Question

Is there a Nash equilbrium in An Auction Game?

Answer 1

An auction is a game where players bid on items, winners are chosen, and then the players pay some amount to the auctioneer.

First Price Auction

A First Price Auction is a type of sealed-bid auction, which means that players bid only once, and do so without knowledge of other players’ bids. In a First Price Auction, players submit bids, b_i , and the player with the highest bid wins the item. The winning player then pays the amount that he bid to the auctioneer, and the other players pay nothing. Players, as usual, attempt to maximize their payoffs. Therefore, the player with the highest bid can increase his payoff by bidding just above the second highest bidder. That way, he still wins the item, but pays as little as possible. Example from class: Students place bids on Professor Joel’s phone. Student 1 bids $75, Student 2 bids $60 and Student 3 bids $10. If it is a First Price Auction, then Professor Joe should award the phone to Student 1 for a price of $75. But Student 1 could have improved his payoff by bidding lower. For instance, if he had bid $61, he still would have won, but would have paid a lower price for the same phone. As the example illustrates, the outcome is not dominant and introduces instabilities into the market because players do not know what other players’ bids are, and thus must speculate. For this reason, First Price Auctions are generally not advisable in large scale systems.

In a FPSBA, each bidder is characterized by his/her monetary valuation of the item for sale.

Suppose Alice is a bidder and her valuation is a. Then, if Alice is rational:

• She will never bid more than a, because bidding more than a can only make her lose net value.
• If she bids exactly a, then she will not lose but also not gain any positive value.
• If she bids less than a, then she may have some positive gain, but the exact gain depends on the bids of the others.

Alice would like to bid the smallest amount that can make her win the item, as long as this amount is less than a. For example, if there is another bidder Bob and he bids and , then Alice would like to bid (where is the smallest amount that can be added, e.g. one cent).

Unfortunately, Alice does not know what the other bidders are going to bid. Moreover, she does not even know the valuations of the other bidders. Hence, strategically, we have a Bayesian game - a game in which agents do not know the payoffs of the other agents.

The interesting challenge in such a game is to find a Bayesian Nash equilibrium. However, this is not easy even when there are only two bidders. The situation is simpler when the valuations of the bidders are i.i.d. random variables, i.e.: there is a known prior distribution and the valuations of the bidders are all drawn from the same distribution.

So, it can be concluded that There is a Nash equilibrium in auction game.