Three-person Game Research Articles

On the Equilibrium in a Queuing System with Retrials and Strategic Arrivals

This paper considers a callback single-server system with an orbit and a First-Come First-Served (FCFS) service discipline. Customers (users, clients) that encounter a busy server are sent into orbit and then have the option to retry service after an exponential period of time. In addition, each customer entering the system uses a strategy and must independently decide when to arrive in the system within a fixed admission period of time so that the expected sojourn time is minimal. We interpret the arrival process as a Nash equilibrium solution of a noncooperative game when the arrival intensity is completely described by an unknown distribution function, and then we propose a way to find an equilibrium for the case when the client’s waiting time for service is obviously limited. The analytical solution for the equilibrium is illustrated numerically for two-person and three-person games.

Equilibrium stacks for a three-person game on a product of staircase-function continuous and finite strategy spaces

Abstract A method of solving a three-person game defined on a product of staircase-function strategy spaces is presented. The spaces can be finite and continuous. The method is based on stacking equilibria of “short” three-person games, each defined on an interval where the pure strategy value is constant. In the case of finite three-person games, which factually are trimatrix games, the equilibria are considered in general terms, so they can be in mixed strategies as well. The stack is any interval-wise combination (succession) of the respective equilibria of the “short” trimatrix games. Apart from the stack, there are no other equilibria in this “long” trimatrix game. An example is presented to show how the stacking is fulfilled for a case of when every “short” trimatrix game has a pure-strategy equilibrium. The presented method, further “breaking” the initial “long” game defined on a product of staircase-function finite spaces, is far more tractable than a straightforward approach to solving directly the “long” trimatrix game would be.

Pareto Equilibrium of Objections and Counterobjections in a Differential Three-Person Game

Pareto Equilibrium of Objections and Counterobjections in a Differential Three-Person Game

Endogenous alliances in survival contests

Esteban and Sákovics (2003) showed in their three-person game that an alliance never appears in a possibly multi-stage contest game for an indivisible prize when allies’ efforts are perfectly substitutable. In this paper, we introduce allies’ effort complementarity in alliances by using a CES effort aggregator function. We consider an open-membership alliance formation game followed by two contests: one played by alliances, and one within the winning alliance. We show that if allies’ efforts are too substitutable or too complementary, there is no meaningful alliance in equilibrium. However, if allies’ efforts are moderately complementary to each other, then competition between two alliances is a subgame perfect equilibrium, which Pareto-dominates the equilibrium in a no-alliance single-stage contest. We also show that if forming more than two alliances is supported in equilibrium, then it Pareto-dominates two-alliance equilibrium. Nevertheless, the parameter space for such an allocation to be supported as an equilibrium shrinks when the number of alliances increases.

Individual Stability of Coalition Structures in Three-Person Games

Individual Stability of Coalition Structures in Three-Person Games

Weak Berge Equilibrium in Finite Three-person Games: Conception and Computation

Abstract The paper proposes the concept of a weak Berge equilibrium. Unlike the Berge equilibrium, the moral basis of this equilibrium is the Hippocratic Oath “First do no harm”. On the other hand, any Berge equilibrium is a weak Berge equilibrium. But, there are weak Berge equilibria, which are not the Berge equilibria. The properties of the weak Berge equilibrium have been investigated. The existence of the weak Berge equilibrium in mixed strategies has been established for finite games. The weak Berge equilibria for finite three-person non-cooperative games are computed.

Automata representation of successful strategies for social dilemmas

In a social dilemma, cooperation is collectively optimal, yet individually each group member prefers to defect. A class of successful strategies of direct reciprocity were recently found for the iterated prisoner’s dilemma and for the iterated three-person public-goods game: By a successful strategy, we mean that it constitutes a cooperative Nash equilibrium under implementation error, with assuring that the long-term payoff never becomes less than the co-players’ regardless of their strategies, when the error rate is small. Although we have a list of actions prescribed by each successful strategy, the rationale behind them has not been fully understood for the iterated public-goods game because the list has hundreds of entries to deal with every relevant history of previous interactions. In this paper, we propose a method to convert such history-based representation into an automaton with a minimal number of states. Our main finding is that a successful strategy for the iterated three-person public-goods game can be represented as a 10-state automaton by this method. In this automaton, each state can be interpreted as the player’s internal judgement of the situation, such as trustworthiness of the co-players and the need to redeem oneself after defection. This result thus suggests a comprehensible way to choose an appropriate action at each step towards cooperation based on a situational judgement, which is mapped from the history of interactions.

Research on the DouDiZhu's playing strategy based on XGBoost

In recent years, imperfect information games have received extensive attention in computer game research. As a three-person imperfect information game in China, DouDiZhu not only has competitive relationship but also has cooperative relationship between the players, which makes the model more complicated, hence it has high research value. In this paper, the DouDiZhu's playing strategy is converted into 182 legal playing multi-classification problems, by extracting the characteristics of winning data of different players, using the XGBoost model and setting reasonable model parameters. In addition, to make the strategy of playing cards more reasonable through phased training, character training, and rule correction methods. In this paper, a new data representation method of DouDiZhu game is proposed. The XGBoost model is innovatively used to solve the poker game problem. By adjusting the parameters and multi model training, a better card playing strategy is obtained. The experimental results show that the playing strategy predicted by the playing strategy model proposed in this paper is basically consistent with the human playing strategy, and has a good playing strategy for different situations. The method in this paper has achieved a third-place excellent result in the Chinese University Student Computer Game Contest 2019.

The Coase Theorem, the Nonempty Core, and the Legal Neutrality Principle

AbstractThe Coase theorem states that where there are externalities and no transaction costs resource allocation is Pareto-optimal and independent of the stakeholders’ legal position. This result has been challenged many times. In the cooperative game approach to resource allocation, the refutation is made by constructing a three-person game which has an empty core under one set of liability rules—which implies that optimal allocations are coalitionally unstable–and a nonempty core under another set. In this example, however, the probability that the core is non-empty is rather high (5/6). Yet, even if coalitionally stable Pareto-optimal arrangements are likely, to establish the plain validity of the Coase theorem it must be shown that the legal neutrality statement also holds. We show that for the three-person cooperative game example mentioned above, the probability that the two assertions of the Coase theorem hold can be as low as 3/8.

Leading-by-example and third-party punishment: Experimental evidence

In this paper we explore how decisions in a sequential three-person game are influenced by either dynamics of roles or group composition. In the game a dictator decides how much of his endowment to transfer to the recipient. The supervisor can punish the dictator and/or transfer to the recipient after learning about the dictator’s decision. We find that transfers by a dictator are highest and stable when the group of three is fixed, no matter whether roles change or not. There is limited support of a leading-by-example effect, that is, only when both role and group composition of subjects are fixed, supervisors give more the more dictators gave, and dictators transfer more in the next period the more supervisors gave in a period. Punishment partially has a disciplining effect on dictators. Finally, we observe that subjects’ actual actions are consistent with their beliefs and expectations.

The Lemke–Howson Algorithm Solving Finite Non-Cooperative Three-Person Games in a Special Setting

We present a brief outline of an approximate method (3LP) proposed by E.G. Golshteyn for solving three-person games in mixed strategies. Similar to the 2LP-algorithm (which approximately solves bimatrix games), the solution procedure consists in the search of a global minimum of the so-called Nash function. By making an exhaustive search of the pairs of the initial strategies the algorithm 2LP (3LP) finds an exact solution of the game if the condition of reciprocal complementarity holds. The numerical experiments show that the 2LP-method successfully competes with the Lemke–Howson (LH) algorithm, which efficiently solves the bimatrix games. Unfortunately, the LH-algorithm cannot be applied to solve arbitrary three-person games. However, we have adapted the Lemke–Howson method to the solution of a special setting called the hexamatrix games. We have also conducted a thorough testing of the LH-algorithm to reveal its advantages and minor points as well.

Play Versus Strategy Method: Behavior and the Role of Emotions in the Ultimatum Game

In the context of a three-person ultimatum game featuring a proposer and two responders, this paper aims to investigate whether there are differences in bargainers’ behavior engendered by the use of the play and the strategy methods and to shed light on the role of emotions in explaining these differences. Although proposers correctly expect responders facing the play and the strategy methods to feel different emotions, our results reveal that they offer the same amount to both responders, on average. The two response methods also yield quantitatively similar acceptance rates. We thus provide further evidence that the play and the strategy methods do not significantly bias the behavior of bargainers.

Метод Лемке–Хаусона для решения конечных бескоалиционных игр трех лиц в частной постановке

Метод Лемке–Хаусона для решения конечных бескоалиционных игр трех лиц в частной постановке

Disability and Economy: A Game Theoretic Approach

This paper takes a game theoretic approach to disability-related issues by constructing a model that studies the case of hereditary deafness on Martha’s Vineyard Island, USA from the seventeenth century to the early years of twentieth, where the island community adjusted itself to the hereditary deafness so that it was not treated as a disability. The model of the present paper has two stages. First of all, there are two types of continua of agents, the deaf and the non-deaf. In the first stage, the non-deaf agents become either bilinguals or monolinguals. In the second stage, agents are classified into deaf people, bilinguals and monolinguals. They are then randomly matched to form a trio to play a three-person bargaining game with infinite horizon, random proposers and language constraints. Two bargaining games are considered. The first one is a majority bargaining game where only two out of three can agree to implement a bargaining outcome. The second one is a unanimity bargaining game where all three agents are required to reach an agreement. The majority game exhibits strategic complementarity, while the unanimity game exhibits strategic substitutability. This paper also takes an inductive approach to examine how prejudice against people with disability may emerge.

The role of frames, numbers and risk in the frequency of cooperation

We report on a simple experiment that addresses three factors in the frequency of cooperation: (1) framing, (2) the number of players and (3) the perceived risk of cooperating. We work with a (two-player) Prisoner’s Dilemma and with a three-player, two-strategy Public Good Game. We also consider a payoff schedule isomorphic with the latter but in a nonsocial setup. These themes have been separately studied by a large number of experimental papers, using diverse methodologies. Our experiment targets them in a common, clear-cut framework that minimizes confusion. We find three strong effects (a) framing; (b) the number of players (there is less cooperation in the three-person games than in the two-person ones); (c) the neutralization of risk (about 50% of participants cooperate when risk is neutralized). Both (a) and (c) go in the expected direction, but, in all three cases, the strength of the effect is surprising. Cooperators appear to be more motivated by efficiency or by Kantian reasoning than by altruism.

On the benefit of cooperation in three-person games

Three-person games in which each player maximizes his payoff function are considered. The question on the benefit of forming a coalition of three players, which is interesting for cooperative game theory, is studied. The aim of the cooperation is that each player increases his guaranteed payoff. Effective sufficient conditions are obtained under which the coalition of the players is beneficial for each of them. The linear case is considered separately. In this case, rather general results are obtained in a constructive form. In the second part of the paper, the question on the benefit of cooperation of three players in the presence of the fourth player—Nature—is studied. The behavior of Nature is assumed to be unpredictable; it may harm any individual player or the coalition of the players. Note that the situation considered in the second part is related to A.V. Kryazhimskii’s talk delivered in the summer of 2014. We obtain constructive conditions under which the union of the players is beneficial in this situation as well.

Toward a Comprehension Challenge, Using Crowdsourcing as a Tool

Human readers comprehend vastly more, and in vastly different ways, than any existing comprehension test would suggest. An ideal comprehension test for a story should cover the full range of questions and answers that humans would expect other humans to reasonably learn or infer from a given story. As a step toward these goals we propose a novel test, the Crowdsourced Comprehension Challenge (C3), which is constructed by repeated runs of a three-person game, the Iterative Crowdsourced Comprehension Game (ICCG). ICCG uses structured crowdsourcing to comprehensively generate relevant questions and supported answers for arbitrary stories, whether fiction or nonfiction, presented across a variety of media such as videos, podcasts, and still images.

An Algorithm to Play Talmud Using the 50-50 Rule

Playing three-person games can be very cumbersome without a good algorithm. This paper works out the algorithm to paly the Talmud bankruptcy game using the 50-50 Rule in a simple Excel spreadsheet. The algorithm can be applied to any estate size, and consistency check is included.

An Algorithm to Play Talmud Using Shapley's Value

Playing three-person games can be very cumbersome without a good algorithm. This paper works out the algorithm to paly the Talmud bankruptcy game using Shapley’s method in a simple Excel spreadsheet. The algorithm can be applied to any estate size, and consistency check is included.

The evolution of the game of baccarat

The game of baccarat has evolved from a parlor game played by French aristocrats in the first half of the 19th century to a casino game that generated over US$41 billion in revenue for the casinos of Macau in 2013. The parlor game was originally a three-person zero-sum game. Later in the 19th century it was simplified to a two-person zero-sum game. Early in the 20th century the parlor game became a casino game, no longer zero-sum. In the mid 20th century, the strategic casino game became a nonstrategic game, with players competing against the house instead of against each other. We argue that this evolution was motivated by both economic and game-theoretic considerations.