Nash Equilibrium
and the
Minimax Strategy
in Zero Sum Games

Two teams are playing football.  The offense can choose from among four strategies, shown as rows in the table.  The defense can choose from three strategies to stop the play.  The payoffs are yards gained by the offense, or yards lost by the defense.  Each yard gained by the offense is a yard lost by the defense.

If Offense chooses the long pass and the Defense runs a Blitz then the Offense is thrown for a loss of  two yards; this is the worst, or minimum, outcome from choosing Long Pass.  The final column of the table shows all of these worst possible outcomes for Offense.  Offense should make the best of these bad outcomes by choosing to run the Short Pass play.

If Defense chooses to defend against the run then the best they can do is give up 2 yards, the worst is give up ten yards. If they defend against the pass then the worst they can do is give up 5.6 yards.   The last row shows the maxima of these bad outcomes. Defense should make the best of a bad situation by choosing that strategy which gives up the least yards of all.   They should choose to defend against the pass.

Run Pass Blitz Min=
Offense Run 2 5 13 2
Short Pass 6 5.6 10.5 5.6
Medium Pass 6 4.5 1 1
Long Pass 10 3 -2 -2


10 5.6 13