# Zero-Sum Games

A **zero-sum game** is a game in which it is impossible for any player to help themselves without hurting another player. The name comes from the fact that in such a situation, the gains and losses of all the players sum to zero. For example, if players A and B are playing a zero-sum game, and player A chooses a strategy that wins him $1 more, then this strategy must cause player B to lose $1 more.

Many simple real world situations can be modeled as zero sum games: for example, dividing up a limited supply of resources among neighboring countries is a zero-sum, because any extra resources one country takes will lead to another country receiving less resources.

## Equilibria in Zero-Sum Games

Two-player zero sum games are extremely nice, because they always have at least one Nash equilibrium as long as mixed strategies are allowed.

Suppose that players A and B play a game where they each write down 0 or 1 on a piece of paper, and receive payoffs according to the following matrix:

\[\begin{array}{ccc} (A/B)& & Player & A \\ & & 0&1\\ Player&0&3/-3&-4/4\\ B&1&-2/2&3/-3. \end{array}\]

Effectively, player A wins when they play the same numbers and player B wins when they play different numbers. Note that this is a zero-sum game, because in any situation, the gains and losses of A and B sum to zero.

Now, if player A plays a mixed strategy where he plays 0 with probability \(p\) and 1 with probability \(1-p\), his expected payoff if player B plays a 0 is \(3p-4(1-p)=7p-4\). If player B plays a 1, his expected payoff is \(-2p+3(1-p)=-5p+3\). At a Nash equilibrium, these two will be equal, so we find \[7p-4=-5p+3\implies p=\frac{7}{12}.\] So player A should play 0 \(\frac7{12}\) of the time and 1 \(\frac5{12}\) of the time.

In general, when there are more than two options available to each player, the Nash equilibrium for the zero-sum game can be found by solving an optimization problem. If \(M\) is the payoff matrix, the problem is to find a vector \(u\) that minimizes \(\sum u_i\) subject to \(u\geq 0\) and \(Mu\geq 1\). Then, rescaling \(u\) to make it a probability vector will give the Nash equilibrium for the zero-sum game, which is guaranteed to exist.

## Real-World Examples

A good example to see how zero-sum games can be used to model real world situations, but fail to account for all complexities, is a simple election. If there are candidates \(A, B, C\), where each receives some number of votes, and the candidate with the highest number of votes wins, then this situation is a zero-sum game. If candidate \(A\) wishes to gain more votes, they must be taken from candidate \(B\) or \(C\). However, this is only true under the assumption that every single person in the population is voting for one of the three candidates--if some voters are abstaining, then a candidate can increase his vote total by attracting abstaining voters, without decreasing any of the other candidates totals.

Other common real-life examples of zero sum games include games like chess and poker, and financial instruments like options and futures (excluding transaction costs). In each of these cases, an increase in one player's payoff corresponds to a decrease in another player's payoff.