Download 2 - BrainMass

2. Suppose that two players are playing the following game. Player 1 can choose either
Top or Bottom, and Player 2 can choose either Left or Right. The payoffs are given in
the following table:
Player 2
Left
Right
Player 1
Top
9
4
2
5
Bottom
1
0
3
1
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) Does player 1 have a dominant strategy, and if so what is it?
Player 1 has no dominant strategy, because he will choose top when player 2 chooses left,
and he will choose bottom when player 2 choose right.
B) Does player 2 have a dominant strategy and if so what is it?
Player 2 has dominant strategy: right, because his payoff is always higher no matter
which choice player 1 chooses. That is, 5>4 and 1>0.
C) Determine the Nash equilibria of this game. If there are no equilibria in pure
strategies, then say so. Otherwise, indicate what the equilibria are by the strategies that
would be chosen (i.e. Top/Left or Bottom/Right).
When the information is known by both players, the Nash equilibrium will be bottomright. Because player 2’s dominant strategy is right, player will choose between top right
and bottom right. Obviously, the payoff is higher for bottom. Thus, the Bottom/Right will
be chosen.
D) If each player plays their maximin strategy, what payoff will each of them receive?
With a maximin strategy, a player determines the worst outcome for each option, then
chooses the option that maximizes the payoff among the worst outcomes. If player 1
chooses top, the worst payoff would occur if player 2 chooses right: player 1’s payoff
would be 2. If player 1 chooses bottom, the worst payoff would occur if player 2 chooses
left: player 1’s payoff would be 1. With a maximin strategy, therefore player 1 chooses
top.
If player 2 chooses left, the worst payoff would occur if player 1 chooses bottom: player
2’s payoff would be 0. If player 2 chooses right, the worst payoff would occur if player 1
chooses bottom: player 2’s payoff would be 1. With a maximin strategy, player 2
therefore chooses right.
So under maximin, the top right will be chosen. player 1’s payoff is 2 and player 2’s
payoff is 5.
E) Now suppose the same game is played with the exception that Player 1 moves first and
Player 2 moves second. Using the backward induction method discussed in class, what
will be the outcome of the game? (Hint: it will be helpful to sketch the game tree.)
1
/
top
2
/ \
left right
(9,4) (2,5)
\
bottom
2
/ \
left right
(1,0) (3,1)
Since player 1 moves first and he knows that player 2’s dominant strategy is right. Player
1 will then compare top/right to bottom/right. Since the payoff of bottom right is higher
for player 1, he will choose bottom. Thus, the Nash equilibrium is again bottom/right.
F) Now suppose the same game is played with the exception that Player 2 moves first
and Player 1 moves second. Using the backward induction method discussed in class,
what will be the outcome of the game? (Hint: it will be helpful to sketch the game tree.)