View source for Quasi-informational extension

''of a non-cooperative game <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765501.png" />''

A [[Non-cooperative game|non-cooperative game]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765502.png" /> for which mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765505.png" />, are given that satisfy the following conditions for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765508.png" />: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q0765509.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655011.png" /> is the composite of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655012.png" /> and the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655013.png" />. A quasi-informational extension of the game <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655014.png" /> can be interpreted as the result of setting up the above scheme of interaction of players in the choice process for their strategies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655016.png" />. The strategies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655017.png" /> correspond to the rules determining the behaviour of player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655018.png" /> in any situation that he or she may encounter. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655019.png" /> associates the rule of behaviour of the players with a realization of them, that is, with the set of strategies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655021.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655022.png" />, while condition 2) expresses the preservation by each player of the old strategies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655023.png" />.

A situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655025.png" /> is the image of the equilibrium situation of some quasi-informational extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655027.png" /> under the corresponding mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655028.png" /> if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655030.png" /> there is a situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655031.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655032.png" /></td> </tr></table>

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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655033.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655034.png" />-person game <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655035.png" />, then one says that in the quasi-informational extension player 1 does not possess (proper) information about the strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655036.png" /> if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655037.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076550/q07655039.png" />. 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.B. Germeier,   "Non-antagonistic games" , Reidel  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Kukushkin,   V.V. Morozov,   "The theory of non-antagonistic games" , Moscow  (1977)  pp. Chapt. 2  (In Russian)</TD></TR></table>