c

Kendra Kilmer June 24, 2008

Section 9.4 - Game Theory and Strictly Determined Games We will focus on two-person zero-sum games.

Definition: A zero-sum game is a game in which the payoff to one party results in an equal loss to the other party.

Definition: A payoff matrix represents the possible moves of each player and the payoffs associated with each case. The entries represent the row player’s net winnings. Example 1: Let’s look at the familiar game of rock, paper, and scissors. Ron and Carl decide to play the game for money. If Ron wins, Carl must pay him one dollar. If Carl wins, Ron must pay him one dollar. If there is a tie, no money is exchanged. Recall, paper beats rock, rock beats scissors, and scissors beats paper. Find the payoff matrix for this game.

Example 2: When a football team has the ball and is planning its next play, it can choose one of several plays. The success of the chosen play depends largely on how well the other team ”reads” the chosen play. Suppose a team with the ball (Team A) can choose from three plays, while the opposition (Team B) has four possible plays. The numbers shown in the payoff matrix represent the yards of gain to Team A. What is the optimal strategy for each team?  15 −3 −4 2  12 9 6 8  −5 −2 3 16 

1

c

Kendra Kilmer June 24, 2008

.

Row Player’s Strategy/Maximin Strategy 1. For each row of the payoff matrix, find the smallest entry in that row. 2. Choose the row for which the entry in step 1 is as large as possible. This row constitutes the row player’s best move. Column Player’s Strategy/Minimax Strategy 1. For each column of the payoff matrix, find the largest entry in that column. 2. Choose the column for which the entry in step 1 is as small as possible. This column constitutes the column player’s best move. Example 3: Let’s return to the previous example and follow the above rules.

2

c

Kendra Kilmer June 24, 2008

Definition: If there is an entry in the payoff matrix that is simultaneously the smallest entry in its row and the largest entry in its column, we call it a saddle point. Definition: If a payoff matrix has a saddle point, we say that the game is strictly determined. Definition: The saddle point is also called the value of the game. • If the value of the game is positive, then the game favors the row player. • If the value of the game is negative, then the game favors the column player. • If the value of the game is zero, then the game is fair. Example 4: For each payoff matrix below (i) determine the maximin and minimax strategies. (ii) Is the game strictly determined? If so, what is the saddle point/value of the game? Who does the game favor?   1 2 −3 a)  −1 2 −2  2 3 −4



 2 4   6  7



3 6 −2 8 −3 5

2  0 b)   1 3

c)



3

c

Kendra Kilmer June 24, 2008

Section 9.5 - Games with Mixed Strategies Definition: If a game is strictly determined, the best strategy for each player is to use a pure strategy (i.e. make the same move over and over). Definition: If a game does NOT have a saddle point (i.e the game is not strictly determined), each player should use a mixed strategy (i.e randomly change up the row or column that is selected.) Example 1: Let’s look at the following payoff matrix:   −1 2 1 0

Question: How often should they select each row/column? Answer:

Let’s suppose the row player selects row 1 1/3 of the time and row 2 2/3 of the time and that the column plyer selects each column 1/2 of the time. Question: What is the expected payoff of the game?

4

c

Kendra Kilmer June 24, 2008

We can allow matrices to simplify the expected value calculation for us: Definition: Given the m × n payoff matrix, A,   • Let P = p1 p2 ... pm be the mixed strategy of the row player where p1 is the probability of selecting row 1, etc...   q1  q2   • Let Q =   ..  be the mixed strategy of the column player where q1 is the probability of selecting column qn 1, etc... Then the expected value of the game is: E = PAQ Example 2: Let’s now use matrices to find the expected payoff of the previous example:

Example 3: Find the expected payoff for the previous game if the row player selects row 1 2/5 of the time and the column player selects column 1 1/4 of the time.

Note: The above two examples were mixed strategies but not the optimal mixed strategies for each player. 5

c

Kendra Kilmer June 24, 2008

Example 4: Let’s look at finding the optimal strategy for the row player:

6

c

Kendra Kilmer June 24, 2008

.

Optimal Mixed Strategy for the Row Player:

Optimal Mixed Strategy for the Column Player:

7

c

Kendra Kilmer June 24, 2008

Example 5: Given the payoff matrix below: (a) Find the optimal strategies for the row and column players. (b) Find the value of the game. Who does the game favor? 

3 −3 −2 1

8