Assignment:
In a first-price auction, each buyer submits his bid in a sealed envelope. The winner of the auction is the buyer who submits the highest bid, and the amount he pays is equal to what he bid. If several buyers have submitted bids equal to the highest bid, a fair lottery is conducted among them to choose one winner, who then pays his bid.
(a) In this situation, does the strategy βi^ of buyer i, in which he bids his private value for the item, weakly dominate all his other strategies?
(b) Find a strategy of buyer i that weakly dominates strategy βi^. Does the strategy under which each buyer bids his private value weakly dominate all the other strategies? Justify your answer.
Provide complete and step by step solution for the question and show calculations and use formulas.