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.     

A test of the gambler's fallacy: Evidence from pari-mutuel games

The “gambler's fallacy” is the belief that the probability of an event is decreased when the event has occurred recently, even though the probability is objectively known to be independent across trials. Clotfelter and Cook (1991, 1993) find evidence of the gambler's fallacy in analysis of data from the Maryland lottery's“Pick 3” numbers game. In the Maryland lottery, the payout to all numbers is equal at $250 on a winning fifty-cent bet, so the gambler's fallacy betting strategy costs bettors nothing. This article looks at the importance of the gambler's fallacy in the New Jersey lottery's three-digit numbers game, a pari-mutuel game where a lower amount of total wagering on a number increases the payout to that number. Results indicate that the gambler's fallacy exists among bettors in New Jersey, although to a lesser extent than among those in Maryland.     

Outsourcing with identical suppliers and shortest-first policy: a laboratory experiment

We study experimentally in the laboratory two 2-player games that mimic a decentralized decision-making situation in which firms repeatedly outsource production orders to multiple identical suppliers. The first game has a unique (inefficient) equilibrium in mixed strategies, while the second game has two (efficient) equilibria in pure strategies and an infinite number of (inefficient) equilibria in mixed strategies. In both games, the optimal social costs can also be obtained via dominated strategies. We find that only in the second game subjects manage to reach an efficient outcome more often when matched in fixed pairs than when randomly rematched each round. Surprisingly, this is because subjects coordinate on dominated strategies (and not an efficient pure strategy equilibrium). We show theoretically that preferences for efficiency cannot explain our experimental results. Inequality aversion, on the other hand, cannot be rejected.     

Backward Induction Is Not Robust: The Parity Problem and the Uncertainty Problem

A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known endpoint in the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed.     

Long-Term Behavior in the Theory of Moves

This paper proposes a revised Theory of Moves (TOM) to analyze matrix games between two players when payoffs are given as ordinals. The games are analyzed when a given player i must make the first move, when there is a finite limit n on the total number of moves, and when the game starts at a given initial state S. Games end when either both players pass in succession or else a total of n moves have been made. Studies are made of the influence of i, n, and S on the outcomes. It is proved that these outcomes ultimately become periodic in n and therefore exhibit long-term predictable behavior. Efficient algorithms are given to calculate these ultimate outcomes by computer. Frequently the ultimate outcomes are independent of i, S, and n when n is sufficiently large; in this situation this common ultimate outcome is proved to be Pareto-optimal. The use of ultimate outcomes gives rise to a concept of stable points, which are analogous to Nash equilibria but consider long-term effects. If the initial state is a stable point, then no player has an incentive to move from that state, under the assumption that any initial move could be followed by a long series of moves and countermoves. The concept may be broadened to that of a stable set. It is proved that every game has a minimal stable set, and any two distinct minimal stable sets are disjoint. Comparisons are made with the results of standard TOM.     

Far-sighted equilibria in 2 × 2, non-cooperative,repeated games

Consider a two-person simultaneous-move game in strategic form. Suppose this game is played over and over at discrete points in time. Suppose, furthermore, that communication is not possible, but nevertheless we observe some regularity in the sequence of outcomes. The aim of this paper is to provide an explanation for the question why such regularity might persist for many (i.e., infinite) periods.Each player, when contemplating a deviation, considers a sequential-move game, roughly speaking of the following form: if I change my strategy this period, then in the next my opponent will take his strategy b and afterwards I can switch to my strategy a, but then I am worse off since at that outcome my opponent has no incentive to change anymore, whatever I do. Theoretically, however, there is no end to such reaction chains. In case that deviating by some player gives him less utility in the long run than before deviation, we say that the original regular sequence of outcomes is far-sighted stable for that player. It is a far-sighted equilibrium if it is far-sighted stable for both players.     

Even Risk-Averters may Love Risk

A decision maker bets on the outcomes of a sequence of coin-tossings. At the beginning of the game the decision maker can choose one of two coins to play the game. This initial choice is irreversible. The coins can be biased and the player is uncertain about the nature of one (or possibly both) coin(s). If the player is an expected-utility maximizer, her choice of the coin will depend on different elements: the nature of the game (namely, whether she can observe the outcomes of the previous tosses before making her next decision), her utility function, the prior distribution on the bias of the coin. We will show that even a risk averter might optimally choose a riskier coin when learning is allowed. We will express most of our results in the language of stochastic orderings, allowing comparisons that are valid for large classes of utility functions.     

An axiomatic analysis of the Nash equilibrium concept

The purpose of this paper is to analyze axiomatically the Nash equilibrium concept. The class of games under study is a (relatively large) subclass of n-person normal form games. Solutions are correspondences which associate to each game a non empty set of strategy vectors of this game. It is shown that if a solution satisfies the axioms Independence of irrelevant alternatives (IIA) and Individual rationality (IR), then all the strategy vectors in this solution are Nash equilibria. This result holds good also if IR is replaced by Strong individual monotonicity (SIM) or Weak principle of fair compromise (WPFC).     

Beyond zero-sum: game theory and national forest management

Following two decades of bitter struggle, the current debate over management of the U.S. national forests is characterized by animosity, controversy, and seemingly intractable gridlock. This view has led some to characterize the current management process a situation of pure conflict or a zero-sum game. In this article, we develop a game-theoretic model based on an argument that national forest policymaking is a game between the U.S. National Forest Service and Environmentalists and that this game is not zero-sum. The model is used to account for outcomes associated with contemporary management policy and to examine some recent changes to the game's rules and how this may affect outcomes. The analysis shows that some changes will have little to no effect on outcomes, while others have a significant potential to do so.     

No Switchbacks: Rethinking Aspiration-Based Dynamics in the Ultimatum Game

Aspiration-based evolutionary dynamics have recently been used to model the evolution of fair play in the ultimatum game showing that incredible threats to reject low offers persist in equilibrium. We focus on two extensions of this analysis: we experimentally test whether assumptions about agent motivations (aspiration levels) and the structure of the game (binary strategy space) reflect actual play, and we examine the problematic assumption embedded in the standard replicator dynamic that unhappy agents who switch strategies may return to a rejected strategy without exploring other options. We find that the resulting “no switchback” dynamic predicts the evolution of play better than the standard dynamic and that aspirations are a significant motivator for our participants. In the process, we also construct and analyze a variant of the ultimatum game in which players can adopt conditional (on their induced aspirations) strategies.     

Endogenizing the order of moves in matrix games

Players often have flexibility in when they move and thus whether a game is played simultaneously or sequentially may be endogenously determined. For 2 × 2 games, we analyze this using an extended game. In a stage prior to actual play, players choose in which of two periods to move. A player moving at the first opportunity knows when his opponent will move. A player moving at the second turn learns the first mover's action. If both select the same turn, they play a simultaneous move subgame.If both players have dominant strategies in the basic game, equilibrium payoffs in the basic and extended games are identical. If only one player has a dominant strategy or if the unique equilibrium in the basic game is in mixed strategies, then the extended game equilibrium payoffs differ if and only if some pair of pure strategies Pareto dominates the basic game simultaneous play payoffs. If so, sequential play attains the Pareto dominating payoffs. The mixed strategy equilibrium occurs only when it is not Pareto dominated by some pair of pure strategies.In an alternative extended game, players cannot observe delay by opponents at the first turn. Results for 2×2 games are essentially the same as with observable delay, differing only when only one player has a dominant strategy.     

Belief formation in a signaling game without common prior: an experiment

Using belief elicitation, the paper investigates the process of belief formation and evolution in a signaling game in which a common prior is not induced. Both prior and posterior beliefs of Receivers about Senders’ types are elicited, as well as beliefs of Senders about Receivers’ strategies. In the experiment, subjects often start with diffuse uniform beliefs and update them in view of observations. However, the speed of updating is influenced by the strength of initial beliefs. An interesting result is that beliefs about the prior distribution of types are updated slower than posterior beliefs, which incorporate Senders’ strategies. In the medium run, for some specifications of game parameters, this leads to outcomes being significantly different from the outcomes of the game in which a common prior is induced. It is also shown that elicitation of beliefs does not considerably change the pattern of play in this game.     

Effective choice in the single-shot Prisoner's Dilemma tournament

This paper discusses the results of a single-shot Prisoner's Dilemma computer tournament. In the single-shot Prisoner's Dilemma tournament each pair of players interacts only once. But players can establish and detect reputations because they know how their current opponent has behaved in previous games with other players. The results show that cooperation is worthwhile, even in single-shot games, provided the outcomes of previous games are common knowledge.Reality is a game, but you shouldn't take a game for reality(Pragmaticist Proverb)     

“囚徒困境”对于访谈研究初涉者的启示

从博弈的角度看，质的访谈是访谈者与被访者之间的博弈游戏，最佳策略是合作，以达成双赢。但对于访谈初涉者，在访谈过程中往往索取很多，给予却很少，结果被访者就成为受损方。不同于拔河游戏的“零和博弈”，一方的损带给另一方的是惠，在质的访谈中，这会成为两败俱伤的“负和博弈”。所以，作为一名访谈研究初涉者，务必走出访谈的“囚徒困境”，摆脱“零和博弈”与“负和博弈”，使访谈成为互惠互利的“正和博弈”。     

Staying power in sequential games

Staying power is the ability of a player to hold off choosing a strategy in a two-person game until the other player has selected his, after which the players are assumed to be able to move and countermove sequentially to ensure their best possible outcomes before the process cycles back to the initial outcome and then repeats itself (rational termination). These rules of sequential play induce a determinate, Paretosuperior outcome in all two-person, finite, sequential games in which the preferences of the players are strict.In 57 of the 78 distinct 2 × 2 ordinal games (73 percent), it makes no difference who the (second-moving) player with staying power is, but in the other 21 games the outcome is power-dependent. In all but one of these games, staying power benefits the player who possesses it.If no player has staying power, the outcomes that result from sequential play and rational termination are called terminal; they coincide with staying power outcomes if they are Pareto-superior. Normative implications of the analysis for rationally justifying cooperation in such games as Prisoners' Dilemma and Chicken, and implementing Pareto-superior outcomes generally, are also discussed.We are grateful to D. Marc Kilgour for very valuable comments on an earlier version of this paper, causing us to rethink and redefine staying power. The earlier version was presented at the Seventeenth North American Conference, Peace Science Society (International), University of Pennsylvania, November 9–11, 1981.     

On Stalnaker's Notion of Strong Rationalizability and Nash Equilibrium in Perfect Information Games

Counterexamples to two results by Stalnaker (Theory and Decision, 1994) are given and a corrected version of one of the two results is proved. Stalnaker's proposed results are: (1) if at the true state of an epistemic model of a perfect information game there is common belief in the rationality of every player and common belief that no player has false beliefs (he calls this joint condition strong rationalizability), then the true (or actual) strategy profile is path equivalent to a Nash equilibrium; (2) in a normal-form game a strategy profile is strongly rationalizable if and only if it belongs to C , the set of profiles that survive the iterative deletion of inferior profiles.     

Strategic uncertainty,coordination failure and emergence: A game theory study on agency‐structure interactions

In the social sciences there is a long standing debate over the primacy of structure or agency in shaping human behavior. Structurationists like Giddens think of individuals' agency and collective features as two sides of the same coin while emergentists including Archer and Elder‐Vass claim that structurationists are blurring an important ontological distinction——they argue for effective social analysis, we should think of structures and individuals as different things because emergence happens when the properties of the collective are not reducible to the properties of the parts that constitute them. This paper contributes to the agency‐structure debate by showing how production and reproduction of emergent properties of social groups sharing same normative commitments (norm circles) can be empirically studied using two behavioral game theory experiments, adding a previously neglected insight that strategic uncertainty can lead to coordination failures of individual behaviors within norm circles, manifested as unintended outcomes of interdependent actions that are difficult to predict. Here the synchronic relation between the unpredictable collective outcomes of each game round and agential reactions of the participants to them defines emergence——the previous outcomes are diachronic inputs but they appear in the new outcome only through their effects on the individual decisions.     

Equilibria for far-sighted players

A new equilibrium concept for non-cooperative games, based on the assumptions that players are rational and far-sighted, is examined. An outcome is extended non-myopically (XNM) stable for a player if that player is assured that no movecountermove sequence he could initiate by departing unilaterally from that outcome would benefit him. The extended non-myopic (XNM) equilibria of a game, the outcomes which are XNM stable for each player, therefore model permanent (enduring) equilibria in an ongoing conflict.Algorithms for the identification of XNM equilibria in a 2 × 2 game are presented. The XNM concepts are then applied to three special classes of games (no-conflict games, games of complete opposition, and strict ordinal games) to compare their predictions of long-term stability with the known properties of games in these classes.Research supported by Natural Sciences and Engineering Research Council of Canada Grant No. A8974.     

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.     

The uncovered set in spatial voting games

In a majority rule voting game, the uncovered set is the set of alternatives each of which can defeat every other alternative in the space either directly or indirectly at one remove. Research has suggested that outcomes under most reasonable agenda processes (both sincere and sophisticated) will be confined to the uncovered set. Most research on the uncovered set has been done in the context of voting games with a finite number of alternatives and relatively little is known about the properties of the uncovered set in spatial voting games.We examine the geometry of the uncovered set in spatial voting games and the geometry of two important subsets of the uncovered set, the Banks set and the Schattschneider set. In particular, we find both general upper and lower limits on the size of the uncovered set, and we give the exact bounds of the uncovered set for situations with three voters. For situations with three voters, we show that the Banks set is identical to the uncovered set.