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## Pure Strategies

We start with a constant-sum game: for every possible outcome of the game, the utility of Player 1 plus the utility of Player 2, adds to a constant. For example, if two firms are competing for market shares, then .

The possible outcomes are best represented in a table.

We can solve Battle of the Networks as follows: from Network 1's point of view, it is better to show a sitcom, whether Network 2 shows a sitcom or sports. The strategy ``Show a Sitcom'' is said to dominate the strategy ``Show Sports'' for Network 1. So Network 1 will show a sitcom. Similarly, Network 2 is better off showing sports whether Network 1 shows a sitcom or sports. In other words, Network 2 also has a dominating strategy. So Network 2 shows sports. The resulting outcome, namely 51% viewer share to Network 1 and 49% to Network 2, is an equilibrium, since neither of the two players in this game can unilaterally improve their outcome. If Network 1 were to switch from sitcom to sports, its viewer share would drop 5%, from 51% to 46%. If Network 2 were to switch from sports to sitcom, its viewer share would also drop, from 49% to 44%. Each network is getting the best viewer share it can, given the competition it is up against.

In a general 2-person game, Strategy i for Player 1 dominates Strategy k for Player 1 if for every j. Similarly, Strategy j for Player 2 dominates Strategy for Player 2 if for every i.

In a general 2-person game, Strategy i for Player 1 and Strategy j for Player 2 is an equilibrium if the corresponding outcome has the following property: is the largest element in column j and is the largest element in row i. When such a pair i,j exists, a pure strategy for each player provides an optimal solution to the game. When no such a pair exists, the players must play mixed strategies to maximize their gains (see Section 1.1.2).

There is an easy way to relate constant-sum games and zero-sum games. Let and be the payoffs of players 1 and 2, respectively, in a constant-sum game: . Now consider a new set of payoffs of the form

Clearly, , so we now have a zero-sum game. Using the relation , we can rewrite the payoffs and as:

These are positive, linear transformations of utility, and so, according to the expected utility theorem (see 45-749), they have no effect on decisions. For the Battle of the Networks example, the new table becomes

It represents each network's advantage in viewer share over the other network. Whether you reason in terms of total viewer shares or in terms of advantage in viewer shares, the solution is the same.

The two-person games we have encountered so far have had unique pure strategy equilibria. However, a two-person zero-sum game may have multiple equilibria. For example, consider the game:

Each player can play indifferently strategy A or C and so there are four pure strategy equilibria, corresponding to the four corners of the above table. Note that these equilibria all have the same payoff. This is not a coincidence. It can be shown that this is always the case. Every equilibrium of a 2-person zero-sum game has the same payoff. For pure strategy equilibria, there is a simple proof: Suppose two equilibria had payoffs (a,-a) and (b,-b) where . If these two solutions lied on the same row or column, we would get an immediate contradiction to the definition of an equilibrium. Let (a,-a) lie in row i and column k and let (b,-b) lie in row j and column . The subtable corresponding to these two rows and columns looks as follows

Since (a,-a) is an equilibrium, we must have (from Player 1) and (from Player 2). Similarly, since (b,-b) is an equilibrium, we must have and . Putting these inequalities together, we get

This implies that a = b, which completes the proof.