Question

Consider 5 bidders whose values are independently and uniformly distributed over [0, 600].

1. Suppose that the values of the bidders are 120,200,240,400 and x. For what values of x, we would have the same selling price in Second-Price Auction (SPA) and FPA? (Hint: there are two solutions!) please explain in more detail,thanks

Answer 1

The first price auction(FPA) in which the highest bidding buyer pays the bidding amount to the seller.

The Second price auction(SPA) in which the highest bidding buyer pays the second highest bidding amount to the seller.

In the second price auction , the dominant straeegy for players is to bid their own valuation .

There are 5 bidders with valuation 120, 200, 240, 400, x. Suppose x>400 then in SPA all players will bid their own valuations and player 5 will win and player 5 will pay the second highest bid that is 400. In FPA truthful bidding is not a dominant strategy as bidders will never get a positive payoff and bidders never bid any price above their valuation. If player 5 knows the valuation of the product of other players he will bid second highest bid with a very small amount addition that is 400+epsilon. so he will win the bid and pay 400. So, in this situation the selling price for FPA and SPA is same.

Suppose 240<x<400 then in SPA all players will bid their own valuations and player 4 will win and player 4 will pay the second highest bid that is x. In FPA truthful bidding is not a dominant strategy as bidders will never get a positive payoff and bidders never bid any price above their valuation. If player 4 knows the valuation of the product of other players he will bid second highest bid with a very small amount addition that is x+epsilon. so he will win the bid and pay x. So, in this situation the selling price for FPA and SPA is same.

Suppose x<240 then in SPA all players will bid their own valuations and player 4 will win and player 4 will pay the second highest bid that is 240. In FPA truthful bidding is not a dominant strategy as bidders will never get a positive payoff and bidders never bid any price above their valuation. If player 4 knows the valuation of the product of other players he will bid second highest bid with a very small amount addition that is 240+epsilon. so he will win the bid and pay 240. So, in this situation the selling price for FPA and SPA is same.