Solutions to some other questions, exercise sheets 1-3
Problem set 1: Exe 1
a) Player 1:M is strictly dominated by T.
After eliminating M; Player 2:R is strictly dominated by C.
b) For P1 M is strictly dominated by T; then R is strictly dominated by C for P2. We have to check the remaining strategy profiles for the NE:
(T, C): P1 has incentive to deviate and play B
(T, L): P2 has incentive to deviate and play C
(B, C): P2 has incentive to deviate and play L
(B, L): P1 has incentive to deviate and play T
So there is no NE in pure strategies.
Problem set 1: Exe 2
Bis strictly dominated by M. R is strictly dominated by C.T is strictly dominated by M.L is strictly dominated by C. Then (M , C) is the pure strategy Nash equilibrium of the game.
Problem set 1: Exe 3
Each player’s payoff is (assume in case of a tie each player has an equal chance of wining the item):
In order to show that bidding bi = vi is every player’s weakly dominant action; suppose xi is another action of this player, she can bid either less or more than her valuation (xi < bi or xi > bi); we will show that both of them are weakly dominated by bi = vi:
i) For xi < bi
/ / / / 0/ / / 0 / 0
then bidding xi < bi is weakly dominated by bi
ii) For xi > bi
/ / 0 / 0 / 0 / 0/ / 0 / / / 0
then bidding xi > bi is weakly dominated by bi
Problem set 1: Exe 4
a) S1 = { x 0 x a } & S2 = { y a y 1 }
&
b) In order to show that (x = y = 0.5) is the unique NE:
i) (x,y) = (0.5,0.5) is a NE: Suppose x = 0.5 then which means y = 0.5 is the best response. Similarly it can be shown that x = 0.5 is the best response to y = 0.5.
ii) (x,y) = (0.5,0.5) is the only NE:Suppose there is an alternative NE (x’,y’) (0.5,0.5) (at least one of the player bid different than 0.5) then . If x’<0.5 regardless of player 2’s action, player 1 can increase her payoff by increasing x’ to 0.5. Similarly for player 2: .If y’>0.5 regardless of player 1’s action, player 2 can increase her payoff by decreasing y’ to 0.5. (x’,y’) (0.5,0.5) cannot be a NE.
c) In this case it is a strictly dominant strategy for player 1 to play x = a because:
but for player 2 there is no best response to x = a, it is because she would prefer to locate herself as close as possible but not equal to a.
Problem set 3: Exe 3
Should check for deviation incentive after any history:
After history (C,C) for player 1:
Player 1 has no incentive to deviate after history (C,C) iff or .
After history (C,C) for player 2:
Player 2 has no incentive to deviate after history (C,C) for any .
Similarly it can be shown that there is no deviation incentive for the players at any other history (out of equilibrium path histories), they simply end up with a lower pay-off at each round.
Problem set 3: Exe 3
a) Suppose player 1 played C in the 1st round, his payoff in the 2nd round would be:
Given that after playing C in the 1st round it is a strictly dominant strategy to play C in the 2nd.
Now suppose player 1 played D in the 1st round, his payoff in the 2nd round would be:
Given that after playing D in the 1st round it is a strictly dominant strategy to play D in the 2nd.
Since the game is symmetric these are true for player 2 as well; then by backward induction to the first round:
Then the only sub-game perfect eq of the game is when DD is played in both rounds.
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