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3. (35%) A single item is offered to 2 players via an auction. Player 1 values the item at $4 and player 2 values the item at $5. Each player i = 1,2 privately submits a bid b_i the auctioneer, which can be either $0,$1,or $2. The player with the higher bid gets the item, but both players must pay their bids. If player i bids strictly more than player j then player i wins the item, but both bidders pay their bids. If bids are equal, then each player receives 1/2 of his/her value, but both players still pay their full bids. The payoff of each player is equal to his/her value of the item in the case of the win, a half of the item's value in the case of the tie (equal bids), and 0 in the case of the loss, minus his/her full bid in either case. a) Write down the game in the matrix form. (Be careful as any mistake here leads to wrong answers below). b) Reduce the game by the iterated elimination of strictly dominated actions. (Note: consider the pure-actions dominance only. No need to prove it, just cross-out the strictly dominated actions in the matrix.) c) Find all mixed-strategy Nash equilibria in the reduced game.

          3. (35%) A single item is offered to 2 players via an auction. Player 1 values the item at $4 and player 2 values the item at $5. Each player i = 1,2 privately submits a bid b_i the auctioneer, which can be either $0,$1,or $2. The player with the higher bid gets the item, but both players must pay their bids. If player i bids strictly more than player j then player i wins the item, but both bidders pay their bids. If bids are equal, then each player receives 1/2 of his/her value, but both players still pay their full bids. The payoff of each player is equal to his/her value of the item in the case of the win, a half of the item's value in the case of the tie (equal bids), and 0 in the case of the loss, minus his/her full bid in either case.
a) Write down the game in the matrix form. (Be careful as any mistake here leads to wrong answers below).
b) Reduce the game by the iterated elimination of strictly dominated actions. (Note: consider the pure-actions dominance only. No need to prove it, just cross-out the strictly dominated actions in the matrix.)
c) Find all mixed-strategy Nash equilibria in the reduced game.

3. (35%) A single item is offered to 2 players via an auction. Player 1 values the item at $4 and player 2 values the item at $5. Each player i = 1,2 privately submits a bid bi the auctioneer, which can be either $0,$1,or $2. The player with the higher bid gets the item, but both players must pay their bids. If player i bids strictly more than player j then player i wins the item, but both bidders pay their bids. If bids are equal, then each player receives 1/2 of his/her value, but both players still pay their full bids. The payoff of each player is equal to his/her value of the item in the case of the win, a half of the item's value in the case of the tie (equal bids), and 0 in the case of the loss, minus his/her full bid in either case.
a) Write down the game in the matrix form. (Be careful as any mistake here leads to wrong answers below).
b) Reduce the game by the iterated elimination of strictly dominated actions. (Note: consider the pure-actions dominance only. No need to prove it, just cross-out the strictly dominated actions in the matrix.)
c) Find all mixed-strategy Nash equilibria in the reduced game.

Added by Daniel D.

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