# game theory

posted by
**Desperate** on
.

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 payoffs from each strategy and formulate

players' indifference 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.