Quasi-informational extension

of a non-cooperative game $ \Gamma = \langle J , \{ S _ {i} \} _ {i \in J } , \{ H _ {i} \} _ {i \in J } \rangle $

A non-cooperative game $ \widetilde \Gamma = \langle J , \{ \widetilde{S} {} _ {i} \} _ {i \in J } , \{ \widetilde{H} {} _ {i} \} _ {i \in J } \rangle $ for which mappings $ \pi : \widetilde{S} \rightarrow S $ and $ c _ {i} : S _ {i} \rightarrow \widetilde{S} _ {i} $, $ i \in J $, are given that satisfy the following conditions for all $ i \in J $, $ s _ {i} \in S _ {i} $, $ \widetilde{s} \in \widetilde{S} $: 1) $ \widetilde{H} _ {i} = H _ {i} \circ \pi $; and 2) $ \pi _ {i} ( \widetilde{s} \| c _ {i} ( s _ {i} ) ) = s _ {i} $, where $ \pi _ {i} $ is the composite of $ \pi $ and the projection $ S \rightarrow S _ {i} $. A quasi-informational extension of the game $ \Gamma $ can be interpreted as the result of setting up the above scheme of interaction of players in the choice process for their strategies $ s _ {i} $ in $ \Gamma $. The strategies $ s _ {i} $ correspond to the rules determining the behaviour of player $ i $ in any situation that he or she may encounter. The mapping $ \pi $ associates the rule of behaviour of the players with a realization of them, that is, with the set of strategies $ s _ {i} $, $ i \in J $, 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 $ \widetilde \Gamma $, while condition 2) expresses the preservation by each player of the old strategies $ s _ {i} \in S _ {i} $.

A situation $ s ^ {*} $ of $ \Gamma $ is the image of the equilibrium situation of some quasi-informational extension $ \widetilde \Gamma $ of $ \Gamma $ under the corresponding mapping $ \pi $ if and only if for any $ i \in J $ and $ \overline{s}\; _ {i} \in S _ {i} $ there is a situation $ s \in S $ such that

$$ H _ {i} ( s ^ {*} ) \geq H _ {i} ( s \| \overline{s}\; _ {i} ) . $$

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 $ \Gamma $ is a $ 2 $- person game $ ( J = \{ 1 , 2 \} ) $, then one says that in the quasi-informational extension player 1 does not possess (proper) information about the strategy $ s _ {2} $ if for each $ \widetilde{s} _ {1} \in \widetilde{S} _ {1} $ there is an $ s _ {1} \in S _ {1} $ such that $ \pi ( \widetilde{s} _ {1} , \widetilde{S} _ {2} ) \supseteq \{ s _ {1} \} \times S _ {2} $. 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)