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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.
 Spring '10
 AsuOzdaglar

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