1. True or false, justify your answer. You may cite any applicable theorems to justify your answer.
(a) A pure-strategy Nash equilibrium does not exist in the Matching Pennies game.
(b) A mixed-strategy Nash equilibrium does not exist in the Matching Pennies game.
(c) A Nash equilibrium always exists in finite games, i.e., games with a finite number of players with finite actions sets.
(d) In a mixed-strategies Nash equilibrium profile, for every player, the expected payoff associated with actions assigned positive probability is the same.
(e) In a mixed-strategies Nash equilibrium profile, for every player, the expected payoff associated with actions assigned zero probability is less than or equal to the expected payoff associated with actions assigned positive probability.
2. Consider the following (2x3) strategic game
| Player 1 |
|
Player 2 |
| Left |
Center |
Right |
| Top |
4,2 |
0,0 |
0,1 |
| Bottom |
0,0 |
2,4 |
1,3 |
Find all the mixed-strategy Nash equilibria of this game by the procedure outlined in the notes using
Proposition 116.2 in your textbook.
3. Consider the following (3x2) game where the matrix only displays the payoff to player 1
|
|
Player 2 |
|
|
L |
R |
| Player 1 |
T |
1 |
1 |
| M |
4 |
0 |
| B |
0 |
3 |
Does (14,14,12) strictly dominate T?
4. Consider the following (2x3) strategic game.
|
|
Player 2 |
|
|
Left |
Center |
Right |
| Player 1 |
Top |
2,2 |
0,3 |
1,2 |
| Bottom |
3,1 |
1,0 |
0,2 |
Find all the mixed-strategy Nash equilibria of this game by first eliminating any strictly dominated actions and then constructing the players' best response functions.
5. (a) Represent in a diagram the two-player extensive game with perfect information in which the terminal histories are (C, E),(C, F),(D, G), and (D, H), the player function is given by P(Φ) = 1 and P(C) = P(D) = 2. Player 1 prefers (C, F) to (D, G) to (C, E) to (D, H) and player 2 prefers (D, G) to (C, F) to (D, H) to (C, E).
(b) List all of player 1's possible strategies.
(c) List all of player 2's possible strategies.
(d) List all possible strategy profiles and their corresponding outcomes and payoffs.
(e) Find all Nash equilibria of the above game.
(f) Find all subgame perfect equilibria of the above game.
6. True or false, justify your answer. You may cite any applicable theorems to justfiy your answer.
(a) The ultimatum game has a finite horizon.
(b) The ultimatum game has finitely many terminal histories.
(c) The ultimatum game is finite.
(d) Backward induction can be applied to solve any extensive game with perfect information.
(e) Every Nash equilibrium of an extensive game is subgame perfect.