Non-zero-sum Games

Non-zero-sum Games#

The Nordstrom textbook gives the following example of a non-zero-sum game:

Battle of the Movies#

Rose and Colin want to go out to a movie. Colin wants to see an action movie, Rose wants to see a comedy. Both prefer to go to a movie together rather than to go alone. We can represent the situation with the payoff matrix:

\[\begin{split}\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|cc}&A&B\\ \hline A&(2,1)&(-1,-1)\\ B&(-1,-1)&(1,2)\end{array}\end{array}\end{split}\]

We notice immediately that, instead of a single value for our outcomes or a pair of values that sum to zero, we have an ordered pair of values that do not fit our definition of zero-sum.

Rose’s Game#

We can analyze the zero-sum game from Rose’s perspective where she focuses on her own payoffs and ignores Colin’s payoffs completely:

\[\begin{split}\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rr}&A&B\\ \hline A&2&-1\\ B&-1&1\end{array}\end{array}\end{split}\]

Colin’s Game#

Colin’s game is also zero-sum where he does the same thing:

\[\begin{split}\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rr}&A&B\\ \hline A&1&-1\\ B&-1&2\end{array}\end{array}\end{split}\]

Nash Equilibrium for Non-zero-sum Games#

If Colin’s optimal strategies arise from Colin playing Rose’s game using his oddments strategy mixture while Rose’s optimal strategies from her playing Colin’s game using her oddments strategy, we will arrive at a point which qualifies as a Nash Equilibrium.

Colin’s oddments strategy in Rose’s game will equalize her expectation. Rose will therefore have no incentive to deviate.

Similarly, Rose’s oddments strategy in Colin’s game will equalize his expectation. Colin has no incentive to deviate either.

To wrap up, a Nash Equilibrium in a non-zero-sum game is an outcome (in mixed strategies) from which neither player can profit by deviating.