Centipede Game Research Articles

Matching Best Replies in the Centipede Game

By applying their Best-Reply Matching (BRM) equilibrium concept to the well known finite centipede game, Droste, Kosfeld & Voorneveld DKV (2003) showed that players continue the game with large probability. In this paper we build on their work by first showing that the BRM equilibria differ whether one works with the normal or with the reduced normal form of a game. This leads us to propose a natural modification to DKV's criterion, which leads to the same set of equilibria regardless of the chosen representation of a game (normal form and reduced normal form). Finally we show that the new concept, in the centipede game, gives rise a larger set of behaviors, which includes the Subgame Perfect Nash equilibrium but also a bounded rational behavior that incites the player to never stop the game with a probability larger than 1/2.

The Evolution of Cooperation in the Centipede Game with Finite Populations

The partial cooperation displayed by subjects in the Centipede Game deviates radically from the predictions of traditional game theory. Even standard, infinite population, evolutionary settings have failed to provide an explanation for this behavior. However, recent work in finite population evolutionary models has shown that such settings can produce radically different results from the standard models. This paper examines the evolution of partial cooperation in finite populations. The results reveal a new possible explanation that is not open to the standard models and gives us reason to be cautious when employing these otherwise helpful idealizations.

Social games: Matching and the play of finitely repeated games

We examine a new class of games, which we call social games, such that players not only choose strategies but also choose with whom they play. A group of players who are dissatisfied with the play of their current partners can join together and play a new equilibrium. This imposes new refinements on equilibrium play in games, and we show how play depends on the relative populations of players in different roles, among other things. We also introduce finitely repeated social games where players may choose to rematch in any period. Some equilibria of fixed-player finitely repeated games cannot be sustained as equilibria in a finitely repeated social game. Conversely, the set of repeated matching equilibria includes some plays that are not part of any subgame perfect equilibrium of the corresponding fixed-player repeated games. We explore existence of finitely repeated matching equilibria, the relationship to renegotiation-proof equilibrium, and show how new predictions are made in trust and centipede games.

Conditional cooperation in a sequential move game

Cooperation is a pervasive social phenomenon, but more often than not economic theories have little to say about its causes and consequences. In this paper, we explore the hypothesis that cooperative behaviour might be motivated by pure self-interest when the “social” payoff in a game is increasing. We report the results of a series of experiments on the Centipede game. The experiments are organised in two subsequent steps. Subjects first participate in a 2-period trust game, randomly matched with unknown partners. We apply the strategy method in order to elicit their social preferences. On the basis of their pre-game behaviour, individuals are divided into three main social groups: selfish individuals, pure altruists and reciprocators. At the second step of the experiment, subjects play a repeated 6-move Centipede game with an increasing final payoff. Each subject plays twice, in a low and in a high-stake Centipede game, and he/she is informed about his/her co-player social preferences. We provide statistical evidence to identify the origin of cooperation within homogeneous and heterogeneous social groups. We construct a Poisson regression model to assess the determinants of the duration of conditional cooperation in the Centipede.

Knowing and Supposing in Games of Perfect Information

The paper provides a framework for representing belief-contravening hypotheses in games of perfect information. The resulting t-extended information structures are used to encode the notion that a player has the disposition to behave rationally at a node. We show that there are models where the condition of all players possessing this disposition at all nodes (under their control) is both a necessary and a sufficient for them to play the backward induction solution in centipede games. To obtain this result, we do not need to assume that rationality is commonly known (as is done in [Aumann (1995)]) or commonly hypothesized by the players (as done in [Samet (1996)]). The proposed model is compared with the account of hypothetical knowledge presented by Samet in [Samet (1996)] and with other possible strategies for extending information structures with conditional propositions.

Modelling Equilibrium Play as Governed By Analogy and Limited Foresight

This paper proposes a bounded rationality approach to model equilibrium play in games. It is based on the observation that decision makers often do not seem to fully distinguish between different but seemingly similar decisions. To capture this, for each player a similarity grouping of decisions is defined based on equality of available actions and analogy of locally foreseen subtrees. The considered equilibrium concept is a (trembling-hand) perfect Nash-equilibrium (Selten, 1975) in which players are required to choose the same behaviour for similar decisions. Based on the approach, it is shown how the Chain Store Paradox (Selten, 1978) can be resolved, and how mixed equilibria in the Centipede Game (Rosenthal, 1981) can be rationalised.

THE CENTIPEDE OF ROSENTHAL

In this short note we extend the very well known Centipede game of Rosenthal to the same extensive games with perfect information. The only difference that here the Centipede games have instead of numbers as payoff functions, they have variables. We introduce and study the relationship between the structure of subgame perfect equilibrium points (see Osborne (1994), Binmore (1994)) and the friendly equilibrium points due to Marchi (2004a) and (2004b). We solve an Asheim's conjecture (private communication).

Finite-Order Beliefs and Welfare-Enhancing Instruments in the Centipede Game

This paper investigates the effectiveness of two instruments designed to defer termination in the centipede game: an insurance against termination by the opponent, and an option to offer the opponent a bonus for not terminating the game. The rational prediction in both cases is passing until close to the end. Empirically, however, only the bonus option is used by the subjects. The results indicate that subjects readily understand the strategic effect of the bonus, which, once offered, renders passing until close to the end the strictly dominant strategy for both players. Yet, they fail to realise the slightly more involved strategic signal entailed in the insurance, namely that passing until close to the end is a strictly dominant strategy for an insured player. In order to further investigate this effect, we propose a simple behavioural model based on level-k thinking and show that it is largely consistent with the data.

Cooperation in an Infinite-Horizon Centipede Game

An infinite-horizon perfect-information centipede game is studied. The unique subgame-perfect Nash equilibrium (SPNE) in pure strategies has each player choose, at each opportunity, to terminate the game. In contrast, mixed strategies can yield equilibrium cooperation described as follows: for all but finitely many periods players randomize between terminating and continuing the game. These continuation probabilities are explicitly calculated. If both players' payoff functions are unbounded, then there exist SPNE exhibiting such cooperation. Conditions are also derived under which such cooperation does not arise.

Backward induction with players who doubt others' faultlessness

We investigate the robustness of the backward-induction outcome, in binary-action extensive-form games, to the introduction of small mistakes in reasoning. Specifically, when a player contemplates the best action at a future decision node, she assigns some small probability to the event that other players may reach a different conclusion when they carry out the same analysis. We show that, in a long centipede game, the prediction that players do not cooperate fails under this perturbation. Importantly, this result does not depend on forward induction or reputation reasoning. It particular, it applies to finite horizon overlapping generations models with fiat money.

Models of the medical consultation: opportunities and limitations of a game theory perspective

The medical consultation is best understood as a two-way social interaction involving interactive decision making. Game theory—a theory based on assumptions of rational choice and focusing on interactive decision making—has the potential to provide models of the consultation that can be used to generate empirically testable predictions about the factors that promote quality of care. Three different game structures—the Prisoner’s Dilemma game, the Assurance game, and the Centipede game—all provide insights into the possible underlying dynamics of the doctor-patient interaction. Further empirical work is needed to uncover the underlying game structures that occur most commonly in medical consultations. Game theory has the potential to provide a new conceptual and theoretical basis for future empirical work on the interaction between doctors and their patients.

Analogy-based expectation equilibrium

In complex situations, agents use simplified representations to learn how their environment may react. I assume that agents bundle nodes at which other agents must move into analogy classes, and agents only try to learn the average behavior in every class. Specifically, I propose a new solution concept for multi-stage games with perfect information: at every node players choose best-responses to their analogy-based expectations, and expectations correctly represent the average behavior in every class. The solution concept is shown to differ from existing concepts, and it is applied to a variety of games, in particular the centipede game, and ultimatum/bargaining games. The approach explains in a new way why players may Pass for a large number of periods in the centipede game, and why the responder need not be stuck to his reservation value in ultimatum games. Some possible avenues for endogenizing the analogy grouping are also suggested.

Analogy-based expectation equilibrium

In complex situations, agents use simplified representations to learn how their environment may react. I assume that agents bundle nodes at which other agents must move into analogy classes, and agents only try to learn the average behavior in every class. Specifically, I propose a new solution concept for multi-stage games with perfect information: at every node players choose best-responses to their analogy-based expectations, and expectations correctly represent the average behavior in every class. The solution concept is shown to differ from existing concepts, and it is applied to a variety of games, in particular the centipede game, and ultimatum/bargaining games. The approach explains in a new way why players may Pass for a large number of periods in the centipede game, and why the responder need not be stuck to his reservation value in ultimatum games. Some possible avenues for endogenizing the analogy grouping are also suggested.

Population Learning of Cooperative Behavior in a Three-Person Centipede Game

We consider mixed populations (N 1/4 21) of genuine (humans) and arti.cial (robots) agents repeatedly interacting in small groups whose composition is changed randomly from round to round. Our purpose is to study the spread of cooperative or non-cooperative behavior over time in populations playing a 3-person centipede game by manipulating the behavior of the robots (cooperative vs. noncooperative) and their proportion in the population. Our results convey a positive message: adding a handful of cooperative robots increases the propensity of the genuine subjects to cooperate over time, whereas adding a handful of non-cooperative agents does not decrease this propensity. If there are enough hard-core cooperative subjects in the population, they not only negate the behavior of the non-cooperative robots but also induce other subjects to behave more cooperatively.

Individual and group decisions in the centipede game: Are groups more “rational” players?

Two experiments compared the Centipede game played either by 2 individuals or by 2 (3-person) groups. The 2 competitors alternate in deciding whether to take the larger portion of an increasing (or constant) pile of money, and as soon as one “takes” the game ends. Assuming that both sides are concerned only with maximizing their own payoffs (and that this is common knowledge), the game theoretic solution, derived by backward induction, is for the first mover to exit the game at the first decision node. Both experiments found that although neither individuals nor groups fully complied with this solution, groups did exit the game significantly earlier than individuals. The study of experimental games has uncovered many instances in which individuals deviate systematically from the game theoretic solution. This study is in accord with other recent experiments in suggesting that game theory may provide a better description of group behavior.

Best-reply matching in games

We study a new equilibrium concept in non-cooperative games, where players follow a behavioral rule called best-reply matching. Under this rule a player matches the probability of playing a pure strategy to the probability that this strategy is a best reply. Kosfeld, Droste, and Voorneveld [Games and Economic Behavior 40 (2002) 270] show that best-reply matching equilibria are stationary states in a simple model of social learning, where newborns adopt a best-reply to recent observations of play. In this paper we analyze best-reply matching in more detail and illustrate the concept by means of well-known examples. For example in the centipede game it is shown that players will continue with large probability.

Deductive Reasoning in Extensive Games

Wejustify the application to extensive games of a model of deductive reasoning based on three key features: 'caution', 'full belief of opponent rationality', and 'no extraneous restrictions on beliefs'. We apply the model to several examples, and show that it yields novel economic insights. The approach supports forward induction, without necessarily promoting backward induction. In many economic contexts decision makers interact and take actions that extend through time. A bargaining party makes an offer, which is observed by the adversary, and accepted, rejected or followed by a counter-offer. Firms competing in markets choose prices, levels of advertisement, or investments with the intent of thereby influencing the future behaviour of competitors. One could add many examples. The standard economic model for analysing such situations is that of an extensive game. This paper is concerned with the following question: what happens in an extensive game if players reason deductively by trying to figure out one another's moves? We have in Asheim and Dufwenberg (2002) (AD) proposed a model for deductive reasoning leading to the concept of 'fully permissible sets', which can be applied to many strategic situations. In this paper we argue that the model is appropriate for analysing extensive games and we apply it to several such games. 1. Motivation There is already a literature exploring the implications of deductive reasoning in extensive games, but the answers provided differ and the issue is controversial. Much of the excitement concerns whether or not deductive reasoning implies backward induction in games where that principle is applicable. We next discuss this issue, since it provides a useful backdrop against which to introduce and motivate our own approach. Consider the 3-stage 'Take-it-or-leave-it', introduced by Reny (1993) (a version of Rosenthal's (1981) centipede game), and shown in Figure 1 together with its pure strategy reduced strategic form (PRSF).' What would 2 do in F1 if called upon to play? Backward induction implies that 2 would choose d, which is consistent with the following idea: 2 chooses d because she 'figures out' that 1 would choose D at * During our work with this paper and its companion piece 'Admissibility and common belief', we have received helpful comments from many scholars, including Pierpaolo Battigalli and Larry Samuelson. We are very grateful to David de Meza and two referees for their constructive and insightful comments. Asheim gratefully acknowledges the hospitality of Harvard University and financial support from the Research Council of Norway. 1 We need not consider what players plan to do at decision nodes that their own strategy precludes them from reaching (Section 3.2).

Equilibrium play and adaptive learning in a three-person centipede game

The two-person centipede game is one of the most celebrated paradoxes of backward induction in complete information extensive form games. An experimental investigation of a three-person centipede game shows that the paradoxical results are strongly affected by the size of the stakes. When the number of players in the game is increased from two to three and the game is played for unusually high stakes with group composition being randomly changed from trial to trial, the paradox is considerably weakened as players approach equilibrium play with multiple iterations of the stage game. When the game is played with low stakes, there is no evidence for equilibrium play or learning across iterations of the stage game. An adaptive learning model that assumes updating of the individual probabilities of choice outperforms alternative static and dynamic models in accounting for the major results observed in the high-stake experiment.

Episodic Learning: Towards the Emergences of Partial Cooperation

We introduce and apply a new learning mechanism, episodic learning. This learning paradigm is shown to be tightly linked to a natural form of reinforcement learning. Surprisingly, though the defect action in the ‘Prisoners Dilemma setting is a dominant strategy (and the only evolutionary stable strategy), when using episodic learning partial cooperation is obtained, which is stable against invasion of individuals with defection as their most frequent action. We further show the emergence of partial cooperation (‘altruism), in the context of the ‘Centipede game. Our results are consistent with the experimental data about partial cooperation obtained by cognitive psychologists.

Effects of financial incentives on the breakdown of mutual trust.

Disagreements between psychologists and economists about the need for and size of financial incentives continue to be hotly discussed. We examine the effects of financial incentives in a class of interactive decision-making situations, called centipede games, in which mutual trust is essential for cooperation. Invoking backward induction, the Nash equilibrium solution for these games is counterintuitive. Our previous research showed that when the number of players in the centipede game is increased from two to three, the game is iterated in time, the players are rematched, and the stakes are unusually high, behavior approaches equilibrium play. Results from the present study show that reducing the size of the stakes elicits dramatically different patterns of behavior. We argue that when mutual trust is involved, the magnitude of financial incentives can induce a considerable difference.