However, ISD makes a weak prediction in many games<\/p>\n\n\n\n
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<\/figure>\n\n\n\nIn order to obtain a sharp prediction, therefore, we need impose restrictions on players’ beliefs<\/mark><\/strong> about their opponents’ behavior. This will lead to the central solution concept of game theory, Nash equilibrium<\/em>.<\/p>\n\n\n\n <\/p>\n\n\n\n DEFINITION 2.10<\/mark>. A strategy profile s* = \\((s_{1}^{*}, s_{2}^{*})\\) is a Nash equilibrium<\/mark><\/strong> if for each player i, \\(s_{i}^{*}\\) is a best response to \\(s_{j}^{*}\\).<\/p>\n\n\n\n This implies,<\/p>\n\n\n\n \\(u_{1}(s_{1}^{*}, s_{2}^{*})\\) \u2265 \\(u_{1}(s_{1}, s_{2}^{*})\\) for every \\(s_{1} \\in S_{1}\\)<\/p>\n\n\n\n and<\/p>\n\n\n\n \\(u_{2}(s_{1}^{*}, s_{2}^{*})\\) \u2265 \\(u_{2}(s_{1}, s_{2}^{*})\\) for every \\(s_{2} \\in S_{2}\\)<\/p>\n\n\n\n Simply put, a Nash equilibrium is the players’ mutual<\/em> best responses<\/p>\n\n\n\n Two underlying assumptions:<\/p>\n\n\n\n (i) Each player has correct beliefs<\/em> about what opponents will do (equilibrium knowledge) The Nash equilibrium first appeared in 1952. If we think about an equilibrium where there is no risk of deviation from economics, the other player must know exactly what they are going to do.<\/mark> The next theorem provides a link between Nash eqbm and ISD<\/p>\n\n\n\n THEOREM 2.11. If a normal-form game is solvable by ISD, the strategy profile surviving the process constitutes a unique<\/em> Nash equilibrium of the game.<\/p>\n\n\n\n Nash eqbm is much more assumption-based than ISD. EXAMPLE 2.13<\/mark> (REVISIT PRISONER’S DILEMMA).<\/p>\n\n\n\nDefinition<\/h3>\n\n\n\n
(ii) Each behaves rationally<\/em> given these beliefs<\/p>\n\n\n\n
But that’s too much of an assumption to make, so we start thinking about Rationality alone.<\/mark>
This is where ISD comes in, and then Rationalizability comes in because there’s not much we can do with ISD after that.<\/mark><\/p>\n\n\n\nJustifications of Equilibrium Knowledge<\/h3>\n\n\n\n
1. Learning or evolution<\/h5>\n\n\n\n
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2. Pre-play communication<\/h5>\n\n\n\n
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3. Coordinatiton of play by a mediator (Contractual relationship, See Section 2.8)<\/h5>\n\n\n\n
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4. Focal points (THOMAS SCHELLING (1960))<\/h5>\n\n\n\n
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Computing Nash Equilibria<\/h3>\n\n\n\n
If ISD solves the problem, the remaining profile is the only Nash equilibrium in the game.<\/mark><\/p>\n\n\n\n
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