Question

Consider a first-price sealed-bid auction as the one analyzed in class. Suppose bidders' valuations are v1=10 and v2=10. Suppose bidder 2 submits a bid b2=10. Then, in a Nash equilibrium in pure strategies bidder 1 must be submitting a bid equal to __ . In this Nash equilibrium, bidder 1's payoff is equal to __ (please, enter numerical values only, for example: 4).

Answer 1

We have an information that bid for bidder 2 is $10 & valuation for bidder 2 is $10 hence the payoff for bidder 2 is 0 (Valuation - Bid) if he wins the bid.

For bidder 1 if he bids $10 that is equivalent to its valuation $10 then both bids match and auctioned product will be divided equally between them. Payoff for both the players is -5. For a bid less than $10 payoff for player 1 is 0 and for a bid more than $10 and less than $15 can have a payoff of - 1 to -4 respectively as he wins the bid of $10.

Hence bidder 1 bids from.$0 to $9 he doesnt win the bid hence nothing to pay therefore payoff is 0

As per my understanding Nash equilibrium for player 1 is $0. Payoff is $0