# game theory

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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.

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