Stability and trembles in extensive-form games

A leading solution concept in the evolutionary study of extensive-form games is Selten (1983) notion of limit ESS. This note demonstrates that a limit ESS does not imply neutral stability, and that it may be dynamically unstable (almost any small perturbation takes the population away). These problems arise due to an implicit assumption that “mutants” are arbitrarily rare relative to “trembling” incumbents. Finally, I present a novel definition that solves this issue and has appealing properties.     

Asymmetric evolutionary games with non-linear pure strategy payoffs

In this paper, I extend the analytical framework of evolutionary game theory to games with two distinct types of players where the type-specific payoff functions are nonlinear on the right—bi-player-type nonlinear (BNL) evolutionary games. That is, asymmetric games where the payoffs for pairwise interactions are influenced by the ambient frequencies of the pertinent strategies of both types of players.While this paper is motivated by my interest in inter- and intra-gender conflicts of interest in sexual reproduction, BNL games also arise just as naturally in most two-species interactions. Thus, the methodology may provide a novel approach to investigating such classical ecological phenomena as prey–predator/host–parasite interactions and interspecies competition for resources.Finally, given the growing tendency to utilize evolutionary games in economic modeling, a case can be made for the current paper being of interest to traditional game-theoretic modelers in connection with situations such as buyer/seller or employer/employee interaction.     

Better-reply dynamics and global convergence to Nash equilibrium in aggregative games

We consider n-person games with quasi-concave payoffs that depend on a player's own action and the sum of all players' actions. We show that a discrete-time, stochastic process in which players move towards better replies—the better-reply dynamics—converges globally to a Nash equilibrium if actions are either strategic substitutes or strategic complements for all players around each Nash equilibrium that is asymptotically stable under a deterministic, adjusted best-reply dynamics. We present an example of a 2-person game with a unique equilibrium where the derivatives of the best-reply functions have different signs and the better-reply dynamics does not converge.     

The evolutionary stability of perfectly competitive behavior

Summary. In a (generalized) symmetric aggregative game, payoffs depend only on individual strategy and an aggregate of all strategies. Players behaving as if they were negligible would optimize taking the aggregate as given. We provide evolutionary and dynamic foundations for such behavior when the game satisfies supermodularity conditions. The results obtained are also useful to characterize evolutionarily stable strategies in a finite population.Received: 29 December 2002, Revised: 27 January 2004, JEL Classification Numbers: C72, D41, D43.An earlier version of this paper was titled The Evolutionary Logic of Feeling Small and was written while the first author was affiliated with the University of Salamanca (Spain). We are indebted to Ken Binmore, Larry Blume, Luis Corchón, Georg Kirchsteiger, Paco Marhuenda, Akihiko Matsui, Diego Moreno, Manfred Nermuth, Klaus Ritzberger, Jörgen Weibull, an anonymous referee, and seminar participants at the universities of Vienna, Carlos III de Madrid, and Salamanca for many helpful comments and suggestions. Obviously, only the authors are responsible for any errors and omissions. We gratefully acknowledge financial support from the Austrian Science Fund (FWF) under Project P15281 as well as from the Austrian Exchange Service (ÖAD) and the Spanish Ministry of Education and Culture under the Spain-Austria Acciones Integradas respective projects HU02-4 and 18/2003.     

Out of your mind: Eliciting individual reasoning in one shot games

We experimentally investigate the fundamental element of the level-k model of reasoning, the level-0 actions and beliefs. We use data from a novel experimental design that allows us to obtain incentivised written accounts of individuals' reasoning. In particular, these accounts allow to infer level-0 beliefs. Level-0 beliefs are not significantly different from 50, and almost 60% of higher level players start their reasoning from a level-0 belief of exactly 50. We also estimate that around one third of the participants play non-strategically. The non-strategic level-0 actions are not uniformly distributed.     

The evolution of conventions in minimum effort games

The literature on minimum effort game has been concerned with a symmetric game with linear payoff functions. The main aim of the present paper is to study the coordination problem arising in a not necessarily symmetric minimum effort game with two players. The sources of asymmetry can be twofold: the productivity of effort and the distribution of the join output. To select among the Pareto ranked equilibria we use the stochastic stability criterion. We show that, for any configuration parameters, the set of stochastically stable equilibria coincides with the set of potential maximizers. We also show that when the disutility of effort is linear, the Pareto dominant equilibrium is stochastically stable provided that the distributive parameter belongs to a well defined range. When the disutility of effort is nonlinear no distributive arguments can be used to successfully affect the selection process. Lastly we prove that the connection between stochastic stability and maximum potential can fail when more than two agents are considered.     

Equilibrium selection through incomplete information in coordination games: an experimental study

We perform an experiment on a pure coordination game with uncertainty about the payoffs. Our game is closely related to models that have been used in many macroeconomic and financial applications to solve problems of equilibrium indeterminacy. In our experiment, each subject receives a noisy signal about the true payoffs. This game (inspired by the “global” games of Carlsson and van Damme, Econometrica, 61, 989–1018, 1993) has a unique strategy profile that survives the iterative deletion of strictly dominated strategies (thus a unique Nash equilibrium). The equilibrium outcome coincides, on average, with the risk-dominant equilibrium outcome of the underlying coordination game. In the baseline game, the behavior of the subjects converges to the theoretical prediction after enough experience has been gained. The data (and the comments) suggest that this behavior can be explained by learning. To test this hypothesis, we use a different game with incomplete information, related to a complete information game where learning and prior experiments suggest a different behavior. Indeed, in the second treatment, the behavior did not converge to equilibrium within 50 periods in some of the sessions. We also run both games under complete information. The results are sufficiently similar between complete and incomplete information to suggest that risk-dominance is also an important part of the explanation.      

Coalition formation in general apex games under monotonic power indices

An apex game consists of one apex player and a set of minor players. We identify two key properties of apex games and use them to introduce the class of general apex games. We derive players' preferences over winning coalitions by applying strongly monotonic power indices on such a game and all its subgames and investigate whether there are core stable coalitions in the induced hedonic coalition formation game. Besides several general results, in particular, we develop conditions on the game for the Shapley–Shubik index, the Banzhaf index, and the normalized Banzhaf index.     

Beliefs and rationalizability in games with complementarities

We propose two characteristics of beliefs and study their role in shaping the set of rationalizable strategy profiles in games with incomplete information. The first characteristic, type-sensitivity, is related to how informative a player thinks his type is. The second characteristic, optimism, is related to how “favorable” a player expects the outcome of the game to be. The paper has two main results: the first result provides an upper bound on the size of the set of rationalizable strategy profiles; the second gives a lower bound on the change of location of this set. These bounds are explicit expressions that involve type-sensitivity, optimism, and payoff characteristics. Our results generalize and clarify the well-known uniqueness result of global games (Carlsson and van Damme, 1993). They also imply new uniqueness results and allow us to study rationalizability in new environments. We provide applications to supermodular mechanism design (Mathevet, 2010b) and information processing errors.     

Stochastic evolutionary stability in extensive form games of perfect information

Nöldeke and Samuelson [Games Econ. Behav. 5 (1993) 425] investigate a stochastic evolutionary model for extensive form games and show that even for games of perfect information with a unique subgame perfect equilibrium, non-subgame perfect equilibrium-strategies may well survive in the long run even when mutation rates tend to zero. In a different model of evolution in the agent normal form of these games Hart [Games Econ. Behav. 41 (2002) 227] shows that under suitable limit-taking, where small mutation rates are accompanied by large population sizes in a particular way, the unique prediction is again the subgame perfect equilibrium. This paper provides a proof of a similar result for the model of Nöldeke and Samuelson.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.     

Evolutionary economics goes mainstream: A review of the theory of learning in games

The Theory of Learning in Games by Fudenberg and Levine surveys a key branch of evolutionary economics from a mainstream perspective. Its publication provides an opportunity to reassess the prospects and goals for evolutionary economics.     

Payoff continuity in incomplete information games: a comment

Kajii and Morris (J. Econ. Theory 82 (1998) 267) provide necessary and sufficient conditions for two priors to be strategically close. The restrictiveness of these conditions establishes that strategic behavior can be highly sensitive to the assumed prior. Their results thus recommend care in the use of priors in economic modelling. Unfortunately, their proof of a central proposition fails for zero probability types. This comment corrects their proof to account for these cases.     

Trust and trustworthiness in games: An experimental study of intergenerational advice

This paper investigates the development of conventions of trust in what we call intergenerational games, i.e., games played by a sequence of non-overplapping agents, who pass on advice on how to play the game across adjacent generations of players. Using the trust game of Berg et al. (1995) as our experimental decision problem, advice seems to decrease the amount of trustthat evolves when this game in played in an inter-generational manner in that it decreases the amount of money sent from Senders to Returners. Ironically, advice increases trustworthinessin that Returners tend to send more back. Further, subjects appear to follows conventions of reciprocity in that they tend to Send more if they think the Returners acted in a “kind” manner, where kind means the Sender sent more money than the receiver expected. Finally, while we find a causal relationship running from trustworthiness to trust, the opposite can not be established. We note that many of our results can only be achieved using the tools offered by inter-generational games. The inter-generational advice offered provides information not available when games are played in their static form. Combining that information with elicited beliefs of the Senders and Returners adds even more information that can be used to investigate the motives that subjects have for doing what they do. Electronic supplementary material Electronic supplementary material is available for this article at and accessible for authorised users. JEL Classification C91 · C72 Resources for this research were provided by National Science Foundation grants SBR-9709962 and SBR-9709079 and by both the Center for Experimental Social Science and the C.V. Starr Center for Applied Economics at New York University. We would like to thank Shachar Kariv for both his comments and research assistance. We also thank Mikhael Shor and Judy Goldberg for research assistance, and Yevgeniy Tovshteyn for computer programming.     

Incentives in market games with asymmetric information: the core

Summary. This paper examines the ex ante core of a pure exchange economy with asymmetric information in which state-dependent allocations are required to satisfy incentive compatibility. This restriction on players' strategies in the cooperative game can be interpreted as incomplete contracts or partial commitment. An example is provided in which the incentive compatible core with nontransferable utility is empty; the game fails to be balanced because convex combinations of incentive compatible net trades can violate incentive compatibility. However, randomization of such strategies leads to ex post allocations which satisfy incentive compatibility and are feasible on average. Hence, convexity is preserved in such a model and the resulting cooperative games are balanced. In this framework, an incentive compatible core concept is defined for NTU games derived from economies with asymmetric information. The main result is nonemptiness of the incentive compatible core. Received: December 26, 2001; revised version: June 11, 2002 RID="*" ID"*" This work was financed, in part, by contract No 26 of the programme “P?le d'attraction interuniversitaire” of the Belgian government, and, in part, by research grant SBR93-09854 from the U.S. National Science Foundation. Much of my thinking about this topic was developed during a wonderful visit to CORE for the 1991–1992 academic year (on sabbatical from the University of Pennsylvania). This paper was originally circulated in December 1991 as CARESS Working Paper #91-38, Center for Analytic Research in Economics and the Social Sciences, Department of Economics, University of Pennsylvania and in February 1992 as CORE Discussion Paper 9221, Center for Operations Research and Econometrics, Université Catholique de Louvain, Louvain-la-Neuve, Belgium. RID="*" ID="*" At the very start of my research, Jean-Fran?ois Mertens was almost a co-author. Fran?ois Forges provided detailed comments at a later stage, during my visit to THEMA, Université Cergy-Pontoise, in Spring 1997. They are entitled to the customary disclaimer.     

One for all and all for one: privatization and Universal Service provision in the postal sector

Universal Service provision has a special role for the public utilities sector in many Organization for Economic Co-operation and Development (OECD) countries. These public utilities have largely been subject to privatization during the last 3 decades. Efficiency effects of privatization are widely documented while the impacts on the quality and accessibility of the Universal Service are not much examined. By using a unique dataset on privatization for 21 countries over the period 1980–2007 for the postal sector, we are able to show that privatization, in particular formal privatization, has led to a decrease in the quality of the Universal Service.     

Hope springs eternal: learning and the stability of cooperation in short horizon repeated games

This paper considers learning rates in finitely repeated prisoners’ dilemmas. If players think their opponents might be relatively cooperative (e.g., tit-for-tat or grim types), they will cooperate in finitely repeated prisoners’ dilemmas (see Kreps et al., J. Econom. Theory 27 (1982) 245). However, if there are actually no cooperative types, players will eventually learn this and cooperation will break down. This paper shows that this learning is extremely slow, so it will take an extremely long time for cooperation to break down.Thus, suppose the world is either “good” or “bad.” The probability of a grim type is δ>0 if the world is good, and zero if the world is bad. Successive generations pair up to play finitely repeated prisoners’ dilemmas. Players observe play in previous generations and use Bayes’ rule to update their prior, π, that the world is good. We show that, if the world is really bad, then π falls per generation on average. Thus, if δ is small, there is less cooperation if the world is good, but cooperation may become more stable. For a representative 19 period repeated prisoners’ dilemma, beliefs fall one percentage point on average after a thousand generations.To derive these learning rates, we must refine existing results on the sensitivity of repeated games to Kreps et al. (1982) type perturbations. Specifically, we show cooperation is possible in perturbed prisoners’ dilemmas repeated O(log(1/δ)) times. This improves significantly on the O(1/δ) results in previous work. The paper thus provides two new reasons why cooperation tends to be stable, even in short horizon repeated games.     

The value and consequences of using public health technology assessments for private payer decision-making in Canada: one size does not fit all

Background: Both public and private insurers provide drug coverage in Canada. All payers are under pressure to contain costs. It has recently been proposed that private plans leverage the public health technology assessment (HTA) evaluation process in their decision-making.

Objectives: The objectives of the current study were to examine use of public health technology assessments (HTAs) for private payer decision-making in the literature, to gather the perspectives of experts from both public and private insurers on this practice, and to summarize which value parameters of public evaluations can be used for private payer decision-making.

Methods: A targeted literature review was conducted to identify publications on the use of public HTA or cost-effectiveness data for private payer decision-making on pharmaceutical reimbursement. Concurrently, a roundtable meeting was organized with invited panelists, including private payer representatives and health economic consultants (total n?=?9). The findings from both were synthesized and expressed in qualitative terms using the PICO framework.

Results: The targeted review identified 20 studies meeting the inclusion criteria, primarily originating from the US and Canada. The panelists felt that, despite some similarities, there were substantial differences between both systems. The PICO framework highlighted the issues with transferability between the two systems. Most of the value parameters were either not applicable, needed to be added, needed to be adjusted, or their applicability to private payer systems needed to be confirmed.

Conclusion: Some components of public HTA may be relevant for private payers, however there are reservations that still exist on whether the HTA process in Canada, designed for a public system, can address the informational needs of private payers. Private insurers need to use caution in assessing which value parameters from public HTAs can be used and which need to be confirmed, ignored, enhanced, or adjusted. One size HTA does not fit all applications.     

Political aspects of household finance: debt,wage bargaining,and macroeconomic (in)stability

The recent literature has shown that income inequality is one of the main causes of borrowing and debt accumulation by working households. This article explores the possibility that household indebtedness is an important cause of rising income inequality. If workers experience rising debt burdens, their cost of job loss may rise if they need labor-market income to continue borrowing and servicing existing debt. This, in turn, will reduce their bargaining power and increase income inequality, inducing workers to borrow more to maintain consumption standards, and so creating a vicious circle of rising inequality, job insecurity, and indebtedness. We believe that these dynamics may have contributed to observed simultaneous increases in income inequality and household debt prior to the recent financial crisis. To explore the two-way interaction between inequality and debt, we develop an employment rent framework that explicitly considers the impact of workers’ indebtedness on their perceived cost of job loss. This is embedded in a neo-Kaleckian macro model in which inequality spurs debt accumulation that contributes to household consumption spending and hence demand formation. Our analysis suggests that (a) workers’ borrowing behavior plays a crucial role in understanding the character of demand and growth regimes; (b) debt and workers’ borrowing behavior play an important role in the labor market by influencing workers’ bargaining power; and (c) through such channels, workers’ borrowing behavior can be a decisive factor in the determination of macroeconomic (in)stability.