EXAMPLE 5.1<\/mark>. Player 1 first moves by choosing one of the three actions: T, M, and B. If player 1 chooses either T or M, then player 2 learns only that player 1 did not choose B, choosing between L and R.<\/p>\n\n\n\n In (B, R) player 2’s behavior is not sequentially rational, because R is strictly dominated<\/mark><\/strong> at his information set<\/p>\n\n\n\n Note that SPE = NE as there is no proper subgame<\/p>\n\n\n\n (T, L), (B, R) is NE<\/mark><\/p>\n\n\n\n <\/p>\n\n\n\n The next example demonstrates why we need another ingredient: player’s beliefs<\/p>\n\n\n\n EXAMPLE 5.2<\/mark>. <\/p>\n\n\n In contrast with the previous example, neither L nor R is strictly dominated<\/p>\n\n\n\n Hence both strategies can be a best response, depending on player 2’s beliefs<\/mark><\/strong><\/p>\n\n\n\n Player 2’s BR at {x,y} = \\(\\begin{cases}L \\quad p = Pr(\\text{player 2 at x}) > \\frac{1}{2} \\\\ \\{L,R\\} \\quad p=\\frac{1}{2} \\\\ R \\quad p < \\frac{1}{2}\\end{cases}\\)<\/mark><\/p>\n\n\n\n Player i’s beliefs are a probability distribution over his decision nodes satisfying<\/p>\n\n\n\n \\(\\sum_{x \\in I}\\mu_i(x)\\) = 1 for each information set \\(I \\subset D_i\\).<\/p>\n\n\n\n e.g., in {a, b, c, d}, II = {{p, 1-p}, {q, 1-q}} = {{a, c}, {b, d}} EXAMPLE 5.3<\/mark>.<\/p>\n\n\n \\(\\mathbb{I}_3\\) = {{a, b}, {c}}.<\/p>\n\n\n\n \\(\\mu_3 = (\\mu_3(a), \\mu_3(b))\\)<\/mark> <\/p>\n\n\n\n <\/p>\n\n\n\n Consider a decision node x in information set I<\/em><\/p>\n\n\n\n Given a strategy profile \\(\\sigma = (\\sigma_1, \\sigma_2)\\), we define<\/p>\n\n\n\n \\(P_\\sigma(x)\\) = the probability of node x being reached under \\(\\sigma\\) Player i<\/em>‘s beliefs \\(\\mu\\) are said to be Bayesian given<\/em> \\(\\sigma\\) if for every x \u2208 I<\/em>.<\/p>\n\n\n\n \\(\\mu_i(x) = \\frac{P_\\sigma(x)}{P_\\sigma(I)}\\) whenever \\(P_\\sigma(I)\\) > 0<\/p>\n\n\n\n EXAMPLE 5.4<\/mark>. Suppose that the 3 players play according to \\(\\sigma\\) in the following game: <\/p>\n\n\n \\(P_sigma(x)\\) = (0.5T + 0.5B, (0.2L + 0.8R, 0.8L + 0.2R), W)<\/p>\n\n\n\n \\(P_\\sigma(x) = P_\\sigma(y)\\) = 0.5 Player 3’s Bayesian beliefs given \\(\\sigma\\) are<\/p>\n\n\n\n \\(\\mu_3(a) = \\frac{P_\\sigma(a)}{P_\\sigma(I)} = \\frac{0.1}{0.9} = \\frac{1}{9}\\), \\(\\mu_3(b) = \\mu_3(c) = \\frac{4}{9}\\)<\/mark><\/p>\n\n\n\n P(A|B) = P(A\u22c2B)\/P(B)<\/mark><\/p>\n\n\n\n When P1 plays B and P2 plays R at node y, P3’s info set I<\/em> is never reached. Recall that in games of perfect information, the principle of sequential rationality requires that each player’s strategy be optimal at every decision node<\/mark><\/strong>, given the strategy profile, regardless of whether the node is reached.<\/p>\n\n\n\n Since some decision nodes are indistinguishable in games of imperfect information<\/em>, we amend the principle as follows:<\/p>\n\n\n\n In games of imperfect<\/mark><\/strong> information, the principle of sequential rationality requires that each player’s strategy be optimal at every information set<\/mark><\/strong>, given the strategy profile and the player’s beliefs<\/mark><\/strong>, regardless of whether the set is reached.<\/p>\n\n\n\n <\/p>\n\n\n\n1 \\ 2<\/td> L<\/td> R<\/td><\/tr> T<\/td> 2, 1<\/td> 0, 0<\/td><\/tr> M<\/td> 0, 2<\/td> 0, 1<\/td><\/tr> B<\/td> 1, 3<\/td> 1, 3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n ![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/06\/image-16.png\")
<\/figure><\/div>\n\n\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/06\/image-17.png\")
<\/figure><\/div>\n\n\n1 \\ 2<\/td> L<\/td> R<\/td><\/tr> T<\/td> 4, 1<\/td> 0, 0<\/td><\/tr> M<\/td> 3, 0<\/td> 0, 1<\/td><\/tr> B<\/td> 2, 2<\/td> 2, 2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n Beliefs<\/h3>\n\n\n\n
The sum of the decision nodes of a and c is 1, b and d are also 1, and beliefs can be defined by their respective probability distributions<\/mark><\/p>\n\n\n\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/07\/image-11-1024x618.png\")
<\/figure><\/div>\n\n\n\n
\n
\\(\\mu_3(a) + \\mu_3(b) = 1\\)<\/mark><\/p>\n\n\n\nBayesian Beliefs<\/h3>\n\n\n\n
\\(P_\\sigma(I)\\) = the probability of info set I<\/em> being reached under \\(\\sigma\\)<\/p>\n\n\n\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/07\/image-12-1024x588.png\")
<\/figure><\/div>\n\n\n
\\(P_\\sigma(a)\\) = 0.1
\\(P_\\sigma(b)\\) = 0.4
\\(P_\\sigma(c)\\) = 0.4
\\(P_\\sigma(I)\\) = 0.9, where I<\/em> = {a, b, c}<\/p>\n\n\n\n
In this case, we say \\(\\mu_3\\) is unrestricted<\/em> and any arbitrary \\(\\mu_3\\) is Bayesian.<\/p>\n\n\n\nSequential Rationality<\/h3>\n\n\n\n