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Unformatted text preview: ther ﬁrm can increase his price and still have 1/3 units of demand
due to the capacity constraint on the other ﬁrm, thus making positive
proﬁts. It can be established using Theorem 2 that there exists a mixed
strategy Nash equilibrium.
Let us next proceed to construct a mixed strategy Nash equilibrium. 17 Game Theory: Lecture 6 Continuous Games Bertrand Competition with Capacity Constraints
We focus on symmetric Nash equilibria, i.e., both ﬁrms use the same mixed
strategy.
We use the cumulative distribution function F (·) to represent the mixed
strategy used by either ﬁrm.
It can be seen that the expected payoﬀ of player 1, when he chooses p1 and
ﬁrm 2 uses the mixed strategy F (·), is given by
p
2
u1 (p1 , F (·)) = F (p1 ) 1 + (1 − F (p1 )) p1 .
3
3
Using the fact that each action in the support of a mixed strategy must yield
the same payoﬀ to a player at the equilibrium, we obtain for all p in the
support of F (·),
p
2
−F (p ) + p = k ,
3 3 for some k ≥ 0. From this we obtain: 3k F (p ) = 2 −
.
p
18 Game Theory: Lecture 6 Continuous Games Bertrand Competition with Capacity Constraints
Note next that the upper support of the mixed strategy must be at p = 1,
which implies that F (1) = 1.
Combining with the preceding, we obtain
⎧
if 0 ≤ p ≤ 1 ,
⎨ 0,
2
1,
2− p
if 1 ≤ p ≤ 1,
F (p ) =
2
⎩
1,
if p ≥ 1. 19 Game Theory: Lecture 6 Continuous Games Uniqueness of a Pure Strategy Nash Equilibrium in
Continuous Games
We have shown in the previous lecture the following result:
Given a strategic form game �I , (Si ), (ui )�, assume that the strategy
sets Si are nonempty, convex, and compact sets, ui (s ) is continuous in
s , and ui (si , s−i ) is quasiconcave in si . Then the game �I , (Si ), (ui )�
has a pure strategy Nash equilibrium. The next example shows that even under strict convexity assumptions,
there may be inﬁnitely many pure strategy Nash equilibria. 20 Game Theory: Lecture 6 Continuous Games Uniqueness of a Pure Strategy Nash Equilibrium
Example
Consider a game with 2 players, Si = [0,...
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 Spring '10
 AsuOzdaglar

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