Game Theory The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. A Mixed Cooperative Dual to the Nash Equilibrium Tue, 15 Sep 2015 09:39:22 +0000 A mixed dual to the Nash equilibrium is defined for -person games in strategic form. In a Nash equilibrium every player’s mixed strategy maximizes his own expected payoff for the other players’ strategies. Conversely, in the dual equilibrium every players have mixed strategies that maximize the remaining player’s expected payoff. Hence this dual equilibrium models mutual support and cooperation to extend the Berge equilibrium from pure to mixed strategies. This dual equilibrium is compared and related to the mixed Nash equilibrium, and both topological and algebraic conditions are given for the existence of the dual. Computational issues are discussed, and it is shown that for each there exists a game for which no dual equilibrium exists. H. W. Corley Copyright © 2015 H. W. Corley. All rights reserved. Hypergame Theory: A Model for Conflict, Misperception, and Deception Wed, 19 Aug 2015 09:06:02 +0000 When dealing with conflicts, game theory and decision theory can be used to model the interactions of the decision-makers. To date, game theory and decision theory have received considerable modeling focus, while hypergame theory has not. A metagame, known as a hypergame, occurs when one player does not know or fully understand all the strategies of a game. Hypergame theory extends the advantages of game theory by allowing a player to outmaneuver an opponent and obtaining a more preferred outcome with a higher utility. The ability to outmaneuver an opponent occurs in the hypergame because the different views (perception or deception) of opponents are captured in the model, through the incorporation of information unknown to other players (misperception or intentional deception). The hypergame model more accurately provides solutions for complex theoretic modeling of conflicts than those modeled by game theory and excels where perception or information differences exist between players. This paper explores the current research in hypergame theory and presents a broad overview of the historical literature on hypergame theory. Nicholas S. Kovach, Alan S. Gibson, and Gary B. Lamont Copyright © 2015 Nicholas S. Kovach et al. All rights reserved. The Fairness of Solidarity Bills under the Solidarity Value of Nowak and Radzik Thu, 23 Apr 2015 07:27:41 +0000 The solidarity value is a variant of the well-known Shapley value in which some sense of solidarity between the players is implemented allowing the disabled to receive help from the fortunate ones. We investigate on how fairly solidarity expenses are shared. We discuss the unwanted side effect of someone paying undue solidarity contributions as far as reversing his condition from a privileged to a needy person. A deeper case study is conducted for two classes of TU games that we obtain by modeling two real world business contexts. Here, we trace all player to player transfers of funds that arise when solidarity actions are processed, and we answer the question of who settles the solidarity bills. Also, we obtain the threshold position of a player below which he gets solidarity help, but above which he instead pays out donation. Lawrence Diffo Lambo and Pierre Wambo Copyright © 2015 Lawrence Diffo Lambo and Pierre Wambo. All rights reserved. An Algorithm for Computing All Berge Equilibria Thu, 29 Jan 2015 15:28:38 +0000 An algorithm is presented in this note for determining all Berge equilibria for an n-person game in normal form. This algorithm is based on the notion of disappointment, with the payoff matrix (PM) being transformed into a disappointment matrix (DM). The DM has the property that a pure strategy profile of the PM is a BE if and only if (0,…,0) is the corresponding entry of the DM. Furthermore, any (0,…,0) entry of the DM is also a more restrictive Berge-Vaisman equilibrium if and only if each player’s BE payoff is at least as large as the player’s maximin security level. H. W. Corley and Phantipa Kwain Copyright © 2015 H. W. Corley and Phantipa Kwain. All rights reserved. The Traveling Salesman Game for Cost Allocation: The Case Study of the Bus Service in Castellanza Sun, 14 Dec 2014 00:10:07 +0000 This paper studies cost allocation for the bus transportation service in Castellanza, a small town (14,000 inhabitants ca.) close to Varese, Italy. Carlo Cattaneo University (LIUC) is one of the promoters and funders of this service, together with the City Council and other private agents. The case study is first analysed as a traveling salesman problem (TSP) to find the optimal route. Then the traveling salesman game (TSG) is introduced, where the bus stops are associated with the players of a cooperative game, thus allowing the study of possible allocations of the total cost among them. The optimal route is found by the Branch and Bound algorithm. The Shapley vector and the separable and nonseparable cost are the methods used to allocate the cost of the optimal route among players. Nicola Besozzi, Luca Ruschetti, Chiara Rossignoli, and Fernanda Strozzi Copyright © 2014 Nicola Besozzi et al. All rights reserved. On Perfect Nash Equilibria of Polymatrix Games Mon, 29 Sep 2014 06:24:34 +0000 When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, the perfectness concept is probably the most famous one. It is known that weakly dominated strategies of two-player games cannot be part of a perfect equilibrium. In general, this undominance property however does not extend to -player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class of -player games, verify the undominance property. Consequently, we prove that every perfect equilibrium of a polymatrix game is undominated and that every undominated equilibrium of a polymatrix game is perfect. This result is used to set a new characterization of perfect Nash equilibria for polymatrix games. We also prove that the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. In addition, we introduce a linear programming formulation to identify perfect equilibria for polymatrix games. These results are illustrated on two small game applications. Computational experiments on randomly generated polymatrix games with different size and density are provided. Slim Belhaiza Copyright © 2014 Slim Belhaiza. All rights reserved. On Taxed Matrix Games and Changes in the Expected Transfer Sun, 31 Aug 2014 07:52:04 +0000 In gambling scenarios the introduction of taxes may affect playing behavior and the transferred monetary volume. Using a game theoretic approach, we ask the following: How does the transferred monetary volume change when the winner has to pay a tax proportional to her win? In this paper we therefore introduce a new parameter: the expected transfer. For a zerosum matrix game with payoff matrix and mixed strategies and of the two players it is defined by . Surprisingly, it turns out that for small fair matrix games higher tax rates lead to an increased expected transfer. This phenomenon occurs also in analogous situations with tax on the loser, bonus for the winner, or bonus for the loser. Higher tax or bonus rates lead to overproportional expected revenues for the tax authority or overproportional expected expenses for the grant authority, respectively. Ingo Althöfer and Marlis Bärthel Copyright © 2014 Ingo Althöfer and Marlis Bärthel. All rights reserved. Extended Games Played by Managerial Firms with Asymmetric Costs Sun, 13 Jul 2014 07:21:13 +0000 Both demand and cost asymmetries are considered in oligopoly model with managerial delegation. It shows that (i) both efficient and inefficient firms with delegation have second move advantage under quantity setting and first move advantage under price competition; (ii) the extended games under both quantity and price competition have subgame equilibria. Lastly, the social welfare of all strategy combinations is considered to find that when the efficient firm moves first and the inefficient firm moves second under price competition, the social welfare can be higher than Bertrand case, if the efficiency gap between the two firms is huge. Leonard F. S. Wang Copyright © 2014 Leonard F. S. Wang. All rights reserved. A Theory of Farsightedness in Committee Games Thu, 03 Apr 2014 14:32:38 +0000 We study the committee decision making process using game theory. A committee here refers to any group of people who have to select one option from a given set of alternatives under a specified rule. Shenoy (1980) introduced two solution concepts, namely, the one-core and a version of bargaining set for committee games. Shortcomings of these solutions concepts are raised and discussed in this paper. These shortcomings are resolved by introducing two new solutions concepts: the farsighted one-core and the bargaining set revised, inspired by an idea of farsightedness initially defined by Rubinstein (1980). It is shown that the farsighted one-core is always non-empty and is better than the one-core. In a well-specified sense, the bargaining set revised is also better than the bargaining set as defined by Shenoy (1980) and it is always non-empty for simple committee games with linear preferences. Other attractive properties are also proved. Alphonse Fodouop Fotso, Joseph Armel Momo Kenfack, and Bertrand Tchantcho Copyright © 2014 Alphonse Fodouop Fotso et al. All rights reserved. Nash Equilibria in Large Games Tue, 18 Mar 2014 08:19:51 +0000 This paper adds to the discussion, in a general setting, that given a Nash-Schmeidler (nonanonymous) game it is not always possible to define a Mas-Colell (anonymous) game. In the two games, the players have different strategic behaviours and the formulations of the two problems are different. Also, we offer a novel explanation for the lack of a Nash equilibrium in an infinite game. We consider this game as the limit of a sequence of approximate, finite games for which an equilibrium exists. However, the limiting pure strategy function is not measurable. Dionysius Glycopantis Copyright © 2014 Dionysius Glycopantis. All rights reserved. Renegotiation Perfection in Infinite Games Wed, 26 Feb 2014 09:06:31 +0000 We study the dynamic structure of equilibria in game theory. Allowing players in a game the opportunity to renegotiate, or switch to a feasible and Pareto superior equilibrium, can lead to welfare gains. However, in an extensive-form game this can also make it more difficult to enforce punishment strategies, leading to the question of which equilibria are feasible after all. This paper attempts to resolve that question by presenting the first definition of renegotiation-proofness in general games. This new concept, the renegotiation perfect set, satisfies five axioms. The first three axioms—namely Rationality, Consistency, and Internal Stability—characterize weakly renegotiation-proof sets. There is a natural generalized tournament defined on the class of all WRP sets, and the final two axioms—External Stability and Optimality—pick a unique “winner” from this tournament. The tournament solution concept employed, termed the catalog, is based on Dutta’s minimal covering set and can be applied to many settings other than renegotiation. It is shown that the renegotiation perfection concept is an extension of the standard renegotiation-proof definition for finite games, introduced by (Benoit and Krishna 1993), and that it captures the notion of a strongly renegotiation-proof equilibrium as defined by (Farrell and Maskin 1989). Julian C. Jamison Copyright © 2014 Julian C. Jamison. All rights reserved. Some Recursive Definitions for Linear Values of Cooperative TU Games Mon, 24 Feb 2014 07:51:08 +0000 We give recursive definitions for the Banzhaf Value and the Semivalues of cooperative TU games. These definitions were suggested by the concept of potential for the Shapley Value due to Hart and Mas-Colell and by some results of the author who introduced the potentials of these values and the Power Game of a given game. Irinel Dragan Copyright © 2014 Irinel Dragan. All rights reserved. A Necessary Condition for Nash Equilibrium in Two-Person Zero-Sum Constrained Stochastic Games Mon, 09 Dec 2013 13:19:51 +0000 We provide a necessary condition that a constrained Nash-equilibrium (CNE) policy pair satisfies in two-person zero-sum constrained stochastic discounted-payoff games and discuss a general method of approximating CNE based on the condition. Hyeong Soo Chang Copyright © 2013 Hyeong Soo Chang. All rights reserved. Allocation Rules for Games with Optimistic Aspirations Tue, 03 Sep 2013 11:04:04 +0000 A game with optimistic aspirations specifies two values for each coalition of players: the first value is the worth that the players in the coalition can guarantee for themselves in the event that they coordinate their actions, and the second value is the amount that the players in the coalition aspire to get under reasonable but very optimistic assumptions about the demands of the players who are not included in the coalition. In this paper, in addition to presenting this model and justifying its relevance, we introduce allocation rules and extend the properties of efficiency, additivity, symmetry, and null player property to this setting. We demonstrate that these four properties are insufficient to find a unique allocation rule and define three properties involving null players and nullifying players that allow the identification of unique allocation rules. The allocation rules we identify are the Midpoint Shapley Value and the Equal Division Rule. Luisa Carpente, Balbina Casas-Méndez, Ignacio García-Jurado, and Anne van den Nouweland Copyright © 2013 Luisa Carpente et al. All rights reserved. Subordinated Hedonic Games Mon, 29 Jul 2013 15:02:57 +0000 Hedonic games are simple models of coalition formation whose main solution concept is that of core partition. Several conditions guaranteeing the existence of core partitions have been proposed so far. In this paper, we explore hedonic games where a reduced family of coalitions determines the development of the game. We allow each coalition to select a subset of it so as to act as its set of representatives (a distribution). Then, we introduce the notion of subordination of a hedonic game to a given distribution. Subordination roughly states that any player chosen as a representative for a coalition has to be comfortable with this decision. With subordination we have a tool, within hedonic games, to compare how a “convenient” agreement reached by the sets of representatives of different groups of a society is “valued” by the rest of the society. In our approach, a “convenient” agreement is a core partition, so this paper is devoted to relate the core of a hedonic game with the core of a hedonic game played by the sets of representatives. Thus we have to tackle the existence problem of core partitions in a reduced game where the only coalitions that matter are those prescribed by the distribution as a set of representatives. We also study how a distribution determines the whole set of core partitions of a hedonic game. As an interesting example, we introduce the notion of hedonic partitioning game, which resembles partitioning games studied in the case where a utility, transferable or not, is present. The existence result obtained in this new class of games is later used to provide a nonconstructive proof of the existence of a stable matching in the marriage model. Juan Carlos Cesco Copyright © 2013 Juan Carlos Cesco. All rights reserved. A Tree Formulation for Signaling Games Tue, 11 Jun 2013 13:20:50 +0000 The paper has as a starting point the work of the philosopher Professor D. Lewis. We provide a detailed presentation and complete analysis of the sender/receiver Lewis signaling game using a game theory extensive form, decision tree formulation. It is shown that there are a number of Bayesian equilibria. We explain which equilibrium is the most likely to prevail. Our explanation provides an essential step for understanding the formation of a language convention. The informational content of signals is discussed and it is shown that a correct action is not always the result of a truthful signal. We allow for this to be reflected in the payoff of the sender. Further, concepts and approaches from neighbouring disciplines, notably economics, suggest themselves immediately for interpreting the results of our analysis (rational expectations, self-fulfilling prophesies). Xeni Dassiou and Dionysius Glycopantis Copyright © 2013 Xeni Dassiou and Dionysius Glycopantis. All rights reserved. Chess-Like Games May Have No Uniform Nash Equilibria Even in Mixed Strategies Wed, 22 May 2013 11:03:26 +0000 Recently, it was shown that Chess-like games may have no uniform (subgame perfect) Nash equilibria in pure positional strategies. Moreover, Nash equilibria may fail to exist already in two-person games in which all infinite plays are equivalent and ranked as the worst outcome by both players. In this paper, we extend this negative result further, providing examples that are uniform Nash equilibria free, even in mixed or independently mixed strategies. Additionally, in case of independently mixed strategies we consider two different definitions for effective payoff: the Markovian and the a priori realization. Endre Boros, Vladimir Gurvich, and Emre Yamangil Copyright © 2013 Endre Boros et al. All rights reserved.