Rational Players Research Articles

Resilient Cooperators Stabilize Long-Run Cooperation in the Finitely Repeated Prisoner's Dilemma

The dynamics of learning in finitely repeated games of cooperation remains an open question in large part because the timescale on which learning takes place is much longer than that of traditional lab experiments. Here we report results of a “virtual lab” experiment in which 94 subjects play up to 400 ten-round games of Prisoners Dilemma over the course of twenty consecutive weekdays. Consistent with previous work, the typical round at which players first defect creeps steadily earlier over the first several days; however, this unraveling process slows after roughly one week and remains stable for the rest of the experiment. Analyzing individual strategies shows that roughly 40% of players resist the temptation to unravel, cooperating conditionally throughout the experiment, even at a significant cost to themselves. We call these players resilient cooperators. Finally, using a standard learning model we predict that the presence of more than a critical fraction of resilient cooperators can permanently stabilize unraveling among a majority of rational players. These results shed new and hopeful light on the long-term dynamics of cooperation, and demonstrate the importance of conducting behavioral experiments on longer timescales than previously contemplated.

Communication-Enhancing Vagueness

I experimentally investigate how vague language changes the nature of communication in a biased strategic information transmission game. Counterintuitively, when both precise and imprecise messages can be sent, in aggregate senders are more accurate and receivers trust them more. I also develop and structurally estimate a model showing that vague messages increase communication between boundedly rational players, especially if some senders are moderately honest. Moderately honest senders avoid stating an outright lie by using vague messages to hedge. Precise messages are then more informative because there are fewer precise lies.

A marginal utility day-to-day traffic evolution model based on one-step strategic thinking

Most deterministic day-to-day traffic evolution models, either in continuous-time or discrete-time space, have been formulated based on a fundamental assumption on driver route choice rationality where a driver seeks to maximize her/his marginal benefit defined as the difference between the perceived route costs. The notion of rationality entails the exploration of the marginal decision rule from economic theory, which states that a rational individual evaluates his/her marginal utility, defined as the difference between the marginal benefit and the marginal cost, of each incremental decision. Seeking to analyze the marginal decision rule in the modeling of deterministic day-to-day traffic evolution, this paper proposes a modeling framework which introduces a term to capture the marginal cost to the driver induced by route switching. The proposed framework enables to capture both benefit and cost associated with route changes. The marginal cost is then formulated upon the assumption that drivers are able to predict other drivers’ responses to the current traffic conditions, which is adopted based on the notion of strategic thinking of rational players developed in behavior game theory. The marginal cost based on 1-step strategic thinking also describes the “shadow price” of shifting routes, which helps to explain the behavioral tendency of the driver perceiving the cost-sensitivity to link/route flows. After developing a formulation of the marginal utility day-to-day model, its theoretical properties are analyzed, including the invariance property, asymptotic stability, and relationship with the rational behavioral adjustment process.

Achieve Fairness in Rational Secret Sharing Protocol

In the setting of ( , ) m n rational secret sharing, n rational players wish to share a secret s , arbitrarily m players can reconstruct it, they always choose the strategies which can bring them more utilities. The security requirement includes privacy, correctness and fairness. Fairness is a central objective of the rational secret sharing scheme, complete fair means either all players get the secret s or none of them get it, but most existing schemes do not meet this nature. In this work, a rational secret sharing protocol is proposed, which neither need particular communication channel, nor assume the existence of honest minority, in addition, the scheme can resist the collusion attack with ( ) k k m  players, and can achieve sequential equilibrium. Theoretical analysis shows that the proposed protocol is complete fair.

Power Control in Networks With Heterogeneous Users: A Quasi-Variational Inequality Approach

This work deals with the power allocation problem in a multipoint-to-multipoint network, which is heterogenous in the sense that each transmit and receiver pair can arbitrarily choose whether to selfishly maximize its own rate or energy efficiency. This is achieved by modeling the transmit and receiver pairs as rational players that engage in a non-cooperative game in which the utility function changes according to each player's nature. The underlying game is reformulated as a quasi variational inequality (QVI) problem using convex fractional program theory. The equivalence between the QVI and the non-cooperative game provides us with all the mathematical tools to study the uniqueness of its Nash equilibrium (NE) points and to derive novel algorithms that allow the network to converge to these points in an iterative manner both with and without the need for a centralized processing. A small-cell network is considered as a possible case study of this heterogeneous scenario. Numerical results are used to validate the proposed solutions in different operating conditions.

Toward a Game Theoretic View of Secure Computation

We demonstrate how Game Theoretic concepts and formalism can be used to capture cryptographic notions of security. In the restricted but indicative case of two-party protocols in the face of malicious fail-stop faults, we first show how the traditional notions of secrecy and correctness of protocols can be captured as properties of Nash equilibria in games for rational players. Next, we concentrate on fairness. Here we demonstrate a Game Theoretic notion and two different cryptographic notions that turn out to all be equivalent. In addition, we provide a simulation-based notion that implies the previous three. All four notions are weaker than existing cryptographic notions of fairness. In particular, we show that they can be met in some natural setting where existing notions of fairness are provably impossible to achieve.

An approach for overlapping and hierarchical community detection in social networks based on coalition formation game theory

With greater availability of data and increasing interaction activities taking place on social media, detecting overlapping and hierarchical communities has become an important issue and one that is essential to social media analysis. In this paper, we propose a coalition formation game theory-based approach to identify overlapping and hierarchical communities. We model community detection as a coalition formation game in which individuals in a social network are modelled as rational players aiming to improve the group's utilities by cooperating with other players to form coalitions. Each player is allowed to join multiple coalitions, and those coalitions with fewer players can merge into a larger coalition as long as the merge operation is beneficial to the utilities of the merged coalitions, thus overlapping and hierarchical communities can be revealed simultaneously. The utility function of each coalition is defined as the combination of a gain function and a cost function. The gain function measures the degree of interactions amongst the players inside a coalition, while the cost function instead represents the degree of the interactions between the players of the coalition and the rest of the network. As game theory provides a formal analytical framework with a set of mathematical tools to study the complex interactions among rational players, applying game theory for detecting communities helps to identify communities more rationally. Some desirable properties of the utility function, such as the non-resolution limit and the non-scaling behavior, have been examined theoretically. To solve the issue of pre-setting the number and size for communities and to improve the efficiency of the detection process, we have developed a greedy agglomerative manner to identify communities. Extensive experiments have been conducted on synthetic and real networks to evaluate the effectiveness and efficiency of the proposed approach which can be applied for real world applications.

Some Problems With Judging Rationality

The gap between game-theoretic predictions and actual choices people make in, for instance, gaming experiments has been over-interpreted as evidence against rationality of players. I consider a version of the ultimatum game and examine its equilibria under different assumptions about players’ preferences. Using standard notions of rationality I show that the discrepancy between the “normative” and the “descriptive” cannot be established by a simple comparison of what is predicted by the equilibrium choices and the actual choices people make.

Profit allocation among rational players in a cooperative game under uncertainty

In this paper, a cooperative game has been considered, where there is a coalition between rational players in an uncertain environment. The uncertainty occurs in terms of the quality of the raw material. The pay-offs of the players are influenced by the degree of satisfaction. Pay-offs are considered as interval numbers. A multisection technique has been applied to obtain a rational solution. In the process, it has been found that the degree of satisfaction of the rational players converges to its optimum value with the convergence of the iterations. A comparative study has been made with methods existing in the literature.

Is poker a skill game? New insights from statistical physics

During last years poker has gained a lot of prestige in several countries and, besides being one of the most famous card games, it represents a modern challenge for scientists belonging to different communities, spanning from artificial intelligence to physics and from psychology to mathematics. Unlike games like chess, the task of classifying the nature of poker (i.e., as “skill game” or gambling) seems really hard and it also constitutes a current problem, whose solution has several implications. In general, gambling offers equal winning probabilities both to rational players (i.e., those that use a strategy) and to irrational ones (i.e., those without a strategy). Therefore, in order to uncover the nature of poker, a viable way is comparing performances of rational vs. irrational players during a series of challenges. Recently, a work on this topic revealed that rationality is a fundamental ingredient to succeed in poker tournaments. In this study we analyze a simple model of poker challenges by a statistical physics approach, with the aim to uncover the nature of this game. As main result we found that, under particular conditions, few irrational players can turn poker into gambling. Therefore, although rationality is a key ingredient to succeed in poker, also the format of challenges has an important role in these dynamics, as it can strongly influence the underlying nature of the game. The importance of our results lies on the related implications, as for instance in identifying the limits within which poker can be considered as a “skill game” and, as a consequence, which kind of format must be chosen to devise algorithms able to face humans.

Complex dynamics and chaos control of duopoly Bertrand model in Chinese air-conditioning market

A dynamic duopoly Bertrand model with quadratic cost function which is closer to reality and different from previous researches is discussed. The model is applied into air-conditioning market where the boundary equilibrium point is locally stable. Numerical simulations illustrate that the stability of Nash equilibrium strongly depends on the speed of adjustment of bounded rational player. The adjustment speeds and the degree of substitutability may undermine the stability of the equilibrium and cause a market structure to behave chaotically. The Lyapunov dimension of the chaos attractor is 1.9585 under some conditions. The stabilization of the chaotic behavior can be obtained by reducing the degree of substitutability. The results have an important theoretical and practical significance to Chinese air-conditioning market.

Tenable Strategy Blocks and Settled Equilibria

When people interact in familiar settings, social conventions usually develop so that people tend to disregard alternatives outside the convention. For rational players to usually restrict attention to a block of conventional strategies, no player should prefer to deviate from the block when others are likely to act conventionally and rationally inside the block. We explore two set-valued concepts, coarsely and finely tenable blocks, that formalize this notion for finite normal-form games. We then identify settled equilibria, which are Nash equilibria with support in minimal tenable blocks. For a generic class of normal-form games, our coarse and fine concepts are equivalent, and yet they differ from standard solution concepts on open sets of games. We demonstrate the nature and power of the solutions by way of examples. Settled equilibria are closely related to persistent equilibria but are strictly more selective on an open set of games. With fine tenability, we obtain invariance under the insertion of a subgame with a unique totally mixed payoff-equivalent equilibrium, a property that other related concepts have not satisfied.

The Intrinsic Quantum Nature of Nash Equilibrium Mixtures

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics (QM), which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics.

Poker as a skill game: rational versus irrational behaviors

In many countries poker is one of the most popular card games. Although each variant of poker has its own rules, all involve the use of money to make the challenge meaningful. Nowadays, in the collective consciousness, some variants of poker are referred to as games of skill, others as gambling. A poker table can be viewed as a psychology lab, where human behavior can be observed and quantified. This work provides a preliminary analysis of the role of rationality in poker games, using a stylized version of Texas Hold'em. In particular, we compare the performance of two different kinds of players, i.e. rational versus irrational players, during a poker tournament. Results show that these behaviors (i.e. rationality and irrationality) affect both the outcomes of challenges and the way poker should be classified.

Dynamics in a Cournot investment game with heterogeneous players

In this paper we concern the investment process in a duopoly game played by heterogeneous players. A discrete and dynamic system is built for the case that a boundedly rational player adjusts its investment decision by the locally marginal profit and a naïve player chooses its strategy according to the opponent’s action in the previous period. By stability analysis of the system, we show that the boundary equilibrium is unstable and obtain the stability conditions for the interior equilibrium. Numerical simulations are used to provide evidence for the influence of the model parameters on the system stability and on the complicated behaviors in the system evolution. It is shown that the system with varying model parameters may drive to chaos and the loss of stability may be caused by period doubling bifurcations or Neimark–Sacker bifurcations. It is also shown that the time-delayed feedback control method can be used to keep the system from instability and chaos. All the numerical simulations show that the capital depreciation rate has great influence on the system evolution: a smaller depreciation rate has a stronger stabilization effect on the survival of the system and makes the system easier to control from chaos.

Optimal cooperation-trap strategies for the iterated rock-paper-scissors game.

In an iterated non-cooperative game, if all the players act to maximize their individual accumulated payoff, the system as a whole usually converges to a Nash equilibrium that poorly benefits any player. Here we show that such an undesirable destiny is avoidable in an iterated Rock-Paper-Scissors (RPS) game involving two rational players, X and Y. Player X has the option of proactively adopting a cooperation-trap strategy, which enforces complete cooperation from the rational player Y and leads to a highly beneficial and maximally fair situation to both players. That maximal degree of cooperation is achievable in such a competitive system with cyclic dominance of actions may stimulate further theoretical and empirical studies on how to resolve conflicts and enhance cooperation in human societies.

Online concealed correlation and bounded rationality

Correlation of players' actions may evolve in the common course of the play of a repeated game with perfect monitoring (“online correlation”). In this paper we study the concealment of such correlation from a boundedly rational player. We show that “strong” players, i.e., players whose strategic complexity is less stringently bounded, can orchestrate the online correlation of the actions of “weak” players, where this correlation is concealed from an opponent of “intermediate” strength. The feasibility of such “online concealed correlation” is reflected in the individually rational payoff of the opponent and in the equilibrium payoffs of the repeated game.This result enables the derivation of a folk theorem that characterizes the set of equilibrium payoffs in a class of repeated games with boundedly rational players and a mechanism designer who sends public signals.The result is illustrated in two models, bounded recall strategies and finite automata.

Natural biased coin encoded in the genome determines cell strategy.

Decision making at a cellular level determines different fates for isogenic cells. However, it is not yet clear how rational decisions are encoded in the genome, how they are transmitted to their offspring, and whether they evolve and become optimized throughout generations. In this paper, we use a game theoretic approach to explain how rational decisions are made in the presence of cooperators and competitors. Our results suggest the existence of an internal switch that operates as a biased coin. The biased coin is, in fact, a biochemical bistable network of interacting genes that can flip to one of its stable states in response to different environmental stimuli. We present a framework to describe how the positions of attractors in such a gene regulatory network correspond to the behavior of a rational player in a competing environment. We evaluate our model by considering lysis/lysogeny decision making of bacteriophage lambda in E. coli.

Multidimensional utility analysis in a two-tier supply chain

Utility analysis is the measure of efficiency in supply chain, which is the function of strategy decided by supply chain players. The competition provided by the coopetitive (cooperation and competition) environment of new manufacturing systems demands analyzing the strategic interaction between two tiers of supply chains such as manufacturer–supplier. However, setting the optimal strategy for rational and intelligent players with conflicting interests is a challenging task for the supply chain managers. In this paper, the multidimensional utility concept is designed and developed for a social outcome with which every player is agreed without causing conflicts. The procurement process of Indian Railways Catering and Tourism Corporation (IRCTC) is studied for the multi-dimensional utility analysis with two parameters: quantity of interest and price per unit. Comparative analysis shows the proposed social utility outperforms the individual utilities of IRCTC and suppliers’ utilities. Based on the results, the framework for the procurement process of IRCTC has been proposed, which includes incentives for the players participating in the supply chain design process.

Supply Contracting with Risk‐Sensitive Retailers under Information Asymmetry: An Exploratory Behavioral Study

In this paper, we explore a two‐echelon supply chain with one upstream seller (supplier) and two downstream retailers selling a newsvendor type of fashion product. The retailers are the same except their degrees of risk sensitivity. In this supply chain, we examine the performance of the buyback contract via behavioral experiments in which the supplier is played by human subjects and the risk‐sensitive retailers are computerized. We study the effect of profit margin and also demand dispersion under different game scenarios. Our behavioral experiments reveal that the human suppliers can be classified into three types, namely type I (the highly consistent and rational players), type II (the highly strategic players), and type III (naïve and highly irrational players). We further generate a few insights: (i) Decision‐makers behave as type II players in the low profit margin scenarios but not in the high profit margin cases; (ii) there are no significant pieces of evidence on the existence of the demand dispersion effect and the game type effect on both type I and type III players; and (iii) the average profits earned by type I and type II decision‐makers are similar, whereas type III decision‐makers are the smallest in almost all cases. We believe that this paper provides many new insights on supply contracting with reference to the behavioral risk of human decision‐makers. Copyright © 2014 John Wiley & Sons, Ltd.