Expected Value Calculations for Mixed Strategies in Game Theory
BSNS5230, J. Wang
Name __________________________
Given a zero-sum game of two players, R and C, without an equilibrium point:
C’s strategies
R’s strategies
1
2
A
8
1
B
4
7
1. Let the mixed strategy for R be {0.6, 0.4}, and mixed strategy for C is {0.3, 0.7}. Calculate the
expected value of each mixed strategy.
C’s mixed strategy
0.3
0.7
A
B
R’s
mixed
strategy
0.6
1
8
4
0.4
2
1
7
Exp. value of each
strategy of R
Exp. value of R’s
mixed strategy
Exp. value of
each strategy of
C
Exp. value of C’s
mixed strategy
The mixed strategies of R and C are equilibrium strategies because the expected values of all
individual strategies are identical.
2. If R took the Maximin strategy and C took the Minimax strategy, as if the game had an
equilibrium point, then:
C’s strategies
(Row Min)
Maximum of
Minimum of each
Row Mins
A
B
strategy of R
(maximin)
R’s
strategies
1
8
4
1
7
2
(Col. Max)
Maximum of each
strategy of C
Minimum of Col.
Max’s
(minimax)
That is, R would take strategy 1 with maximin payoff 4, and C would take strategy 2 with
minimax penalty 7.
The expected payoff / penalty with the equilibrium mixed strategies as in (1) is __________,
which is ____________ (better off, worse off) than R’s maximin and _____________ (better off,
worse off) than C’s minimax.
3. If player C unilaterally leaves his equilibrium mixed strategy to {0.1, 0.9}, then the expected
value of each mixed strategy becomes:
C’s mixed strategy
0.1
0.9
A
B
R’s
mixed
strategy
0.6
1
8
4
0.4
2
1
7
Exp. value of each
strategy of R
Exp. value of
each strategy of
C
Exp. value of C’s
mixed strategy
Comparing to the equilibrium mixed strategies:
Who is better off with this change? ____________________
Who is worse off with this change? ____________________
Exp. value of R’s
mixed strategy
4. If player C unilaterally leaves his equilibrium mixed strategy to {1, 0}, then the expected value of
each mixed strategy becomes:
C’s mixed strategy
R’s
mixed
strategy
1
A
0
B
0.6
1
8
4
0.4
2
1
7
Exp. value of each
strategy of R
Exp. value of R’s
mixed strategy
Exp. value of
each strategy of
C
Exp. value of C’s
mixed strategy
Comparing to the equilibrium mixed strategies:
Who is better off with this change? ____________________
Who is worse off with this change? ____________________
5. If player R unilaterally leaves his equilibrium mixed strategy to {0.2, 0.8}, then the expected
value of each mixed strategy becomes:
C’s mixed strategy
0.3
0.7
A
B
R’s
mixed
strategy
0.2
1
8
4
0.8
2
1
7
Exp. value of each
strategy of R
Exp. value of
each strategy of
C
Exp. value of C’s
mixed strategy
Comparing to the equilibrium mixed strategies:
Who is better off with this change? ____________________
Who is worse off with this change? ____________________
Exp. value of R’s
mixed strategy
6. If both players leave their equilibrium mixed strategies: R changes to {0.5, 0.5}, and C changes
to {0.8, 0.2}, then the expected value of each mixed strategy becomes:
C’s mixed strategy
0.8
0.2
A
B
R’s
mixed
strategy
0.5
1
8
4
0.5
2
1
7
Exp. value of each
strategy of R
Exp. value of R’s
mixed strategy
Exp. value of
each strategy of
C
Exp. value of C’s
mixed strategy
Comparing to the equilibrium mixed strategies:
Who is better off with this change? ____________________
Who is worse off with this change? ____________________
7. If both players leave their mixed strategies: R changes to {0.2, 0.8}, and C changes to {0, 1},
then the expected value of each mixed strategy becomes:
C’s mixed strategy
R’s
mixed
strategy
0
A
1
B
0.2
1
8
4
0.8
2
1
7
Exp. value of each
strategy of R
Exp. value of
each strategy of
C
Exp. value of C’s
mixed strategy
Comparing to the equilibrium mixed strategies:
Who is better off with this change? ____________________
Who is worse off with this change? ____________________
Exp. value of R’s
mixed strategy