Lecture Notes 6: Zero-Sum Games
In a zero-sum game, the payoffs for any strategy pair sum to zero. In other words, one player’s gain is exactly equivalent to the other player’s loss. Zero-sum games are strictly competitive, meaning that the total payoff in the game is always the same. Players just fight over how to split it up. A great deal of early work in game theory dealt with zero-sum games.
Zero-sum games can be analyzed using the techniques we have already developed. There is nothing about zero sum games that prevents us from using any of the standard tools for simultaneous games. However, the special structure of zero-sum games allows us to go further and to develop some special results that are unique to zero-sum games. The basic idea is that, in terms of optimization, any choice that is the best choice for a particular player is by definition the worst choice from the perspective of his opponent.
Minmax and Maxmin
Consider the zero-sum game below.
L C R
T -4,4 1,-1 -2,2
M 1,-1 0,0 6,-6
B 3,-3 2,-2 3,-3
Suppose that player 1 chooses 𝑇𝑇. The lowest payoff that he can get is −4. He should assume that this is exactly what he will get if he chooses 𝑇𝑇. Since the game is zero-sum, whatever is the worst payoff for player 1 is by definition the best payoff for player 2. In other words, when player 1 chooses 𝑇𝑇, player 2 will then maximize his own payoff, which is equivalent to minimizing player 1’s payoff.
Along similar lines, when player 1 chooses 𝑀𝑀 the worst payoff he can get is 0 .When player 1 chooses 𝐵𝐵, the worst payoff he can get is 2. Let’s indicate on the game matrix the worst possible payoff for any of player 1’s strategy choices.
L C R
T -4,4 1,-1 -2,2 min = −4
M 1,-1 0,0 6,-6 min = 0
Being rational, player 1 will choose the strategy that gives him the maximum of these minimum payoffs. This is called the maxmin payoff and is equal to 2 for player 1.
Let’s ask the same question for player 2.
• When player 2 chooses 𝐿𝐿, her worst payoff is −3
• When player 2 chooses 𝐶𝐶, her worst payoff is −2
• When player 2 chooses 𝑅𝑅, her worst payoff is −6
L C R
T -4,4 1,-1 -2,2 min = −4
M 1,-1 0,0 6,-6 min = 0
B 3,-3 2,-2 3,-3 min = 2
min = −3 min = −2 min = −6
Again, being rational, player 2 will choose the strategy giving her the best of these possible minimum payoffs, so her maxmin payoff is equal to −2.
The two maxmin payoffs coincide at (𝐵𝐵, 𝐶𝐶), which is the Nash Equilibrium of the game. We say that the game’s maxmin payoffs are (2, −2).
Let’s think about this a different way. Take the perspective of player 1. When he chooses strategy
𝑇𝑇, he might ask himself what is the best payoff that player 2 can get in response to 𝑇𝑇. Inspection of the game matrix shows that this is equal to 4, when player 2 chooses 𝐿𝐿.
In a similar way, when player 1 chooses 𝑀𝑀, player 2’s best possible payoff is 0. When player 1 chooses 𝐵𝐵, player 2’s best possible payoff is −2. Let’s indicate on the game matrix the best payoff for player 2 given any strategy choice by player 1.
L C R
T -4,4 1,-1 -2,2 max = 4
M 1,-1 0,0 6,-6 max = 0
B 3,-3 2,-2 3,-3 max = −2
Because the game is zero-sum, we can say for sure that player 1 will act to minimize player 2’s payoff. By definition, giving player 2 the lowest possible payoff is equivalent to player 1 giving himself the highest possible payoff. Player 1 therefore chooses 𝐵𝐵 in order to “minmax” player 2.
We can ask the same questions from the perspective of player 2.
• When player 2 chooses 𝐿𝐿, player 1’s best payoff is 3
• When player 2 chooses 𝐶𝐶, player 1’s best payoff is 2
• When player 2 chooses 𝑅𝑅, player 1’s best payoff is 6
We indicate these minmax payoffs for player 1 on the game matrix.
L C R
T -4,4 1,-1 -2,2 max = 4
M 1,-1 0,0 6,-6 max = 0
B 3,-3 2,-2 3,-3 max = −2
max = 3 max = 2 max = 6
Player 2 chooses strategy 𝐶𝐶 in order to “minmax” player 1 and impose the lowest possible payoff on him. These minmaxes coincide at (𝐵𝐵, 𝐶𝐶), so the minmax payoff vector in this game is (2, −2).
Results on Zero Sum Games
You may have noticed some correspondences in the example above. Here are three results that always hold for zero-sum games. Together, they are called the minmax theorem.
• Minmax and maxmin payoffs are equivalent.
• Mutual minmaxing is the Nash Equilibrium.
• The Nash Equilibrium payoff is unique for both players and is equal to the minmax payoff vector.
then tries to minimize this maximum payoff that player 2 can get. Because the game is zero-sum, these two ways of thinking about the problem are equivalent.
The relationship between minmaxing and Nash Equilibrium is also straightforward. Player 1 assumes that player 2 will optimize, but given this optimization tries to hold player 2’s payoff as low as possible. By definition, holding down player 2’s payoff as low as possible maximizes player 1’s own payoff, meaning that player 1 has no profitable deviation – as required for Nash Equilibrium. The same can be said for player 2 holding down player 1’s payoff as low as possible, so mutual minmaxing is equivalent to Nash Equilibrium for the special case of zero-sum games.
The last result says that, even though a zero-sum game might have more than one Nash Equilibrium, the payoff in every Nash Equilibrium is the same, and equal to the minmax payoff. For example, if player 1 has a strategy that can hold player 2’s payoff down to 10, then there can never be an equilibrium where player 2 gets a payoff higher than 10. To do so would require player 1 to accept a lower payoff since the game is zero-sum. Similarly, when player 2 can use a minmax strategy to hold player 1’s payoff down to −10, there cannot be any equilibrium where player 1 gets any payoff higher than −10, because doing so would require player 2 to accept a lower payoff. Again, this is a consequence of the special structure of zero-sum games.
Minmaxing with a Mixed Strategy
To start with a simple motivating example, consider the game below (sometimes called “matching pennies”)
L R
T 1,-1 -1,1 B -1,1 1,-1
How does player 2 minmax player 1 in this game? If player 2 plays 𝐿𝐿, then player 1 will play 𝑇𝑇 and get a payoff of 1. If player 2 plays 𝑅𝑅, then player 1 will play 𝑅𝑅 and still get a payoff of 1. Either
𝐿𝐿 or 𝑅𝑅, played as a pure strategy, will allow player 1 to get a payoff of 1, meaning that player 2 is stuck with a payoff of −1.
But player 2 can make player 1 worse off by using a mixed strategy and keeping him guessing.
Suppose that player 2 randomizes and uses the strategy 1
2𝐿𝐿 + 1
2𝑅𝑅. Then player 1 is actually
indifferent between 𝑇𝑇 and 𝐵𝐵 and receives an expected payoff of 0 either way. Summarizing, player
2 makes player 1 worse off by using the strategy 1
2𝐿𝐿 + 1
2𝑅𝑅 than she can by using any pure strategy.
player 1 minmaxes player 2 by using the mixed strategy given by 1
2𝑇𝑇 + 1
2𝐵𝐵. Using the minmax
theorem, mutual minmaxing is the Nash Equilibrium. You can verify yourself using the standard way to calculate Nash Equilibria in mixed strategies that the only equilibrium of this game is the
mixed equilibrium �1
2𝑇𝑇 + 1 2𝐵𝐵,
1 2𝐿𝐿 +
1 2𝑅𝑅�.
Let’s consider one more example to demonstrate the technique more generally.
L R
T 5,-5 8,-8 B 9,-9 2,-2
Suppose we restrict ourselves to pure strategies. Consider first how player 1 minmaxes player 2.
• When player 1 plays 𝑇𝑇, the maximum payoff that player 2 can obtain is −5
• When player 1 plays 𝐵𝐵, the maximum payoff that player 2 can obtain is −2
Using a pure strategy, player 1 minmaxes player 2 by playing 𝑇𝑇 and player 2’s minmax payoff is −5.
How does player 2 minmax player 1 using pure strategies?
• When player 2 plays 𝐿𝐿, the maximum payoff that player 1 can obtain is 9
• When player 2 plays 𝑅𝑅, the maximum payoff that player 1 can obtain is 8
Restricted to pure strategies, player 2 minmaxes player 1 by playing 𝑅𝑅 and player 1’s minmax payoff is 8.
These do not coincide at the same strategy, so these are not actually the minmaxes. Rather, the minmax strategies for this game are in mixed strategies.
To solve for this explicitly, think about how player 2 can minmax player 1. To see what minmaxing means using mixed strategies, let’s graph player 1’s payoffs as a function of player 2’s strategies.
By playing 𝑇𝑇, player 1’s payoff is:
Π𝑇𝑇 = 5𝑝𝑝𝐿𝐿+ 8𝑝𝑝𝑅𝑅
= 5𝑝𝑝𝐿𝐿+ 8(1 − 𝑝𝑝𝐿𝐿)
By playing 𝐵𝐵, player 1’s payoff is:
Π𝐵𝐵 = 9𝑝𝑝𝐿𝐿+ 2𝑝𝑝𝑅𝑅
= 9𝑝𝑝𝐿𝐿+ 2(1 − 𝑝𝑝𝐿𝐿)
= 2 + 7𝑝𝑝𝐿𝐿
The diagram below shows player 1’s payoff from each strategy, as a function of player 2’s strategy.
You can solve for the crossing point explicitly by equating the two.
Π𝑇𝑇 = Π𝐵𝐵
8 − 3𝑝𝑝𝐿𝐿 = 2 + 7𝑝𝑝𝐿𝐿
𝑝𝑝𝐿𝐿 = 0.6
To find the minmax, the idea is that, for whatever strategy player 2 chooses, player 1 will choose the strategy that maximizes his own payoff. For example, when player 2 plays 𝐿𝐿 with probability
𝑝𝑝𝐿𝐿 ∈ [0,0.6) then player 1 gets a higher payoff by choosing 𝑇𝑇. But if player 2 plays 𝐿𝐿 with
probability 𝑝𝑝𝐿𝐿 ∈ (0.6,1], then player 1 gets a higher payoff by choosing 𝐵𝐵.
From the diagram, it is clear that player 2 minmaxes player 1 by playing 0.6𝐿𝐿 + 0.4𝑅𝑅. This attains the lowest possible payoff for player 1, given that player 1 chooses an optimal strategy in response.
We can do the same exercise for player 2. By playing 𝐿𝐿, player 2’s payoff is:
Π𝐿𝐿 = −5𝑝𝑝𝑇𝑇+ −9𝑝𝑝𝐵𝐵
= −9 + 4𝑝𝑝𝑇𝑇
By playing 𝑅𝑅, player 2’s payoff is:
Π𝑅𝑅 = −8𝑝𝑝𝑇𝑇+ −2𝑝𝑝𝐵𝐵
= −2 − 6𝑝𝑝𝑇𝑇
Looking at the upper envelope of these payoff functions, it is clear that player 1 minmaxes player 2 by using the strategy 0.7𝑇𝑇 + 0.3𝐵𝐵.
To summarize, player 2 minmaxes player 1 by using the strategy 0.6𝐿𝐿 + 0.4𝑅𝑅. Player 1 minmaxes player 2 by using the strategy 0.7𝑇𝑇 + 0.3𝐵𝐵. Because the game is zero-sum, these are the strategies that each player should use. Minmaxing your opponent means getting the highest payoff for yourself in a zero-sum game.
Problems
1. Consider the zero-sum game below.
a. What are the minmax payoffs for each player in this game? (Hint: the players minmax each other using pure strategies).
b. A friend claims to you that this game has a mixed-strategy Nash Equilibrium generating payoffs (Π1, Π2) = (4.5, −4.5). Do you think she is right? Explain.
W X Y Z
A 8,-8 1,-1 -2,2 -4,4 B 3,-3 3,-3 0,0 6,-6 C 6,-6 4,-4 5,-5 4,-4 D 0,0 -2,2 -6,6 -6,6
2. Consider the zero-sum game below.
a. Verify that the minmax solution is not in pure strategies.
b. Graph player 2’s payoffs from 𝑋𝑋, 𝑌𝑌 and 𝑍𝑍 as a function of 𝑝𝑝𝐴𝐴. (You need an accurate graph – use graph paper and a straight edge).
c. What strategy should player 1 use to minmax player 2? d. Which strategies does player 2 use in this minmax solution?
e. Find the mixture of strategies in (d) that makes player 1 willing to mix.
f. Combine all of the above and state the mutual minmax strategies in this game. Because the game is zero-sum, this is the game’s Nash Equilibrium.
X Y Z
A 2,8 8,2 4,6
