Question

Consider 2nd price Sealed-Bid auction, where n bidders participate in the purchase of one antique.Each contestant pays for antiques (WTPs) to meet:

\( v_{1}>v_{2}>\ldots>v_{n} \)
The contestant{i}writes down the bidb_{i}\geq 0, and the highest bidder is the auction winner.If two or more bidders make the highest bid, the item will be taken away by the higher bidder.If player{i}wins the auction, he takes the payoff ofv_{i}-maxb_{j}, but if not, the payoff is zero
All competitors have shown that this is Wickley Dominat's strategy, and his payment pricev_{i}is the bid.In other words,show thatb_{i}=v_{i}is weakly dominant strategy for each player i " 1, 2, ..., n.

Answer 1

To show that bidding the true willingness-to-pay (WTP) \(b_i = v_i\) is a weakly dominant strategy for each player i in a second-price sealed-bid auction, we need to demonstrate that it is the best response regardless of what other players do. In other words, regardless of the bids made by other players, bidding the true WTP maximizes the expected payoff for each player.

Let's consider player i and analyze the two possible scenarios:

1. Player i has the highest bid: In this case, player i wins the auction and obtains the antique. The payoff for player i is \(v_i - \max(b_j)\), where \(b_j\) represents the bids of other players. If player i bids \(b_i = v_i\), their payoff becomes \(v_i - \max(b_j) = v_i - v_{\text{max}}\), where \(v_{\text{max}}\) is the maximum bid among the other players. Since \(v_i\) is player i's true WTP and \(v_i > v_{\text{max}}\), the payoff is positive, resulting in a higher payoff compared to any other bid.

2. Player i does not have the highest bid: In this case, player i does not win the auction and receives a payoff of zero. No matter what bid player i places, their chances of winning do not change because the winner is determined solely by the highest bid. Therefore, bidding \(b_i = v_i\) does not decrease the probability of winning for player i.

Considering both scenarios, we can conclude that bidding the true WTP \(b_i = v_i\) is a weakly dominant strategy for each player i in a second-price sealed-bid auction. It ensures that players maximize their expected payoffs regardless of the bids made by other players.