A Mixed Cooperative Dual to the Nash Equilibrium

A mixed dual to the Nash equilibrium is defined for n-person games in strategic form. In a Nash equilibrium every player’s mixed strategy maximizes his own expected payoff for the other n-1 players’ strategies. Conversely, in the dual equilibrium every n-1 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 n>2 there exists a game for which no dual equilibrium exists.

Hypergame Theory: A Model for Conflict, Misperception, and Deception

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.

The Fairness of Solidarity Bills under the Solidarity Value of Nowak and Radzik

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.

An Algorithm for Computing All Berge Equilibria

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.

The Traveling Salesman Game for Cost Allocation: The Case Study of the Bus Service in Castellanza

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.

On Perfect Nash Equilibria of Polymatrix Games

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 n-player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class of n-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.

On Taxed Matrix Games and Changes in the Expected Transfer

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 A and mixed strategies p and q of the two players it is defined by ET(A;p,q)=∑‍∑‍piqj|aij|. 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.

Extended Games Played by Managerial Firms with Asymmetric Costs

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.