Adversarial risk analysis for auctions using non-strategic play and level-k thinking: A general case of n bidders with regret

Abstract

First-price sealed-bid auctions have been modeled using a utility function that takes into account bidders’ regret in case of winning or losing the auctioned item. However, that modeling does not consider bidders’ wealth which is an important determinant of bidders’ behavior in these auctions. In this paper, we apply an adversarial risk analysis approach for these auctions and find solutions using non-strategic play and level-k thinking solution concepts assuming n bidders. We define new regret parameters and a modified utility function to incorporate the effect of bidders’ wealth on their bidding behavior. In our modeling, we assume that the auctioned item is a normal item and has a reserve price. We give numerical examples to illustrate our methodology for each solution concept.

Keywords:

• First-price sealed-bid auctions
• adversarial risk analysis
• Bayesian game theory
• decision theory

Notes

1 In FPSB auctions, bidders submit their bids in sealed envelopes to the auctioneer. Those envelopes are then opened and the bidder with the highest bid wins the auction and pays the amount equal to the bid.

2 An item with positive income elasticity is defined as a normal item in economic theory, i.e., the demand for a normal item rises with an increase in income and falls with a decrease in income (see, e.g., Fisher 1990; Goeree, Holt, and Palfrey 2002; Piros and Pinto 2013; Perloff 2015; Baisa 2017, for more details)

3 In a random mth-price sealed-bid auction, each bid is rank-ordered from highest to lowest; the auctioneer selects a random number, the m in the mth-price sealed-bid auction, uniformly-distributed between 2 and n (n bidders); and the auctioneer sells one auctioned item to each of the ($m-1$) highest bidders at the mth-price.

4 The price at which the quantity demanded of an item or service is equal to the quantity supplied and no surplus or shortage exists in the market.

5 The participants make increasingly higher bids and stop bidding when they are not prepared to pay more than the current highest bid. This continues until no participant is prepared to make a higher bid; the highest bidder wins the auction at the final amount bid.

6 In Dutch auctions, the price is set by the auctioneer at a level sufficiently high to deter all bidders, and is progressively lowered until a bidder is prepared to buy at the current price auctioned item.