Chapter 20Violating Equivalence

1.Preferences for Risk2.Asymmetric Valuations3.Relaxing Independence4.Differential Information5.Collusion and Entry

1. Preferences for Risk

We now relax the conditions for revenue equivalence to apply, seeking to show the effects on bidding behavior and auction revenue. First we investigate what happen if bidders are not risk neutral.

When does revenue equivalence fail?

Bidders might be risk averse or risk loving rather than risk neutral.

The private valuations of bidders might be drawn from probability distributions that are not identical.

The theorem does not apply when bidders receive signals about the value of the object to them that are correlated with each other.

If bidders are differentially informed, the format of the auction also matters.

Attitudes towards risk in second price sealed bid auctions with private values

It remains a weakly dominant strategy for each player to bid his or her valuation.

The optimal bidding strategy for the second price sealed bid auction (and also the Japanese and English auctions) is independent of a bidder's attitude towards risk and uncertainty when private values are drawn from a common probability distribution.

Attitudes towards risk in first price sealed bid auctions with private values

Bidding your valuation guarantees exactly zero surplus.

If you place a lower bid than your valuation your expected surplus initially increases until it reaches the maximum for a risk neutral bidder, and then falls, but the variance of the surplus increases as well.

A risk averse gambler is willing to trade a lower expected value to reduce the amount of uncertainty, he accordingly bids higher than a risk neutral bidder.

Comparing first and second price sealed bid auctions

Revenue generated by a second price auction is independent of the bidders' preferences over uncertainty, since bidding is unaffected.

The revenue generated by the first price auction is the same as the revenue generated by a second price auction when bidders are risk neutral.

Therefore risk averse bidders generate more revenue in a first price auction than they would in a second price auction, and they generate more revenue in a first price auction than do risk neutral bidders.

Theorem on optimal bidding players with constant relative risk aversion

Suppose there are two bidders with constant relative risk aversion. In a first price auction the utility the nth bidder obtains from winning is

We show the optimal bidding function is

where

a probability distribution function that is first order stochastically dominated by F(v).

vn b

F v Fv1/

bvn vn 1F vn

v

vn F vdv

Corollary: risk averse players make higher bids in first price auctions Since for all v < vn and it follows

that

and vice versa if the bidder is risk loving. Thus risk averse players “buy insurance” against losing the auction.

b risk aversevn vn v

vn F vF vn

dv

vn v

vn FvFvn

1/dv

vn 1Fvn

v

vn FvFvn

1/ 1

Fvdv

vn 1Fvn

v

vnFvdv

b risk neutralvn

Fv Fvn 1

Proof of theorem

Let b(v) denote the optimal bidding function as a mapping from a player’s valuation and consider the optimization problem

which is solved by setting v = vn. The first order condition for this problem is

or

a probability distribution function that is first order stochastically dominated by F(v).

vv , v max Fvvn bv

F vvn bv Fvvn bv 1b v 0

F vvn bv Fvb v 0

Concluding the proof

We rewrite the line on the previous slide as:

Rescaling:

By definition this is:

Or, upon integrating from v to vn we obtain the desired result using the facts that

F ww F

wbw F wb w

1Fw1/ 1F ww 1Fw1/ 1F wbw Fw1/ b w

F ww F wbw Fwb w

v

vn F wwdw vnF vn

v

vn F wwdw

v

vnF wbw F wb wdw F vn bvn

2. Asymmetric Valuations

What happens if the private valuations of bidders are not drawn from the same probability distribution function?

Asymmetric valuations

In a private valuation auctions suppose the bidders have different uses for the auctioned object, and this fact is common knowledge to every bidder.

Each bidder knows the probability distributions from which the other valuations are drawn, and uses that information when making her bid.

In that case the revenue equivalence theorem is not valid, and the auctioneer's prefers some types of auctions over others.

An example of asymmetry

Instead of assuming all bidders appear the same to the seller and to each other, suppose that bidders fall into two recognizably different classes.

Instead of there being a single distribution F(v) from which the bidders draw their valuations, there are two cumulative distributions, F₁(v) and F₂(v) with probability p₁ and p₂ respectively. Thus

F(v) = p₁F₁(v) + p₂F₂(v)

Bidders of type i ∈ {1,2} draw their valuations independently from the distribution Fi(v).

A conceptual experiment

Suppose each bidder sees his valuation, but does not immediately learn whether he comes from the high or low probability distribution. She only know v comes from F(v).

At that point the bidding strategy cannot depend on which probability distribution the valuation comes from. She forms hew bid b(v).

Then each bidder is told what type of person they are, type i ∈ {1,2}, that is the probability distribution their valuations were drawn from F₁(v) and F₂(v).

How should he revise his bid?

Intuition

Suppose F₁(v) and F2(v) have the same support, but F1(v) first order stochastically dominates F2(v), that is F₁(v) 6F2(v) .

When a bidder learns that his valuation is drawn from F2(v) (respectively F₁(v)), he deduces the other one is more likely to draw a higher (lower) valuation than himself, realizes the probability of winning falls (rises), so adjusts his bid upwards (downwards).

The intuition is in first price auctions to bid aggressively from weakness and vice versa.

3. Relaxing Independence

The revenue equivalence theorem applies to situations where the valuation of each is bidder is independently distributed. This is not always a valid assumption, because how a bidder values the object on the auction block for his own use might depend on information that another bidder has.

An example: Value of the object not known to bidders

Consider a new oil field tract that drillers bid for after conducting seismic their individual explorations.

The value of the oil field is the same to each bidder, but unknown. The nth bidder receives a signal sn which is distributed about the common value v, where

sn = v + n

and n E[v| sn] – v is independently distributed across bidders.

Notice that each drilling company would have more precise estimates of the common valuation from reviewing the geological survey results of their rivals.

The expected value of the item upon winning the auction

If the nth bidder wins the auction, he realizes his signal exceeded the signals of everybody else, that is

sn ≡ max{s₁,…,sN}

so he should condition the expected value of the item on this new information.

His expected value is now the expected value of vn conditional upon observing the maximum signal:

E[vn| sn ≡ max{s₁,…,sN}]

This is the value that the bidder should use in the auction, because he should recognize that unless his signal is the maximum he will receive a payoff of zero.

The Winner’s Curse

Conditional on the signal, but before the bidding starts, the expectation of the common value is

We define the winner’s curse as

Although bidders should take the winner's curse into account, there is widespread evidence that novice bidders do not take this extra information into account when placing a bid.

Ev|sn Esn n |sn sn maxs1, , sN

maxs1, , sN Ev|sn

Symmetric valuations We relax independence and consider the class of

symmetric valuations, which have two defining features:

1. All bidders have the same utility function.

2. Each bidder only cares about the collection of signals received by the other bidders, not who

received them.

Thus we may write the valuation of bidder n as:

uns1 , , sn 1 , sn , sn 1 , sN usn , s1 , , sn 1 , sn 1 , , sN

usn , sN, , sn 1 , sn 1 , , s1

Revenue Comparisons forSymmetric Auctions

We can rank the expected revenue generated in symmetric equilibrium for auctions where valuations are also symmetric.

There are two basic results. In a symmetric auction:

1. The expected revenue from a Japanese auction is higher than what an English auction yields.

2. The expected revenue from an English auction exceeds a first price sealed bid auction.

4. Differential Information

We have discussed several types of information structures in auctions, but one important case we have not touched yet, is when some bidders know more about the common value of the object than other bidders do.

Bidding with differential information

An extreme form of dependent signals occurs when one bidder know the signal and the others do not.

How should an informed player bid?

What about an uninformed player?

Second price sealed bid auctions

The arguments we have given in previous lectures imply the informed player optimally bids the true value.

The uninformed player bids any pure or mixed distribution. If he wins the auction he pays the common value, if he loses he pays nothing, and therefore makes neither gains or losses on any bid.

This implies the revenue from the auction is indeterminate.

Perspective of the less informed bidder in a first price auction

Suppose the uninformed bidder always makes the same positive bid, denoted b. This is an example of a pure strategy.

Is this pure strategy part of a Nash equilibrium?

The best response of the informed bidder is to bid a little more than b when the value of the object v is worth more than b, and less than b otherwise.

Therefore the uninformed bidder makes an expected loss by playing a pure strategy in this auction. A better strategy would be to bid nothing.

A theorem on first price sealed bid auctions

The argument in the previous slide shows that the uninformed bidder plays a mixed strategy in this game.

One can show that when the auctioned item is worth v the informed bidder bids:

(v) = E[V|V v]

in equilibrium, and that the uninformed bidder chooses a bid at random from the interval [0, E[V]] according to the probability distribution H defined by

H(b) = Prob[(v) b]

Return to the uninformed bidder

If the uninformed player bids more than E[V], then his expected return is negative, since he would win the auction every time v < E[V] but less frequently when v > E[V].

We now show that if his bid b < E[V], his expected return is zero, and therefore any bid b < E[V] is a best response to the informed player’s bid.

If the uniformed bids less than E[V] and loses the auction, his return is zero. If he bids less than E[V] and wins the auction, his return is

E[V| (V) < b] – b = E[V| V < -1(b) ] – b = (-1(b) ) – b = 0

Return to the informed bidder

Since the uninformed player bids less than E[v] with unit probability, so does the informed player.

Noting that (w) varies from v to E[v], we prove it is better to bid (v) rather than (w). Given a valuation of v, the expected net benefit from bidding (w) is:

H((w))[v - (w)] = Pr{V w}[v - (w)] = F(w)[v - (w)]

Differentiating with respect to w, using derivations found on the next slide, yields F’(w)[v - w] which is positive for all v > w and negative for all v < w, and zero at v = w. Therefore bidding (v) is optimal for the informed bidder with valuation v.

The derivative

Noting

it follows from the fundamental theorem of calculus that

and so the derivative of F(w)[v - (w)] with respect to w is :

Fw w FwEV|V w v

wtF tdt

ddw

Fw w Fww

F wv w Fw w F wv ddw

Fw w

F wv F ww

5. Collusion and Entry

Collusion and entry deterrence are also considerations that auctioneers should account for. The auctioneer also affects the number of bidders by promoting the auction.

The field of bidders

The last part of study on auctions discusses two issues outside the bidding process itself that nevertheless may affect the outcome of the auction.

They are:1. Collusion amongst bidders2. Determining the number of bidders

The degree of collusion and entry deterrence may affect how the auctioneer and the bidders rank different types of auctions that are revenue equivalent, or even strategically equivalent.

Collusion

Collusion between bidders is only possible if there is a mechanism for determining within a designated bidding ring which bidder has the highest valuation, and then ensuring members of the ring do not break the collusive agreement in the bidding.

We focus on the second point. Sometimes determining which member has the highest valuation is trivial (for example in common value auctions), and sometimes it can be resolved through an auction within the ring for the right to present the only serious bid to the auctioneer.

Evidence about collusion in auctions

Section 1 of the Sherman Act pertains to trusts and illegal constraints to trade.

Over three quarters of the criminal cases filed in the 1980s under its provisions were in auctions markets.

Given the difficulty in finding sufficient grounds to prosecute this illegal activity, one can safely conclude that collusive behavior in auctions is often a serious concern for the seller.

What are the gains from collusive behavior?

Consider a bidding ring of R members out of a total of N bidders.

The goal of the ring is to internally solve which of the players have the highest valuation, and then make one (serious) bid instead of R bids.

With fewer effective bidders in the auction, the expected price of the object is lower.

If all the bidders are able to collude, meaning N = R, the auction reduces to a game of bilateral bargaining between the auctioneer and the single bidder.

Bidding rings in second price sealed bid auctions

For example, suppose the player with the highest valuation in the ring bids it, the other bidders in the ring submit the auctioneer’s reservation price, and the bidders outside the ring respond optimally but individually by bidding their true valuations.

Then the ring benefits from colluding if:

1. The high bid from the ring wins the auction

2. The second highest valuation of bidders in the ring exceeds the highest valuation of all the bidders outside the ring.

Enforcing collusion in first and second price sealed bid auctions

One reason why second price auctions are rarely used in practice is because they are more susceptible to collusion.

In a first price sealed bid auction, a member of the bidding ring must submit a bid over the ring’s low price to win the auction. Providing his valuation exceeds the ring’s bid, there is an opportunity to make profits by deviating from the ring’s decision. This makes collusive bidding harder to enforce.

Contrast this with a second price auction. In order to win the auction a bidder in the ring who breaks the collusive agreement must pay the reported value of the ring. The agreement by the ring is self enforcing!

Ascending auctions encourage communication amongst the bidders

A key issue for firms attempting to collude is agreeing how to share the spoils.

In a multiunit ascending auction, bidders can use the early stages when prices are still low to signal their views about who should win which object, and then when consensus has been reached, tacitly agree to stop pushing prices up.

By contrast, bidders cannot easily achieve the same coordination in simultaneous sealed-bid or descending auctions, in which each player simultaneously makes at most a single “best and final” offer to each object.

Low reservation prices encourage collusion

Reducing the reserve price increase the potential gains from joint-bidding or colluding, because the gains from colluding are greater.

Therefore the auctioneer should set a reservation price that is higher than the opportunity cost of failing to sell the auctioned item if it reduces the probability that bidders will collude.

Entry and the provision of information about the auctioned object

Should the auctioneer encourage more players to enter the auction?

Potential bidders could, for example, be encouraged to bid in an auction, by providing them with services that help them to value the object for sale.

Note that simply paying people to participate in the auction would not achieve any useful purpose, because anyone could accept the payment and then make a very low bid.

Encouraging entry in private valuation auctions

If another bidder enters a private valuation auction, the the level of the highest or the second highest valuation might increase.

In either case, the revenue from the auction would increase.

Therefore the expected revenue from holding a private valuation auction increases with the number of bidders.

Encouraging entry to avoid collusion

The lower the number of bidders, the easier it is for them to reach a collusive agreement.

Thus another reason to encourage entry is to reduce the probability for facing a bidding cartel.

Summary and synthesisWhen comparing different auction mechanisms, we should consider the following questions:

1. Are the auctions strategically equivalent?2. Are the auctions revenue equivalent?3. Are the bidders risk neutral?4. Are bidders with private valuations drawing from

the same probability distribution? 5. Are the valuations of bidders affiliated with each

other? 6. Do some bidders have more information than

others?7. Is collusion between bidders likely?8. Should information about the auction be given to

bidders?