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)