a group of strangers via the “contagious strategy” as defined in Kandori (1992). Over an infinite horizon, the players anonymously and randomly meet each other and play a binary trust game. In order to provide the investors with proper incentives to follow the contagious strategy, there is a sufficient condition that requires that there exists an outside option for the investors. Moreover, the investorsʼ payoff from the outside option must converge to the payoff from trust and reciprocity as the group size goes to infinity. We show that this sufficient condition is also a necessary condition to sustain any sequential equilibrium in which the trustees adopt the contagious strategy. Our results imply that a contagious equilibrium only supports trust if trust contributes almost nothing to the investorsʼ payoffs.     

How sensitive are bargaining outcomes to changes in disagreement payoffs?

We use a human-subjects experiment to investigate how bargaining outcomes are affected by changes in bargainers’ disagreement payoffs. Subjects bargain against changing opponents, with randomly drawn asymmetric disagreement outcomes that vary over plays of the game, and with complete information about disagreement payoffs and the cake size. We find that subjects only respond about half as much as theoretically predicted to changes in their own disagreement payoff and to changes in their opponent’s disagreement payoff. This effect is observed in a standard Nash demand game and a related unstructured bargaining game, in both early and late rounds, and is robust to moderate changes in stake sizes. We show theoretically that standard models of expected utility maximisation are unable to account for this under-responsiveness, even when generalised to allow for risk aversion. We also show that quantal-response equilibrium has, at best, mixed success in characterising our results. However, a simple model of other-regarding preferences can explain our main results.     

Sowing Doubt Optimally in Two-Person Repeated Games

Consider a two-person repeated game, where one of the players, P1, can sow doubt, in the mind of his opponent, as to what P1's payoffs are. This results in a two-person repeated game with incomplete information. By sowing doubt, P1 can sometimes increase his minimal equilibrium payoff in the original game. We prove that this minimum is maximal when only one payoff matrix, the negative of the payoff matrix of the opponent, is added (the opponent thus believes that he might play a zero-sum game). We obtain two formulas for calculating this maximal minimum payoff. Journal of Economic Literature Classification Numbers: C7, D8.     

A theory of mutual migration of polluting firms

Abstract.  Suppose that governments care about their tax revenue and local firms have some say in environmental regulations. Then, the level of employment and environmental compliance may be negotiated. We find that firms located in different countries can improve their threat‐point payoffs by mutual migration. This in turn affects the negotiated output/employment and environmental regulations, which causes profits to increase if the firm's threat‐point payoff is higher than that of the local government. The model predicts that pollution‐intensive firms or firms with highly inelastic demands are more likely to move out. Increases in the government's valuation of the environment, or in the degree of globalization also cause mutual migration of dirty firms. The effect of a government caring about consumer surplus leads to a lower pollution tax, reducing firms' incentives to move out. JEL classification: F2, Q0