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# Why is this a reasonable notion no player can

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Unformatted text preview: 2 B2(s1) 1 / 4 1 / 4 1/2 s1 1 B2(s1) 1 / 4 1 / 4 1/2 1 s1 1 1 One round of elimination yields S1 = [0, 1/2], S2 = [0, 1/2] 2 2 Second round of elimination yields S1 = [1/4, 1/2], S2 = [1/4, 1/2] It can be shown that the endpoints of the intervals converge to the intersection Most games not solvable by iterated strict dominance, need a stronger equilibrium notion 24 Game Theory: Lecture 2 Nash Equilibrium Pure Strategy Nash Equilibrium Deﬁnition (Nash equilibrium) A (pure strategy) Nash Equilibrium of a strategic game �I , (Si )i ∈I , (ui )i ∈I � is a strategy proﬁle s ∗ ∈ S such that for all i∈I ∗ ∗ ui (si∗ , s−i ) ≥ ui (si , s−i ) for all si ∈ Si . Why is this a “reasonable” notion? No player can proﬁtably deviate given the strategies of the other players. Thus in Nash equilibrium, “best response correspondences intersect”. Put diﬀerently, the conjectures of the players are consistent: each ∗ player i chooses si∗ expecting all other players to choose s−i , and each player’s conjecture is veriﬁed in a Nash equilibrium. 25 Game Theory: Lecture 2 Nash Equilibrium Reasoning about Nash Equilibrium This has a “steady state” type ﬂavor. In fact, two...
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