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ECN214 Games and Strategies – 2015
Questions and Answers
Part A
1.
a)
Holding the opponents’ strategy choices fixed, a strategy is a best response for a player if it yields a
weakly greater payoff than any other strategy. A strategy profile is a Nash equilibrium if each
player’s strategy is a best response to his or her opponents’ strategies.
To find all the Nash equilibria in pure strategies, we consider each player’s Best Response to each of
the other player’s actions. First, we consider the player choosing from l and r’s best responses. Their
payoffs from these actions are underlined below:
L r
u 2, 6 9, 3
m 4, 9 5, 7
d 4, 7 6, 8
Next, we consider the other player’s best responses. Their payoffs are also underlined below:
l r
u 2, 6 9, 3
m 4, 9 5, 7
d 4, 7 6, 8
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This demonstrates that there is one unique Nash equilibrium in pure strategies. This is the one
where one player plays m and the other plays l. Neither play can benefit from deviating, and
therefore this is a NE.
b) One outcome Pareto dominates another if it is weakly preferred by all players and strictly
preferred by at least one.
The payoff profiles which are Pareto dominated are highlighted below:
l r
u 2, 6 9, 3
m 4, 9 5, 7
d 4, 7 6, 8
This is because there are other payoff profiles which result in a payoff which is at least as large for
one player, and larger for the other.
c) The strategies which are weakly dominated for each player are:
l r
u 2, 6 9, 3
m 4, 9 5, 7
d 4, 7 6, 8
One strategy weakly dominates another if it yields (1) a weakly greater payoff for any strategy choice
by the opponent; and (2) a strictly greater payoff for some strategy choice by the opponent.
d)
A mixed strategy for a player is a probability distribution over his or her pure strategies. To find a
mixed strategy Nash equilibrium, we have to find the probabilities such that each player is
indifferent between playing their strategies. Therefore, their expected return from each strategy
must be equal.
Player 1 has 3 possible strategies, but they will never choose to play m because m is weakly
dominated by the strategy d. Therefore, their expected return from each of their two possible
strategies are:
𝐸[𝑢] = 2𝜎 + 9(1 − 𝜎)
𝐸[𝑑] = 4𝜎 + 6(1 − 𝜎)
Setting these two equal:
2𝜎 + 9(1 − 𝜎) = 4𝜎 + 6(1 − 𝜎)
3 = 5𝜎
3/5 = 𝜎
Similarly, player 2 has two potential strategies, neither of which is dominated. Their expected return
from the two possible strategies are:
