4. Two player nonzero sum games

In the last Section we considered the situation where the gain of one player was the loss of the other.

This is however, far too restrictive for many games, especially for games in economics and politics, where both players can win something or both players can lose something.

In this section we will consider how to solve games that are not zero sum, which is in general quite hard.

But before we do we will briefly consider the problem of how to decide between different Nash equilibria.

For zero sum games we do not have to worry about different choices of Nash equilibria. Each player can independently choose their own optimal strategy, and the final value is always the same.

For general games we have seen that neither of these is the case. So how do we decide between different Nash equilibria?

Definition 4.1

The outcome of a game is Pareto optimal if there is no other outcome which would either give each player a higher payoff or would give one player the same payoff and the other a higher payoff. An outcome is non-Pareto optimal otherwise.

If a Nash equilbrium is Pareto optimal then it would be a good candidate for a preferred equilibrium. However this is often not the case, and even if it is then there may be other Pareto optimal equilibria.

Example 4.2

The coordination game is given by the bimatrix

Strategy 1 Strategy 2
Strategy 1 \((1,1)\) \((0,0)\)
Strategy 2 \((0,0)\) \((1,1)\)

The idea is that the two players have a choice of two options, and simply want to agree on the one to pick.

For example, two manufacturing companies that work together need to decide whether to configure their machinery in metric or imperial measurements; two platoons in the same army need to decide whether to attack an enemy’s left flank or right flank; two people trying to find each other in a crowded mall need to decide whether to wait at the north end of the mall or at the south end.

This is easy if the players can communicate in advance, but what if that is not possible.

For example, suppose two drivers are approaching each other at night on an undivided country road. Each driver has to decide whether to move over to the left or the right.

If the drivers coordinate, making the same choice of side, then they pass each other, but if they fail to coordinate, then they get a severely low payoff due to the resulting collision.

Fortunately, social convention can help the drivers decide what to do in this case: if this game is being played in the U.S., convention strongly suggests that they should move to the right, while in England, convention strongly suggests that they should move to the left.

In other words, social conventions, while often arbitrary, can sometimes be useful in helping people coordinate among multiple equilibria.

Example 4.3

The battle of the sexes is given by the bimatrix

Strategy 1 Strategy 2
Strategy 1 \((1,2)\) \((0,0)\)
Strategy 2 \((0,0)\) \((2,1)\)

The name comes from the following (potentially sexist) example.

A husband and wife want to see a movie together, and they need to choose between a romantic comedy and an action movie.

They want to coordinate on their choice (so they can go to the same film) but each have a different preference.

In a Battle of the Sexes game, it can be hard to predict the equilibrium that will be played using either the payoff structure or some purely external social convention.

Rather, it helps to know something about conventions that exist between the two players themselves, suggesting how they resolve disagreements when they prefer different ways of coordinating.

We have have seen that there might be serious problems with an equilibrium as solution concept for nonzero-sum games.

An equilibrium outcome is certainly desirable due to its stability properties (and Nash proved that one always exists).

However, there might be multiple equilibria that are non-equivalent and non-exchangeable giving rise to coordination problems. Even if there is an unique equilibrium it might not be Pareto optimal.

Therefore let us start by take a different view on nonzero sum games.

Example 4.4

Consider the game

Strategy 1 Strategy 2
Strategy 1 \((2,0)\) \((1,3)\)
Strategy 2 \((0,1)\) \((3,0)\)

This game has no Nash equilibrium points in pure strategies (i.e. there is no pair \((a,b)\) in which \(a\) is the largest in the column and \(b\) the largest in the rows).

But we know that the largest amount that player 1 can be guaranteed to receive is obtained by assuming that player 2 is actually trying to minimize player 1’s payoff.

That means, in an nonzero-sum game with two players with the corresponding payoff matrices \(A\) and \(B\) we can consider the two games arising from each matrix separately.

Matrix \(A\) describes the zero sum game player 1 plays against player 2 (player 1 plays the row strategies, ie is the maximizer and player 2 plays the column strategies, ie is the minimizer). The value of the game \(A\) is the guaranteed amount for player 1.

Similarly, the amount that player 2 can be guaranteed to receive is obtained by assuming that player 1 is actually trying to minimize player 2’s payoff.

For player 2 the zero sum game is \(B^T\) because the player who chooses the row strategies is always maximizing. Consequently, player 2 can be guaranteed that she will receive the value of the game \(B^T\).

Definition 4.5

Consider the bimatrix game given by the pair \((A,B)\). The safety value for player 1 is \(v(A)\), i.e. the value of the zero sum game with payoff matrix \(A\).

The safety value for player 2 is \(v(B^T)\), i.e. the value of the zero sum game with payoff matrix \(B^T\).

From our definition of max-min strategies for bimtrix games it is easy to see that if \(A\) has a Nash equilibrium \((\underline{x}^A,\underline{y}^A)\) then \(\underline{x}^A\) is a max-min strategy for player 1.

Similarly, if \(B^T\) has a Nash equilibrium \((\underline{x}^{B^T},\underline{y}^{B^T})\) then \(\underline{x}^{B^T}\) is a max-min strategy for player 2.

Example 4.4 (continued)

Consider the game

Strategy 1 Strategy 2
Strategy 1 \((2,0)\) \((1,3)\)
Strategy 2 \((0,1)\) \((3,0)\)

Example completed by hand in the lecture

A pair of strategies \((\underline{x},\underline{y})\) is called individually rational if \[\mathbb E_1(\underline{x},\underline{y})\geq v(A)\quad\text{and}\quad \mathbb E_2(\underline{x},\underline{y})\geq v(B^T).\]

A player should expect their payoff to be individually rational, as this is what they are guaranteed to get even if their opponent is trying to minimise their payoff.

The safety values are the amount each player can get by using their own individual level maxmin-strategies.

If \((\underline{x}^*,\underline{y}^*)\) is a Nash equilibrium for the nonzero-sum game \((A,B)\) then

\[{\mathbb E}_1(\underline{x}^*,\underline{y}^*)\geq v(A)\] and \[{\mathbb E}_{2}(\underline{x}^*,\underline{y}^*)\geq v(B^T).\] because otherwise one of the players would do better to play their max-min strategy (which contradicts the best response condition).

So any Nash equilibrium must be individually rational.