An item is up for auction. Player 1 values the item at 3 while player 2 values t

Question

An item is up for auction. Player 1 values the item at 3 while player 2 values the item at 5. Each player can bid either 0,1, or 2. If player i bids more than player j then i wins the good and pays his bid, while the loser does not pay. If both players bid the same amount then a fair coin is tossed to determine who the winner is, and the winner gets the good and pays his bid while the loser pays nothing.

Consider player 1. Which, if any, of his strategies are strictly dominated?

Now consider player 2. Which, if any, of his strategies are strictly dominated?

Explanation / Answer

v1 = 3

v2 = 5

Bids = Bi = 0,1,2

i = 1, 2

For player 1,

If player2 plays 0, player 1 will play 1

If player2 plays 1, player 1 will play 1 or 2

If player2 plays 2, player 1 will play 2

So there is no strictly dominated startegy for 1.

For player2;

If player 1 plays 0, player 2 will play 1

If player 1 plays 1, player2 will play 2

If player 1 plays 2, player2 will play 2

Nash equilibrium will be for both players to bid the maximum i.e. 2.

Player 2 0 1 2 0 1/2(3), (1/2)5 0, 4 0,3 Player 1 1 2,0 (1/2)2,(1/2)4 0,3 2 1,0 1,0 (1/2)1, (1/2)3 Player 2 0 1 2 0 1.5,2.5 0, 4 0,3 Player 1 1 2,0 1,2 0,3 2 1,0 1,0 0.5 , 1.5

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