In the AFC championship game between the Indianapolis Colts and the New England Patriots in 2007, the Colts had a fourth down and inches play. Rather than punt the ball and turn it over to their opponent, the Colts decided to go for a first down. Suppose the Colts have two play options: a fullback running up the middle or a screen pass to a wide receiver. The Patriots also have two play options: setting up to defend against the run or setting up to defend against the screen pass. The coaches of the two teams simultaneously choose their plays. If the Colts run the ball and the Patriots set up to defend against the run, then the Colts' payoff is and the Patriots' payoff is 1. If the Colts pass and the Patriots set up to defend against the pass, then the Colts' payoff is and the Patriots receive 2. If the Colts run and the Patriots set up to defend against the pass, the Colts' payoff is 6 and the Patriots' is If the Colts pass and the Patriots set up to defend against the run, the Colts' payoff is 10 and the Patriots' is
a. Show the payoff matrix for this simultaneousmoves game. What is the Nash equilibrium? Is it a pure-strategy or mixed-strategies Nash equilibrium?
b. Now suppose instead that if the Colts pass and the Patriots set up to defend again the pass, the Colts' payoff is 8 and the Patriots' -8. is Write the payoff matrix for this simultaneous-moves game. What is the Nash equilibrium? Does this Nash equilibrium involve pure or mixed strategies?