Graphical Method for Solving Mixed Strategy Game

The graphical method is another technique used in game theory to solve mixed strategy games. In this method, the expected payoffs for each player are plotted on a graph, and the Nash equilibrium can be found where the two payoff curves intersect.

To use the graphical method to solve a two-player mixed strategy game, follow these steps:

Create a matrix representing the game and calculate the expected payoff for each player for each possible combination of strategies.

Plot the expected payoffs on a graph, with the payoffs for one player on the x-axis and the payoffs for the other player on the y-axis. Label the axes accordingly.

Draw a line from the highest expected payoff for one player to the lowest expected payoff for the other player, passing through the point where the two payoffs are equal. This line is called the indifference curve.

Repeat step 3 for the other player, drawing a line from their highest expected payoff to their lowest expected payoff, passing through the point where the two payoffs are equal. This is the second indifference curve.

Find the intersection point of the two indifference curves. This is the Nash equilibrium, which represents the optimal mixed strategy for each player.

For example, consider the following two-player game:

markdown

| L | M | R |

—————-

U | 2 | 1 | 3 |

—————-

D | 1 | 3 | 2 |

—————-

To use the graphical method to solve this game, we first calculate the expected payoffs for each player for each possible combination of strategies:

markdown

| L | M | R |

—————-

U |2/3|1/3| |

—————-

D |1/3|2/3| |

—————-

Player 1’s expected payoffs:

– If Player 2 chooses L: (2+1)/2 = 1.5

– If Player 2 chooses M: (1+3)/2 = 2

– If Player 2 chooses R: (3+2)/2 = 2.5

Player 2’s expected payoffs:

– If Player 1 chooses U: (2+1)/2 = 1.5

– If Player 1 chooses D: (1+3)/2 = 2

– If Player 1 chooses R: (3+2)/2 = 2.5

Next, we plot the expected payoffs on a graph, with the payoffs for Player 1 on the x-axis and the payoffs for Player 2 on the y-axis:

rust

Player 2’s Payoff

| 1.5 2.0 2.5

————————-

2.5| X

2 | X

1.5|X

|

|——————

1.5 2.0 2.5

Player 1’s Payoff

We draw the first indifference curve passing through the point (2,2), which is the point where Player 1’s expected payoff is equal to Player 2’s expected payoff.

Then, we draw the second indifference curve passing through the point (2,2), which is the point where Player 2’s expected payoff is equal to Player 1’s expected payoff.

The intersection of the two indifference curves is at point (2,2), which is the Nash equilibrium. This means that each player should play their strategies with equal probability.

In summary, the graphical method is a useful technique for solving mixed strategy games by plotting the expected payoffs on a graph and finding the intersection of the indifference curves