Answer: Rational players not only avoid dominated strategies but also avoid strategies that are never a best response<\/mark><\/strong><\/p>\n\n\n\n
Applying this idea iteratively, we obtain the set of rationalizable<\/em> strategies<\/p>\n\n\n\n DEFINITION 2.7<\/mark>. Suppose player i has a belief \\(\\theta_{i} \\in \\Delta S_{j}\\) about player j’s behavior.<\/p>\n\n\n\n Strategy \\(s_{i}\\) is a best response<\/mark><\/strong> to belief \\(\\theta_{i}\\) if<\/p>\n\n\n\n \\(u_{i}(s_{i}, \\theta_{i})\\) \u2265 \\(u_{i}(s_{i}^{\\prime}, \\theta_{i})\\) for every \\(s_{i}^{\\prime}\\ \\in S_{i}\\).<\/p>\n\n\n\n For any belief \\(\\theta_{i}\\) of player i, we denote by \\(BR_{i}(\\theta_{i})\\) the set of best responses.<\/p>\n\n\n\n Observe that the set of best response is a function of the player’s belief<\/p>\n\n\n\n EXAMPLE 2.12. Suppose player 1 forms a belief \\(\\theta_{1}\\) = (1\/3, 1\/2, 1\/6) on player 2’s behavior in the following game:<\/p>\n\n\n\n Under the belief \\(\\theta_{1}\\), player 1’s expected payoff from each action is<\/p>\n\n\n\n \\(u_{1}(U, \\theta_{1})\\) = \\(2 \\cdot \\frac{1}{3} + 0 \\cdot \\frac{1}{2} + 4 \\cdot \\frac{1}{6} = \\frac{2}{3} + \\frac{2}{3} = \\frac{4}{3}\\)<\/mark><\/p>\n\n\n\n \\(u_{1}(M, \\theta_{1})\\) = \\(3 \\cdot \\frac{1}{3} + 0 \\cdot \\frac{1}{2} + 1 \\cdot \\frac{1}{6} = 1 \\cdot \\frac{1}{6} = \\frac{7}{6}\\)<\/mark><\/p>\n\n\n\n \\(u_{1}(D, \\theta_{1})\\) = \\(1 \\cdot \\frac{1}{3} + 3 \\cdot \\frac{1}{2} + 2 \\cdot \\frac{1}{6} = \\frac{1}{3} + \\frac{3}{2} + \\frac{1}{3} = \\frac{13}{6}\\)<\/mark><\/p>\n\n\n\n Hence player 1’s set of best responses is \\(BR_{1}(\\theta_{1})\\) = {D}<\/p>\n\n\n\n Given \\(\\theta_{2}\\) = (1\/2, 1\/4, 1\/4), what is player 2’s best response?<\/p>\n\n\n\n \\(u_{2}(L, \\theta_{2})\\) = \\(6 \\cdot \\frac{1}{2} + 3 \\cdot \\frac{1}{4} + 1 \\cdot \\frac{1}{4} = 3 + 1 = 4\\)<\/mark><\/p>\n\n\n\n \\(u_{2}(C, \\theta_{2})\\) = \\(4 \\cdot \\frac{1}{2} + 0 \\cdot \\frac{1}{4} + 5 \\cdot \\frac{1}{4} = 2 + \\frac{5}{4} = \\frac{13}{4}\\)<\/mark><\/p>\n\n\n\n \\(u_{2}(R, \\theta_{2})\\) = \\(4 \\cdot \\frac{1}{2} + 5 \\cdot \\frac{1}{4} + 3 \\cdot \\frac{1}{4} = \\frac{8+5+3}{4} = \\frac{16}{4} = 4\\)<\/mark><\/p>\n\n\n\n Observe that a player’s set of best responses relies upon the player’s beliefs<\/p>\n\n\n\n Playing a best response is not in itself a strategic act<\/p>\n\n\n\n It is formation of beliefs<\/mark><\/strong> that captures the important strategic component in games<\/p>\n\n\n\n There is a close relations between dominance and best response. To see this, let<\/p>\n\n\n\n \\(B_{i}\\) = {\\(s_{i} \\in S_{i}\\) | there exists a belief \\(\\theta_{i}\\) such that \\(s_{i} \\in BR_{i}(\\theta_{i})\\)}<\/p>\n\n\n\n \\(UD_{i}\\) = {\\(s_{i} \\in S_{i}\\) | \\(s_{i}\\) is not strictly dominated}.<\/p>\n\n\n\n \\(u_{1}(U, p)\\) = 6p + 0(1-p) = 6p 6p = 5 – 3p \u21d4 9p = 5 \u21d4 p = 5\/9<\/mark><\/p>\n\n\n\n p > \\(\\frac{5}{9}\\) \u2192 \\(BR_{1}(p)\\) = {U} P1’s belief \\(\\theta_{1}\\) can be represented by a scalar variable p \u2208 [0, 1]<\/p>\n\n\n\n Strategy D is never a best response to player 1 for all p<\/em>. So \\(B_{1}\\) = {U, M}<\/mark><\/strong>.<\/p>\n\n\n\n One can easily see that U and M are undominated. What about strategy D? xU + (1-x)M = \\(\\sigma_{1}\\) When x = 1\/3, \\(u_{1}(\\sigma_{1}, p)\\) = 10\/3, which is greater than 3, the payoff of \\(u_{1}\\)(D, p) Strategies that are not dominated by player 1 \\(UD_{1}\\) = {U, M} = \\(B_{1}\\)<\/mark><\/p>\n\n\n\n Observe that if a strategy is strictly dominated, then it is never a best response:<\/p>\n\n\n\n \\(B_{i} \\subseteq UD_{i}\\) for all i<\/p>\n\n\n\n The converse is not true in general, but it turns out that the converse is also true at least in 2-player games<\/p>\n\n\n\n Therefore,<\/p>\n\n\n\n strictly dominated \u21d4 never a best response in 2-player games.<\/p>\n\n\n\n DEFINITION 2.8<\/mark>. Rationalizable strategies<\/mark><\/strong> are those that remain after we iteratively remove all strategy that are never a best response to allowable beliefs.<\/p>\n\n\n\n THEOREM 2.9 (BERNHEIM (1984), PEARCE (1984))<\/mark>. In two-player games, a strategies is rationalizable if and only if it survives ISD.<\/p>\n\n\n\n In games with more than two-players,<\/p>\n\n\n\n the set of rationalizable strategies \u2282 the set of strategies that survives ISD.<\/p>\n\n\n\n To see this relation, consider the following example:<\/p>\n\n\n\n P3’s strategy C (the middle matrix) is not<\/em> strictly dominated so survives ISD<\/p>\n\n\n\n On the other hand, C is never a best response to any beliefs \\(\\theta_{3}\\), implying that the strategy is not rationalizable<\/p>\n\n\n\n <\/p>\n\n\n\n1 \\ 2<\/td> L<\/td> C<\/td> R<\/td><\/tr> U<\/td> 2, 6<\/td> 0, 4<\/td> 4, 4<\/td><\/tr> M<\/td> 3, 3<\/td> 0, 0<\/td> 1, 5<\/td><\/tr> D<\/td> 1, 1<\/td> 3, 5<\/td> 2, 3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n Best Responses and Beliefs<\/h3>\n\n\n\n
\n
Dominance and Best Response<\/h3>\n\n\n\n
1 \\ 2<\/td> L (p)<\/td> R (1-p)<\/td><\/tr> U<\/td> 6, 3<\/td> 0, 1<\/td><\/tr> M<\/td> 2, 1<\/td> 5, 0<\/td><\/tr> D<\/td> 3, 2<\/td> 3, 1<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\\(u_{1}(M, p)\\) = 2p + 5(1-p) = -3p + 5
\\(u_{1}(D, p)\\) = 3p + 3(1-p) = 3<\/mark><\/p>\n\n\n\nf(p) = 6p, f(p) = -3p+5, f(p) = 3, p=(0,1)\nx-axis: p, y-axis: u1<\/code><\/pre>\n\n\n\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/06\/image-1-1024x533.png\")
<\/figure>\n\n\n\n
p < \\(\\frac{5}{9}\\) \u2192 \\(BR_{1}(p)\\) = {M}
p = \\(\\frac{5}{9}\\) \u2192 \\(BR_{1}(p)\\) = {U, M}<\/mark><\/p>\n\n\n\n
To examine if there is a mixed strategy strictly better than D, let \\(\\sigma_{1}\\) = (x, 1-x, 0)
and compute P1’s expected payoff from \\(\\sigma_{1}\\) given his belief p<\/em>:<\/p>\n\n\n\n
\\(u_{1}(\\sigma_{1}, p)\\) = x \u2219 \\(u_{1}\\)(U, p) + (1-x)\\(u_{1}\\)(M, p)
= x \u2219 6p + (1-x)(5-3p)
= 5(1-x) – 3p + 9xp<\/mark><\/p>\n\n\n\n
so strategy D is strictly dominated by mixed strategies U and M.<\/mark><\/p>\n\n\n\nRationalizability**<\/h3>\n\n\n\n
Rationalizability in 3-player games<\/h3>\n\n\n\n
<\/td> W<\/td> E<\/td> <\/td> W<\/td> E<\/td> <\/td> W<\/td> E<\/td><\/tr> T<\/td> 9, 9, 5<\/td> 0, 8, 2<\/td> T<\/td> 9, 9, 4<\/td> 0, 8, 0<\/td> T<\/td> 9, 9, 1<\/td> 0, 8, 2<\/td><\/tr> B<\/td> 8, 0, 2<\/td> 7, 7, 1<\/td> B<\/td> 8, 0, 0<\/td> 7, 7, 4<\/td> B<\/td> 8, 0, 2<\/td> 7, 7, 5<\/td><\/tr> <\/td> L<\/td> <\/td> <\/td> C<\/td> <\/td> <\/td> R<\/td> <\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n