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.
Subsection 1.2.1 Example: Matching Pennies
¶Suppose each player has two choices: Heads (H) or Tails (T). If they choose the same letter, 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.
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.
| Player 2 | |||
| Head | Tail | ||
| Player 1 | Head | ||
| Tail | |||
Definition 1.2.2.
A payoff is the amount a player receives for given outcome of the game.
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 \((1, -1)\text{.}\)
We fill in the matrix with the appropriate payoff vectors in Table 1.2.3
| Player 2 | |||
| H | T | ||
| Player 1 | H | \((1, -1)\) | \((-1, 1)\) |
| T | \((-1, 1)\) | \((1, -1)\) | |
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.
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.
Subsection 1.2.2 Understanding the Players
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 wisest choice for themselves.
It is important to note that each player also knows that his or her opponent is also self-interested and perfectly logical!
Exercise 1.2.4. Preferred payoffs.
Which payoff does the player prefer: 0, 2, or -2?
Which payoff does the player prefer: -2, -5, or -10?
Which payoff does the player prefer: -1, -3, or 0?
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.
Example 1.2.5. A \(2\times 2\) Game.
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 1.2.6.
| Player 2 | |||
| A | B | ||
| Player 1 | A | \((100, -100)\) | \((-10, 10)\) |
| B | \((0, 0)\) | \((-1, 11)\) | |
In the following exercise, we will try to determine what each player should do.
Exercise 1.2.7. 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)?
Example 1.2.8. A \(3\times 3\) Game.
Suppose there are two players with the game matrix given in Table 1.2.9.
| Player 2 | ||||
| X | Y | Z | ||
| A | \((1000, -1000)\) | \((-5, 5)\) | \((-15,15)\) | |
| Player 1 | B | \((200, -200)\) | \((0, 0)\) | \((-5,5)\) |
| C | \((500, -500)\) | \((20, -20)\) | \((-25,25)\) | |
In the following exercise, we will try to determine what each player should do.
Exercise 1.2.10. 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)?
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.