ZERO-SUM GAMES

ECON2112

1. Introduction

So far we have analyzed normal form games in a broad sense. Nonetheless,soetime we may be interested in analyzing specific classes of games where somecondition is satisfied. For instance, two-player games have properties that do notnecessarily hold when we increase the number of players.

Here we analyze two-player zero-sum games. A zero-sum game is a game where,for each pure strategy profile, the sum of the payoffs of all the players is equal to 0.While zero-sum games have importance of their own, this lecture is mainly usefulto introduce some important concepts and terminology.

2. Zero-sum game

Let us start giving a formal definition.

Definition 1 (zero-sum Games). A game G = {N, {Si}iN , {ui}iN} is a zero-sumgame if for every s = (s1, . . . , sn) S

u1(s) + u2(s) + + un(s) = 0.Notice that the definition also implies that for every mixed strategy =

(1, . . . , n) we have that U1() + U2() + + Un() = 0.Most of the study of zero-sum games has been devoted to two-player zero-sum

games.

Definition 2 (Two-player zero-sum game). A two-player zero-sum game G ={S1, S2, u1, u2} is a two-player game where for every (s1, s2) S1 S2

u1(s1, s2) = u2(s1, s2).The previous definition also implies that for every mixed strategy profile we

have that U1() = U2().Example 1 (The Matching Pennies Game). The typical example of a zero-sumgame is the matching game.

heads tailsheads 1,1 1, 1

tails 1, 1 1,1

For every pure strategy profile s = (s1, s2), it holds that u1(s1, s2) = u2(s1, s2).Example 2 (Rock, Paper, Scissors). The well known rock, raper, scissors is an-other illustrative example of zero-sum game.

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Rock Paper ScissorsRock 0, 0 1, 1 1,1

Paper 1,1 0, 0 1, 1Scissors 1, 1 1,1 0, 0

Again, every pure strategy profile s = (s1, s2) satisfies u1(s1, s2) = u2(s1, s2).

The theory of two-person zero-sum games was developed by von Neumann andMorgenstern in their book Theory of Games and Economic Behavior (1944) andconstitutes the foundation of the work of Nash. John Nash constructed the generalframework that was covered in the past lectures.

3. Minmax Strategies.

The specific formulation of two-player zero-sum games facilitates their study.In the next section, we will see that players can guarantee themselves their Nashequilibrium payoff by choosing their minmax strategy. Loosely speaking, a playersminmax strategy is a strategy that minimizes the maximum damage that the op-ponents could inflict him. Formally,

Definition 3 (Minmax Strategy). Given a game G, i is a minmax strategy ofplayer i if

i arg maxi

(minsi

Ui(i, si)

).

It is always easier to understand the idea behind the concept if we only considerpure strategies.

Example 3. Let us compute the minmax (pure) strategies of the following game.

L RT 3, 5 1, 2B 2, 3 4, 4

If player 1 plays T the worst thing that could happen to him is that player 2plays R which would give him a payoff equal to 1. If player 1 plays T the worstthing that could happen to him is that player 2 plays L which would give him apayoff equal to 2. Player 1s minmax (pure) strategy is, therefore, B. You cancheck that player 2s minmax (pure) strategy is L.

However, we know that we cannot prevent players from randomizing. If player 1played 12T +

12R the worst thing (in fact, the only thing) that could happen to him

is that he obtains an expected payoff equal to 52 which is larger than 2. Therefore,

player 2s minmax strategy is 12T +12R. An analogous argument can be made for

player 2 that shows that his minmax strategy is 12L+12R.

We now present the Minmax Theorem. It receives this name because the strate-gies that the theorem pins down are minmax strategies.

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4. Minmax Theorem

Theorem 1 (Minmax Theorem). Every two-player zero-sum game has a uniquevalue V and optimal strategies for both players 1 and

2 such that

U1(1 , 2) V for every 2 2

U2(1, 2) V for every 1 1.

The reason why the strategies are called optimal is that if player 1 (respectively,player 2) plays 1 (resp.

2) he guarantees himself an expected payoff equal to V

(resp. equal to V ) independently of what player 2 (resp. player 1) plays. Let usproceed to prove the Theorem.

Proof. Consider an arbitrary two-player zero-sum game G. We know that everygame has a Nash equilibrium. Let = (1, 2) be one Nash equilibrium of G.

Since is a Nash equilibrium we have that

U1(1, 2) U1(1, 2) for every 1 1.In a two-player zero-sum game U1() = U2(). Therefore, we can write the

previous inequality as

U2(1, 2) U2(1, 2) for every 1 1,and multiplying both sides by (1) yields

U2(1, 2) U2(1, 2) for every 1 1.Hence, if player 2 plays the equilibrium strategy 2 he can guarantee himself his

equilibrium payoff U2(1, 2) whatever is the strategy played by player 1. Therefore,let us assign U2(1, 2) = V and 2 = 2 .

We can now write the Nash equilibrium conditions for player 2,

U2(1, 2) U2(1, 2) for every 2 2V U2(1, 2) for every 2 2.

Since U2(1, 2) = U1(1, 2) we have thatV U1(1, 2) for every 2 2.

Multiplying both sides by (1) we obtainV U1(1, 2) for every 2 2.

Which implies that 1 is the optimal strategy 1 for player 1.

It remains to prove that the value V is unique. Suppose that we have two Nashequilibria and that yield, respectively, a expected payoff to player 1 equal toV and V . Without loss of generality let us assume the V > V . Therefore:

U1(1, 2) > U1(1, 2).

Since (1, 2) is a Nash equilibrium it must hold that player 1 does not want to

deviate to 1, i.e. U1(1, 2) U1(1, 2). This, together with the last inequality

implies that

U1(1, 2) > U1(1, 2).

We use again the fact that for every , U1() = U2(), and write the previousinequatily in terms of player 2s payoffs

U2(1, 2) > U2(1, 2),

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multiplying both sides by (1) yieldsU2(1, 2) < U2(1,

2).

Which contradicts the fact that (1, 2) is a Nash equilibrium because player 2would have an incentive to deviate to 2. Consequently, the value V is unique.

One consequence of the previous theorem is that the set of Nash equilibria in atwo-player zero-sum game is interchangeable. That is, if (1, 2) and (

1, 2) are

Nash equilibria, then (1, 2) and (

1, 2) are also Nash equilibria.

(Nash defined a game as solvable if the set of Nash equilibria is interchangeable.Therefore, using the terminology of Nash, we obtained that every two-player zero-sum game is solvable.)

5. Constant Sum Games

A constant sum game is game G with the property that for every s =(s1, . . . , sn) S we have that u1(s) + + un(s) = c for some constant c.

Every constant sum game is equivalent to a zero-sum game (see affine trans-formations in the lecture devoted to Decision Theory). Consequently there is noadded generality from considering values of c different from 0.

Appendix A. Tutorial Questions

Exercise 1. Find all minmax pure strategies of the following game

A B C D EF 2, 3 6, 1 7, 0 7, 2 1, 2G 7, 3 8, 2 4, 2 2, 0 9, 4H 2, 1 9, 5 3, 2 1, 8 3, 5I 1, 8 4, 3 1, 0 5, 9 8, 2J 0, 5 2, 6 2, 3 4, 6 7, 3

Exercise 2. Find all Nash equilibria in pure strategies of the game in Exercise 1.

Exercise 3. Find the unique equilibrium of the following game. Compare thegame with the matching pennies game. Why do they have the same set of Nashequilibria?

L RT 2,3 0,1B 0,1 2,3

Exercise 4. Find the set of Nash equilibria of the following game. Compare thegame with the Battle of the Sexes game. Why do they have the same set of Nashequilibria?

L RT 9, 1 1, 0B 1, 0 3, 4

Exercise 5. Find one zero-sum game (there are many) that has the same set ofNash equilibria than the following game.

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L M R

T 3, 4 5, 2 4, 3

B 7, 0 1, 6 72 ,72