Consider the following strategic-form game G1(z) where z > 0 is a parameter.
Figure 1: Strategic-form game G1(z)
Column: C D
3z + 3, 2z + 2
3, 6
9, 2
3z, 2z
Row:
(a) Rescale and normalize the payoffs for each player in G1(z) (holding constant the underlying preferences over A) so that both Row and Column get a payoff of zero if (b, D) is chosen, and one if (C, D) is chosen.
(b) What are the conditions on the parameter of the game z for the Row player to have a strictly dominated action? (Hint: Your answer to part (a) will make subsequent calculations easier if you correctly rescale!)
(c) What are the conditions on the parameter of the game z for the Column player to have a weakly dominated action?
(d) What are the conditions on the parameter z for (b, C) to be a pure-strategy Nash equilibrium of this game?
(e) For what values of z is there a mixed strategy Nash Equilibrium for G1(z) where both players randomize over both of their actions with positive probability?
(f) When it exists, what is the mixed-strategy for each player in this equilibrium? (Hint: If your game in part (a) should have enough symmetry that solving for one player's mixture is enough to figure out the Nash!)