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This paper considers extended prisoners' dilemma games in which a second pure strategy equilibrium in the stage game allows for mutual cooperation in all but the last round of the finitely repeated game as an equilibrium outcome. We distinguish a strict and a weak extension of the prisoners' dilemma game in a long and a short horizon treatment. A comparison with the corresponding finitely repeated prisoners' dilemma games shows that the strict additional equilibrium increases cooperation rates while the weak does not. This result is robust to the variation of the time horizon.

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... Using a backward-induction argument, it is easy to show that the unique subgame-perfect equilibrium in repeated and multistage games is to implement Nash equilibrium controls at each stage of the finite game. This clear-cut theoretical result has not always received empirical support, and in fact, experiments show that cooperation may be realized, at least partially, in finitehorizon games (see, e.g., Angelova et al. 2013). The literature has came out with different ways to cope with the difficulties in enforcing cooperation in finite-horizon dynamic games. ...

In many instances, players find it individually and collectively rational to sign a long-term cooperative agreement. A major concern in such a setting is how to ensure that each player will abide by her commitment as time goes by. This will occur if each player still finds it individually rational at any intermediate instant of time to continue to implement her cooperative control rather than switch to a noncooperative control. If this condition is satisfied for all players, then we say that the agreement is time consistent. This chapter deals with the design of schemes that guarantee time consistency in deterministic differential games with transferable payoffs.

... Indeed, experiments show that cooperation may be realized, at least partially, in finite-horizon games (see, e.g. [1]). ...

It is a challenge to sustain cooperation in a finite-horizon dynamic game. Since players generally have an incentive to deviate to their noncooperative strategies in the last stage, a backward induction argument leads them to defect from cooperation in all stages. In this paper, we propose two payment schemes having some desirable properties, namely, individual rationality and stability, which ensure that the players cooperate throughout the entire planning horizon. The setup and the results are general, that is, they do not rest on particular specifications of the payoff functionals or the state dynamics. We illustrate our results with a linear-quadratic dynamic game of pollution control.

Purpose
The purpose of this paper is to examine whether managers punish more and work harder in teams with peer monitoring when it is less costly to punish in a two-period, one-shot horizon.
Design/methodology/approach
An experiment is conducted in a two-period horizon with two treatments. The structure of performance measures makes it costless or costly to punish in the second period.
Findings
The results find punishing, contingent on first-period strategies, was significantly greater when it was costless compared to costly, as expected. Working, which is analogous to cooperating in prisoner dilemma games, was also significantly greater in the first and second periods when punishing was costless.
Practical implications
This paper is informative about the potential benefits of performance measures in dynamic team environments, which can be challenging and costly to develop. It adds insight into the design of self-discipline and tasks in teams which might help increase productivity.
Originality/value
This paper is related to the research on indefinite horizons, which attributes increases in cooperation to the existence of subgame perfect strategies to cooperate and potential gains from future cooperation. In comparison, this study examines the effects of the existence of subgame perfect strategies to work in isolation from the potential gains from future interactions. In addition, it examines whether their potential benefits depend on the cost of punishing when punishing is subgame perfect in a one-shot horizon.

Decentralized and impersonal exchange is fundamental to contemporary economies, where many interactions take place among individuals with low levels of information about their counterpart. We review the experimental literature about markets with frictions, where strangers interact in pairs formed at random in economies of indefinite duration. We focus on the impact of communication on the efficiency of the outcome and report results of a new experiment.

Cooperation in prisoner's dilemma games can usually be sustained only if the game has an infinite horizon. We analyze to what extent the theoretically crucial distinction of finite vs. infinite-horizon games is reflected in the outcomes of a prisoner's dilemma experiment. We compare three different experimental termination rules in four treatments: a known finite end, an unknown end, and two variants with a random termination rule (with a high and with a low continuation probability, where cooperation can occur in a subgame-perfect equilibrium only with the high probability). We find that the termination rules do not significantly affect average cooperation rates. Specifically, employing a random termination rule does not cause significantly more cooperation compared to a known finite horizon, and the continuation probability does not significantly affect average cooperation rates either. However, the termination rules may influence cooperation over time and end-game behavior. Further, the (expected) length of the game significantly increases cooperation rates. The results suggest that subjects may need at least some learning opportunities (like repetitions of the supergame) before significant backward induction arguments in finitely repeated game have force.

This paper conductsan experiment to investigate the economic effect of public disclosureswithin a multi-move adaptation of the Prisoner''s Dilemma game.The game, which has multiple equilibria, is characterized by:(1) a stochastic endpoint, (2) random, repeated pairings withanonymous partners, and (3) public disclosures concerning thecurrent partner''s previous strategies. In the experiment, cooperationis improved by the disclosures. In addition, subjects cooperatemore frequently when encountering a player who has tended tocooperate in the past, and less frequently when encounteringa player who has tended to defect in the past. Delayed disclosureleads to levels of cooperation only slightly less than thoseobtained with timely disclosure.

z-Tree (Zurich Toolbox for Ready-made Economic Experiments) is a software for developing and conducting economic experiments. The software is stable and allows programming almost any kind of experiments in a short time. In this article, I present the guiding principles behind the software design, its features, and its limitations. Copyright Economic Science Association 2007

The financial crisis of 2008, which started with an initially well-defined epicenter focused on mortgage backed securities (MBS), has been cascading into a global economic recession, whose increasing severity and uncertain duration has led and is continuing to lead to massive losses and damage for billions of people. Heavy central bank interventions and government spending programs have been launched worldwide and especially in the USA and Europe, with the hope to unfreeze credit and boltster consumption. Here, we present evidence and articulate a general framework that allows one to diagnose the fundamental cause of the unfolding financial and economic crisis: the accumulation of several bubbles and their interplay and mutual reinforcement has led to an illusion of a ``perpetual money machine'' allowing financial institutions to extract wealth from an unsustainable artificial process. Taking stock of this diagnostic, we conclude that many of the interventions to address the so-called liquidity crisis and to encourage more consumption are ill-advised and even dangerous, given that precautionary reserves were not accumulated in the ``good times'' but that huge liabilities were. The most ``interesting'' present times constitute unique opportunities but also great challenges, for which we offer a few recommendations.

Theoretical work starting with Stigler (1964) suggests that collusion may be difficult to sustain in a repeated game with secret price cuts and demand uncertainty. Compared to equilibria in games of perfect information, trigger-strategy equilibria in this context result in lower payoffs because punishments occur along the equilibrium path. We tested the theory in a series of economic experiments. Consistent with the theory, treatments with imperfect information were less collusive than treatments with perfect information. However, in the imperfect-information treatments, players seemed to settle on the static Nash outcome rather than using trigger strategies. Players did resort to punishments for undercutting in perfect-information treatments, and this sometimes led to successful collusion afterward.

1 Introduction.- 1.1 Informal Description of Games and Game Theory.- 1.2 Dynamic Programming.- 1.3 Subgame Perfect Equilibria.- 1.4 Sequential Equilibria and Perfect Equilibria.- 1.5 Perfect, Proper and Persistent Equilibria.- 1.6 Essential Equilibria and Regular Equilibria.- Notes.- 2 Games in Normal Form.- 2.1 Preliminaries.- 2.2 Perfect Equilibria.- 2.3 Proper Equilibria.- 2.4 Essential Equilibria.- 2.5 Regular Equilibria.- 2.6 An "Almost all" Theorem.- Notes.- 3 Matrix and Bimatrix Games.- 3.1 Preliminaries.- 3.2 Perfect Equilibria.- 3.3 Regular Equilibria.- 3.4 Characterizations of Regular Equilibria.- 3.5 Matrix Games.- Notes.- 4 Control Costs.- 4.1 Introduction.- 4.2 Games with Control Costs.- 4.3 Approachable Equilibria.- 4.4 Proper Equilibria.- 4.5 Perfect Equilibria.- 4.6 Regular Equilibria.- Notes.- 5 Incomplete Information.- 5.1 Introduction.- 5.2 Disturbed Games.- 5.3 Firm Equilibria.- 5.4 Perfect Equilibria.- 5.5 Weakly Proper Equilibria.- 5.6 Strictly Proper Equilibria and Regular Equilibria.- 5.7 Proofs of the Theorems of Sect. 5.5.- Notes.- 6 Extensive Form Games.- 6.1 Definitions.- 6.2 Equilibria and Subgame Perfectness.- 6.3 Sequential Equilibria.- 6.4 Perfect Equilibria.- 6.5 Proper Equilibria.- 6.6 Control Costs.- 6.7 Incomplete Information.- Notes.- 7 Bargaining and Fair Division.- 7.1 Introduction.- 7.2 Divide and Choose.- 7.3 Auction Methods.- 7.4 Bargaining Problems and Bargaining Solutions.- 7.5 The Nash Negotiation Game.- 7.6 The Rubinstein/Binmore Model.- 7.7 The Crawford/Moulin Model.- 7.8 Bargaining Games with Variable Threat Point.- Notes.- 8 Repeated Games.- 8.1 Introduction.- 8.2 Preliminaries.- 8.3 Infinitely Repeated Games Without Discounting.- 8.4 Infinitely Repeated Games with Discounting: Nash Equilibria.- 8.5 Infinitely Repeated Games with Discounting: Subgame Perfect Equilibria.- 8.6 Finitely Repeated Games: Nash Equilibria.- 8.7 Finitely Repeated Games: Subgame Perfect Equilibria.- 8.8 Renegotiation-Proof Equilibria.- Notes.- 9 Evolutionary Game Theory.- 9.1 Introduction.- 9.2 Evolutionarily Stable Strategies.- 9.3 Strategic Stability of ESS.- 9.4 Population Dynamics.- 9.5 Asymmetric Contests: Examples and the Model.- 9.6 Asymmetric Contests: Results.- 9.7 Contests in Extensive Form: Definitions.- 9.8 Contests in Extensive Form: Results.- Notes.- 10 Strategic Stability and Applications.- 10.1 Equivalence of Games.- 10.2 Requirements for Strategic Stability.- 10.3 Stable Equilibria.- 10.4 Signalling Games: Introduction.- 10.5 Signalling Games: Dominance, Intuitive Arguments and Stability.- 10.6 Spence's Job Market Signalling Model.- 10.7 The Chain Store Paradox.- 10.8 Repeated Games.- Notes.- References.- Survey Diagrams.

In an experiment, players’ ability to learn to cooperate in the repeated prisoner’s dilemma was substantially diminished when the payoffs were noisy, even though players could monitor one another's past actions perfectly. In contrast, in one-time play against a succession of opponents, noisy payoffs increased cooperation, by slowing the rate at which cooperation decays. These observations are consistent with the robust observation from the psychology literature that partial reinforcement (adding randomness to the link between an action and its consequences while holding expected payoffs constant) slows learning. This effect is magnified in the repeated game: When others are slow to learn to cooperate, the benefits of cooperation are reduced, which further hampers cooperation. These results show that a small change in the payoff environment, which changes the speed of individual learning, can have a large effect on collective behavior. And they show that there may be interesting comparative dynamics that can be derived from careful attention to the fact that at least some economic behavior is learned from experience. Economics

Using a symmetric two-player prisoners’ dilemma as base game, each player receives a signal for the number of rounds to be
played with the same partner. One of these signals is the true number of rounds R while the other is R − 5. Thus both players know that the game has a finite end. They both know that the opponent knows this, but the finite end
is not commonly known. As a consequence, both mutual defection and mutual cooperation until the second last round are subgame
perfect equilibrium outcomes. We find experimental evidence that many players do in fact cooperate beyond their individual
signal round.
KeywordsPrisoners’ dilemma–Continuation probability–Uncertainty–Experiment

The concept of a perfect equilibrium point has been introduced in order to exclude the possibility that disequilibrium behavior is prescribed on unreached subgames [SELTEN 1965 and 1973]. Unfortunately this definition of perfectness does not remove all difficulties which may arise with respect to unreached parts of the game. It is necessary to reexamine the problem of defining a satisfactory non-cooperative equilibrium concept for games in extensive form. Therefore a new concept of a perfect equilibrium point will be introduced in this paper2). In retrospect the earlier use of the word "perfect" was premature. Therefore a perfect equilibrium point in the old sense will be called "subgame perfect". The new definition of perfectness has the property that a perfect equilibrium point is always subgame perfect but a subgame perfect equilibrium point may not be perfect. It will be shown that every finite extensive game with perfect recall has at least one perfect equilibrium point. Since subgame perfectness cannot be detected in the normal form, it is clear that for the purpose of the investigation of the problem of perfectness, the normal form is an inadequate representation of the extensive form. It will be convenient to introduce an "agent normal form" as a more adequate representation of games with perfect recall.

We consider stability of Selten's perfect equilibrium point against slight imperfections of rationality of players. As its stability is not sufficient, we strengthen the perfectness concept and define astrictly perfect equilibrium point. We provide sufficient conditions for this equilibrium point.

A usual criticism of the theory of infinitely repeated games is that it does not provide sharp predictions since there may be a multiplicity of equilibria. To address this issue, we present experimental evidence on the evolution of cooperation in infinitely repeated prisoner's dilemma games as subjects gain experience. We show that cooperation may prevail in infinitely repeated games, but the conditions under which this occurs are more stringent than the subgame perfect conditions usually considered or even a condition based on risk dominance. (JEL C71, C73)

Selten's concept of perfect equilibrium for normal form games is reviewed, and a new concept of proper equilibrium is defined. It is shown that the proper equilibria form a nonempty subset of the perfect equilibria, which in turn form a subset of the Nash equilibria. An example is given to show that these inclusions may be strict.

In general, the result of the elimination of weakly dominated strategies depends on order. We definenice weak dominance. Under nice weak dominance, order does not matter. We identify an important class of games under which nice weak dominance and weak dominance are equivalent, and so the order under weak dominance does not matter. For all games, the result of iterative nice weak dominance is an upper bound on the result from any order of weak dominance. The results strengthen the intuitive relationship between backward induction and weak dominance and shed light on some computational problems relating to weak dominance.Journal of Economic LiteratureClassification Number: C72.

In a game of a finite number of repetitions of a Cournot-type model of an industry, if firms are satisfied to get close to (but not necessarily achieve) their optimal responses to other firms' sequential strategies, then in the resulting noncooperative “equilibria” of the sequential market game, (1) if the lifetime of the industry is large compared to the number of firms, there are equilibria corresponding to any given duration of the cartel, whereas (2) if the number of firms is large compared to the industry's lifetime, all equilibria will be close (in some sense) to the competitive equilibrium.

A common observation in experiments involving finite repetition of the prisoners' dilemma is that players do not always play the single-period dominant strategies (“finking”), but instead achieve some measure of cooperation. Yet finking at each stage is the only Nash equilibrium in the finitely repeated game. We show here how incomplete information about one or both players' options, motivation or behavior can explain the observed cooperation. Specifically, we provide a bound on the number of rounds at which Fink may be played, when one player may possibly be committed to a “Tit-for-Tat” strategy.

Contemporary political theory often assumes that individuals cannot make credible commitments where substantial temptations exist to break them unless such commitments are enforced by an external agent. One such situation may occur in relation to common pool resources, which are natural or man-made resources whose yield is subtractable and whose exclusion is nontrivial (but not necessarily impossible). Examples include fisheries, forests, grazing ranges, irrigation systems, and groundwater basins. Empirical evidence, however, suggests that appropriators in common pool resources develop credible commitments in many cases without relying on external authorities. We present findings from a series of experiments exploring (1) covenants alone (both one-shot and repeated communication opportunities); (2) swords alone (repeated opportunities to sanction each other); and (3) covenants combined with an internal sword (one-shot communication followed by repeated opportunities to sanction each other).

The last decade has seen a steady increase in the application of concepts from noncooperative game theory to such diverse fields as economics, political science, law, operations research, biology and social psychology. As a byproduct of this increased activity, there has been a growing awareness of the fact that the basic noncooperative solution concept, that of Nash equilibrium, suffers from severe drawbacks. The two main shortcomings of this concept are the following: (i) In extensive form games, a Nash strategy may prescribe off the equilibrium path behavior that is manifestly irrational. (Specifically, Nash equilibria may involve incredible threats), (ii) Nash equilibria need not be robust with respect to small perturbations in the data of the game. Confronted with the growing evidence to the detriment of the Nash concept, game theorists were prompted to search for more refined equilibrium notions with better properties and they have come up with a wide array of alternative solution concepts. This book surveys the most important refinements that have been introduced. Its objectives are fourfold (i) to illustrate desirable properties as well as drawbacks of the various equilibrium notions by means of simple specific examples, (ii) to study the relationships between the various refinements, (iii) to derive simplifying characterizations, and (iv) to discuss the plausibility of the assumptions underlying the concepts.

For stategic-form games with communication, acceptable correlated equilibria are defined as correlated equilibria that are stable when every player has an infinitesimal probability of trembling to any of his feasible actions. A set of acceptable actions is defined for each player, and it is shown that a correlated equilibrium is acceptable if and only if all unacceptable actions have zero probability. The unacceptable actions can be found by computing certain vectors called codomination systems, which extend the concept of dominated actions. Predominant correlated equilibria are defined by iterative elimination of unacceptable actions and are shown to exist.

In this paper we introduce the Online Recruitment System for Economic Experiments (ORSEE). With this software experimenters have a free, convenient and very powerful tool to organize their experiments and sessions in a standardized way. Additionally, ORSEE provides subject pool statistics, a laboratory calendar, and tools for scientific exchange. A test system has been installed in order to visually support the reader while reading the paper.

The authors, two of the most prominent game theorists of this generation, have devoted a number of years to the development of the theory presented here, and to its economic applications. They propose rational criteria for selecting one particular uniformly perfect equilibrium point as the solution of any noncooperative game. And, because any cooperative game can be remodelled as a noncooperative bargaining game, their theory defines a one-point solution for any cooperative game as well. By providing solutions - based on the same principles of rational behavior - for all classes of games, both cooperative and noncooperative, both those with complete and with incomplete information, Harsanyi and Selten's approach achieves a remarkable degree of theoretical unification for game theory as a whole and provides a deeper insight into the nature of game-theoretic rationality. The book applies this theory to a number of specific game classes, such as unanimity games; bargaining with transaction costs; trade involving one seller and several buyers; two-person bargaining with incomplete information on one side, and on both sides. The last chapter discusses the relationship of the authors' theory to other recently proposed solution concepts, particularly the Kohberg-Mertens stability theory.

In equal punishment games like in ultimatum games first a proposer suggests how to split the pie, i.e. a positive monetary reward. Unlike in ultimatum games, the responder can decide among many (for proposer and responder) equal penalty payments. To exclude negative payoffs, punishment was bounded from above by the sum of the offer and the (for proposer and responder) same show up-fee, our only treatment variable. Although inequality aversion predicts zero-punishments, we observe positive punishments which however, decrease with experience, Initial fairness, 1/3 of initial offers were equal splits, is often substituted in the repetition by greed. Whereas greed is sticky, fairness seems to be an initial inclination but unstable.

While there is an extensive literature on the theory of infinitely repeated games, empirical evidence on how "the shadow of the future" affects behavior is scarce and inconclusive. I simulate infinitely repeated prisoner's dilemma games in the lab with a random continuation rule. The experimental design represents an improvement over the existing literature by including sessions with finite repeated games as controls and a large number of players per session (which allows for learning without contagion effects). I find that the shadow of the future matters not only by significantly reducing opportunistic behavior, but also because its impact closely follows theoretical predictions.

We use a laboratory experiment to study the extent to which investors’ choices are affected by limited loss deduction in income taxation. We first compare investment behavior in the no tax baseline to a tax control setting, in which the income from investments is taxed. We find that investors significantly reduce their risk-taking as predicted by theory. Next we compare the baseline investment choices to choices under three different types of income taxation. We observe that risk-taking is significantly increased with partial and with capped loss deduction, but is unaffected by a tax system that allows no loss deduction. Since in all these treatments the after tax outcomes of the prospects were identical, we conjecture that investors have a positively biased perception of partial and capped loss deduction that promotes their willingness to take risks.

We study subgame perfect equilibria of finitely repeated games. We prove a limit "folk theorem" for these games. Under weak conditions, any feasible and individually rational payoff vector of the one-shot game can be approximated by the average payoff in a perfect equilibrium of a repeated game with a sufficiently long horizon.

- van Damme