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The weakly lower semicontinuity condition on the

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Unformatted text preview: in xi ∗ over a subset X−i ⊂ X−i , if for all xi there exists λ ∈ [0, 1] such that, for all ∗ x−i ∈ X−i , λ lim inf f (xi� , x−i ) + (1 − λ) lim inf f (xi� , x−i ) ≥ f (xi , x−i ). xi� ↑ i x xi� ↓xi 14 Game Theory: Lecture 6 Continuous Games Discontinuous Games Theorem (2) [Dasgupta and Maskin] Let Si be a closed interval of R. Assume that ui is continuous except on a subset S ∗∗ (i ) of the set S ∗ (i ) defined in Eq. (1). Assume also that ∑n=1 ui (s ) is upper semicontinuous and that ui (si , s−i ) i is bounded and weakly lower semicontinuous in si over the set {s−i ∈ S−i | (si , s−i ) ∈ S ∗∗ (i )}. Then the game has a mixed strategy Nash equilibrium. The weakly lower semicontinuity condition on the utility functions implies that the function ui does not jump up when approaching si either from below or above. Loosely, this ensures that player i can do almost as well with strategies near si as with si , even if his opponents put weight on the discontinuity points of ui . 15 Game Theory: Lecture 6 Continuous Games Bertrand Competition with Capacity Constraints Consider two firms that charge prices p1 , p2 ∈ [0, 1] per unit of the same good. Assume that there is unit demand and all customers choose the firm with the lower price. If both firms charge the same price, each firm gets half the demand. All demand has to be supplied. The payoff functions of each firm is the profit they make (we assume for simplicity that cost of supplying the good is equal to 0 for both firms). 16 Game Theory: Lecture 6 Continuous Games Bertrand Competition with Capacity Constraints We have shown before that (p1 , p2 ) = (0, 0) is the unique pure strategy Nash equilibrium. Assume now that each firm has a capacity constraint of 2/3 units of demand: Since all demand has to be supplied, this implies that when p1 < p2 , firm 2 gets 1/3 units of demand). It can be seen in this case that the strategy profile (p1 , p2 ) = (0, 0) is no longer a pure strategy Nash equilibrium: Ei...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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