1. Two art dealers, A and B are bidding for an artwork in an auction. Art dealer A values the work at $10 (million), and art dealer B values the work at $8 (million). The rules of the auction are as follows. Each dealer simultaneously bids some dollar amount for the artwork, and the highest bidder wins the item, and has to pay the bid made by the other bidder. In case of a tie, dealer A gets the artwork. (Such an auction is called a second-price auction; a generalization of this is used to sell spots for ads to advertisers on many websites.)
Assume each dealer values an outcome by the dollar amount they have in that outcome. Assume further that dealer A can only bid in multiples of 3 million, and dealer B can only bid in multiples of 2 million. No dealer will ever bid more than their value for the artwork.
(a) Specify the strategic form of the preceding auction.
(b) Does the game have any strictly dominated strategies? Give reasons. (c)State the weakly dominant strategies for each player, if
there are any.
2. Consider the following two player game between players I and II:
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A
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B
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C
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D
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a
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(1,1)
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(4,5)
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(3,2)
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(2,4)
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b
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(3,2)
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(5,0)
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(4,2)
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(3,1)
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c
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(3,3)
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(2,2)
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(2,4)
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(1,5)
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d
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(0,2)
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(3,3)
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(1,4)
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(4,5)
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(a) Find all strictly dominated strategies.
(b)Find all weakly dominated strategies.
(c) Is this game (strict) dominance solvable? If yes, what is the solution of the game? If not, write down the reduced game that remains at the end of ISD.
3. Consider the following three player game among players I, II, and III. The representa- tion is as follows: Player I chooses the row, Player II chooses the column, and Player III chooses the matrix. This choice is made simultaneously. The corresponding entry gives the payoffs to each player, with the first component being player I's payoff, second component being player II's payoff, and the third component being player III's payoff.
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A
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B
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C
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a
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(1,2,2)
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(2,3,1)
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(3,3,7)
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b
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(3,2,4)
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(3,5,1)
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(4,0,4)
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c
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(2,3,9)
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(4,4,5)
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(2,3,5)
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|
|
A
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B
|
C
|
|
a
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(2,3,4)
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(0,1,4)
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(2,3,2)
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|
b
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(4,1,3)
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(1,3,0)
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(3,8,3)
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|
c
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(2,2,5)
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(4,5,3)
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(2,6,2)
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(a) Find all strictly dominated strategies.
(b)Is this game (strict) dominance solvable?