Maximum Payoff Research Articles

Words, Deeds, and Lies: Strategic Behaviour in Games with Multiple Signals

We report the results of an experiment in which subjects play games against changing opponents. In one treatment, “senders” send “receivers” messages indicating intended actions in that round, and receivers observe senders' previous-round actions (when matched with another receiver). In another treatment, the receiver additionally observes the sender's previous-round message to the previous opponent, enabling him to determine whether the sender had lied. We find that allowing multiple signals leads to better outcomes when signals are aligned (all pointing to the same action), but worse outcomes when signals are crossed. Also, senders' signals tend to be truthful, though the degree of truthfulness depends on the game and treatment, and receivers' behaviour combines elements of pay-off maximization and reciprocity.

AN ALTERNATIVE APPROACH TO EXPLAINING BARGAINING BEHAVIOUR IN SIMPLE SEQUENTIAL GAMES

. For simple sequential games, the payoff structure has certain bargaining power implication. Intuitively, certain actions may have the effect of the carrot and the stick as sanctioning means. Fair outcomes can be alternatively viewed as the consequence of their presence. Experimental data on various games are used to test the hypothesis of carrot and stick behaviour. We show that our stick and carrot heuristic neatly captures puzzling phenomena in a wide class of simple sequential games. The results support the view that punishing subjects are not worse off than myopic pay-off maximizers as long as subjects are involved in a repeated strangers scheme.

For Better or Forever: Formal versus Informal Enforcement

This article contrasts supporting partnerships through relational contracting and supporting partnerships through formal legal institutions. A large population of players interact in bilateral relationships. Efficiency requires cooperation, but cheating yields a higher short‐term payoff. There is a positive probability that the maximum feasible payoff available to a partnership decreases. Opportunistic behavior makes it impossible to realize the efficient outcome. A legal system can lead to efficient contracting. Without such a system, productive relationships arise in equilibrium if it is costly to initiate new relationships. This type of relational contracting tends to make partnerships last longer than is efficient.

What To Maximize If You Must

The assumption that decision makers choose actions to maximize their preferences is a central tenet in economics. This assumption is often justified either formally or informally by appealing to evolutionary arguments. In contrast, this paper shows that in almost every game, payoff maximization cannot be justified by appealing to such arguments. We show that in almost every game, for almost every distortion of a player's actual payoffs, some extent of this distortion is beneficial to the player because of the resulting effect on opponents' play. Consequently, such distortions will not be driven out by any evolutionary process involving payoff-monotonic selection dynamics, in which agents with higher actual payoffs proliferate at the expense of less successful agents. In particular, under any such selection dynamics, the population will not converge to payoff-maximizing behavior. We also show that payoff-maximizing behavior need not prevail even when preferences are imperfectly observed.

Characterization of equilibrium in pay-as-bid auctions for multiple units

Equilibrium bidding strategies under most multi-unit auction rules cannot be obtained as closed form expressions. Research in multi-unit auctions has, therefore, depended on implicit characterization of equilibrium strategies using the first-order conditions of the bidders’ expected payoff maximization problem. In this paper we consider the pay-as-bid auction with diminishing marginal values for two units and show that any symmetric equilibrium in continuous strategies has the necessary properties to allow such a characterization. Moreover, any increasing solution to the system of differential equations that is used to characterize the equilibrium strategies describes an equilibrium strategy.

Evidential Reasoning in the Prisoner’s Dilemma

Classic game theory considers defection to be the rational choice in the Prisoner's Dilemma. Although defection maximizes a player's gain regardless of the opponent's choice, many players cooperate. We suggest that cooperation can be explained in part by expectations of reciprocal behavior, that is, by the belief that players facing the same situation probably will come to the same decision. In Experiment 1, expectations of reciprocity were experimentally manipulated. As predicted, cooperation increased monotonically with these expectations. In Experiment 2, experimentally manipulated expectations of opponent cooperation were not associated with higher rates of cooperation. These findings are interpreted in light of a general model of payoff maximization, and implications for other models (e.g., social value orientation) are discussed.

False and Reluctant Friends? National Money Laundering Regulation, International Compliance and Non-Cooperative Countries

In the aftermath of September 11th, growing attention has been paid to the role of Non-Cooperative Countries and Territories (NCCT) in money laundering and terrorist financing. The paper, applying the new political economy approach in the law and economics field, provides a model to describe, through the policymaker payoff maximization, the relationships between specific country features and endowments, on the one hand, and lax financial regulations, on the other hand. Given that in the real world relatively lax regulation means a non-cooperative attitude in the international fight against money laundering and terrorist financing we empirically test the above theoretical relationship in the case of the NCCT jurisdictions. The policy consequences on the pros and cons of international blacklisting procedures are discussed.

Population dynamics with a stable efficient equilibrium

We propose a game-theoretic dynamics of a population of replicating individuals. It consists of two parts: the standard replicator one and a migration between two different habitats. We consider symmetric two-player games with two evolutionarily stable strategies: the efficient one in which the population is in a state with a maximal payoff and the risk-dominant one where players are averse to risk. We show that for a large range of parameters of our dynamics, even if the initial conditions in both habitats are in the basin of attraction of the risk-dominant equilibrium (with respect to the standard replication dynamics without migration), in the long run most individuals play the efficient strategy.

Optimal guessing: Choice in complex environments

In this paper, we extend the analysis of our earlier work on boundedly rational learning in an i.i.d. setting [Easley and Rustichini, Econometrica 67 (1999) 1157–1184] to complex decision problems. We show that the axioms from our earlier analysis can be applied in this more complex setting, and along with some new axioms, they asymptotically yield expected utility maximization. Perhaps most important is our demonstration of a simple procedure that insures expected payoff maximization no matter what Markov process the underlying process on states follows. We view this result as providing a positive learning result for all worlds in which learning is possible.

Security Issuance and Investor Information

Although the commoditisation of illiquid asset exposures through securitisation facilitates the disciplining effect of capital markets on the risk management, private information about securitised debt as well as complex transaction structures could possibly impair the fair market valuation. In a simple issue design model without intermediaries we maximise issuer proceeds over a positive measure of issue quality, where a direct revelation mechanism (DRM) by profitable informed investors engages endogenous price discovery through auction-style allocation preference as a continuous function of perceived issue quality. We derive an optimal allocation schedule for maximum issuer payoffs under different pricing regimes if asymmetric information requires underpricing. In particular, we study how the incidence of uninformed investors at varying levels of valuation uncertainty and their function of clearing the market effects profitable informed investment. We find that the issuer optimises own payoffs at each valuation irrespective of the applicable pricing mechanism by awarding informed investors the lowest possible allocation (and attendant underpricing) that still guarantees profitable informed investment. Under uniform pricing the composition of the investor pool ensures that informed investors appropriate higher profit than uninformed types. Any reservation utility by issuers lowers the probability of information disclosure by informed investors and the scope of issuers to curtail profitable informed investment.

Individual Decision Making in a Negative Externality Experiment

The experimental treatments analysed in this paper are simple in that there is a unique Nash equilibrium resulting in each player having a dominant strategy. However, the data show quite clearly that subjects do not always choose this strategy. In fact, when this dominant strategy is not a “focal” outcome it does not even describe the average decision adequately. It is shown that average individual decisions are best described by a decision error model based on a censored distribution as opposed to the truncated regression model which is typically used in similar studies. Moreover it is shown that in the treatments where the dominant strategy is not “focal” dynamics are important with average subject decisions initially corresponding to the “focal” outcome and then adjusting towards the Nash prediction. Overall, 66.7% of subjects are consistent with Payoff Maximization, 27.8% are consistent with an alternate preference maximization and 5.6% are random.

Equilibrium selection in evolutionary games with random matching of players

We discuss stochastic dynamics of populations of individuals playing games. Our models possess two evolutionarily stable strategies: an efficient one, where a population is in a state with the maximal payoff (fitness) and a risk-dominant one, where players are averse to risks. We assume that individuals play with randomly chosen opponents (they do not play against average strategies as in the standard replicator dynamics). We show that the long-run behavior of a population depends on its size and the mutation level.

Stochastic Darwinian equilibria in small and large populations

We establish necessary and sufficient conditions for the stability of stochastic Darwinian dynamics in quadratic games. Each player's strategy adjusts through mutation and selection shocks, and stability is independent of the rates at which these shocks arrive. Given stability, we characterize the midpoint of the nondegenerate ergodic distribution. In small populations, some equilibria correspond to relative payoff maximization, but others are unanticipated by existing static concepts. In the large population limit of a finite population, the set of stable Nash equilibria strictly includes all equilibria stable under myopic best reply, but some strict Nash equilibria are highly unstable. The stability result shows, for the first time, that large finite populations converge to Nash play even if they do not understand the game and strategies are so numerous that most are never played. The large population stability condition is related to risk dominance and, separately, to the static CSS condition.

The Maximum Efficient Equilibrium Payoff in the Repeated Prisoners' Dilemma

We describe the maximum efficient subgame perfect equilibrium payoff for a player in the repeated Prisoners' Dilemma, as a function of the discount factor. For discount factors above a critical level, every efficient, feasible, individually rational payoff profile can be sustained. For an open and dense subset of discount factors below the critical value, the maximum efficient payoff is not an equilibrium payoff. When a player cannot achieve this payoff, the unique equilibrium outcome achieving the best efficient equilibrium payoff for a player is eventually cyclic. There is an uncountable number of discount factors below the critical level such that the maximum efficient payoff is an equilibrium payoff. Journal of Economic Literature Classification Numbers: C72, C73.

Evolution of Social Behavior: Individual and Group Selection

How selfish does our evolutionary history suggest that humans will be? We explore models in which groups are formed and dissolved and where reproduction of individuals is determined by their payoffs in a game played within groups. If groups are formed “randomly ” and reproductive success of group founders is determined by a multiperson prisoners' dilemma game, then selfish behavior will prevail over maximization of group payoffs. However, interesting models exist in which “group selection ”sustains cooperative behavior. Forces that support cooperative behavior include assortative matching in groups, group longevity and punishment-based group norms.

The dynamics of self-referential paradoxical games

Paradoxical games are non-constant sum, non-negotiable conflicts, in which two contestants (players) blackmail each other, acting as components of a nonlinear dynamical system characterized by time varying probabilities of cooperation. Such games are 'paradoxical' in the sense that both players could win or lose simultaneously and are called 'self-referential' if the parameters of the system depend explicitly on the contestants' probabilities of cooperation. Previously studied twocontestant models with constant parameters were found to be 'conservative', possessing two centres around which the game can oscillate forever. In this paper, we first study the case where all parameters are allowed to vary and find that the dynamics becomes 'dissipative', possessing a single fixed point attractor of moderate equal gains. On this attractor large subsets of initial conditions (strategies) converge as t M X and attain constant cooperation probabilities. If both contestants cooperate 'equally disregarding' the other's tendency to do so, the attractor moves closer to the state of 'maximum pay-off', where both parties cooperate with probability 1. However, in the asymmetric case, where one of the players 'takes less into account' the other's tendency to cooperate, it is the more 'indifferent' player who profits the most! Partially self-referential games, in which only the 'gain' or 'loss' factors due to defection vary by a small parameter k , lead to the state of full cooperation (and maximum gain), as the stable fixed points of the k = 0 case either become repellors, or are eliminated via a pitchfork and a saddle-node bifurcation.

When will payoff maximization survive?

Survival of payoff maximization is the usual as if-justification for assuming rational economic agents. An indirect evolutionary analysis allows for stimuli which are not directly related to reproductive success although they affect behavior. One first determines the solution for all possible constellations of stimuli, and then the evolutionarily stable stimuli. Our general analysis confirms the special results of former studies that payoff maximization in case of commonly known stimuli requires either that own success does not depend on other's behavior or that other's behavior is not influenced by own stimuli. When stimuli are private information, one can derive similar necessary conditions.

The Maximum Efficient Equilibrium Payoff in the Repeated Prisoners' Dilemma

We describe the maximum efficient subgame perfect equiligrium payoff for a player in the repeated Prisoners' Dilemma, as a function of the discount factor. For discount factors above a critical level, every efficient, feasible, individually rational payoff profile can be sustained. For an open and dense subset of discount factors below the critical value, the maximum efficient payoff is not an equilibrium payoff. When a player cannot achieve this payoff, the unique equilibrium outcome achieving the best efficient equilibrium payoff for a player is eventually cyclic. There is an uncountable number of discount factors below the critical level such that the maximum efficient payoff is an equilibrium payoff.

A Generalized Adaptive Dynamics Framework can Describe the Evolutionary Ultimatum Game

Adaptive dynamics describes the evolution of games where the strategies are continuous functions of some parameters. The standard adaptive dynamics framework assumes that the population is homogeneous at any one time. Differential equations point to the direction of the mutant that has maximum payoff against the resident population. The population then moves towards this mutant. The standard adaptive dynamics formulation cannot deal with games in which the payoff is not differentiable. Here we present a generalized framework which can. We assume that the population is not homogeneous but distributed around an average strategy. This approach can describe the long-term dynamics of the Ultimatum Game and also explain the evolution of fairness in a one-parameter Ultimatum Game.

Ultimatum Bargaining Behavior in Israel, Japan, Slovenia, and the United States: A Social Utility Analysis

We explain the main features of the results of the four-country ultimatum bargaining experiments of Roth et al. (1991), Amer. Econom. Rev.81, 1068–1095) by a social utility model. The specification of social utility of a player has two parts: a linear combination of the monetary payoffs of the proposer and the responder and payoff uncertainty. We find that, on average, responders have negative regard for proposers' earnings in all countries. Proposers have negative regard for responders' monetary earnings in countries where responders have high negative regard for proposers' earnings (USA and Slovenia). In countries where responders have low negative regard for proposers' earnings (Israel and Japan), proposers are expected payoff maximizers. Journal of Economic Literature Classification Numbers: A13, C19, C44, C72, C92, D63, D64.