# Dominant Strategy

Suppose that two players are playing the following game. Player A can choose either Top or Bottom, and Player B can choose either Left or Right. The payoffs are given in the following table:

Player B

Player A Left Right

Top 2 5 1 4

Bottom 0 1 3 8

where the number on the left is the payoff to Player A, and the number on the right is the payoff to Player B.

A) (4 points) Does player A have a dominant strategy, and if so what is it?

B) (4 points) Does player B have a dominant strategy and if so what is it?

C) (16 points) For each of the following say True if the strategy combination is a Nash equilibrium, and False if it is not a Nash equilibrium:

i) Player A plays Top and Player B plays Left

ii) Player A plays Bottom and Player B plays Left

iii) Player A plays Top and Player B plays Right

iv) Player A plays Bottom and Player B plays Right

Suppose that two players are playing the following game. Player A can choose either Top or Bottom, and Player B can choose either Left or Right. The payoffs are given in the following table:

Player B

Player A Left Right

Top 2 5 1 4

Bottom 0 1 3 8

where the number on the left is the payoff to Player A, and the number on the right is the payoff to Player B.

A) (4 points) Does player A have a dominant strategy, and if so what is it?

B) (4 points) Does player B have a dominant strategy and if so what is it?

C) (16 points) For each of the following say True if the strategy combination is a Nash equilibrium, and False if it is not a Nash equilibrium:

i) Player A plays Top and Player B plays Left

ii) Player A plays Bottom and Player B plays Left

iii) Player A plays Top and Player B plays Right

iv) Player A plays Bottom and Player B plays Right

If each player plays her maximum strategy what will be the outcome of the game? (Give your answer in terms of the strategies each player chooses—for example, "Player A plays Bottom and Player B plays Right"

Now suppose the same game is played with the exception that Player A moves first and Player B moves second. Draw the game tree associated with this situation. Using the backward induction method discussed in the online class notes, what will be the outcome of the game?

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#### Solution Preview

Let me first remind you that dominant strategy means that a player is ALWAYS better playing this strategy no matter what the other guy does.

A) player A does not have a dominant strategy (DS).

Reason: Top is not a DS because if B plays Right, then Top gives player A 1 while bottom gives player A 3, so top is not always the best choice. You may verify that neither is Bottom a DS

B) player B does ...

#### Solution Summary

The expert finds dominant strategy and Nash equilibrium for payoffs of players.