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Lecture 9: Errors in Decisions Games against p-intelligent players In real decision situations, we observe that decisions of players are not optimal. Reasons: Different levels of information. Lack of time (e.g., stock exchange). One of the modeling approaches: games against a pintelligent players. Definition: A player behaving with probability p like a normatively intelligent player and with probability 1-p like a random mechanism will be called a p-intelligent player. For p=0, player behaves as a random mechanism, for p=1, he is an perfectly intelligent player. It is not reasonable to apply the same strategies against the p-intelligent players as against intelligent ones. Approach P: optimal strategy of normatively intelligent player is the row maximizing the mean value payoff (only pure strategies are used). Approach M: intelligent player applies mixed strategies. Let us have matrix (constant-sum game): 3 3 3 3 7 1 7 7 3 1 −1 2 8 0 8 8 Player 1 (intelligent) observes strategy of player 2 s(p) = py* + (1 − p) r y – Nash equilibrium strategy of player 2, r – rectangular (uniform) distribution. The equilibrium strategies are: x* = (1, 0, 0, 0), y* = (0, 1, 0, 0). If player 2 is p-intelligent, player 1 expects that player 2 is going to use the strategy: s(p) = p(0, 1, 0, 0) + (1−p)(1/4, 1/4, 1/4, 1/4) = = (0.25−0.25p, 0.25+0.75p, 0.25−0.25p, 0.25−0.25p). We can show that: the first row is an optimal strategy if p > 5/9; the second row is an optimal strategy if 1/3 < p < 5/9; the fourth row is an optimal strategy if p < 1/3.