csc304: algorithmic game theory and mechanism design fall 2016

CSC304: Algorithmic Game Theory andMechanism Design

Fall 2016

Allan Borodin (instructor)Tyrone Strangway and Young Wu (TAs)

October 17, 2016

1 / 18

Lecture 10

AnnouncementsI have posted the first two questions for Assignment 2 which is dueOctober 28. I expect to post some more questions during the week.

Todays agendaWe continue the discussion of auctions.

I We continue the discussion of the Vickery 2nd price mechanism forselling a single item.

I We compare the Vickery auction and the 1st price auction.I We will use today to give a very fast overview of some of the important

results in Chapter 14 of the KP text. This is not an easy chapter toread and we will try to come back to these results as we proceed withour discussion of auctions in this and other chapters.

2 / 18

The English (ascending price) auction

While the second price auction may seem a little strange at first, it is insome sense equivalent to the familar English ascending price auction.

In the English auction, the auctioneer starts with a price (i.e. a reserveprice) and then continues to ask who wants to raise the bid? The final(highest) bid wins.

Suppose the auctioneer always asks if there is anyone who wants to raisethe current bid by some small ε. Then this ascending price auction isessentially producing the same outcome (assuming buyers do not changetheir valuation given other bids) in terms of who wins the item and what ispaid. These are referred to as two different implementations of the sameoutcome.

But how do these two implementations differ?

The Dutch descending price auction (i.e. keep decreasing the price by asmall amount until one buyer accepts the price) is in the same way“equivalent” to a 1st price auction .

3 / 18

The Dutch descending price auction (i.e. keep decreasing the price by asmall amount until one buyer accepts the price) is in the same way“equivalent” to a 1st price auction . 3 / 18

Revenue equivalence

We should have seen (if my iphone 4 sale worked as it should have) thatthe buyers strategies are different for 1st price and 2nd price auctions.Namely, if you believed that the 2nd price auction was truthful, then youwould bid your true value whereas for a 1st price auction you should have“shaved” your bid as otherwise your utility is guaranteed to be 0 whetheryou win or lose the auction. We can discuss why it didn’t turn out as thetheory would imply. For the 2nd price (resp. 1st price) auction, theaverage bid/value ratio was .632 (resp. .721).

Which mechanism is best for me (the seller) in terms of my revenue?

Surpising fact: If all bidders are independently drawing their values fromsame uniform distribution U[0, h] then the expected revenue (inequilibrium) is the same for the first and second price auctions!

Even more surprising: This revenue equivalence holds more generally givensome technical (but intuitive) conditions.

4 / 18

Revenue equivalence

4 / 18

Revenue equivalence

4 / 18

The revenue equivalence for 1st and 2nd priceauction and the uniform distribution U[0, 1].

This was discussed in tutorial so lets just do this quickly and informally.

In a second price equilibrium, we can assume buyers bid truthfully andpay the second (highest) bid (= value). For the uniform distributionU[0, 1], we claim that the expectation of the kth lowest value is k

The expectation of the second highest value is then n−1n+1 . This implies

that the expected revenue is n−1n+1 .

For a first price auction, the buyer will bid (at equilibrium) a fractionn−1n of his value and the highest value has expectation n

n+1 so that

the expected bid (and price) for the winning buyer is n−1n

nn+1 = n−1

5 / 18

Revenue Equivalence Theorem

Here is a statement given in Vijay Krishna’s “Auction Theory” text. (Seealso, the informal statement in Wikipedia and a formal statement inTheorem 14.4.2).

The Revenue Equivalence Theorem for selling a single item

Suppose that values are independently and identically distributed and allbidders are risk neutral. Then any symmetric and increasing equilibrium ofany standard auction, such that the expected payment of a bidder withvalue zero is zero, yields the same expected revenue to the seller.

What do all the technical conditions mean?

6 / 18

Revenue Equivalence Theorem

Here is a statement given in Vijay Krishna’s “Auction Theory” text. (Seealso, the informal statement in Wikipedia and a formal statement inTheorem 14.4.2).

The Revenue Equivalence Theorem for selling a single item

Suppose that values are independently and identically distributed and allbidders are risk neutral. Then any symmetric and increasing equilibrium ofany standard auction, such that the expected payment of a bidder withvalue zero is zero, yields the same expected revenue to the seller.

What do all the technical conditions mean?

6 / 18

Revenue equivalence theorem continued

The theorem assumes all bidders are drawing their value from the samecontinuous distribution. It isn’t said explicitly but like the presentation inthe KP text, the proof uses the fact that the payment rule and theallocation probability are differentiable (as well as increasing) functions ofone’s value.

Since every buyer is drawing from the same distribution, it is reasonable toassume that in an equilibrium, all players will be playing the same strategyso that we are considering a symmetric equilbrium.

A standard auction is one that will always allocate the item to a playerwith the highest bid.

Intuitively, A risk neutral buyer is simply trying to maximize their utilitywithout taking into consideration either minimizing the probabiity of lowutility (i.e. a risk averse buyer) or maximizing the probabilty of high utility(i.e. a risk seeking buyer).

7 / 18

The revenue theorem continued

A buyer with value 0 for the item is unlikely want to offer any payment asthat would lead to negative utility. More generally, we just need somenormalizing condition; e.g. if the distribution has suppport [h1, h2] thenthe expected payment p(h1) = p′ for some fixed p′ independent of theparticular auction.

We emphasize that there is a wide range of possible auctions thatfit these conditions, including the 1st and 2nd (i.e. Vickery) priceauctions as well as say a third price auction, the all pay auction andmany others.

The all pay auction allocates to the highest bidder but charges all buyerstheir bid. As the text notes, this is an auction implicitly used when biddingfor contracts where there is a cost in making the bid.

We also emphasize that there is a wide range of possible valuationdistributions although to provide simple examples, one oftenconsiders uniform distributions U[h1, h2] and more specifically U[0, 1].

8 / 18

Sketch of proof (sections 14.4.1 and 14.4.2)

We assume that the bidders are drawing their values i.i.d from adistribution with strictly increasing cummulative distribution function F .We are considering a symmetric equilibrium for bidding β = βi for all i . Weneed to define the probability ai (v) that the buyer i wins the auction whenbidding β(vi ) and then paying pi (vi ) assuming that all the other players jare playing the same strategy β(vj). We will drop the subscript on theprobability a and payment p since all buyers are playing the same strategy.

The proof comes in two parts, where the first part (i) proves a necessarycondition for β to be an equilibrium and the resulting probability a(vi ) andpayment p(vi ). The second part verifies that this does indeeed yield anequilibrium.

9 / 18

Necessary condition for a symmetric equilibrium

Since we are assuming a standard auction where the highest bid wins, andF is strictly increasing, it is reasonable that β(v) is also strictly increasingand then any bidding strategy is equivalent to bidding β(w) for some wwhen given vi .

We have

a(w) = Prob[β(w) maxj 6=i

β(Vj)] = Prob[β(w) maxj 6=i

Vj ] = F n−1

which gives utility u(w |vi ) = via(w)− p(w)

For β(vi ) to be an equilibrium, the derivative via′(w)− p′(w) of the utility

must be 0 at w = vi . That is, p′(vi ) = via′(vi ) for all vi .

Using (for simplicity), the particular normalization p(0) = 0, andintegrating, we get p(vi ) =

∫ vi0 v · a′(v)dv

so that integrating parts, we get p(vi ) = via(vi )−∫ vi0 a(w)dw

10 / 18

We have

Vj ] = F n−1

10 / 18

We have

Vj ] = F n−1

10 / 18

We have

Vj ] = F n−1

10 / 18

Completing the proof of revenue equivalence

Note that since a(v) = F n−1, this probability only depends on thedistribution and does not depend on the auction as long as the winner isthe buyer with the highest valuation which will be the case since this is asymmetric auction and the highest bid corresponds to the highest value).Furthermore, since p(v) only depends on v and a(v), the price is also justa function of the distribution. Finally, the price p(v) for each buyerdetermines the expected revenue from each buyer which determines thetotal expected revenue. Hence we obtain revenue equivalence,

It only remains to verify that the bidding strategy β(v) is an equilibriumwhich is done in section 14.4.2. That is, the proof shows that the utility ofbidding β(v) given value v dominates bidding β(w) for any w 6= v .

11 / 18

Completing the proof of revenue equivalence

Note that since a(v) = F n−1, this probability only depends on thedistribution and does not depend on the auction as long as the winner isthe buyer with the highest valuation which will be the case since this is asymmetric auction and the highest bid corresponds to the highest value).Furthermore, since p(v) only depends on v and a(v), the price is also justa function of the distribution. Finally, the price p(v) for each buyerdetermines the expected revenue from each buyer which determines thetotal expected revenue. Hence we obtain revenue equivalence,

It only remains to verify that the bidding strategy β(v) is an equilibriumwhich is done in section 14.4.2. That is, the proof shows that the utility ofbidding β(v) given value v dominates bidding β(w) for any w 6= v .

11 / 18

Can the revenue equivalence theorem be extended?

For a one item auction when all buyers are drawing values from an i.i.ddistribution, the revenue equivalence theorem is quite general.

As the KP text says, it can be extended to deal with randomized pricingand reserve pricing.

And as the text also says, it can be extended to the case of k identicalitems (priced the same) and “unit demand” buyers.

It is natural to ask if the theorem can be extended beyond the continuousi.i.d case or for more than one item.

I believe (but have to verify) that it can be extended to i.i.d. discretedistributions although in the Economics literature, it seems that oneis always considering continuous distributions.

The theorem does not extend in general to the asymmetric case! Nordoes it extend to the case of muitiple items.

12 / 18

Can the revenue equivalence theorem be extended?

For a one item auction when all buyers are drawing values from an i.i.ddistribution, the revenue equivalence theorem is quite general.

As the KP text says, it can be extended to deal with randomized pricingand reserve pricing.

And as the text also says, it can be extended to the case of k identicalitems (priced the same) and “unit demand” buyers.

It is natural to ask if the theorem can be extended beyond the continuousi.i.d case or for more than one item.

I believe (but have to verify) that it can be extended to i.i.d. discretedistributions although in the Economics literature, it seems that oneis always considering continuous distributions.

The theorem does not extend in general to the asymmetric case! Nordoes it extend to the case of muitiple items.

12 / 18

An example of the asymmetric case

The following is an example (Krishna, Example 4.3) shows how theexpected revenue of a 1st price auction can exceed that of a 2nd priceauction.

Let the value of buyer 2 be drawn from U[0, 11−α ] while the value of buyer

1 is drawn from U[0, 11+α ] some 0 < α < 1.

Krishna shows that the equilbrium prices in a 2nd price auction is less thanif α = 0 (i.e. the distribution U[0, 1]) whereas for a 1st price auction theequibrium prices are higher than for the U[0, 1] distribution.

As the KP text explains, buyer 1 (with a smaller possible maximumvaluation) has to bid more agressively than bidder 2. This suggests thatthere will be valuations v1 < v2 such that β1(v1) > β2(v2) so that buyer 1wins the auction.

13 / 18

Bayes-Nash equlibrium and the price of anarchy

While revenue equivalence need not hold in asymmetric distributions, thereis a Bayes- Nash equilibrium and it is characterized in Theorem 14.6.1.

Aside: It is easy to misread Theorem 14.6.1 to think that it is suggestingrevenue equivalence.

Since a 2nd price auction insures truthfulness, we can assume that buyersbid their values and hence the auction is won by the buyer with thehighest value. This insures that the 2nd price auction is optimal withrespect to social welfare (i.e. price of anarchy POA = 1).

In a 1st price auction, the buyer with the highest value may not win theauction. So is there any bound on the POA for a 1st price auction?

Theorem: In a Bayes-Nash equilibrium for a 1st price auction, POA ≥ 1/2;that is, the winning bidder obtains at least 1/2 of the maximum valueamongst all the buyers.

14 / 18

Individual rationality

In auctions, we often assume that a buyer is individually rational (IR) anddoes not expect to lose money. That is, the expected utility E[vi − pi ] of abuyer is non negative. The expectation is taken over the distributions ofthe buyers and for randomized auctions, also over the randomness in theauction’s payment rule. (So far, our examples of auctions are alldeterministic.)

Given that the payment is an expected payment, there are three types ofIR that should be distinguished. In what follows we are still assumingcommon knowledge of everyones distributions and bidding strategies butnot assuming i.i.d or any other assumptions about the distributions.Following the notationn in KP, we let bi (vi ) = β(vi ) be the bid when thebuyer knows his value vi and has common knowledge about the otherbuyers.

15 / 18

Individual rationality

In auctions, we often assume that a buyer is individually rational (IR) anddoes not expect to lose money. That is, the expected utility E[vi − pi ] of abuyer is non negative. The expectation is taken over the distributions ofthe buyers and for randomized auctions, also over the randomness in theauction’s payment rule. (So far, our examples of auctions are alldeterministic.)

Given that the payment is an expected payment, there are three types ofIR that should be distinguished. In what follows we are still assumingcommon knowledge of everyones distributions and bidding strategies butnot assuming i.i.d or any other assumptions about the distributions.Following the notationn in KP, we let bi (vi ) = β(vi ) be the bid when thebuyer knows his value vi and has common knowledge about the otherbuyers.

15 / 18

Three types if IR

The following distinctions for individual rationality also apply to incentivecompatability (i.e. truthfulness) of a mechanism that we will discuss later.

ex-ante IRE[ui [Vi |V−i ] ≥ 0]; that is, the expectation, is over everyonesdistrbutions including player i .

ex-interim IRE[ui [bi (vi )|V−i ] ≥ 0]; that is, the expectation, is over the otherbuyers’ distrbutions but knowing ones own valuation.

ex-post IRui (bi (vi )|bi ) ≥ 0; that is, when the auction is finished and everonestrue values are revealed, buyer i is guaranteed to have non-negativeutility.

16 / 18

The Revelation Principle: a simple but useful insight

We have just touched on a few of the many types of auctions one canhave even when there is only one item for sale. We know that someauctions may not be truthfuli and tryin to determine good bidding strategyin a Bayes-Nash equilibirum may be complicated.

Since we are considering a Bayesian setting, the notion of truthfulness wewant is called Bayes-Nash incentive compatability (BIC) which we willdefine to mean that given ones true value, and distributional knowledge ofeveryone elses strattegies, it is a dominant strategy (in expectation) to betruthful.

The following result is really a simple observation but still quite useful. Itbasically says that any complication of ones bidding strategy in an auctioncan be transfered to a more complicated auction.

The revelation principle

If A is an auction achieving a Bayes-Nash euilibrium, then there is a directauction (i.e. a sealed bid auction) A′ which is BIC and has the same set ofwinners and payments as the auction A.

17 / 18

The Revelation Principle: a simple but useful insight

We have just touched on a few of the many types of auctions one canhave even when there is only one item for sale. We know that someauctions may not be truthfuli and tryin to determine good bidding strategyin a Bayes-Nash equilibirum may be complicated.

Since we are considering a Bayesian setting, the notion of truthfulness wewant is called Bayes-Nash incentive compatability (BIC) which we willdefine to mean that given ones true value, and distributional knowledge ofeveryone elses strattegies, it is a dominant strategy (in expectation) to betruthful.

The following result is really a simple observation but still quite useful. Itbasically says that any complication of ones bidding strategy in an auctioncan be transfered to a more complicated auction.

The revelation principle

If A is an auction achieving a Bayes-Nash euilibrium, then there is a directauction (i.e. a sealed bid auction) A′ which is BIC and has the same set ofwinners and payments as the auction A. 17 / 18

Reserve prices and the Myerson optimal auction forone item

As stated before, the seller may value the item more than the buyers andthe most direct way to prevent this is to have a reserve price. We havealready noted that revenue equivalence holds even if we have a reserveprice. Even if the auctioneer has little or no value for the item, she canstill use a reserve price if that will increase her expected revenue.

When should a seller set a reserve price above her actual valuation?This depends on the tradeoff betwen the probability of an increased saleprice and the probability that the item doesn’t sell.

So is there a way to optimize one’s revenue in a single item auction?

Surprinsgly, for independent buyer distributions that are regular,there is an optimal auction where the seller just needs to post anappropriate reserve price and then use a 2nd price auction.

18 / 18

When should a seller set a reserve price above her actual valuation?

This depends on the tradeoff betwen the probability of an increased saleprice and the probability that the item doesn’t sell.

18 / 18

csc304: algorithmic game theory and mechanism design fall 2016

Documents