A two player game is called a zero-sum game if the sum of the payoffs to each player is constant for all possible outcomes of the game. More specifically, the coordinates in each payoff vector must add up to the same value for each payoff vector. Such games are sometimes called constant-sum games instead.
Consider a poker game in which each player comes to the game with $100. If there are five players, then the sum of money for all five players is always $500. At any given time during the game, a particular player may have more than $100, but then another player must have less than $100. One player’s win is another player’s loss.
Consider the cake division game. We want to find the payoff matrix for this game. It is important to determine what each player’s options are first. How can the “cutter” cut the cake? How can the “chooser” pick her piece? The payoff matrix is given in Table 2.1.4.
In order to better see that this game is zero-sum (or constant-sum), we could give values for the amount of cake each player gets. For example, half the cake would be 50%, a small piece might be 40%. Then we can rewrite the matrix with the percentage values in Table 2.1.5
In each outcome, the payoffs to each player add up to 100 (or 100%). In more mathematical terms, the coordinates of each payoff vector add up to 100. Thus the sum is the same, or constant, for each outcome.
We can see from the matrix in Table 2.1.5 that Player 2 will always choose the larger piece, thus Player 1 does best to cut the cake evenly. The outcome of the game is the strategy pair denoted [Cut Evenly, Larger Piece], with resulting payoff vector \((50, 50)\text{.}\)
But why are we going to call these games “zero-sum” rather than “constant-sum”? We can convert any zero-sum game to a game where the payoffs actually sum to zero.
Consider the above poker game where each player begins the game with $100. Suppose at some point in the game the five players have the following amounts of money: $50, $200, $140, $100, $10. Then we could think of their gain as -$50, $100, $40, $0, -$90. What do these five numbers add up to?
But let’s make sure we understand what these numbers mean. For example, a payoff of \((0,0)\) does not mean each player gets no cake, it means they don’t get any more cake than the other player. In this example, each player gets half the cake (50%) plus the payoff.
When the game matrix is in the form of Example 2.1.7, it is easy to recognize a zero-sum game since each payoff vector has the form \((a, -a)\) (or \((-a, a)\)).
Two candidates, Arnold and Bainbridge, are facing each other in a state election. They have three choices regarding the issue of the speed limit on I-5: they can support raising the speed limit to 70 MPH, they can support keeping the current speed limit, or they can dodge the issue entirely. The next three examples present three different payoff matrices for Arnold and Bainbridge.
What should Arnold choose to do? What should Bainbridge choose to do? Be sure to explain each candidate’s choice. And remember, a player doesn’t just want to win, he wants to get THE MOST votes. For example, you could assume these are polling numbers and that there is some margin of error, thus a candidate prefers to have a larger margin over his opponent.
Does Arnold need to consider Bainbridge’s strategies in order to decide on his own strategy? Does Bainbridge need to consider Arnold’s strategies in order to decide on his own strategy? Explain your answer.
Does Arnold need to consider Bainbridge’s strategies in order to decide on his own strategy? Does Bainbridge need to consider Arnold’s strategies in order to decide on his own strategy? Explain your answer.
Does Arnold need to consider Bainbridge’s strategies in order to decide on his own strategy? Does Bainbridge need to consider Arnold’s strategies in order to decide on his own strategy? Explain your answer.
In each of the above scenarios, is there any reason for Arnold or Bainbridge to change his strategy? If there is, explain under what circumstances it makes sense to change strategy. If not, explain why it never makes sense to change strategy.
Chances are, in each of the exercises above, you were able to determine what each player should do. In particular, if both players play your suggested strategies, there is no reason for either player to change to a different strategy.
You determined that Player 2 should choose to play B, and thus, Player 1 should play B (i.e., we have the strategy pair [B, B]). Why is this an equilibrium pair? If Player 2 plays B, does Player 1 have any reason to change to strategy A? No, she would lose 10 instead of 1. If Player 1 plays B, does Player 2 have any reason to change to strategy A? No, she would gain 0 instead of 1. Thus neither player benefits from changing strategy, and so we say [B, B] is an equilibrium pair.
For now, we can use a “guess and check” method for finding equilibrium pairs. Take each outcome and decide whether either player would prefer to switch. Remember, Player 1 can only choose a different row, and Player 2 can only choose a different column. In our above example there are four outcomes to check: [A, A], [A, B], [B, A], and [B, B]. We already know [B, B] is an equilibrium pair, but let’s check the rest. Suppose the players play [A, A]. Does Player 1 want to switch to B? No, she’d rather get 100 than 0. Does Player 2 want to switch to B? Yes! She’d rather get 10 than \(-100\text{.}\) So [A, A] is NOT an equilibrium pair since a player wants to switch. Now check that for [A, B] Player 1 would want to switch, and for [B, A] both players would want to switch. Thus [A, B] and [B, A] are NOT equilibrium pairs. Now you can try to find equilibrium pairs in any matrix game by just checking each payoff vector to see if one of the players would have wanted to switch to a different strategy.
Generally, when we define a game matrix for a game, our rows and columns will be named with the strategy choices for the players. However, mathematically, we can just think of the matrix of payoff vectors without the row and column labels. In this case, we often identify the payoff vector itself as an equilibrium or equilibrium point. In Table 1.2.5, for example, we would say the payoff vector \((-1, 1)\) is an equilibrium point.
Are the strategy pairs you determined in the three election scenarios, Table 2.1.10, Table 2.1.12, and Table 2.1.14, equilibrium pairs? In other words, would either player prefer to change strategies? (You don’t need to check whether any other strategies are equilibrium pairs.)
Use the “guess and check” method to determine any equilibrium points for the following payoff matrices. It can also be helpful to identify the associated row and column for the equilibrium pair.
Consider the game ROCK, PAPER, SCISSORS (Rock beats Scissors, Scissors beat Paper, Paper beats Rock). Construct the payoff matrix for this game. Does it have an equilibrium pair? Explain your answer.
Two television networks are battling for viewers for 7 pm Monday night. They each need to decide if they are going to show a sitcom or a sporting event. Table 2.1.17 gives the payoffs as percent of viewers.
This game is an example of what economists call Competitive Advantage. Two competing firms need to decide whether or not to adopt a new type of technology. The payoff matrix is in Table 2.1.18. The variable \(a\) is a positive number representing the economic advantage a firm will gain if it is the first to adopt the new technology.
We’ve seen how to describe a zero-sum game and how to find equilibrium pairs. We’ve tried to decide what each player’s strategy should be. Each player may need to consider the strategy of the other player in order to determine his or her best strategy. But we need to be careful, although our intuition can be useful in deciding the best strategy, we’d like to be able to be more precise about finding strategies for each player. We’ll learn some of these tools in the next section.
\((2, -2)\) is an equilibrium point since if Player 2 chose column 2, Player 1 does not prefer row 1 with a payoff of 0. Similarly, if Player 1 chose row 2, Player 2 does not prefer column 1 with a payoff of \(-3\text{.}\)
\((1, -1)\)
\((1, -1)\) is not an equilibrium point since if Player 2 chose column 1, Player 1 prefers row 2 with a payoff of 3. Similarly, if Player 1 chose row 1, Player 2 prefers column 2 with a payoff of \(0\text{.}\)
\((0, 0)\)
\((0, 0)\) is not an equilibrium point since if Player 2 chose column 2, Player 1 prefers row 2 with a payoff of 2. Note, Player 2 doesn’t want to switch, but we only need one player to want to change strategies for the vector to not be an equilbrium.
\((3, -3)\)
\((3, -3)\) is not an equilibrium point since if Player 1 chose row 2, Player 2 prefers column 2 with a payoff of \(-2\text{.}\) Note, Player 1 doesn’t want to switch, but we only need one player to want to change strategies for the vector to not be an equilbrium.
