### Mixed Extension of Hypergames and Its Applications to Inspection Games

*Yasuo Sasaki, Norimasa Kobayashi, Kyoichi Kijima*

#### Abstract

In the present paper, we extend hypergame models by introducing mixed strategies and show that the mixed extension enables us to deal with hypergames with cardinal utilities, while the literature has dealt only with hypergames with ordinal utilities. We then illustrate some unique features of mixed-strategy equilibria of hypergames (hyper Nash equilibrium [4]), particularly our three theorems, and conduct comparative static analysis of equilibria due to change in misperceptions about cardinal utilities. Finally, we examine these findings in the framework of inspection games [1].

Hypergame theory [2] is a framework that deals with agents with misperceptions about the situations in which they are involved. Although this framework has been extended in several ways with various solution concepts [2, 3], we begin our research with the most basic model – simple hypergames and hyper Nash equilibria as its solution concept. In a simple hypergame, it is assumed that each agent perceives the situation subjectively in a form of a complete information game to construct a subjective game. The whole game, hypergame, is the list of the subjective games of all agents. A hyper Nash equilibrium is defined as a profile of each agent’s Nash strategy in his/her subjective game [4].

There only have existed hypergame models with pure strategies in the literature so far, so that we introduce mixed strategies to hypergame framework, to be exact, we consider mixed extension of every subjective game, and analyze the features of mixed-strategy hyper Nash equilibria.

Our first result is an existence theorem. That is, every finite hypergame with mixed strategies has at least one mixed-strategy hyper Nash equilibrium. This is a natural generalization of Nash's theorem [5] about noncooperative games.

Subsequent analysis focuses only on cases where agents misperceive the other’s utilities but perceive the other components of the situation correctly (perturbed hypergame situation), because analysis of this situation is our motivation to introduce this model.

Then, we illustrate derivation procedure of mixed-strategy hyper Nash equilibria and our second result, a derivation theorem. This theorem says that in two-agent hypergames, when there exists a unique mixed-strategy hyper Nash equilibrium, the equilibrium coincides with the Nash equilibrium in a game in which each agent’s utilities are his/her utilities in the opponent’s subjective game. It enables us to simplify calculation of the hyper Nash equilibrium.

The last part of the analysis is devoted to comparative statics. A hyper Nash equilibrium may change with change of misperceptions about the other’s utilities. In the literature, comparative statics is only carried out with respect to drastic change in misperceptions in terms of ordinal utilities. In this paper, we discuss effects of small change in misperceptions about cardinal utilities. We point out that mixed-strategy hyper Nash equilibria move continuously with the change in misperception, the small change in cardinal utilities that does not affect ordinal utilities.

Finally, we apply the concepts to inspection games [1] as an example. An inspection game is a two-agent game in which there exists only one mixed-strategy Nash equilibrium. We analyze the modified hypergame in the class of perturbed hypergame situation and illustrate the intuitive implications regarding our analysis. We also refer to the possibility of ex-ante analysis to manipulate the opponent’s misperception and obtain the better future based on this model.

References

[1] Avenhaus, R., M. Canty, D.M. Kilgour, B. von Stengel and S. Zamir, Inspection Games in Arms Control, European Journal of Operations Research, 90 (1996), 383-394.

[2] P.G.Bennett, M.R.Dando, Complex strategic analysis: a hypregame study of the fall of France, Journal of the Operational Research Society 30(1) (1979), 23-32.

[3] T. Inohara, Interperceptional Equilibrium as a Generalization of Nash Equilibrium in Games with Interperception, IEEE Transactions on Systems, Man, and Cybernetics 30(6)(2000), 625-638.

[4] K. Kijima, An Intelligent Poly-agent Learning Model and Its Application, Information and Systems Engineering 2 (1996), 47-61.

[5] Nash, J.F., Equilibrium points in N person games, Proceedings of the National Academy of Sciences 36 (1950), 48-49.

Hypergame theory [2] is a framework that deals with agents with misperceptions about the situations in which they are involved. Although this framework has been extended in several ways with various solution concepts [2, 3], we begin our research with the most basic model – simple hypergames and hyper Nash equilibria as its solution concept. In a simple hypergame, it is assumed that each agent perceives the situation subjectively in a form of a complete information game to construct a subjective game. The whole game, hypergame, is the list of the subjective games of all agents. A hyper Nash equilibrium is defined as a profile of each agent’s Nash strategy in his/her subjective game [4].

There only have existed hypergame models with pure strategies in the literature so far, so that we introduce mixed strategies to hypergame framework, to be exact, we consider mixed extension of every subjective game, and analyze the features of mixed-strategy hyper Nash equilibria.

Our first result is an existence theorem. That is, every finite hypergame with mixed strategies has at least one mixed-strategy hyper Nash equilibrium. This is a natural generalization of Nash's theorem [5] about noncooperative games.

Subsequent analysis focuses only on cases where agents misperceive the other’s utilities but perceive the other components of the situation correctly (perturbed hypergame situation), because analysis of this situation is our motivation to introduce this model.

Then, we illustrate derivation procedure of mixed-strategy hyper Nash equilibria and our second result, a derivation theorem. This theorem says that in two-agent hypergames, when there exists a unique mixed-strategy hyper Nash equilibrium, the equilibrium coincides with the Nash equilibrium in a game in which each agent’s utilities are his/her utilities in the opponent’s subjective game. It enables us to simplify calculation of the hyper Nash equilibrium.

The last part of the analysis is devoted to comparative statics. A hyper Nash equilibrium may change with change of misperceptions about the other’s utilities. In the literature, comparative statics is only carried out with respect to drastic change in misperceptions in terms of ordinal utilities. In this paper, we discuss effects of small change in misperceptions about cardinal utilities. We point out that mixed-strategy hyper Nash equilibria move continuously with the change in misperception, the small change in cardinal utilities that does not affect ordinal utilities.

Finally, we apply the concepts to inspection games [1] as an example. An inspection game is a two-agent game in which there exists only one mixed-strategy Nash equilibrium. We analyze the modified hypergame in the class of perturbed hypergame situation and illustrate the intuitive implications regarding our analysis. We also refer to the possibility of ex-ante analysis to manipulate the opponent’s misperception and obtain the better future based on this model.

References

[1] Avenhaus, R., M. Canty, D.M. Kilgour, B. von Stengel and S. Zamir, Inspection Games in Arms Control, European Journal of Operations Research, 90 (1996), 383-394.

[2] P.G.Bennett, M.R.Dando, Complex strategic analysis: a hypregame study of the fall of France, Journal of the Operational Research Society 30(1) (1979), 23-32.

[3] T. Inohara, Interperceptional Equilibrium as a Generalization of Nash Equilibrium in Games with Interperception, IEEE Transactions on Systems, Man, and Cybernetics 30(6)(2000), 625-638.

[4] K. Kijima, An Intelligent Poly-agent Learning Model and Its Application, Information and Systems Engineering 2 (1996), 47-61.

[5] Nash, J.F., Equilibrium points in N person games, Proceedings of the National Academy of Sciences 36 (1950), 48-49.

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