Observe that pi is not lhc in general. Matching Pennies provides an example(you should give the details) We are ready to state our main result today Theorem 0.4 For any finite game G, its mixed extension has a Nash equilibrium Interpretation There are two independent issues we need to discuss: the meaning of Nash equilibrium per se,e.g.regardless of the "trick"used to guarantee existence in finite games, and th he meanl of randomization Nash equilibrium For the purposes of this subsection, let us focus on Definition 2, so we need not worry about MiXing. There are at least two main approaches to game theory The eductive approach: we assume that players are rational(e. g. Bayesian rational and capable of thinking strategically. That is, we assume that players think hard before playing a game, evaluating their options in light of conjectures about their ents behavior. Solution concepts are characterized by assumptions about the of reasoning followed by the players The learning approach: we assume that players interact several times, i.e. play the same game over and over. They may be assumed to be rational to different extents the crucial idea is that their beliefs are"empirical", i.e. crucially based on their play experience. Solution concepts then reflect steady states of the ensuing dynamical process. As should be clear by now, the "informal equation" Rationality Assumptions about Beliefs= Solution Concepts conveys the main message of the eductive approach. Let us first evaluate Nash equilibrium form this particular viewpoint ash equilibrium is characterized by the assumption that players, beliefs are correct that is, every player believes that her opponents will choose precisely that action which according to the solution concept, they will in fact choose. One must be very clear about this. Consider the simple game in Figure 1Observe that ρ Γ i is not lhc in general. Matching Pennies provides an example (you should give the details). We are ready to state our main result today: Theorem 0.4 For any finite game G, its mixed extension has a Nash equilibrium. Interpretation There are two independent issues we need to discuss: the meaning of Nash equilibrium per se, e.g. regardless of the “trick” used to guarantee existence in finite games, and the meaning of randomization. Nash equilibrium For the purposes of this subsection, let us focus on Definition 2, so we need not worry about mixing. There are at least two main approaches to game theory. • The eductive approach: we assume that players are rational (e.g. Bayesian rational) and capable of thinking strategically. That is, we assume that players think hard before playing a game, evaluating their options in light of conjectures about their opponents’ behavior. Solution concepts are characterized by assumptions about the specific line of reasoning followed by the players. • The learning approach: we assume that players interact several times, i.e. play the same game over and over. They may be assumed to be rational to different extents; the crucial idea is that their beliefs are “empirical”, i.e. crucially based on their play experience. Solution concepts then reflect steady states of the ensuing dynamical process. As should be clear by now, the “informal equation” Rationality + Assumptions about Beliefs = Solution Concepts conveys the main message of the eductive approach. Let us first evaluate Nash equilibrium form this particular viewpoint. Nash equilibrium is characterized by the assumption that players’ beliefs are correct: that is, every player believes that her opponents will choose precisely that action which, according to the solution concept, they will in fact choose. One must be very clear about this. Consider the simple game in Figure 1. 5