On the predictive efficiency of the core solution in side-payment games

This paper reports the first cross-study competitive test of thecore solution in side-payment games where the core is nonempty and nonunique (i.e., larger than a single point). The core was tested against five alternative theories including the Shapley value, the disruption nucleolus, the nucleolus, the 2-center, and the equality solution. A generalized Euclidean distance metric which indexes the average distance between an observed payoff vector and the entire set of predicted payoff vectors (Bonacich, 1979) was used as the measure of goodness-of-fit. Analysis of data assembled from six previously reported studies (encompassing a total of 1,464 observations over 56 3-person and 4-person side-payment games) showed the core to predict less accurately than the Shapley value, disruption nucleolus, and nucleolus solutions (p < 0.01). These findings are consistent with previous empirical results that show the core to have a low level of predictive accuracy in side-payment games.This research was supported by grants SOC-7726932 and SES-8015528 from the National Science Foundation. Data analysis was performed at the Madison Academic Computing Center.     

Mechanism robustness in multilateral bargaining

This paper discusses the relationship between coalitional stability and the robustness of bargaining outcomes to the bargaining procedure. We consider a class of bargaining procedures described by extensive form games, where payoff opportunities are given by a characteristic function (cooperative) game. The extensive form games differ on the probability distribution assigned to chance moves which determine the order in which players take actions. One way to define mechanism robustness is in terms of the property of no first mover advantage. An equilibrium is mechanism robust if for each member the expected payoff before and after being called to propose is the same. Alternatively one can define mechanism robustness as a property of equilibrium outcomes. An outcome is said to be mechanism robust if it is supported by some equilibrium in all the extensive form games (mechanisms) within our class. We show that both definitions of mechanism robustness provide an interesting characterization of the core of the underlying cooperative game.     

Economic Darwinism

We define an evolutionary process of “economic Darwinism” for playing the field, symmetric games. The process captures two forces. One is “economic selection”: if current behavior leads to payoff differences, behavior yielding lowest payoff has strictly positive probability of being replaced by an arbitrary behavior. The other is “mutation”: any behavior has at any point in time a strictly positive, very small probability of shifting to an arbitrary behavior. We show that behavior observed frequently is in accordance with “evolutionary equilibrium”, a static equilibrium concept suggested in the literature. Using this result, we demonstrate that generally under positive (negative) externalities, economic Darwinism implies even more under- (over-)activity than does Nash equilibrium.     

Learning strategic environments: an experimental study of strategy formation and transfer

I present an experiment on learning about a game in an initially unknown environment. Subjects play repeatedly simple 2 × 2 normal-form coordination games. I compare behavioral learning algorithms for different feedback information. Minimal feedback only informs about own payoffs, while additional feedback informs about own payoffs and the opponent’s choice. Results show that minimal feedback information leads to a myopic learning algorithm, while additional feedback induces non-myopic learning and increases the impulse with which players respond to payoff differences. Finally, there is evidence for a strategy transfer across games which differ only according to the relabel of actions, but not according to permutation in the payoff matrix.     

Markov interactions in a class of dynamic games

This paper contributes to the understanding of economic strategic behaviors in inter-temporal settings. Comparing the MPE and the OLNE of a widely used class of differential games it is shown: (i) what qualifications on behaviors a markov (dynamic) information structure brings about compared with an open-loop (static) information structure, (ii) what is the reason leading to intensified or reduced competition between the agents in the long run. It depends on whether agents’ interactions are characterized by markov substitutability or markov complementarity, which can be seen as dynamic translations of the ideas of strategic substitutability and strategic complementarity (Bulow et al. 1985, Journal of Political Economy 93:488–511). In addition, an important practical contribution of the paper for modelers is to show that these results can be directly deduced from the payoff structure, with no need to compute equilibria first. I dedicate this paper to Philippe Michel, who introduced me to the literature on differential games.     

On Loss Aversion in Bimatrix Games

In this article three different types of loss aversion equilibria in bimatrix games are studied. Loss aversion equilibria are Nash equilibria of games where players are loss averse and where the reference points—points below which they consider payoffs to be losses—are endogenous to the equilibrium calculation. The first type is the fixed point loss aversion equilibrium, introduced in Shalev (2000; Int. J. Game Theory 29(2):269) under the name of ‘myopic loss aversion equilibrium.’ There, the players’ reference points depend on the beliefs about their opponents’ strategies. The second type, the maximin loss aversion equilibrium, differs from the fixed point loss aversion equilibrium in that the reference points are only based on the carriers of the strategies, not on the exact probabilities. In the third type, the safety level loss aversion equilibrium, the reference points depend on the values of the own payoff matrices. Finally, a comparative statics analysis is carried out of all three equilibrium concepts in 2 × 2 bimatrix games. It is established when a player benefits from his opponent falsely believing that he is loss averse.     

Using turn taking to achieve intertemporal cooperation and symmetry in infinitely repeated 2 × 2 games

Turn taking is observed in many field and laboratory settings captured by various widely studied 2 × 2 games. This article develops a repeated game model that allows us to systematically investigate turn-taking behavior in many 2 × 2 games, including the battle of the sexes, the game of chicken, the game of common-pool-resources assignment, and a particular version of the prisoners’ dilemma. We consider the “turn taking with independent randomizations” (TTIR) strategy that achieves three objectives: (a) helping the players reach the turn-taking path, (b) resolving the question of who takes the good turn first, and (c) deterring defection. We determine conditions under which there exists a unique TTIR strategy profile that can be supported as a subgame-perfect equilibrium. We also show that there exist conditions under which an increase in the “degree of conflict” of the stage game leads to a decrease in the expected number of periods in reaching the turn-taking path.     

On the $$\gamma $$γ-core of asymmetric aggregative games

Theory and Decision - This paper analyzes the core of cooperative games generated by asymmetric aggregative normal-form games, i.e., games where the payoff of each player depends on his strategy...     

FORWARD INDUCTION IN GAMES WITH AN OUTSIDE OPTION

We provide eductive foundations for the concept of forward induction, in the class of games with an outside option. The formulation presented tries to capture in a static notion the rest point of an introspective process, achievable from some restricted preliminary beliefs. The former requisite is met by requiring the rest point to be a Nash equilibrium that yields a higher payoff than the outside option. With respect to the beliefs, we propose the Incentive Dominance Criterion. Players should consider one action more likely than another whenever the former is better than getting the outside option for more conjectures over his rival's actions. We apply this model to the case where the subgame is a coordination game with a conflict between payoff dominance and risk dominance. Our results provide support for dominance solvability, but not for Van Damme's notion of forward induction. We show how the forward induction logic helps to select the Pareto dominant equilibrium. This is the case whenever player 1's act of giving up the outside option reverses the incentive dominance relations among 1's pure actions in the subgame.     

Game-theoretic analyses of coalition behavior

Coalitions are frequently more visible than payoffs. The theory of n-person games seeks primarily to identify stable allocations of valued resources; consequently, it gives inadequate attention to predicting which coalitions form. This paper explores a way of correcting this deficiency of game-theoretic reasoning by extending the theory of two-person cooperative games to predict both coalitions and payoffs in a three-person game of status in which each player seeks to maximize the rank of his total score. To accomplish this, we analyze the negotiations within each potential two-person coalition from the perspective of Nash's procedure for arbitrating two-person bargaining games, then assume that players expect to achieve the arbitrated outcome selected by this procedure and use these expectations to predict achieved ranks and to identify players' preferences between alternative coalition partners in order to predict the probability that each coalition forms. We test these payoff and coalition predictions with data from three laboratory studies, and compare the results with those attained in the same data by von Neumann and Morgenstern's solution of two-person cooperative games, Aumann and Maschler's bargaining set solution for cooperative n-person games, and an alternative model of coalition behavior in three-person sequential games of status.

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Population monotonic path schemes for simple games

A path scheme for a game is composed of a path, i.e., a sequence of coalitions that is formed during the coalition formation process and a scheme, i.e., a payoff vector for each coalition in the path. A path scheme is called population monotonic if a player’s payoff does not decrease as the path coalition grows. In this study, we focus on Shapley path schemes of simple games in which for every path coalition the Shapley value of the associated subgame provides the allocation at hand. Obviously, each Shapley path scheme of a game is population monotonic if and only if the Shapley allocation scheme of the game is population monotonic in the sense of Sprumont (Games Econ Behav 2:378–394, 1990). We prove that a simple game allows for population monotonic Shapley path schemes if and only if the game is balanced. Moreover, the Shapley path scheme of a specific path is population monotonic if and only if the first winning coalition that is formed along the path contains every minimal winning coalition. We also show that each Shapley path scheme of a simple game is population monotonic if and only if the set of veto players of the game is a winning coalition. Extensions of these results to other efficient probabilistic values are discussed.     

Different carrots and different sticks: do we reward and punish differently than we approve and disapprove?

This paper reports lab data from four games in order to analyze and compare the motivations behind monetary punishment and reward and their non-monetary counterparts, disapproval and approval, an important question given that both types of punishment/rewards affect cooperation and norm compliance. The results in our games support the hypothesis that a motivation akin to reciprocity plays the key role for approval and disapproval whereas payoff comparisons play the key role for monetary rewards and punishment.     

The utility of gambling

A tiny utility of gambling is appended to an expected utility model for a risk-averse individual. It is shown that the model can explain small payoff gambles, large prize lotteries, and patterns of risk-seeking in the experimental evidence that are puzzling from the viewpoint of standard theory. At the same time, the model maintains expected utility theory's ability to explain insurance purchase, portfolio diversification, and other risk-averting behavior. The tiny utility of gambling could equally well be appended to models of risky choice other than the expected utility model.     

On Hobbes’s state of nature and game theory

Hobbes’s state of nature is often analyzed in two-person two-action non-cooperative games. By definition, this literature only focuses on duels. Yet, if we consider general games, i.e., with more than two agents, analyzing Hobbes’s state of nature in terms of duel is not completely satisfactory, since it is a very specific interpretation of the war of all against all. Therefore, we propose a definition of the state of nature for games with an arbitrary number of players. We show that this definition coincides with the strategy profile considered as the state of nature in two-person games. Furthermore, we study what we call rational states of nature (that is, strategy profiles which are both states of nature and Nash equilibria). We show that in rational states of nature, the utility level of any agent is equal to his maximin payoff. We also show that rational states of nature always exist in inessential games. Finally, we prove the existence of states of nature in a class of (not necessarily inessential) symmetric games.     

Uncertain discount and hyperbolic preferences

This paper studies the interaction between savagean uncertainty and time preferences. We introduce a variation of the discounted subjective expected utility model, where time preferences are state dependent. Before uncertainty is resolved, the individual is unsure about the discount factor that will be used, even when evaluating certain payoffs. The model can account for the present bias and diminishing impatience, even if the future is discounted geometrically. The present bias disappears when the immediate payoff becomes uncertain. Although preferences are not stationary, choices may be time consistent.     

The Value of a Probability Forecast from Portfolio Theory

A probability forecast scored ex post using a probability scoring rule (e.g. Brier) is analogous to a risky financial security. With only superficial adaptation, the same economic logic by which securities are valued ex ante – in particular, portfolio theory and the capital asset pricing model (CAPM) – applies to the valuation of probability forecasts. Each available forecast of a given event is valued relative to each other and to the “market” (all available forecasts). A forecast is seen to be more valuable the higher its expected score and the lower the covariance of its score with the market aggregate score. Forecasts that score highly in trials when others do poorly are appreciated more than those with equal success in “easy” trials where most forecasts score well. The CAPM defines economically rational (equilibrium) forecast prices at which forecasters can trade shares in each other’s ex post score – or associated monetary payoff – thereby balancing forecast risk against return and ultimately forming optimally hedged portfolios. Hedging this way offers risk averse forecasters an “honest” alternative to the ruse of reporting conservative probability assessments.     

Hierarchies achievable in simple games

A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, achievable in simple games. It is proved that only four hierarchies are non-achievable in simple games. Moreover, it is also proved that all achievable hierarchies are already obtainable in the class of weakly linear games. Our results prove that given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.     

Groups can make a difference: voting power measures extended

The voting power of a voter—the extent to which she can affect the outcome of a collective decision—is often quantified in terms of the probability that she is critical. This measure is extended to a series of power measures of different ranks. The measures quantify the extent to which a voter can be part of a group that can jointly make a difference as to whether a bill passes or not. It is argued that the series of these measures allow for a more appropriate assessment of voting power, particularly of a posteriori voting power in case the votes are stochastically dependent. Also, the new measures discriminate between voting games that cannot be distinguished in terms of the probability of only criticality.     

The Evolution of Coding in Signaling Games

Signaling games with reinforcement learning have been used to model the evolution of term languages (Lewis 1969, Convention. Cambridge, MA: Harvard University Press; Skyrms 2006, “Signals” Presidential Address. Philosophy of Science Association for PSA). In this article, syntactic games, extensions of David Lewis’s original sender–receiver game, are used to illustrate how a language that exploits available syntactic structure might evolve to code for states of the world. The evolution of a language occurs in the context of available vocabulary and syntax—the role played by each component is compared in the context of simple reinforcement learning.     

Risk-adjusted martingales and the design of “indifference” gambles

In the probability literature, a martingale is often referred to as a “fair game.” A martingale investment is a stochastic sequence of wealth levels, whose expected value at any future stage is equal to the investor’s current wealth. In decision theory, a risk neutral investor would therefore be indifferent between holding on to a martingale investment, and receiving its payoff at any future stage, or giving it up and maintaining his current wealth. But a risk-averse decision maker would not be indifferent between a martingale investment and his current wealth level, since he values uncertain deals less than their mean. A risk seeking decision maker, on the other hand, would readily accept a martingale investment in exchange for his current wealth, and would repeat this investment any number of times. These ideas lead us to introduce the notion of a “risk-adjusted martingale”; a stochastic sequence of wealth levels that a rational decision maker with any attitude toward risk would value constantly with time, and would be indifferent between receiving its pay-off at any future stage, or giving it up and maintaining his current wealth level. We show how to construct such risk-adjusted investments for any decision maker with a continuous monotonic utility function. The fundamental result we derive is that a pay-off structure of an investment (i) is a risk-adjusted martingale and (ii) can be represented by a lattice if and only if the pay-off functions are invariant transformations of the given utility function.