People will choose the strategy that maximizes their payoff the most. Bayesian rationality requires players to form beliefs<\/mark><\/strong> about how opponents will act<\/p>\n\n\n\n What this means First, each player has prior beliefs about all the events involved in the payoff. Consider a two-player game<\/p>\n\n\n\n \\(\\theta_{1}(s_{2})\\) = player 1’s assessment that player 2 chooses \\(s_{2}\\), and \\(\\sum_{s_{i} \\in S_{i}}\\theta_{1}(s_{2})\\) = 1<\/p>\n\n\n\n theta is a function, s is a pure strategy<\/mark><\/p>\n\n\n\n e.g., \\(S_{2}\\) = {L, C, R} – with probabilities 30, 60, and 10, respectively \\(u_{1}(s_{1},\\theta_{1}) = \\sum_{s_{i} \\in S_{i}}\\theta_{2}u_{1}(s_{1}, s_{2})\\)<\/p>\n\n\n\n (\\Delta S_{2}) is the sum of all pure and mixed strategies that player 2 can choose.<\/mark><\/p>\n\n\n\n Player 1 believes that the probability of player 2 choosing \\(s_{2}\\) is \\(\\theta_{1}(s_{2})\\). EXAMPLE 2.4<\/mark>.<\/p>\n\n\n\n Suppose \\(\\theta_{1}\\) = (1\/4, 1\/2, 1\/4).<\/p>\n\n\n\n \\(u_{1}(U, \\theta_{1}) = u_{1}(U, L)\\theta_{1}(L) + u_{1}(U, M)\\theta_{1}(M) + u_{1}(U, R)\\theta_{1}(R)\\)<\/p>\n\n\n\n = 8 \u2219 \\(\\frac{1}{4}\\) + 0 \u2219 \\(\\frac{1}{2}\\) + 4 \u2219 \\(\\frac{1}{4}\\) = 3.<\/p>\n\n\n\n The sum of all of these is the expected payoff.<\/mark><\/p>\n\n\n\n Similary, the expected payoff from playing a mixed strategy \\(\\sigma_{1}\\) = (1\/2, 0, 1\/2) is<\/p>\n\n\n\n \\(u_{1}(\\sigma_{1}, \\theta_{1}) = u_{1}(U, \\theta_{1})\\sigma_{1}(U) + u_{1}(C, \\theta_{1})\\sigma_{1}(C) + u_{1}(D, \\theta_{1})\\sigma_{1}(D)\\)<\/p>\n\n\n\n = 3 \u2219 \\(\\frac{1}{2}\\) + \\(\\frac{5}{4}\\) \u2219 0 + \\(\\frac{17}{4}\\) \u2219 \\(\\frac{1}{2}\\) = \\(\\frac{29}{8}\\)<\/p>\n\n\n\n \u2235 \\(u_{1}(D, \\theta_{1}) = 5 \u2219 \\frac{1}{4} + 2 \u2219 \\frac{1}{2} + 8 \u2219 \\frac{1}{4} = \\frac{(5+4+8)}{4} = \\frac{17}{4}\\)<\/mark><\/p>\n\n\n\n Given a belief \\(\\theta_{i}\\), a Bayesian-rational player will choose a strategy \\(s_{i}\\) that maximizes her expected utility, put another way, player 1 would choose the optimal strategy,<\/p>\n\n\n\n \\(\\sigma_{1}^* \\in argmax_{\\sigma_{1} \\in \\Delta S_{1}} u_{1}(\\sigma_{1}, \\theta_{1})\\)<\/p>\n\n\n\n Player 1’s optimal strategy is which strategy (sigma) maximizes player 1’s expected payoff given player 1’s beliefs (theta).<\/mark> In the previous example, pure strategy D yields the highest expected utility given \\(\\theta_{1}\\). The first of the three elements of a normal-form game, the player has little to say about it.<\/mark><\/p>\n\n\n\n The second element, strategy, has pure and mixed strategies, of which the mixed strategy is based on independence. We want to calculate the expected payoff for a given belief and choose the strategy that maximizes the expected payoff.<\/mark><\/p>\n\n\n\n <\/p>\n\n\n\n
Game theory assumes that all players are Bayesian rational.<\/mark><\/p>\n\n\n\n
All information is complete, the only uncertainty is not knowing what the other player will do.
Each player’s actions affect the other player’s payoff.<\/mark><\/p>\n\n\n\n\n
Specifically, for each \\(s_{2} \\in S_{2}\\).<\/li>\n<\/ul>\n\n\n\n
Then, one of the beliefs the player can have, \\(\\theta_{1} = \\frac{3}{10}L + \\frac{6}{10}C + \\frac{1}{10}R\\)
\\(\\theta_{1}(2)\\) = 30%
So if you add everything up, you get 1<\/mark><\/p>\n\n\n\n\n
In reality, if player 2 uses the \\(s_{2}\\) strategy, player 1’s payoff will be \\(u_{1}(s_{1}, s_{2})\\).
The probability of occurrence multiplied by the actual realized payoff and summed over all possible \\(S_{2}\\) is the expected payoff.<\/mark><\/p>\n\n\n\n<\/td> L<\/td> M<\/td> R<\/td><\/tr> U<\/td> 8, 1<\/td> 0, 2<\/td> 4, 0<\/td><\/tr> C<\/td> 3, 3<\/td> 1, 2<\/td> 0, 0<\/td><\/tr> D<\/td> 5, 0<\/td> 2, 3<\/td> 8, 1<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
Bayesian-rational requires that the prediction is that player 1 will eventually choose this.<\/mark><\/p>\n\n\n\n
Therefore, if player 1 is a Bayesian rational DM with \\(\\theta_{1}\\), she would choose D.<\/p>\n\n\n\nSummary<\/h4>\n\n\n\n
Using the payoff function, a Bayesian-rational player can say which strategy to choose, depends on the beliefs about how the opponent will move.<\/mark><\/p>\n\n\n\n