Chapter 19Equivalence

1.Types of Auctions2.Strategic Equivalence3.Revenue Equivalence4.Optimal Bidding

1. Types of Auctions

To begin this chapter we describe the main kinds of auctions, and pose some questions about auctions of interest to business men and women.

Why study auctions?

1. Studying auctions is the simplest way of approaching the question of price formation.

2. Auctions serve the dual purpose of eliciting preferences and allocating resources between competing uses.

3. A less fundamental but more practical reason for studying auctions is that the value of goods exchanged each year by auction is huge.

Auction mechanisms

There are 5 standard types of auctions for auctioning a single item which are widely used and analyzed:

1. First-price sealed-bid2. Second-price sealed-bid3. English4. Japanese 5. Dutch

as well as several other types we will investigate.

Sealed bid auctions

Each bidder in a sealed bid auction submits a price or bid to the auctioneer simultaneously.

The highest bidder receives the auctioned item.

Sealed bid auctions only differ in how much bidders pay.

We investigate three variations, first price, second price, and all pay.

First price sealed bid auction

In a first price sealed bid auction, the highest bidder pays the amount she bid in exchange for the object up for auction

Suppose there are N bidders. Let bn denote the bid by the nth player and let vn denote how much she values the auctioned item. Also rank the bid from the highest to the lowest as b(1) through b(N). In a first price auction, un, the net payoff to the nth player is defined as:

The auctioneer receives:u0 b 1

un vn b 1 if bn b 1

0 if bn b 1

Second price sealed bid auctionEach bidder in a second price sealed bid auction submits a price to the auctioneer simultaneously. The bidder submitting the highest price pays the second highest price submitted. The other bidders neither pay nor receive anything.

Following the same notation as in the first price sealed bid auction the net payoff to bidder n is:

The auctioneer receives:

un vn b 2 if bn b 1

0 if bn b 1

u0 b 2

All-pay sealed bid auctions

In an all-pay sealed bid auction, each bidder pays what she bids, and the highest bidder wins the auction.

The net payoff to the nth bidder is defined as:

The auctioneer receives:

un vn b 1 if bn b 1

bn if bn b 1

u0 n 1N bn

Examples of all-pay auctions

More generally an all-pay auction is a paradigm for modeling competitions of various kinds, not a common institution for literally conducting auctions.

For example supply contracts are like all-pay auctions. Bidders expend considerable resources preparing a proposal, but only one bidder is awarded the contract.

Similarly research teams in the same field use resources competing with each other, but the first team to make a discovery benefits disproportionately in the rewards from their discovery through patenting, first mover advantages, and so on.

Comparing the revenue from sealed bid auctions

Notice that:

So if all bidders adopted the same bidding strategy for the three auctions, then the second price sealed bid auction would yield least revenue of the three, and the all pay auction would yield the most.

But would a potential buyer bid more if the winner pays less than their own bid? And would she be bid as much if she had to pay her bid regardless of whether hers is the winning bid or not?

On reflection it is unclear which of the three auctions yields more revenue to the auctioneer!

n 1N bn b 1 b 2

Descending auctions (Dutch auctions)

The auctioneer begins by offering the item at a very high price which he confidently believes exceeds the willingness to pay of any bidder.

Then he continuously lowers it until one bidder announces that she is willing to pay the current price.

At that point the auction ends, the bidder buying the item at the lowest price offered.

Payoffs in descending auctionsTo formally describe the payoffs from this game, let at denote the auctioneer's ask price in period t.

Denote by tn the time at which the nth bidder accepts the bid if no other bidder has submitted an order by then, meaning the auction has not ended yet.

Analogous to our ranking of bids in sealed bid auctions, let t(k) denote the kth earliest, which implies t(¹) ≤ t(²) ≤ … ≤ t(N).

The player's payoffs can be then defined as:un

vn a t 1 if tn t1

0 if tn t1

Ascending auctionsIn ascending auctions, the auctioneer raises the price as long as more than one person is willing to pay the current price. The winner pays the lowest price at which every other bidder has dropped out of the auction.

Let rt denote the auctioneer's request price in period t, and now let tn denote the time at which the nth bidder will drop out of the auction. Also let t(k) denote the kth earliest time, which implies t(¹)

≤ t(²) ≤ … ≤ t(N).

The net payoff to the nth bidder is then:un vn rt N 1 if tn tN

0 if tn tN

English auction

We will study two types of ascending auctions, English auctions and Japanese auctions. The feature differentiating these two auctions is how much the bidders observe as the auction proceeds.

In an English auction bidders compete against each other by successively raising the price at which they are willing to pay for the auctioned object.

The bidding stops when nobody is willing to raise the price any further, and the item is sold to the person who has bid the highest price, at that price.

How much do bidders observe in English auctions?

During the auction a bidder might be able to observe a sample of bidders who make bids, and thus update his beliefs about the value of the item as the auction progresses.

The most restrictive assumption is that the bidders do not observe the identity of the other people making bids, and that to win the auction, a bidder must continuously indicate his willingness to pay successively higher prices.

This simplification implies that as the auction progresses, a bidder willing to pay for the auctioned item at the current quote knows only that at least one other bidder has also signaled.

Japanese auction

Everyone willing to pay the current price for the auctioned indicates this to the auctioneer. Those who are not willing to pay the current price rt must leave the auction and cannot reenter.

The auctioneer raises the price until the second last bidder drops out of contention, and the winner is assigned the item at that price.

In contrast to an English auction, every bidder sees which which bidders have dropped out of the auction as the auctioneer raises the bid price.

2. Strategic Equivalence

By definition the strategic form solutions to strategically equivalent auctions are the same. This section provides provides several examples of strategically equivalent auctions.

Strategic equivalence

The introduction showed there are many ways of auctioning an item to interested buyers. However many auctions are closely related to each other.

Recall that a strategy is a complete description of instructions to be played throughout the game, and that the strategic form of a game is the set of alternative strategies to each player and their corresponding expected payoffs from following them. Two games are strategically equivalent if they share the same strategic form.

In strategically equivalent auctions, the set of bidding strategies that each potential bidders receive, and the mapping to the bidder’s payoffs, are the same.

Descending auctions are strategically equivalent to first-price auctions

During the course of a descending auction no information is received by bidders.

Each bidder sets his reservation price before the auction, and submits a market order to buy if and when the limit auctioneer's limit order to sell falls to that point.

Dutch auctions and first price sealed bid auctions share strategic form, and hence yield the same realized payoffs if the initial valuation draws are the same.

Rule 1: Pick the same reservation price in Dutch auction that you would submit in a first price auction

Second-price versus ascending auctions

When there are only 2 bidders, the two ascending auction mechanisms (English and Japanese) are strategically equivalent to the second price sealed bid auction (because no information is received during the auction).

All three auctions are strategically equivalent are (almost) strategically equivalent if all the players have independently distributed valuations (because the information conveyed by the other bidders has no effect on a bidder’s valuation).

In common value auctions the 3 mechanisms are not strategically equivalent if there are more than 2 players.

Rule 2: If there are only two bidders, or if valuations are independently distributed, choose the same reservation price in English, Japanese and second price auctions.

Summary

Rule 1: Pick the same reservation price in a Dutch auction that you would submit in a first price sealed bid auction.

Rule 2: In private value auctions, or if there are only two bidders, choose the same reservation price for an English or a Japanese auction that you would submit in a second price sealed bid auction.

3. Revenue Equivalence

Revenue equivalent auctions generate the same expected revenue. Thus strategic equivalence implies revenue equivalence, but not vice versa. This section explores sufficient conditions for auctions to be revenue equivalent.

Relaxing strategic equivalence

In strategically equivalent auctions, the strategic form solution strategies of the bidders, and the payoffs to all the players are identical.

This is a very strong form of equivalence. Can we show that such players might be indifferent to certain auctions which lack strategic equivalence?

Revenue equivalence defined

The concept of revenue equivalence provides a useful tool for exploring this question.

Two auction mechanisms are revenue equivalent if, given a set of players their valuations, and their information sets, the expected surplus to each bidder and the expected revenue to the auctioneer is the same.

Revenue equivalence is a less stringent condition than strategic equivalence.

Thus two strategic equivalent auctions are invariably revenue equivalent, but not all revenue equivalent auctions are strategic equivalent .

Why study revenue equivalence ?

If the auctioneer and the bidders are risk neutral, studying revenue equivalence yields conditions under which the players are indifferent between auctions that are not strategically equivalent.

Exploiting the principle of revenue equivalence can sometimes give bidders a straightforward way of deriving their solution bid strategies.

Preferences and Expected Payoffs

Let:U(vn) denote the expected value of the nth

bidder with valuation vn bidding according to his equilibrium strategy when everyone else does too.

P(vn) denote the probability the nth bidder will win the auction when all players bid according to their equilibrium strategy.

C(vn) denote the expected costs (including any fees to enter the auction, and payments in the case of submitting a winning bid).

An Additivity Assumption

We suppose preferences are additive, symmetric and private, meaning:

U(v) = P(v) v - C(v)

So the expected value of participating in the auction is additive in the expected benefits of winning the auction and the expected costs incurred.

A revealed preference argument Suppose the valuation of n is vn and the valuation of

j is vj.

The surplus from n bidding as if his valuation is vj is U(vj), the value from participating if his valuation is vj, plus the difference in how he values the expected winnings compared to a bidder with valuation vj, or (vn – vj)P(vn).

In equilibrium the value of n following his solution strategy is at least as profitable as deviating from it by pretending his valuation is vj. Therefore:

U(vn) > U(vj) + (vn – vj)P(vj)

Revealed preference continued

For convenience, we rewrite the last slide on the previous page as:

U(vn) - U(vj) > (vn – vj)P(vj)

Now viewing the problem from the jth bidder’s perspective we see that by symmetry:

U(vj) > U(vn) + (vj – vn)P(vn)

which can be expressed as:

(vn– vj)P(vn) > U(vn) - U(vj)

A fundamental equality

Putting the two inequalities together, we obtain:

(vn – vj) P(vn)> U(vn) - U(vj) > (vn – vj) P(vj)

Writing:vn = vj + dv

yields

which, upon integration, yields

dvvPvdU

nv

vvn dvvPvUvU

Revenue equivalence

This equality shows that in private value auctions, the expected surplus to each bidder does not depend on the auction mechanism itself providing two conditions are satisfied:

1. In equilibrium the auction rules award the bid to the bidder with highest valuation.

2. The expected value to the lowest possible valuation is the same (for example zero).

Note that if all the bidders obtain the same expected surplus, the auctioneer must obtain the same expected revenue.

A theorem Assume each bidder:

- is a risk-neutral demander for the auctioned object;

- draws a valuation independently from a common, strictly increasing probability distribution function.

Consider auction mechanisms where - the buyer with the highest valuation always

wins- the bidder with the lowest feasible signal

expects zero surplus.

Then the same expected revenue is generated by the auctions, and each bidder makes the same expected payment as a function of her valuation.

4. Optimal Bidding

We apply the revenue equivalence theorem to solve for the optimal bidding rules for several types of private value auctions.

Steps for deriving expected revenue

The expected revenue from any auction satisfying the conditions of the theorem, is the expected value of the second highest bidder.

To obtain this quantity, we proceed in two steps:

1. derive the probability distribution of the second highest valuation,

2. obtain its density and integrate to find the mean.

Second price sealed bids In a sealed bid auction, the strategy of each

player n is to submit a bid, which we label by bn.

In a second price sealed bid auction, when a bidder knows his own valuation, there is a very general result available about how he should bid, which does not depend at all on what he knows about the valuations of the other players, or what they know about their own valuations.

It is a weakly dominant strategy to bid his valuation vn.

An immediate corollary of this general result is that if every bidder knows his own valuation, the unique solution to the game is for each bidder n to submit his or her true valuation. We establish this claim by showing that bidding vn weakly dominates bidding above or below it.

Bidding in a second-price auction

Bidding your own valuation is a weakly dominant strategy in second price sealed bid auctions.

The logic supporting this result, weak dominance, extends beyond second price auctions with perfect foresight to any auction where a bidder knows her own valuation, that is regardless of the information available to the other bidders, and regardless of how they bid.

This important result also applies to ascending auctions.

Rule 3 : In a second price sealed bid auction, bid your valuation if you know it.

Proving the third rule Suppose you bid above your valuation, win the

auction, and the second highest bid also exceeds your valuation. In this case you make a loss. If you had bid your valuation then you would not have won the auction in this case. In every other case your winnings are identical. Therefore bidding your valuation weakly dominates bidding above it.

Suppose you bid below your valuation, and the winning bidder places a bid between your bid and your valuation. If you had bid your valuation, you would have won the auction and profited. In every other case your winnings are identical. Therefore bidding your valuation weakly dominates bidding below it.

Combining the two parts of the proof, we conclude bidding your valuation is a weakly dominant strategy.

Probability distribution of the second highest valuation

Since any auction satisfying the conditions for the theorem can be used to calculate the expected revenue, we select the second price auction.

The probability that the second highest valuation is less than x is the sum of the the probabilities that:

1. all the valuations are less than x, or: F(x)N

2. N-1 valuations are less than x and the other one is greater than x. There are N ways of doing this so the probability is:

NF(x)N-1[1 - F(x)]

The probability distribution for the second highest valuation is therefore: NF(x)N-1 - (N - 1) F(x)N

Expected revenue from Private Value Auctions

The probability density function for the second highest valuation is therefore:

N(N –1)F(x)N-2 [1 - F(x)]F‘(x)

Therefore the expected revenue to the auctioneer, or the expected value of the second highest valuation is:

Using the revenue equivalence theorem to derive optimal bidding functions

We can also derive the solution bidding strategies for auctions that are revenue equivalent to the second price sealed bid auction.

Consider, for example a first price sealed bid auctions with independent and identically distributed valuations.

The revenue equivalence theorem implies that each bidder will bid the expected value of the next highest bidder conditional upon his valuation being the highest.

Bidding in a first price sealed bid auction

The truncated probability distribution for the next highest valuation when vn is the highest valuation is:

In a symmetric equilibrium to first price sealed bid auction, we can show that a bidder with valuation vn bids:

Comparison of bidding strategies

The bidding strategies in the first and second price auctions markedly differ.

In a second price auction bidders should submit their valuation regardless of the number of players bidding on the object.

In the first price auction bidders should shave their valuations, by an amount depending on the number of bidders.

The derivation The probability of the remaining valuations

being less than w when the highest valuation is v(1) is:

Therefore the probability density for the second highest valuation when vn = v(1) is:

This implies the expected value of the second highest valuation, conditional on vn = v(1) is:

Integrating by parts we obtain the bidding function

Prv2 w|v1 Fv1 1NFwN 1

bv Fvn 1N vFvN 1v v

v vn Fvn 1N v

vn FvN 1dv

vn Fvn 1N v

vn FvN 1dv

N 1Fvn 1N v

vn vFvN 2F wdv

N 1Fvn 1NFwN 2F w

An example: the uniform distribution

bvn vn Fvn 1N v

vn FvN 1dv

vn vn v 1N v

vnv v N 1dv

v /N vN 1/N

Fv v vv v

Suppose valuations are uniformly distributed within a closed interval, with probability distribution:

Then:

Bidding function for the uniform distribution

Thus in the case of the uniform distribution the equilibrium bid of the player with valuation v is to bid a weighted average of the lowest possible valuation and his own, where the weights are respectively 1/N and (N-1)/N:

bv v /N vN 1/N

All pay sealed bid auction with private values

The revenue equivalence theorem implies that the amount bidders expect to pay in an all-pay auction as in all other auctions satisfying the conditions of the theorem.

In contrast to a first or second price sealed bid auctions where only the winner bidder pays his bid or the second highest bid in an all pay auction losers also pays their bids.

The amount paid by the nth bidder is certain, and not paid with the probability of winning the auction, that is F(vn)N-1.

By the revenue equivalence theorem the amount each bidder expects to pay in the first two auctions, upon seeing their valuation, equals the amount the bidder actually does pay in all pay auction.

Bidding in all pay auction

The previous slide implies that in an all pay auction a bidder with vn bids the product of F(vn)N-1 and the amount he would bid in a first price auction.

This is:

The uniform distribution revisited

Fv v vv v

If valuations are uniformly distributed within in a closed interval, with probability distribution:

then:ball payvn Pr n wins auction b first pricevn

v vv v

N 1 vN

N 1Nv