Game Theory: Algorithms and Applications

Running head: GAME THEORY: ALGORITHMS AND APPLICATIONS
GAME THEORY: ALGORITHMS AND APPLICATIONS
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Author Note

GAME THEORY: ALGORITHMS AND APPLICATIONS2
Question 1:
The payoff matrix of the two players rolling two integers x1 and x2 by player 1 and player 2
respectively.
x2 <= x1 x1<=x2
x1<=3x2 (1,-1) (-1,1)
x2<= 3x1 (-1,1) (1,-1)
Now, in this particular strategy there is no pure Nash equilibrium as by choosing any strategy
one player win or other player loses. Hence, the best strategy in this case will be to mix or
randomize the between the strategies such that no player gets exploited.
Now, let player 1 chooses x1 in x2 <= x1 <= 3x2 with probability p and hence, (1-p) is the
probability of choosing outside the range. Now, E(player 1) = p*1 + (1-p)*(-1). Now, the
Nash equilibrium condition is when player 1 neither gain nor loses.
Hence, p*1 + (1-p)*(-1) = 0 => 2p = 1 => p = 0.5
This is same for player 2 also.
Thus in the mixed Nash equilibrium condition player 1 chooses x1 with 50% and player 2
chooses x2 with 50% probability.
2.
The pay-off matrix for the Co-ordination game is given below.
Payoff Slack-off Effort
Slack-off (0,0) (0,-c)
Effort (-c,0) (1-c,1-c)