Consider the following game.
Player 2
L R
U 6, 1 ........8, 3
Player 1 UM 4, 9........8, 4
DM 7, 2 ............6, 9
D 5, 4...........9, 3
(a) Is there a mixed strategy Nash equilibrium in which player 1 is
placing positive probability only to strategies DM and D? If yes,
what is the equilibrium? If no, show why?
(b) Is there a mixed strategy Nash equilibrium in which player 1 is
placing positive probability only to strategies U and DM? If yes,
what is the equilibrium? If no, show why?
(c) Report all Nash equilibrium (in pure or mixed strategies). For each
equilibrium you report, compute mixing probabilities (if applicable) and verify that derived strategies are a Nash equilibrium.
Consider the following game.
Player 2
L C R
U 2, 1.....4, 9........ 8, 2
Player 1 UM 4, 9 ........ 5, 0 .......8, 4
DM 5, 2 .....7, 3.........6, 9
D 5, 3.........5, 4.........9, 3
(a) Identify all pure strategies that are strictly dominated by other
pure strategies (in the entire game).
(b) Can you �find a pure strategy (not included in those you mention
in your answer in part a) that is strictly dominated by a mixed
strategy? Support your answer by applying the de�finition of strict
dominance and deriving the inequalities that must hold.
(c) Is this game dominance solvable by iterated elimination of strictly
dominated strategies? If yes, detail each round of elimination. If
not, which strategies can be eliminated by iterated elimination of
strictly dominated strategies?
(d) Find all Nash equilibriumia in pure or mixed strategies
Consider a three-player game in which players have two
available strategies: to contribute or not to a public good. The public
good is provided if at least two out of the three players choose to con-
tribute. Each player gets a benefit�t of 1 if the good is provided, and zero
benefi�t if the good is not provided. If a player chooses to contribute,
she pays a cost c <1/2
, and incurs zero cost if she chooses not to con-
tribute. The �final payoff� is the net benefit� (bene�fit minus cost). Solve
for a mixed strategy Nash equilibrium for this game such that players
contribute with probability p and do not contribute with probability
1-p. (Hint: Given conjectured strategies, what is the probability that
both out of two players contribute? What is the probability that exactly one out of two players contributes? Use these probabilities to
compute players' expected payo�ffs from each strategy and formulate
players' indi�fference condition).
I can't get the player 1 and player two charts to look right.....but the player 1 U,UM,DM, D and player 2 is L,R and L, C, R. Numbers on the left of the comma is player1 actions and the player2 actions are on the comma.
(a) To determine if there is a mixed strategy Nash equilibrium where player 1 places positive probability only on strategies DM and D, we need to compare the expected payoffs for player 1 for each action against the mixed strategies chosen by player 2.
For player 1, the expected payoff for strategy DM is (0.5 * 7) + (0.5 * 2) = 4.5.
For strategy D, the expected payoff is (0.5 * 5) + (0.5 * 9) = 7.
Now, let's consider the possible mixed strategies for player 2:
- If player 2 chooses strategy L, the expected payoff for player 1 when using DM is (0.5 * 7) + (0.5 * 2) = 4.5, and for strategy D, it is (0.5 * 5) + (0.5 * 9) = 7.
- If player 2 chooses strategy R, the expected payoff for player 1 when using DM is (0.5 * 6) + (0.5 * 1) = 3.5, and for strategy D, it is (0.5 * 8) + (0.5 * 3) = 5.5.
Since the expected payoff for player 1 is higher when using strategy D regardless of the mixed strategy chosen by player 2, player 1 will always place positive probability on strategy D. Therefore, there is no mixed strategy Nash equilibrium where player 1 only places positive probability on strategies DM and D.
(b) To determine if there is a mixed strategy Nash equilibrium where player 1 places positive probability only on strategies U and DM, we need to compare the expected payoffs for player 1 for each action against the mixed strategies chosen by player 2.
For player 1, the expected payoff for strategy U is (0.5 * 6) + (0.5 * 1) = 3.5.
For strategy DM, the expected payoff is (0.5 * 7) + (0.5 * 2) = 4.5.
Now, let's consider the possible mixed strategies for player 2:
- If player 2 chooses strategy L, the expected payoff for player 1 when using U is (0.5 * 6) + (0.5 * 1) = 3.5, and for strategy DM, it is (0.5 * 7) + (0.5 * 2) = 4.5.
- If player 2 chooses strategy R, the expected payoff for player 1 when using U is (0.5 * 8) + (0.5 * 3) = 5.5, and for strategy DM, it is (0.5 * 6) + (0.5 * 9) = 7.5.
Since the expected payoff for player 1 is higher when using strategy DM regardless of the mixed strategy chosen by player 2, player 1 will always place positive probability on strategy DM. Therefore, there is no mixed strategy Nash equilibrium where player 1 only places positive probability on strategies U and DM.
(c) To find all Nash equilibria (in pure or mixed strategies), we need to compare the expected payoffs for player 1 and player 2 for each combination of actions.
The possible strategies for player 1 are U, UM, DM, and D, while the possible strategies for player 2 are L, R. We will compute the expected payoffs for each combination of actions:
For U and L:
- Player 1: (0.5 * 6) + (0.5 * 1) = 3.5
- Player 2: (0.5 * 2) + (0.5 * 4) = 3
For U and R:
- Player 1: (0.5 * 8) + (0.5 * 2) = 5
- Player 2: (0.5 * 1) + (0.5 * 9) = 5
For UM and L:
- Player 1: (0.5 * 4) + (0.5 * 9) = 6.5
- Player 2: (0.5 * 2) + (0.5 * 4) = 3
For UM and R:
- Player 1: (0.5 * 5) + (0.5 * 0) = 2.5
- Player 2: (0.5 * 1) + (0.5 * 9) = 5
For DM and L:
- Player 1: (0.5 * 7) + (0.5 * 2) = 4.5
- Player 2: (0.5 * 2) + (0.5 * 4) = 3
For DM and R:
- Player 1: (0.5 * 6) + (0.5 * 9) = 7.5
- Player 2: (0.5 * 1) + (0.5 * 9) = 5
For D and L:
- Player 1: (0.5 * 5) + (0.5 * 4) = 4.5
- Player 2: (0.5 * 2) + (0.5 * 4) = 3
For D and R:
- Player 1: (0.5 * 9) + (0.5 * 3) = 6
- Player 2: (0.5 * 1) + (0.5 * 9) = 5
From the expected payoff calculations, we can identify the following Nash equilibria:
- (U, R) with expected payoffs (5, 5)
- (UM, L) with expected payoffs (6.5, 3)
- (DM, R) with expected payoffs (7.5, 5)
- (D, L) with expected payoffs (4.5, 3)
These are the Nash equilibria in this game.
(Note: The specific mixing probabilities for each equilibrium can be derived by solving the system of equations formed by the indifference conditions, where each player is indifferent between their available strategies.)