8.1
Let p be the probability that Row plays Up, and q the probability that Column plays Left. Row’s
expected payoffs from each strategy are
EU
U
= 4q – (1 – q) = 5q – 1
EU
D
= q + 2(1 – q) = 2 – q
If q = 1/2, Row is indifferent between the two strategies, if q > 1/2, Up gives a higher expected payoff,
and if q < 1/2, Down gives a higher expected payoff. Row’s best response function is therefore
Similarly, Column’s expected payoffs from each strategy are
EU
L
= 1 – p
EU
R
= 2p – (1 – p) = 3p – 1
If p = 1/2, Column is indifferent between the two strategies, if p > 1/2, Right gives a higher expected
payoff, and if p < 1/2, Left gives a higher expected payoff. Column’s best response function is
We can now graph these functions to create a bestresponse diagram and show the mixedstrategy
Nash equilibrium, (p=1/2, q=1/2):
0.5
BR(q)
0.5
BC(p)
NE
p
q
The players’ expected payoffs can be found by plugging the equilibrium values of p and q back into
one of the expected payoff equations used to find the equilibrium strategies:
EU
Row
= EU
U
= EU
D
= 2 – q = 2 – 1/2 = 1.5
EU
Column
= EU
R
= EU
L
= 1 – p = 1 – 1/2 = 0.5
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View Full Document8.2
(a)
Using bestresponse analysis, we find two purestrategy NE: (2, A) and (1, C).
(b)
Before calculating the equilibrium mix, it’s always best to see if any pure strategies are strictly
dominated, and can therefore be eliminated (this greatly simplifies the analysis). In this case, D is
dominated by C for column. B is never a best response for Column, but it is not strictly dominated by
either A or C. In this sort of situation, it’s worth checking to see if B is strictly dominated by a mixed
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 Summer '08
 Bonanno,G
 Game Theory, Cutler, higher expected payoff, strictly dominant strategy

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