Question

1. Consider the following (very simplified) version of a simultaneous-move game that occurs in football. The Offense can do two types of plays: a run or a pass. The Defense can defend the run or defend the pass. If the Defense guesses correctly that a run is coming, the Offense gets 0 yards on average (no gain), and if the Defense guesses correctly a pass is coming, the Offense gets 3 yards on average. However, if the Defense doesn’t guess correctly, the Offense stands to gain 6 yards on average if they run the ball and 7 in the case of a pass. The payoffs for the Offense are the yards they gain on average (0, 6, 7, or 3); the payoffs for the Defense are minus that amount (0, -6, -7, or -3), so the best of these four payoffs for the Defense is 0 (while the best for the Offense is 7).

   (a) Are there any pure strategy equilibria in this game? If so, find them showing your work (b) Are there any mixed strategy equilibria in this game? If so, find them showing your work

Answer 1

(a) Given below the strategic form of this 2×2 game in matrix form

Let's check for pure strategy Nash equilibrium by underlying the payoff method. In this method we need to underline the higher payoff of a player when other player is playing a particular strategy.

Let's first underline the payoff for offence.

When defence is guessing pass (GP), the offence can either play pass or run. If offence plays pass it gets the payoff of 3 and if it plays run then it gets the payoff of 6. Hence we underline 6.

When defence is guessing run(GR), the offence again can either play run or pass. If it plays pass it gets the payoff of 7 and if it plays run then it gets the payoff of 0. Hence we underline 7 that is the higher payoff.

Now let's underline the payoff for defence.

When offence is playing pass. The defence can wither guess pass or guess run. If defence guesses pass it gets the payoff of -3 and if it guesses run then it gets the payoff of -7. Hence we underline -3. The higher of the payoffs for player defence.

When offence is playing run. The defence can again play either guess pass or guess run. If it guesses run it gets payoff of 0 and if it guesses pass then it gets the payoff of -6. Hence we underline 0 the higher of the payoffs for player defence.

As we can see there is no strategy profiles whose both payoffs gets underlined and hence there is no pure strategy Nash equilibrium.

(b)Now let's check for mixed strategy Nash equilibrium. For finding mixed strategy Nash equilibrium we need to find some positive probabilities for each players with which they play both strategies with positive probability. The probabilities is set such that it makes player indifferent between playing either strategy.

Below I have shown how to calculate mixed strategy Nash equilibrium.

these probabilities are mixed strategy Nash equilibrium of this game.