Quasi-informational extension

of a non-cooperative game

A non-cooperative game for which mappings and , , are given that satisfy the following conditions for all , , : 1) ; and 2) , where is the composite of and the projection . A quasi-informational extension of the game can be interpreted as the result of setting up the above scheme of interaction of players in the choice process for their strategies in . The strategies correspond to the rules determining the behaviour of player in any situation that he or she may encounter. The mapping associates the rule of behaviour of the players with a realization of them, that is, with the set of strategies , , that will be chosen by the players adhering to the given rules. Condition 1) of the definition of a quasi-informational extension is then the definition of the pay-off function of the new game , while condition 2) expresses the preservation by each player of the old strategies .

A situation of is the image of the equilibrium situation of some quasi-informational extension of under the corresponding mapping if and only if for any and there is a situation such that

The notion of a quasi-informational extension is particularly widely used in the theory of games with a hierarchy structure (cf. Game with a hierarchy structure), where the informal problem of optimizing an informational scheme is transformed into the problem of constructing a quasi-informational extension of a given game providing the first player with an optimum result. One also considers classes of quasi-informational extensions satisfying conditions that express some or other restrictions on the information available to the players. For example, if is a -person game , then one says that in the quasi-informational extension player 1 does not possess (proper) information about the strategy if for each there is an such that . The best of the quasi-informational extensions satisfying this condition is, for example, "game G3" , whereas the best of the quasi-informational extension is "game G2" .

References

[1]	Yu.B. Germeier, "Non-antagonistic games" , Reidel (1986) (Translated from Russian)
[2]	N.S. Kukushkin, V.V. Morozov, "The theory of non-antagonistic games" , Moscow (1977) pp. Chapt. 2 (In Russian)