MATHEMATICS

Mathematics



Mathematics

Question 4

Dominated strategies

The payoff matrix facilitates elimination of dominated strategies, and it is usually used to illustrate this concept. For example, in the prisoner's dilemma (to the right), we can see that each prisoner can either "cooperate" or "defect". If exactly one prisoner defects, he gets off easily and the other prisoner is locked up for good. However, if they both defect, they will both be locked up for longer. One can determine that Cooperate is strictly dominated by Defect. One must compare the first numbers in each column, in this case 0 > -1 and -2 > -5. This shows that no matter what the column player chooses, the row player does better by choosing Defect. Similarly, one compares the second payoff in each row; again 0 > -1 and -2 > -5. This shows that no matter what row does, column does better by choosing Defect. This demonstrates the unique Nash equilibrium of this game is (Defect, Defect).

In this case there are two pure strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%,100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player is (50%, 50%).

Player 2 chooses '0'

Player 2 chooses '1'

Player 2 chooses '2'

Player 2 chooses '3'

Player 1 chooses '0'

0, 0

2, -2

2, -2

2, -2

Player 1 chooses '1'

-2, 2

1, 1

3, -1

3, -1

Player 1 chooses '2'

-2, 2

-1, 3

2, 2

4, 0

Player 1 chooses '3'

-2, 2

-1, 3

0, 4

3, 3

Equilibria in a Payoff Matrix

There is an easy numerical way to identify Nash equilibria on a payoff matrix. It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the duplet of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium.

Option A

Option B

Option C

Option A

0, 0

25, 40

5, 10

Option B

40, 25

0, 0

5, 15

Option C

10, 5

15, 5

10, 10

A Payoff Matrix - Nash Equilibria in bold

We can apply this rule to a 3×3 matrix:

Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash Equlibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A) 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B) 25 is the maximum of the second column and 40 is the maximum of the first row. Same for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and ...