Traveler's Dilemma Research Articles

Sampling dynamics applied to the traveler's dilemma reveals non-Nash behavior

The traveler's dilemma is characterized by a strict Nash equilibrium that is Pareto-inefficient. Even though a coordinated choice of more efficient options would be preferred by both players, any individual deviation from Nash behavior will induce a lower payoff. Yet experimental behavior reported in the literature demonstrates that deviations from Nash behavior toward Pareto-better outcomes are typical. An application of the sampling dynamics to this dilemma further justifies the non-Nash behavior. Due to the fact, that the sampling dynamics is not payoff-monotonic, even dominated strategies can survive the selection process. It is shown that for a generalized traveler's dilemma an interior equilibrium attracts almost all trajectories, given that there are sufficiently many options. The limit for infinitely many options is derived.

Solution of Traveler’s Dilemma

Aim: The aim of the article is to show: 1) that the reasoning of perfectly rational players presented in 1994 by the author of the Traveler's Dilemma Kaushik Basu is incorrect and therefore leads to wrong conclusions, 2) how the reasoning of these players should look like and what solution it leads to. Design / Research methods: Logical analysis. Conclusions / findings: Perfectly rational Traveler’s Dilemma players should use, according to game theory, so-called retrograde (iterative) induction. This is wrong, as in the set of Traveler’s Dilemma games results the principle of transitivity is not met. We believe that perfectly rational players will achieve a better result when they make a random decision from a suitably limited set of decisions. After applying this method of decision making, perfectly rational players will achieve a result similar to those obtained by real players in experiments. Thus, the paradox described in the theory of games disappears, that perfectly rational players achieve worse results than real players Originality / value of the article: A new way of making decisions in the Traveler’s Dilemma game. Implications of the research: A new way of making decisions in other games similar to the Traveler’s Dilemma may allow to find new solutions in these games. Limitations of the research: The described decision-making method can potentially be used in decision-making situations when the following five conditions are met: 1) the set of possible decisions of each player is greater than 2, 2) the winning matrix is known to both players and both know the purpose of their choices, 3) when it is played once with an unknown opponent, 4) when both players have to make their decision without knowing the opponent's choice, 5) when there is no decision, which is a stable balance point or when it is, but its choice means that the player does not achieve a satisfying result.

Cooperating over losses and competing over gains: A social dilemma experiment

Evidence from studies in international relations, the politics of reform, collective action and price competition suggests that economic agents in social dilemma situations cooperate more to avoid losses than in the pursuit of gains. To test whether the prospect of losses can induce cooperation, we let experimental subjects play the traveler's dilemma in the gain and loss domain. Subjects cooperate substantially more over losses. Furthermore, our results suggest that this treatment effect is best explained by reference-dependent risk preferences and reference-dependent strategic sophistication. We discuss the implications of our results and relate our findings to other experimental games played in the loss domain.

Interaction of reasoning ability and distributional preferences in a social dilemma

In a within subjects design we evaluate distributional preferences and reasoning ability to explain choices in the Traveler's Dilemma. We recruit subjects from economics and non-economics majors to have a high variance of preferences and abilities. We find that economists follow the efficiency criterion while non-economists follow maximin. Economists also show a better reasoning ability. We, therefore, confirm the self-selection hypothesis of choosing a major. An equilibrium of an incomplete information version of the Traveler's Dilemma explains the behavior we observe. Subjects with low reasoning ability make choices away from equilibrium. Thus, (non)cooperative behavior might be misinterpreted if subjects’ reasoning ability is not taken into account.

Translucent Players: Explaining Cooperative Behavior in Social Dilemmas

In the last few decades, numerous experiments have shown that humans do not always behave so as to maximize their material payoff. Cooperative behavior when non-cooperation is a dominant strategy (with respect to the material payoffs) is particularly puzzling. Here we propose a novel approach to explain cooperation, assuming what Halpern and Pass call translucent players. Typically, players are assumed to be opaque, in the sense that a deviation by one player in a normal-form game does not affect the strategies used by other players. But a player may believe that if he switches from one strategy to another, the fact that he chooses to switch may be visible to the other players. For example, if he chooses to defect in Prisoner's Dilemma, the other player may sense his guilt. We show that by assuming translucent players, we can recover many of the regularities observed in human behavior in well-studied games such as Prisoner's Dilemma, Traveler's Dilemma, Bertrand Competition, and the Public Goods game.

Cooperating Over Losses and Competing Over Gains: A Social Dilemma Experiment

Evidence from studies in international relations, the politics of reform, collective action and price competition suggests that economic agents in social dilemma situations cooperate more to avoid losses than in the pursuit of gains. To test whether the prospect of losses can induce cooperation, we let experimental subjects play the traveler's dilemma in the gain and loss domain. Subjects cooperate substantially more over losses. Our experimental design allows us to show that this treatment effect is best explained by reference-dependent risk preferences and reference-dependent strategic sophistication. We discuss policy implications and relate our findings to other experimental games played in the loss domain.

Social preferences are not enough: Accounting for anomalous behavior in a complex mixed-motive game

In some economic games, participants systematically display behavior that departs from the model of payoff maximization, and this is usually attributed to social preferences. In this paper we focus on a new interactive context, a mixed-motive game called "Alternative Traveler's Dilemma" (ATD). In this context, most participants choose strictly dominated strategies. Preliminary studies suggest the reason for such a tendency is that participants have social preferences, usually competitive in nature. The question is whether some cognitive factors should also be included in models that account for the tendency at stake. More specifically, we investigate whether participants neglect payoff maximization, that is, whether they fail to notice that, by pursuing some competitive goal, they are not maximizing their payoffs. We report the results of an experiment that supports this hypothesis. We conclude that in order to explain anomalous behavior in the ATD, and similar games, we need to study the cognitive factors that bound participants' strategies and understanding of the game.

Implementation without incentive compatibility: Two stories with partially informed planners

We consider two stories, Typhoon by Conrad and Traveler's Dilemma by Basu (1994, 2007), as implementation problems under incomplete information without incentive compatibility, but where the planner has some private information regarding the state. If social choice functions do not satisfy incentive compatibility, full implementation is unattainable via existing approaches. For each story, we construct a direct mechanism that relies on the planner's private information. We provide a sufficient condition on players' beliefs regarding the state under which every player has a unique rationalizable action, namely, telling the truth. Thus, in these stories, the planner's information can bypass the lack of incentive compatibility.

Translucent Players: Explaining Cooperative Behavior in Social Dilemmas

In the last few decades, numerous experiments have shown that humans do not always behave so as to maximize their material payoff. Cooperative behavior when non-cooperation is a dominant strategy (with respect to the material payoffs) is particularly puzzling. Here we propose a novel approach to explain cooperation, assuming what Halpern and Pass 2013 call Typically, players are assumed to be opaque, in the sense that a deviation by one player does not affect the strategies used by other players. But a player may believe that if he switches from one strategy to another, the fact that he chooses to switch may be visible to the other players. For example, if he chooses to defect in Prisoner's Dilemma, the other player may sense his guilt. We show that by assuming translucent players, we can recover many of the regularities observed in human behavior in well-studied games such as Prisoner's Dilemma, Traveler's Dilemma, Bertrand Competition, and the Public Goods game.

Dynamics of spatial traveler's dilemma games

By examining payoff structures, we elucidate the traveler's dilemma (TD) game. It can be said that the TD game as the 2-player & multi-strategy game is a variant of the donor & recipient game, which itself is a subclass of the 2-player & 2-strategy (2 × 2) prisoner's dilemma, in the sense that the Nash equilibrium and fair Pareto optimum appear on mutual defection and mutual cooperation, respectively. Exploring spatial TD games, we observed that the network reciprocity of TD games can be universally scaled with a single dilemma parameter as can be performed in usual 2 × 2 games. Furthermore, we observed a local peak event reflecting a resurging cooperation fraction with increasing dilemma strength after degradation to a low level. We successfully demonstrate that this is caused by the specific dynamics of TD games created by their specific payoff structure.

Evolutionary dynamics of the traveler's dilemma and minimum-effort coordination games on complex networks.

The traveler's dilemma game and the minimum-effort coordination game are social dilemmas that have received significant attention resulting from the fact that the predictions of classical game theory are inconsistent with the results found when the games are studied experimentally. Moreover, both the traveler's dilemma and the minimum-effort coordination games have potentially important applications in evolutionary biology. Interestingly, standard deterministic evolutionary game theory, as represented by the replicator dynamics in a well-mixed population, is also inadequate to account for the behavior observed in these games. Here we study the evolutionary dynamics of both these games in populations with interaction patterns described by a variety of complex network topologies. We investigate the evolutionary dynamics of these games through agent-based simulations on both model and empirical networks. In particular, we study the effects of network clustering and assortativity on the evolutionary dynamics of both games. In general, we show that the evolutionary behavior of the traveler's dilemma and minimum-effort coordination games on complex networks is in good agreement with that observed experimentally. Thus, formulating the traveler's dilemma and the minimum-effort coordination games on complex networks neatly resolves the paradoxical aspects of these games.

An Application of Evolutionary Game Theory to Social Dilemmas: The Traveler's Dilemma and the Minimum Effort Coordination Game

The Traveler's Dilemma game and the Minimum Effort Coordination game are two social dilemmas that have attracted considerable attention due to the fact that the predictions of classical game theory are at odds with the results found when the games are studied experimentally. Moreover, a direct application of deterministic evolutionary game theory, as embodied in the replicator dynamics, to these games does not explain the observed behavior. In this work, we formulate natural variants of these two games as smoothed continuous-strategy games. We study the evolutionary dynamics of these continuous-strategy games, both analytically and through agent-based simulations, and show that the behavior predicted theoretically is in accord with that observed experimentally. Thus, these variants of the Traveler's Dilemma and the Minimum Effort Coordination games provide a simple resolution of the paradoxical behavior associated with the original games.

Individual and group behaviour in the traveler's dilemma: An experimental study

We provide an experimental test of the traveler's dilemma using individual and group data. Our investigation aims to assess whether individual decisions differ significantly from group decisions. Experimental findings reported in this paper show that: (1) groups are always more rational – i.e. their claims are closer to the Nash equilibrium; (2) the size of the penalty/reward influences convergence to the equilibrium both when decisions are taken individually or in groups; and (3) groups are more sensitive to the size of the penalty/reward.

Selfishness Level of Strategic Games

We introduce a new measure of the discrepancy in strategic games between the social welfare in a Nash equilibrium and in a social optimum, that we call selfishness level. It is the smallest fraction of the social welfare that needs to be offered to each player to achieve that a social optimum is realized in a pure Nash equilibrium. The selfishness level is unrelated to the price of stability and the price of anarchy and is invariant under positive linear transformations of the payoff functions. Also, it naturally applies to other solution concepts and other forms of games. We study the selfishness level of several well-known strategic games. This allows us to quantify the implicit tension within a game between players' individual interests and the impact of their decisions on the society as a whole. Our analyses reveal that the selfishness level often provides a deeper understanding of the characteristics of the underlying game that influence the players' willingness to cooperate. In particular, the selfishness level of finite ordinal potential games is finite, while that of weakly acyclic games can be infinite. We derive explicit bounds on the selfishness level of fair cost sharing games and linear congestion games, which depend on specific parameters of the underlying game but are independent of the number of players. Further, we show that the selfishness level of the $n$-players Prisoner's Dilemma is c/(b(n-1)-c), where b and c are the benefit and cost for cooperation, respectively, that of the n-players public goods game is (1-c/n)/(c-1), where c is the public good multiplier, and that of the Traveler's Dilemma game is (b-1)/2, where b is the bonus. Finally, the selfishness level of Cournot competition (an example of an infinite ordinal potential game), Tragedy of the Commons, and Bertrand competition is infinite.

Evolution of Cooperation in Spatial Traveler's Dilemma Game

Traveler's dilemma (TD) is one of social dilemmas which has been well studied in the economics community, but it is attracted little attention in the physics community. The TD game is a two-person game. Each player can select an integer value between and () as a pure strategy. If both of them select the same value, the payoff to them will be that value. If the players select different values, say and (), then the payoff to the player who chooses the small value will be and the payoff to the other player will be . We term the player who selects a large value as the cooperator, and the one who chooses a small value as the defector. The reason is that if both of them select large values, it will result in a large total payoff. The Nash equilibrium of the TD game is to choose the smallest value . However, in previous behavioral studies, players in TD game typically select values that are much larger than , and the average selected value exhibits an inverse relationship with . To explain such anomalous behavior, in this paper, we study the evolution of cooperation in spatial traveler's dilemma game where the players are located on a square lattice and each player plays TD games with his neighbors. Players in our model can adopt their neighbors' strategies following two standard models of spatial game dynamics. Monte-Carlo simulation is applied to our model, and the results show that the cooperation level of the system, which is proportional to the average value of the strategies, decreases with increasing until is greater than the critical value where cooperation vanishes. Our findings indicate that spatial reciprocity promotes the evolution of cooperation in TD game and the spatial TD game model can interpret the anomalous behavior observed in previous behavioral experiments.

Cooperative Equilibria in Iterated Social Dilemmas

The implausibility of the extreme rationality assumptions of Nash equilibrium has been attested by numerous experimental studies with human players. In particular, the fundamental social dilemmas such as the Traveler's dilemma, the Prisoner's dilemma, and the Public Goods game demonstrate high rates of deviation from the unique Nash equilibrium, dependent on the game parameters or the environment in which the game is played. These results inspired several attempts to develop suitable solution concepts to more accurately explain human behaviour. In this line, the recently proposed notion of cooperative equilibrium, based on the idea that players have a natural attitude to cooperation, has shown promising results for single-shot games. In this paper, we extend this approach to iterated settings. Specifically, we define the Iterated Cooperative Equilibrium (ICE) and show it makes statistically precise predictions of population average behaviour in the aforementioned domains. Importantly, the definition of ICE does not involve any free parameters, and so it is fully predictive.

Stochastic evolutionary dynamics resolve the Traveler's Dilemma

Behavior in social dilemmas is often inconsistent with the predictions of classical game theory: people (and a wide variety of other organisms) are more cooperative than might be expected. Here we consider behavior in one such social dilemma, the Traveler's Dilemma, that has received considerable attention in the economics literature but is little known among theoretical biologists. The rules of the game are as follows. Two players each choose a value between R and M, where 0<R<M. If the players choose the same value, both receive that amount. If the players choose different values v1 and v2, where v1<v2, then the player choosing v1 receives v1+R and the player choosing v2 receives v1−R. While the players would maximize their payoffs by both choosing the largest allowed value, M, the Nash equilibrium is to choose the smallest allowed value, R. In behavioral experiments, however, people generally choose values much larger than the minimum and the deviation from the expected equilibrium decreases with R. In this paper, we show that the cooperative behavior observed in the Traveler's Dilemma can be explained in an evolutionary framework. We study stochastic evolutionary dynamics in finite populations with varying intensity of selection and varying mutation rate. We derive analytic results showing that strategies choosing high values can be favored when selection is weak. More generally, selection favors strategies that choose high values if R is small (relative to M) and strategies that choose low values if R is large. Finally, we show that a two-parameter model involving the intensity of selection and the mutation rate can quantitatively reproduce data that from a Traveler's Dilemma experiment. These results demonstrate the power of evolutionary game theory for explaining human behavior in contexts that are challenging for standard economic game theory.

Experiments with the Traveler's Dilemma: Welfare, Strategic Choice and Implicit Collusion

This paper investigates behavior in the Traveler's Dilemma game and isolates deviations from textbook predictions caused by differences in welfare perceptions and strategic miscalculations. It presents the results of an experimental analysis based on a 2x2 design where the own and the other subject's bonus-penalty parameters are changed independently. We find that the change in own bonus-penalty alone entirely explains the effect on claims of a simultaneous change in one's own and the other's bonus-penalty. An increase in the other subject's bonus-penalty has a significant negative effect on claims when the own bonus-penalty is low, whereas it does not have a significant effect when the own bonus-penalty is high. We also find that expected claims are inconsistent with actual claims in the asymmetric treatments. Focusing on reported strategies, we document substantial heterogeneity and show that changes in choices across treatments are largely explained by risk aversion.

Travelers’ types

This paper uses subjects’ diverse self-reported justifications to explain discrepancies between observed heterogeneous behavior and the unique equilibrium prediction in a one-shot traveler's dilemma experiment. Principal components analysis suggests that iterative reasoning, aspiration levels, competitive behavior, attitudes towards risk and penalties and focal points may be behind different choices. Such reasons are coherent with same subjects’ behavior in other tests and experiments in which these particular issues are prominent, and thus, we identify “types” of subjects. Overall, we conclude that subjects’ self-justifications in complex strategic situations contain informational value which may be used to predict behavior in other situations of economic importance.

Strategic Choice of Preferences: The Persona Model

We introduce a modification to the two-timescale games studied in the evolution of preferences (EOP) literature. In this modification, the strategic process occurring on the long timescale is learning by an individual across his or her lifetime, not natural selection operating on genomes over multiple generations. This change to the longer timescale removes many of the formal difficulties of EOP models and allows us to show how two-timescale games can provide endogenous explanations for why humans sometimes adopt interdependent preferences and sometimes exhibit logit quantal response functions. In particular, we show that our modification to EOP explains experimental data in the Traveler's Dilemma. We also use our modification to show how cooperation can arise in nonrepeated versions of the Prisoner's Dilemma (PD). We then show that our modification to EOP predicts a crowding out phenomenon in the PD, in which introducing incentives to cooperate causes players to stop cooperating instead. We also use our modification to predict a tradeoff between the robustness and the benefit of cooperation in the PD.