Are “Anti-Folk Theorems” in repeated games nongeneric?

Folk Theorems in repeated games hold fixed the game payoffs, while the discount factor is varied freely. We show that these results may be sensitive to the order of limits in situations where players move asynchronously. Specifically, we show that when moves are asynchronous, then for a fixed discount factor close to one there is an open neighborhood of games which contains a pure coordination game such that every Perfect equilibrium of every game in the neighborhood approximates to an arbitrary degree the unique Pareto dominant payoff of the pure coordination game.     

Repeated two-person zero-sum games with unequal discounting and private monitoring

We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.     

Constrained mobility and the evolution of efficient outcomes

We study an evolutionary model akin to the one studied in Anwar (2002) where a set of agents use myopic best response learning to i) determine their action in a 2 × 2 coordination game and ii) to choose on which of multiple islands to interact. We focus on the case where the number of agents maximally allowed on each islands is constrained. We extend Anwar’s original analysis by considering the case when there may be more than two islands. We find that if the constraints are such that one island may be empty, universal coordination on the payoff dominant action is possible in the long run. If the constraints are such that all islands will be full, then for relatively mild constraints, and apart from special cases, the coexistence of conventions will occur, with one island coordinating on the risk dominant action and all remaining islands coordinating on the payoff dominant action. For relatively stringent constraints all agents will play the risk dominant action.     

Implementability via protective equilibria

We present a notion of non-cooperative strategic equilibrium for games generated by social choice functions, and fully characterize the class of those functions which are directly implementable under this equilibrium concept. Correct preference revelation turns out to always be such an equilibrium for the games generated by this class of implementable functions.     

Simple collective equilibria in stopping games

At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.     

Robust games: theory and application to a Cournot duopoly model

In this paper, the robust game model proposed by Aghassi and Bertsimas (Math Program Ser B 107:231–273, 2006) for matrix games is extended to games with a broader class of payoff functions. This is a distribution-free model of incomplete information for finite games where players adopt a robust-optimization approach to contend with payoff uncertainty. They are called robust players and seek the maximum guaranteed payoff given the strategy of the others. Consistently with this decision criterion, a set of strategies is an equilibrium, robust-optimization equilibrium, if each player’s strategy is a best response to the other player’s strategies, under the worst-case scenarios. The aim of the paper is twofold. In the first part, we provide robust-optimization equilibrium’s existence result for a quite general class of games and we prove that it exists a suitable value \(\epsilon \) such that robust-optimization equilibria are a subset of \(\epsilon \)-Nash equilibria of the nominal version, i.e., without uncertainty, of the robust game. This provides a theoretical motivation for the robust approach, as it provides new insight and a rational agent motivation for \(\epsilon \)-Nash equilibrium. In the last part, we propose an application of the theory to a classical Cournot duopoly model which shows significant differences between the robust game and its nominal version.     

On a theorem of Schmeidler

Over a decade ago D. Schmeidler (1973) introduced a concept of non-cooperative equilibrium for games with a continuum of agents and, under a restriction on the payoff functions, established the existence of an equilibrium in pure strategies. The purpose of this note is to reformalize the model and the equilibrium notion of Schmeidler in terms of distributions rather than measurable functions. We shall see how once the definitions are available we get (pure strategy) equilibrium existence theorems quite effortlessly and under general conditions. A number of remarks contain applications to, among others, incomplete information games.     

Risk aversion in a model of endogenous growth

Despite the evidence on incomplete financial markets and substantial risk being borne by innovators, current models of growth through creative destruction predominantly model innovators’ as risk neutral. Risk aversion is expected to reduce the incentive to innovate and we might fear that without insurance innovation completely disappears in the long run. The present paper introduces risk averse agents into an occupational choice model of endogenous growth in which insurance against failure to innovate is not available. We derive a clear negative relationship between the level of risk aversion and long run growth. Surprisingly, we show that in an equilibrium there exists a cut-off value of risk aversion below which the growth rate of the mass of innovators tends to a strictly positive constant. In this case, innovation persists on the long run and consumption per capita grows at a strictly positive rate. On the other hand, for levels of risk aversion above the cut-off value, the economy eventually stagnates.     

Multilateral negotiations and formation of coalitions

This paper analyses properties of games modeling multilateral negotiations leading to the formation of coalitions in an environment with widespread externalities. The payoff generated by each coalition is determined by an exogenous partition function (the parameter space). We show that in almost all games, except in a set of measure zero of the parameter space, the Markov perfect equilibrium value of coalitions and the state transition probability that describe the path of coalition formation is locally unique and stable. Therefore, comparative statics analysis are well-defined and can be performed using standard calculus tools. Global uniqueness does not hold in general, but the number of equilibria is finite and odd. In addition, a sufficient condition for global uniqueness is derived, and using this sufficient condition we show that there is a globally unique equilibrium in three-player superadditive games.     

Truth-telling and trust in sender–receiver games with intervention: an experimental study

Recent experimental studies find excessive truth-telling in sender–receiver games. We show that this phenomenon is robust to the random intervention of a truthful regulator. In addition, intervention significantly increases the excessive trust of receivers while the overall percentage of truthful messages received does not change much with or without intervention. We offer a theoretical explanation for the behavior of senders and receivers, using a logit agent quantal response equilibrium (logit-AQRE) model incorporating a non-monetary lying cost for senders (like Peeters et al. in Scand J Econ 115(2):508–548, 2013). We show that our experimental findings are all consistent with the predictions of this model. Moreover, we find that the lying cost is significantly higher under intervention, implying that truthful intervention is beneficial for receivers and justified as a tool for policy makers acting on behalf of informationally inferior parties.     

Perfect and limit admissible perfect equilibria in discontinuous games

We compare the properties of several notions of trembling-hand perfection within classes of compact, metric, and possibly discontinuous games, and show that in the presence of payoff discontinuities, standard notions of trembling-hand perfection fail a weakening of admissibility termed limit admissibility. We also provide conditions ensuring the existence of a limit admissible perfect equilibrium.     

Bounded memory and permanent reputations

A recent result in repeated incomplete information games is that after an arbitrarily long history, any equilibrium of the continuation game must be an equilibrium of the complete information game. This result implies that reputation is a short-run phenomenon. We study a particular class of reputation games and show that bounded memory may lead to permanent reputations. In fact, for a particular range of parameters, a bounded memory player may never be able to learn anything at all.     

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

Do Smaller Classes Always Improve Students’ Long‐run Outcomes?

We exploit the strict class size rule in Norway and matched individual and school register information for 1982–2011 to estimate long‐run causal effects on income and educational attainment. Contrary to recent evidence from the US and Sweden, we do not find any significant average effect on long‐run outcomes of reduced class size. We further use the large register data set and quasi‐experimental strategy to estimate whether the class size effect depends on external conditions facing students and schools, such as teacher quality, extent of upper secondary school choice, school district size, local fiscal constraints and labour market conditions. Overall, we find that the class size effect does not depend on these factors measured at the school district level. The absence of class size effects on long‐run outcomes in Norway is consistent with earlier findings for short‐run outcomes, using comparable data and empirical strategies.     

Popularity of reinforcement-based and belief-based learning models: An evolutionary approach

In an evolutionary model, players from a given population meet randomly in pairs each instant to play a coordination game. At each instant, the learning model used is determined via some replicator dynamics that respects payoff fitness. We allow for two such models: a belief-based best-response model that uses a costly predictor, and a costless reinforcement-based one. This generates dynamics over the choice of learning models and the consequent choices of endogenous variables. We report conditions under which the long run outcomes are efficient (or inefficient) and they support the exclusive use of either of the models (or their co-existence).     

On the equivalence of location choice models: Conditional logit, nested logit and Poisson

It is well understood that the two most popular empirical models of location choice - conditional logit and Poisson - return identical coefficient estimates when the regressors are not individual specific. We show that these two models differ starkly in terms of their implied predictions. The conditional logit model represents a zero-sum world, in which one region’s gain is the other regions’ loss. In contrast, the Poisson model implies a positive-sum economy, in which one region’s gain is no other region’s loss. We also show that all intermediate cases can be represented as a nested logit model with a single outside option. The nested logit turns out to be a linear combination of the conditional logit and Poisson models. Conditional logit and Poisson elasticities mark the polar cases and can therefore serve as boundary values in applied research.     

Solving dynamic public insurance games with endogenous agent distributions: Theory and computational approximation

We make two contributions in this paper. First, we extend the characterization of equilibrium payoff correspondences in history-dependent dynamic policy games to a class with endogenously heterogeneous private agents. In contrast to policy games involving representative agents, this extension has interesting consequences as it implies additional nonlinearity (i.e., bilinearity) between the game states (distributions) and continuation/promised values in the policymaker’s objective and incentive constraints. The second contribution of our paper is in addressing the computational challenges arising from this payoff-relevant nonlinearity. Exploiting the game’s structure, we propose implementable approximate bilinear programming formulations to construct estimates of the equilibrium value correspondence. Our approximation method respects the property of upper hemicontinuity in the target correspondence. We provide small-scale computational examples as proofs of concept.     

What restrictions do Bayesian games impose on the value of information?

In a Bayesian game players play an unknown game. Before the game starts some players may receive a signal regarding the specific game actually played. Typically, information structures that determine different signals, induce different equilibrium payoffs. In two-person zero-sum games the equilibrium payoff measures the value of the particular information structure which induces it. We pose a question as to what restrictions Bayesian games impose on the value of information. We provide answers for two kinds of information structures: symmetric, where both players are equally informed, and one-sided, where only one player is informed.     

Optimal risk taking in an uneven tournament game with risk averse players

We analyze the optimal choice of risk in a two-stage tournament game between two players that have different concave utility functions. At the first stage, both players simultaneously choose risk. At the second stage, both observe overall risk and simultaneously decide on effort or investment. The results show that those two effects which mainly determine risk taking – an effort effect and a likelihood effect – are strictly interrelated. This finding sharply contrasts with existing results on risk taking in tournament games with symmetric equilibrium efforts where such linkage can never arise. Conditions are derived under which this linkage leads to a reversed likelihood effect so that the favorite (underdog) can increase his winning probability by increasing (decreasing) risk which is impossible in a completely symmetric setting.     

Generic finiteness of equilibrium payoffs for bimatrix games

It is shown that in any affine space of payoff matrices the equilibrium payoffs of bimatrix games are generically finite.