# Explain the proposition that it does not matter to a seller which type of auction is used.

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Introduction

Name: Upesh Patel

Course: ES3600

Tutor: Martin Currie

Microeconomics Essay

Explain the proposition that it does not matter to a seller which type of auction is used.

Some of the most exciting advances in recent microeconomic theory have been in the modelling of strategic behaviour under asymmetric information. One of the fundamental aspects of this area is the theory of bidding systems in auctions. An auction is a formal institutional arrangement for allocating scarce commodities among competing bidders. According to McAfee and McMillan (1989), “some theoretical study of auctions is warranted,” seeing as they are increasingly used in the exchange of goods in modern times.

Many advances have been made in creating theories to explain auctions. When discussing theories on auctions, we focus on the case of N bidders competing to buy a single unit of a commodity. Economic theorists have identified four primary auction mechanisms, the first of which is the English auction. This is the type most people are familiar with. An English auction is an open one, which involves ascending bids, and the auction ends when no higher bids are forthcoming. The good goes to the bidder who makes the final bid at a price equal to that bid. The Dutch auction is also an open auction, but involves descending bids, where the price is reduced until some bidder indicates their willingness to pay the going price. First Price Sealed Bidding auctions differ from the auctions mentioned above because they are “closed” auctions.

Middle

I will now go on to analyse the outcomes of each of the four different auctions. I will begin with the English auction. Assuming infinitesimally small increments are possible, bidder i has a dominant strategy: raise the standing bid if and only if i’s valuation is more than the bid made by bidder k. To understand why this is the case, we need to realise that as the auction progresses, bidders drop out when the standing bid reaches their valuations. Thus, when the bidder with the highest valuation submits a bid equal to the second highest valuation, the bidding process stops. Therefore, the highest bidder receives the good and pays a price equal to the second-highest valuation, which we denote Pe = v2. (where Pe denotes the equilibrium price for the English auction, and v2 is the second highest valuation.) The expected price to the seller is N-1/N+1. (Obtained from the equation for the gth order statistic.)

The English auction bears a strong resemblance to the case of Second Price Sealed Bidding.

Conclusion

To determine the outcome of the auctions, theorists have focused on Nash equilibria. Each bidder is employing their optimal bidding strategy given their beliefs about the bidding strategy of others, where their beliefs are confirmed in equilibrium. With N bidders, a symmetric Nash equilibrium for our uniform distribution involves: bi(vi)=N-1/N(vi), for i=1…N. since each bidders bid is increasing in their own valuation, the item goes to bidder 1 at a price N-1/N(vi). The expected price in these auctions is: N-1 / N x E(vi) = (N-1/N) x (N/N+1) = N-1/N+1. As we can see, all four auctions result in the same expected price to the seller. In general, the actual prices will not be the same for all auctions, since the equilibrium price for the Dutch and FPSB auctions does not equal the equilibrium price for the Dutch an SPSB auctions. However, the fact that the expected prices are the same is nonetheless an important result. We have thus confirmed the revenue equivalence theorem. Under the stipulated assumptions, the four primary types of auction are “equivalent” in that they would all yield the same expected price to the seller, with the item going to bidder who values it most highly. This theorem explains why it would not matter to a seller which type of auction is used.

This student written piece of work is one of many that can be found in our AS and A Level Probability & Statistics section.

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