Zero-Sum Matrix Games

Prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

The diagram below shows an example of a matrix game with players Row (R) and Column (C).

 Column I Column II Row 1 (20, -20) (-5, 5) Row 2 (-8, 8) (2, -2)

Figure 1

The game is played as follows. Row gets to choose either Row 1 or Row 2 to "play," while Column gets to play either Column I or Column II. Row and Column make their choices without consulting each other. The entries in the matrix (table) give the playoffs to the players. For simplicity we will assume the payoffs are in money - pennies. Thus, if Row plays Row 2 and Column plays Column II then Row wins 2 pennies and Column loses 2 pennies. Row's payoff is the first entry in the pair shown in the matrix and Column's payoff is the second entry. In this game the sum of the entries in each pair adds to zero, so the game is known as a zero-sum game. We will think of Row's gains as coming at Column's expense and Column's gains as coming at Row's expense. Thus, the sum of the payoffs to the two players adds to zero, which is why games of this kind are known as zero-sum games.

It is common for the information involving zero-sum matrix games to be displayed more succinctly, by just showing the payoff pairs as a single number given from Row's point of view. Thus, in Figure 2, -11 means that Column would get 11 cents while Row would lose 11 cents (when Row plays Row 1 and Column plays Column II), and the 3 means a gain for Row and a loss for Column (when Row plays Row 1 and when Column plays Column I).

 Column I Column II Row 1 3 -11 Row 2 -2 10

Figure 2

In many cases such game matrices are shown in the even more compressed form shown in Figure 3.

 3 -11 -2 10

Figure 3

In Figure 3, again, the payoffs are shown from Row's point of view.

We will assume that the games described here are "supervised" by an impartial umpire who makes sure that the rules are followed and the payoffs are made.

How would you play this game if you had to play it only once? How would you play the game if you had to play it over and over again, some finite number of times? To try to get a feel for games of this kind you might try playing the games below with someone, each game being repeated, say, 10 times, though this may not be enough to get an empirical feel for the way the "long term" outcome of playing the game affects Row and Column's earnings. Of course, if Row wins in the long run then Column will lose.

Activity 1:

Play the game shown 10 times.

 Column I Column II Row 1 3 -11 Row 2 -2 10

Activity 2:

Play the game shown 10 times.

 Column I Column II Row 1 -12 -1 Row 2 15 0

Activity 3:

Play the game shown 10 times.

 Column I Column II Row 1 -3 11 Row 2 2 -10

Activity 4:

Play the game shown 4 times.

 Column I Column II Row 1 100 -10 Row 2 -10 1

Activity 5:

Play the game shown 10 times.

 Column I Column II Row 1 2 0 Row 2 20 -2

Activity 6:

Play the game shown 10 times.

 Column I Column II Row 1 3 -2 Row 2 -15 10

Question 1

After playing games of this kind, each several times, what insights do you have into how to play 2 person zero-sum games based on the entries in the payoff matrix?

Question 2

After playing games of this kind, each several times, what meaning would you give to saying that a particular game of this kind is fair?

Question 3

What topics in the K-12 curriculum are supported by having students learn about zero-sum games? (In particular, you can look at this from the point of view of what is taught as part of the Common Core State Standards in Mathematics.)

Question 4

Given a particular zero-sum game such as those above, what is the optimal way to play if you are Row (or Column) if:

a. you play the game only once?
b. you play the game over and over again, a finite number of times?

Question 5:

How can you tell if a particular 2x2 (2 players, 2 choices of action by each player) zero-sum game is "fair?"