Differential Games and Control Theory by Emilio O. Roxin

By Emilio O. Roxin

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This is important since at this stage of the game, the final outcome depends not only on player 2's choice between c and d but also on the action chosen by player 1 at his second information set. Speaking in terms of preferences, player 2 is asked to revise his preference relation on his set of strategies after observing that player l's behavior has contradicted his initial conjecture Ill' but it is assumed that player 2's utility function U2 remains unchanged during this revision; only the conjecture III is revised.

Recall that Hi(Ai) is the collection of player i information sets not avoided by Ai and Hi(Si) is the collection of player i information sets not avoided by the strategy Si. 26 CHAPTER 2. 5 Let Ai be a local mixed conjecture. Then, the mixed conjecture induced by Ai is the mixed conjecture Ili E ~(Si) given by lli(Si) = { TIhEHi(Si) Aih(si(h)), 0, if Hi(si) ~ Hi(Ai) otherwise. It may be checked that if Ili is the mixed conjecture induced by Ai, then Ai is the mixed conjecture induced by Ili, and vice versa.

Consider an information set h E Hi. By perfect recall, there is a unique sequence hl, h 2 , ... , hK of player i information sets that leads to h. Here, hk follows h k- l for every k, h follows h K , and there are no player i information sets between h k- l and hk or between hK and h. At every hk, let ak E A(hk) be the unique action at hk that leads to h. Let (ji be a behavioral conjecture which possibly avoids h. By (jilh we denote the behavioral conjecture which at every hk assigns probability one to the action ak leading to h, and which coincides with (j i at all h E Hi \ {hI, ...

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