Game Matrices and Payoff Vectors

Section 1.2 Game Matrices and Payoff Vectors

We need a way to describe the possible choices for the players and the outcomes of those choices. For now, we will stick with games that have only two players. We will call them Player 1 and Player 2.

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Subsection 1.2.1 Setting up a Payoff Matrix

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We begin with an example of the game of Matching Pennies.

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Suppose each player has two choices: Heads (H) or Tails (T). If they choose the same side of the coin, then Player 1 wins $1 from Player 2. If they don’t match, then Player 1 loses $1 to Player 2. We can represent all the possible outcomes of the game with a matrix.

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Player 1’s options will always correspond to the rows of the matrix, and Player 2’s options will correspond to the columns. See Table 1.2.1.

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Table 1.2.1. A game matrix showing the strategies for each player

		Player 2
		Head	Tail
Player 1	Head
	Tail

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Definition 1.2.2.

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A payoff is the amount a player receives for a given outcome of the game.

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Now we can fill in the matrix with each player’s payoff. Since the payoffs to each player are different, we will use ordered pairs where the first number is Player 1’s payoff and the second number is Player 2’s payoff. The ordered pair is called the payoff vector. For example, if both players choose H, then Player 1’s payoff is $1 and Player 2’s payoff is -$1 (since he loses to Player 1). Thus the payoff vector associated with the outcome H, H is

(1, - 1) .

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We fill in the matrix with the appropriate payoff vectors in Table 1.2.3

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Table 1.2.3. A game matrix showing the payoff vectors

		Player 2
		H	T
Player 1	H	$(1, - 1)$	$(- 1, 1)$
	T	$(- 1, 1)$	$(1, - 1)$

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It is useful to think about different ways to quantify winning and losing. What are some possible measures of value? For example, we could use money, chips, counters, votes, points, amount of cake, etc.

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Remember, a player always prefers to win the MOST points (money, chips, votes, cake), not just more than her opponent. If you want to study a game where players simply win or lose (such as Tic Tac Toe), we could just use “1” for a win and “-1” for a loss.

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Subsection 1.2.2 Revisiting the Assumptions

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Recall that we said there are two major assumptions we must make about our players:

Our players are self-interested. This means they will always prefer the largest possible payoff. They will choose a strategy which maximizes their payoff.
Our players are perfectly logical. This means they will use all the information available and make the choice that results in the largest payoff for themselves.

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It is important to note that each player also knows that his or her opponent is also self-interested and perfectly logical!

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Activity 1.2.1. Preferred payoffs.

Which payoff does a player prefer: 0, 2, or -2?
Which payoff does a player prefer: -2, -5, or -10?
Which payoff does a player prefer: -1, -3, or 0?

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It may be strightforward to decide the best payoff for a player out of a list of values, and it would be great if a player could just determine the biggest value in the table and choose that strategy. However, when there are two players a player may have to choose a strategy more carefully, since Player 1 can only choose the row, and Player 2 can only choose the column. Thus, the outcome of the game depends on BOTH players.

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Example 1.2.4. A $2 \times 2$ Game.

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Suppose two players are playing a game in which they can choose A or B with the payoffs given in the game matrix in Table 1.2.5.

Table 1.2.5. Payoff matrix for Activity 1.2.2

		Player 2
		A	B
Player 1	A	$(100, - 100)$	$(- 10, 10)$
	B	$(0, 0)$	$(- 1, 1)$

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In the following activity, we will try to determine what each player should do.

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Activity 1.2.2. Finding strategies.

Just by quickly looking at the matrix, which player appears to be able to win more than the other player? Does one player seem to have an advantage? Explain.
Determine what each player should do. Explain your answer.
Compare your answer in (b) to your answer in (a). Did the player you suggested in (a) actually win more than the other player?
According to your answer in (b), does Player 1 end up with the largest possible payoff (for Player 1) in the matrix?
According to your answer in (b), does Player 2 end up with the largest possible payoff (for Player 2) in the matrix?
Do you still think a player has an advantage in this game? Is it the same answer as in (a)?

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Example 1.2.6. A $3 \times 3$ Game.

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Suppose there are two players with the game matrix given in Table 1.2.7.

Table 1.2.7. Payoff matrix for Activity 1.2.3

		Player 2
		X	Y	Z
	A	$(1000, - 1000)$	$(- 5, 5)$	$(- 15, 15)$
Player 1	B	$(200, - 200)$	$(0, 0)$	$(- 5, 5)$
	C	$(500, - 500)$	$(20, - 20)$	$(- 25, 25)$

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In the following activity, we will try to determine what each player should do.

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Activity 1.2.3. More practice finding strategies.

Just by quickly looking at the matrix, which player appears to be able to win more than the other player? Does one player seem to have an advantage? Explain.
Determine what each player should do. Explain your answer.
Compare your answer in (b) to your answer in (a). Did the player you suggested in (a) actually win more than the other player?
According to your answer in (b), does Player 1 end up with the largest possible payoff (for Player 1) in the matrix?
According to your answer in (b), does Player 2 end up with the largest possible payoff (for Player 2) in the matrix?
Do you still think a player has an advantage in this game? Is it the same answer as in (a)?

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This chapter has introduced you to who the players are and how to organize strategies and payoffs into a matrix. In the next chapter we will study some methods for how a player can determine his or her best strategy.

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Reading Questions 1.2.3 Check Your Understanding

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1.

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Which payoff does a player prefer: 3, 1, or -5?

$3$
Correct. 3 is the largest payoff.
$1$
1 is not the largest payoff.
$- 5$
-5 is not the largest payoff.

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2.

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Which payoff does a player prefer: -5, -1, or -10?

$- 1$
Correct. -1 is the largest payoff.
$- 5$
-5 is not the largest payoff.
$- 10$
$- 10$ is not the largest payoff.

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3.

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Which payoff does a player prefer: 1/2, 0, or 5/6?

$5 / 6$
Correct. 5/6 is the largest payoff.
$0$
0 is not the largest payoff.
$1 / 2$
$1 / 2$ is not the largest payoff.

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4.

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In the game matrix, Player 1 will always choose

the row.
Correct. Player 1’s strategies are represented by the rows in the matrix.
the column
Player 2’s strategies are represented by the columns in the matrix.

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5.

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For the payoff vector

(5, - 5),

Player 1’s payoff is

$5$
Correct. Player 1’s payoff is always the first coordinate.
$- 5$
Player 2’s payoff is the second coordinate.

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6.

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Consider the following game matrix.

Table 1.2.8.

		Player 2
		C	D
Player 1	A	$(5, - 5)$	$(- 1, 1)$
	B	$(2, - 2)$	$(- 3, 3)$

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What column should Player 2 choose?

C
Incorrect.
D
Correct. Player 2 always does better by choosing D, no matter what Player 1 does.

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7.

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Using the game matrix Table 1.2.8.

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What row should Player 1 choose?

A
Correct. Player 1 always does better by choosing A, no matter what Player 2 does.
B
Incorrect.

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8.

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