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Applied Microeconomic ModelsProf. Ugo Colombino

Asymmetric information: Auctions and Negotiations

Three developments in economics and in technology converge in generating a lot of interest for the topics of auctions and negotiations.i) The explosion of Internet implies that a growing mass of

transactions are made on the web.ii) The progress in computer science and artificial intelligence makes it

possible to manage those transactions at least in part through computerized procedures.

iii) Economists realized that many recipes coming from their theory, and especially game theory and “mechanism design” theory, was useful in specifying the procedures for efficiently managing transactions and in allocating property rights various resources (commons, pollution permits, public utilities etc.). The expression “Mechanism Design” was used – possibly for the first time – in a 1973 paper by Leo Hurwicz (Nobel Prize 2007). By “mechanism” we mean here a set of rules or procedures that define how a group of economic subjects make decisions and interact.

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AuctionsWe consider a situation where a seller wants to sell one item. There are many potential buyers (also called bidders). Each potential buyer knows her own – and only her own – valuation of the item. The seller does not know the buyers’ individual valuations.This setting is known as an independent-private-value auction.What follows is valid also for the so-called reverse auction, where a buyer wants to buy one item from many potential sellers. Typically the negotiation between the seller and the buyers is managed by a mediator (the auctioneer).

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Objectives of an auctionIn designing an auction we might pursue two main purposes:1) Maximizing the seller’s revenueand/or 2) Assigning the item to the buyer who values the item

most.Although these goals can be pursued with many different types of auctions, particularly interesting are the mechanisms wherea) reveal their true evaluation of the item, i.e. Truth-

Telling mechanisms (TT) ;b) telling the truth is a dominant strategy for all the

candidate buyers.

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The Revelation PrinciplePoint (a) – truthful revelation – is important because of the Revelation Principle. This theorem says that whatever outcome we can get with a Non-TT mechanism, there is a TT mechanism that leads to the same outcome. If we find a TT procedure that leads to an optimal (in some sense) outcome, then we are sure that that’s the best that we can do (even with a Non-TT procedure).

The Revelation Principle was stated and proved – in various versions – around 1978 independently by many researchers, among them Roger Myerson and Eric Maskin (both 2007 Nobel Prize winners together with Leo Hurwicz).

Point (b) – dominant strategy equilibrium – is obviously important because it is a very robust equilibrium.

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A TT mechanism with dominant strategy equilibrium

In a 1961 paper, William Vickrey (1996 Nobel Prize) proposed and analyzed the following mechanism (since then known a Vickrey’s auction).Each bidder communicates to the auctioneer her own bid, in one round. The bid is private information: it is communicated – so to speak – in a “sealed envelope”. The highest bid wins, and the winner pays the second-highest bid.

The mechanism is also called Second-Price-Sealed-Bid (SPSB) auction.

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Result 1: The Vickrey’s Auction leads to truth-telling as a dominant strategy.Consider bidder 1. Call v1 her true valuation of the item and b1 her bid (not necessarily equal to v1).Let b2 the highest bid of the other bidders.The expected gain of bidder 1 is(v1 – b2 )Prob(b1 ≥ b2) + (0)Prob(b1 < b2) = (v1 – b2)Prob(b1 ≥ b2).

• If v1 > b2 then bidder 1 maximizes her expected gain by setting b1 = v1: this way, Prob(b1 ≥ b2) = 1 (highest possible value).

• If v1 < b2 then again bidder 1 maximizes her expected gain by setting b1 = v1: this way, Prob(b1 ≥ b2) = 0 (lowest possible value).

• If v1 = b then the expected gain is always 0, so any choice (including b1=v1) is optimal.

Therefore b1 = v1 is a dominant strategy.

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Result 2: The Vickrey’s Auction leads to an efficient result, since the winner will be the bidder with the highest (true) valuation.

Result 3: The Vickrey’s Auction does not maximize the seller’s revenue

Procedures similar to the Vickrey’s Auction have been traditionally used since mid-1800 in transactions between stamps collectors and are now used by eBay online market. Slots for advertisements on Google and Yahoo search pages are also sold with more sophisticated versions of Vickrey’s Auction.

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Comparing Vickrey’s auction with FPSB auction

A very common form of auction is the First-Price-Sealed-Bid (FPSB) auction. FPSB) works as the SPSB (or Vickrey’s) auction, except for the fact that the winner pays her own bid instead of the second-highest bid.Write down the expected gain of bidder 1:

(v1 – b1)Prob(b1 ≥ b2),

The crucial difference is that here the term (v1 – b1) is affected by b1. Certainly there is no point in bidding higher than v1. However there might be an incentive to bid lower than v1: this would increase the expected gain as long as Prob(b1 ≥ b2) remains sufficiently high, and in turn this will depend on b2.In conclusion the optimal bid b1 depends on the bidder 1’s expectations on b2: therefore there is no dominant strategy.

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References- A useful survey book – both on theoretical results and on practical issues concerning auction design – is: Paul Klemperer, Auction Theory and Practice.It can be downloaded free from Klemperer’s web page:www.paulklemperer.org .- Two very readable papers tell the fascinating story of the spectrum auctions in the US, also with much attention to institutional details:J. McMillan, Selling Spectrum Rights, Journal of Economic Perspectives,

Vol. 8, No. 3, 1994.P. McAfee and J. McMillan, Analyzing the Airwave Auction, Journal of

Economic Perspective, Vol. 10, No. 1, 1996.- An introduction – but more difficult – both to the theory and to the

practical design is provided by:P. Milgrom, Auctions and Bidding: A primer, Journal of Economic

Perspectives, Vol. 3, 1989.

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Bilateral Negotiations

We now consider a different scenario for a transaction, namely bilateral negotiations.

Imagine a random encounter between a potential Seller and a potential Buyer.

Just as an example, think of two consumers or two firms “meeting” on an internet platform were items (stamps, custom guitars, jobs, electronic components, pollution permits etc.) are exchanged.

For the sake of simplicity we assume that there are only two types of sellers: those who value 0 the item they own (weak sellers) and those who value it 2 (strong sellers).

Analogously, there are two types of buyers, those who value 1 the item (strong buyers) and those who value it 3 (weak buyers).

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When a seller and a buyer meet, they do not know what type is the other party.

The buyer only knows that the seller is type 0 with probability 0.5 or type 2 with probability 0.5.

The seller only knows that the buyer is type 1 with probability 0.5 or type 3 with probability 0.5.

The situation is represented in Table 1.

The Table also indicates in which cases the exchange would be efficient.

Clearly the exchange would be efficient when the buyer’s valuation is greater than seller’s valuation.

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Table 1

Buyer’s valuation

Probability of Seller’s type1 3

Seller’s valuation

0 Exchange efficient

Exchange efficient

0.5

2 Exchange not efficient

Exchange efficient

0.5

Probability of Buyer’s type0.5 0.5

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We will consider a class of procedures where:

1) The buyer and the seller simultaneously announce the value (not necessarily the true one) they assign to the item; we call S and B respectively the seller’s and the buyer’s announcements.

2) If B ≥ S the exchange takes place at some price in the interval [S, B], otherwise the exchange does not take place.

If both the seller and the buyer announce their true valuations, such a procedure guarantees that all the mutually beneficial exchanges take place and all those that are not beneficial do not take place. Truth-telling leads to a fully efficient outcome.

Note that under the truth-telling hypothesis, the proportion of encounters that end with an efficient transaction is 0.75.

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However, is truth-telling a sensible assumption?

Probably it is as far as the “strong” sellers and buyers are concerned: they have nothing to gain from pretending they are “weak”.

But let’s consider a weak seller. She gives value 0 to the item she owns. At end she is willing to sell it for any price ≥ 0, but she might be temped to declare S = 2 just to get a better price. Then, if she does so and meets a strong buyer who truthfully announce B = 1, the exchange will not take place although in principle it would be efficient to exchange the item.

Consider also a weak buyer. She gives value 3 to the item, but she might be tempted to announce 1 in order to pay less. If she does so and meets a strong seller, again a potentially efficient transaction will not take place.

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Can we design a procedure (a mechanism) such that the buyer and the seller are induced to tell the truth?

We will exemplify such a design problem with the help of Table 3.

For example, if the seller announces 0 and the buyer announces 3, then the exchange takes place with the seller selling the item to the buyer for a price = p. If the seller announces 2 and the buyer announces 3, the item will be exchanged at a price equal to 3 – y. And so on. The Mechanism-design problem will consists in specifying y and p such that telling the truth is a Nash-equilibrium (of course this is a more modest – but simpler – aim as compared with obtaining a truth-telling equilibrium with dominant strategies).

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Table 3

Buyer’s valuation

Probability of Seller’s type1 3

Seller’s valuation

0 0 + y p 0.5

2 * 3 - y 0.5

Probability of Buyer’s type0.5 0.5

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Participation constraints.

Assuming 0 ≤ p ≤ 3, we have the following:

A weak seller will require y ≥ 0.

A strong seller will require y ≤ 1.

A weak buyer will require y ≥ 0.

A strong buyer will require y ≤ 1.

We conclude that the participation constraints require:

0 ≤ y ≤ 1

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Incentive-Compatibility ConstraintsAs before, we only need to care about weak players. We want to set y and p such that the expected gain from telling the truth (assuming that the other party tells the truth) is at least as large the expected gain from lying.

Weak seller (0):(0.5)y + (0.5)p ≥ (0.5)(0) + (0.5)(3 – y)

or p ≥ 3 – 2y.

Weak buyer (3):(0.5)(3 – p) + (0.5)(3 – 3 + y) ≥ (0.5)(3 – y) + (0.5)(0) or p ≤ 2y.

Therefore: 3 – 2y ≤ p ≤ 2y.

This in turn requires 3 – 2y ≤ 2y, that is y ≥ 3/4.

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As long as we set y ≥ 3/4, we can find a value feasible value of p.

For example, if we put y = 1, then any p such that 1 ≤ p ≤ 2 will do the job, i.e:

any buyer and any seller will accept to play the game and they will find it optimal to tell the truth if they expect the other party will also tell the truth; as a consequence, all the potentially efficient transactions will take place.

However this is just an example.

Is it always possible to design such a mechanism? Unfortunately the answer is no.

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Myerson-Satterthwaite’s Theorem

In a 1983 paper the two authors prove that in general is impossible to find a mechanism that leads to full efficiency under all circumstances and results in a balanced budget (i.e. payment done by the buyer = payment received by the seller).

As “mechanism designers” we would like to find a procedure that always work and is not dependent, say, on the proportion of weak and strong players or on the players’ valuations.

Unfortunately, the above theorem proves that there is no such thing. We might be lucky enough to find a mechanism that does the job under some circumstances, but not all circumstances.

It remains true, however, that there are mechanisms that works better than others.

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The Vickrey Mechanism

The Vickrey’ auction procedure has been generalized to many other contexts (including negotiations) by Varian & MacKie Mason (1994).

This more general version is also called the Generalized Vickrey Auction (GVA). It is also related to the Clarke tax studied in relation to the eliciting of the willingness-to-pay for a public good.

In what follows we will see a very simple example of its application to a negotiation.

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The GVA works as follows:

1) The buyer and the seller simultaneously announce the value they assign to the item; we call S and B respectively the seller’s and the buyer’s announced values.

2) If (and only if) B ≥ S the exchange takes place. The buyer pays S and the seller receives B.

The procedure has a very nice property and also a bad one. The nice property is that it leads to truth-telling as a dominant-strategy equilibrium. The bad property is that if the exchange take place there is a deficit = B – S.

First we try to understand the nice property.Then we see how bad is the bad property and how it could be taken care of.

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GVA leads to truth-telling dominant strategy equilibrium

Let’s consider the buyer.

If vB ≥ S, her expected gain is (vB – S)Prob(B ≥ S).

Clearly she can maximize that by telling the truth, i.e. announcing B = vB, so that Prob(B ≥ S)= 1.

If vB < S, her expected gain is (vB – S)Prob(B ≥ S).

Notice that in this case vB – S < 0.

So she can maximize the expected gain (i.e. making it = 0) by announcing B= vB, so that Prob(B ≥ S) = 0.

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Let’s turn now to the seller.

If B ≥ vS, her expected gain is (B – vS)Prob(B ≥ S).

Clearly she can maximize it by telling the truth, i.e. by announcing S = vS, so that Prob(B ≥ S)= 1.

If B < vB, her expected gain is (B –vS)Prob(B ≥ S).

Notice that in this case B – vS < 0.

So she can maximize the expected gain (i.e. making it = 0) by setting S= vS, so that Prob(B ≥ S) = 0.

We conclude that telling the truth is the best choice both for the buyer and the seller, whatever the other party does. Telling the truth is the dominant strategy.

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GVA leads to a budget deficitSuppose a seller and a buyer meet, with vB > vS.The exchange is potentially efficient.Under the GVA the seller sells her item and receives a payment B, so her net benefit is B – vS. The buyer gets the item and pays S, so her net gain is vB – S.However, under GVA they both will tell the truth, i.e. B = vB and S = vS , so that the net gains are: Seller: vB - vS

Buyer: vB - vS .The buyer pays vS but the seller receives vB. But since vB > vS there is a budget deficit = vB - vS.Suppose the buyer and the seller are asked in advance to pay a fee to enter the negotiation, fee = (vB - vS)/2. Then at the end the net gain for both of them is = (vB - vS)/2 and the budget is balanced.The problem here is that in general the fee must be set in advance and it cannot depend on players’ announced values (otherwise the players will strategically take it into account). So the fee might turn out to be too low (and leave a deficit) or too high (and maybe discourage participation in advance, thus loosing some potential efficiency). Some researchers in the area are studying iterative procedures to set an appropriate fee. Alternatively, if the interest in the efficiency of the transactions is sufficiently large and diffuse, the deficit might be covered by general taxes.

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GVA with fees

Various ideas have been explored in order to overcome the problem of budget deficit with the GVA mechanism.

An idea considers entry fees.

Suppose the buyer and the seller are asked in advance to pay a fee to enter the negotiation, fee = (vB - vS)/2. Then at the end the net gain for both of them is = (vB - vS)/2 and the budget is balanced.

The problem here is that in general the fee must be set in advance and it cannot depend on players’ announced values (otherwise the players will strategically take it into account). So the fee might turn out to be too low (and leave a deficit) or too high (and maybe discourage participation in advance, thus loosing some potential efficiency). Some researchers in the area are studying iterative procedures to set an appropriate fee. Alternatively, if the interest in the efficiency of the transactions is sufficiently large and diffuse, the deficit might be covered by general taxes.

A paper by Anderson et al. (1999) studies systematically the issue of fees in the context of negotiation procedures.

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References

Anderson, Birgean and MacKie-Mason (1999): Bilateral Negotiation With Fees. Available from http://deepblue.lib.umich.edu/handle/2027.42/50451

Hurwicz (1973): The Design of Mechanisms for Resources Allocations, American Economic Review, 63, pp. 1-30.

MacKie-Mason and Varian, (1994), Generalized Vickrey Auctions. Ann Arbor, MI, Dept. of Economics, University of Michigan. Available from http://www-personal.umich.edu/~jmm/papers/gva3.pdf.

Myerson and Satterthwaite (1083): Efficient Mechanisms for Bilateral Trade, Journal of Economic Theory, 29, pp. 265-81.

Vickrey (1961): Counterspeculation, auctions, and competitive sealed tenders, Journal of Finance, 16, pp. 8-37.