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# GAMES AND DECISIONS ECO 4400

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This 15 page Class Notes was uploaded by Elmore Funk on Thursday September 17, 2015. The Class Notes belongs to ECO 4400 at Florida State University taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/205450/eco-4400-florida-state-university in Economcs at Florida State University.

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Date Created: 09/17/15

ECO 4400 Supplemental Handout All About Auctions Tim Salmon Florida State University March 2004 1 Introduction The basic auction problem that this set of notes attempts to solve involves one agent called the seller who has a single indivisible item that he wants to try to sell He possesses some value for the object U0 2 0 and is happy selling it to someone else so long as the purchase price or revenue 7 is greater than U0 or 7 2 U0 He should be able to do better than just U0 though and since he is greedy he wants to get as much money as he can How can he do that We will be going through the steps to evaluate a number of different alternative mechanisms he might consider and discuss the good and bad points to each One possibility is that we go to our principles of micro story of monopoly pricing lf seller knew the real demand curve he faced then he can solve his problem quite easily He nds the price that makes SupplyDemand One unit so he sets the price equal to the value of the top person on the demand curve done Problem is most sellers won t know the real demand curve If a seller wants to try this they must create a sort of expected demand curve based on his beliefs about the distribution of values among the pool of potential buyers He could set a monopolistic price based on this demand but it may not net him much money unless his information is very good Foreshadowing of a number of results we are going to see one way of seeing an auction is as a way of discovering that demand curve ln general auction institutions that are better at revealing the underlying demand curve are going to tend to perform better so this is a really nice image to set in your mind now Before moving on to evaluating different mechanisms there are a few things that must be explained First what do we mean by function better A couple of possibilities When we are thinking of the seller s problem he cares about one thing maximizing 7 7 U0 or just maximizing 7 since U0 is xed and can not be changed If the seller is a government organization he might care about maximizing e iciency E iciency of an auction is computed by calculating the value generated by the auction in terms of utility to the holders of the objects at the end of the auction V and dividing it by the maximum value the auction could have generated V or VV An auction that is 100 e icient is one that achieves the maximum possible value For the single unit auctions that we will be talking about most of the time a perfectly e icient outcome is one in which the buyer who values the item most receives it at the end quick example U1 10 U2 9 What is ef ciency if 2 gets it 90 as the maximum attainable value is 10 but only a value of 9 was achieved Second what exactly is an auction It is a set of rules that de ne how bids are submitted how those bids determine the allocation Le who gets the item and how much each participant has to pay based on their bids and the allocation There are four classic forms H Sealed Bid First Price All bidders submit bids in sealed envelopes so that other bidders do not know each others bids at the time of submission winner is determined by nding the highest bid and payment is decided by having only the winner pay their bid 10 English Auction Ascending Clock Auction Bids are submitted sequentially in an ascending manner until bidding stops the last person to submit a bid wins the item he pays what he bid 3 Dutch Auction Descending ClockPrice Auction Price starts high and descends until one bidder declares he will buy item is assigned to him he pays what he bid 4 Second Price Sealed Bid Auction Vickrey Again bidders submit sealed bids the item is allocated to the one who submitted the highest bid the winner pays the price of the second highest bid 2 Optimal Bidding Strategies The rst step in evaluating which of these four auction formats our seller might want to use involves understanding how bidders will be have in each That means we need to solve for the optimal or equilibrium bidding strategies for each auction format In each case we are going to be nding a BayesiNash equilibrium of the game We will take a simple environment and then investigate the optimal bidding strategy under each of the institutions The environment we will begin with is referred to as the private values case or more formally the symmetric independent private values SIPV case The characteristics of this environment are 0 single indivisible unit up for auction 0 each bidder knows his own value for the item and only his all bidders are indistinguishable valuations are independently and identically distributed iid and are continuous random variables 0 all agents are risk neutral N7 number of bidders participating in auction xed exogenously m 7 is the value of bidder 239 For convenience we will simply label bidders such that U1 gt U2 gt gt UN Each m is drawn from a uniform distribution on the range 0 1 This means that bidders have an equal chance of drawing a value anywhere between 0 and 1 Further the probability of drawing a value less than say 7 is 7 The probability of drawing a value greater than 7 is 177 and that works for any 33 6 01 br The bid submitted by bidder 239 All of these auctions are going to be Bayesian games So what we need to do to solve them is derive bid functions that map from the type or value space to bids or actions This means deriving what bid a bidder would place for any possible value they might have 21 English Auction Let us do the simple one rst What is the equilibrium bidding strategy Proposition 1 In an Symmetric Independent Private Values SIPV environment bv v for alli is a NE of the ascending auction Proof Think of b as the point that bidder is going to be willing to drop out of the auction What this means is that the bidder with the highest value v1 will end up paying a price equal to the point at which the bidder with the second highest value drops out b v2 All we need to do is verify that no bidder will do better by altering their bid Can anyone do better by choosing bv39 gt v No If I would not have won but I does then we know the bidder lost money as there was another bid bj gt b and this is what i will pay If does not win either then no help What about lt vi If bf would not have won certainly won t so no effect If bf would have won will help The only way possible is by lowering amount paid Will it No It will however lower the probability of winning I So in the simple SIPV case the strategy is to just stay in the auction until price hits your value and drop out NOTE This is a dominant strategy eq as it is optimal regardless of what your opponents do 2 2 Vickrey Auction This auction is de ned as follows Each bidder submits a bid bi The assignment rule is 2 Arg MW 1 l The payment rule is p madly 2 1 Proposition 2 In an SIPV environment b vi v for alli is a NE of a Viekrey auction It is clear that the same proof as before works here Also note that this too is a dominant strategy equilibrium It should be clear to see that the 2nd price and English are what is called strategically equivalent since the same strategy is a NE for both It should also be clear that the revenue expected by the seller between these two institutions is identical How do we nd that expected revenue First since the bidding strategy is the same for both cases it should be obvious that the expected revenue will be the same for both To nd what that is that is done by nding the expectation of the second highest value drawn from the distribution To do that we use something called an order statistic In general nding the order statistic of a distribution is a tricky thing We will ignore that trickiness part of why we are using the uniform distribution In this case letting k refer to the order statistic we want or rather to nd the expected k th highest draw among n values n 7 k 1 Emml n 1 3 The second order statistic is then n 7 1 Ema n1 4 3 If we assume that the values of bidders in the English or second price auction are distributed uniformly on 0 1 then this gives us the expected revenue to the seller Why does it only depend on n If you only have one person you will get 0 revenue as he does not have to outbid anyone If you have two how likely is it that both have a high value say 9 or larger The probability of one of them have a value above 9 is 1 and the value of both is 12 001 lfl have three people what is the probability that 2 out of the 3 have values greater than 9 This one is more tricky to work out Think of the people as persons A B and C There are three different combinations for which two of these guys have values greater than 9 It could be A and B B and C or A and C The probability of two of them having draws above 9 is still 01 but we get three different ways that could happen That means the probability of at least 2 having values above 9 is 3 9r 01 003 1 The idea being that the more people you have the more likely you are to have two people draw high values and result in a high price Consequently if we only have two bidders at least one of them are still likely to have only mediocre a value so the expected revenue is 13 If however we have 9 bidders the expected revenue is 8 as there is a much better chance of two bidder having drawn high values This lls in a few bits of information in our quest to gure out which mechanism our seller should pick We know what the optimal bidding strategy is under the English and second price auction We know that they are strategically equivalent and we know that our expected revenue is n 7 1 23 Sealed Bid First Price How do we solve this game Well rst we need to set up the problem uBV viii if bgtbvjf0rallj7 i 0 else ignore the possibility of ties and look for a symmetric solution ie all bidders have the same bid function So what the bidder wants to do is maximize his expected utility 5 mbaxv 7 bPr0bwin 6 I m not going to go through the general solution to this as it is very complex but 1 will explain the intuition which is not so complex and go over a simpli ed derivation of the solution In setting up the intuition we need to look at what the bidder is really trying to do His rst interest is that he wants to win which means he should bid as high as possible Problem is that in the event that he does win he does not want to have bid very much So there are two components to his expected utility probability of winning and value of wining and both go in opposite directions based upon an increase or decrease in bi If you get too greedy in terms of your value of a win you lower your probability of winning but if you raise your bid too high you make it rather joyless to win The problem comes down to guring out how two balance these two things appropriately If we know the distributions of values and assume that everyone else is a good game theorist then what we do is take the derivative of this function set that equal to 0 and solve for b One detail to that is what the value is for Pr 0bwin You win if your bid is higher than everyone else s so this is the probability that your bid is higher than everyone else s The general version of this is 1For those who know something about combinatorics this isn t quite right but it gets the basic idea across tough so let s take a shortcut that we could solve for if we wanted Let s assume that everyone is going to bid some constant fraction of their value or bi km In that case if 1 want b1 b1 pr0bb1 gt b2 pr0bb1 gt keg prom gt 122 E If we want to look at this generally then if there are two other bidders the probability that my bid is greater than both is 12 and for n 7 1 other bidders it is quot 1 So if we assume a total of n bidders the problem becomes 5 WWM W1 n FOC 1 1 b quot7 b n n 7 1 lt1 lt1 o Solution is 1 1 n 7 Tm 127 Ev So what this is saying is that in a rst price auction you do not want to bid exactly your value because if you do there is no point in winning What you want to do is to place a bid in which you shade your value or you bid some fraction of your value Further the fraction by which you shade your value is dependent only on the number of other bidders First let s note what additional bidders do to your bid For a given value do you bid more or less with additional bidders 2 bidders gt 12 3 bidders your bid 23 etc so with additional bidders you raise your bid Why Because with more bidders around you think there is a better chance of some drawing a high value If there is only you and some other guy there is a pretty decent chance that he drew a low value and will be bidding low If there are 10 other guys chances are pretty good that someone else drew a higher value Consequently you don t want to run the risk of losing that comes from bidding low 24 Dutch Auction For the dutch auction the price starts high and descends until the rst person jumps in To to determine a strategy a bidder must decide at what price he will jump in How will he decide this Well should he jump in at his value No He should let the price drop a little more so that he can make some surplus So he has to trade off surplus against probability of winning and since he wants to maximize his EU we have mbaxv 7 bPr0bwin 9 looks pretty similar to sealed bid rst price auction These two are strategically equivalent as it is solved in the exact same manner as above yielding the exact same bidding strategy I quotfol What we now want to do is be able to compare the revenue between the rst two and second two auctions We therefore need to nd the expected revenue for the rst price and Dutch In this case the revenue is determined not based on the bid submitted by the second highest bidder but based on the expected value of the highest bidder less any shading Need to compute the expectation of the highest bid Since I quot7711 we need n 7 1 Em nEMM um 5 taking note of the trick shown earlier that quot 1H1 we have n1 7 mmquotnip an Elrl 12 Note that if you flip back a few pages in your notes what is the expected revenue from a second price or English auction Same thing This is no coincidence and is one of the central results in auction theory and one of the most powerful and useful results we will develop this semester known as revenue equivalence 3 Revenue Equivalence Theorem 3 Assume each of N RiskiNeutral bidders has a privately known value that is indepen7 dently drawn from a common distribution Fv that is strictly increasing and atomless on v Suppose that no buyer wants more than one of the k available identical indivisible objects Then any auction mechanism in which 1 the objects always go to the k buyers with the highest values or the probability that a bidder with value v wins an item is the same and 2 any bidder with value v eacpects 0 surplus Theorem 4 yields the same eacpected revenue and results in a buyer with value v making the same eacpected payment Proof Consider any mechanism for allocating an item among n bidders In equilibrium it must be the case the the surplus to a bidder with value v for following the equilibrium strategy is greater than for doing anything else de nition of equilibrium so wuzwvwevmv as where Sv is the surplus obtained by following the strategy for value v The RHS is the surplus that the bidder would get if he had value v but followed the strategy for 5 135 is the probability of winning if they bid as if they had a That is if his value is 10 v and S105 we want to know what his expected surplus would be if he chose to bid as if his value were 40v eq bid of 20 He would get the same surplus a a bidder would get if he won 20 the difference between the two values v 7 a 730 times the probability of having to pay up Another way of writing this is by letting a v dv Sv 2 Sn 110 idvPv dv 14 If there is someone else who actually has the value a v dv it must also be the case that Sv dv 2 Sv dvPv 15 or if he is of type vdv he must prefer choosing Sv dv as opposed to what a v bidder would do Sv Rewriting and doing some fancy algebra implies Sv dv 7 Sv P d gt UT U dv gt Pa 16 The first inequality comes from solving first eq for P1 11 and second from solving second for P1 Take the limit of this as d1 gt 0 and we have simply 85W 8U 71 17 The left and right sides approach each other so middle must be equal Middle is simply the formula for the derivative of 51 We can then integrate both sides between E and 1 51 52 Pvd 18 This simply states that the expected surplus of any given value 1 is equal to the surplus of the very base value 5 plus some cumulative function based only on the probability of winning So if we know probability of winning P1 then we know the slope of the function and if we know 5g we know where it starts Thus we know every point along the function and it will look something like the graph below Any two mechanisms that have the same 5g and the same P1 have the same 51 function This means that any bidder has the same surplus in any mechanism and the seller expects the same revenue in either mechanism Note that all of the designs we have discussed have the bidder with the highest value winning with probability one in equilibrium and all will imply that the guy with the lowest value receives a surplus of zero Thus l The proofis really not all that dif cult but let us just look at a graphical proof of it What this is saying is quite elegant and simple Lets graph value against expected surplus What this says is then obvious If two institutions give the same 5g and the probability of winning is constant across both mechanisms given the same 1 that is what second condition really states then what has to be the case is that 51 goes up in the exact same manner So all you need to know is 59 and P1 and the surplus as a function of value can be drawn as below This will then hold for every single mechanism that has the same 5g and P1 If you look across the mechanisms we just talked about Anyone who drew a value of 0 was losing and a boundary condition you really need to solve the first price is that b0 0 and the guy with the highest draw always wins Thus revenue will be equivalent across the 4 institutions Sv is expected surplus I Slope is l PW l i I 5D n g l l i 39 I 39 I I l So which auction mechanism should the seller choose Doesn t matter a bit Any one will yield the same revenue This is an incredibly powerful result and useful result A bit later I will show how it can be used to give us insight into things that would normally be very di cult to analyze and that most people probably think auction theory couldn t tell us anything about In practice this result seems kind of hard to believe How might it go wrong in certain situa tions First thing to note is that if an environment in the real world matches this one it will hold If it does not hold in reality as it does not in many situations it must be the case that there are differences between our hypothesized environment and real one Let s propose some modi cations to the environment and see if they matter 4 How to Break RE In the following examples we will only consider the second price vs rst price auctions for simplicity 41 Risk Aversion 411 Risk Averse buyers Theorem 5 In a rst price sealed bid auction for any two bidders if bidder 2 is risk aierse and bidder 1 is risk neutral then b2 2 b1V I won t go through the derivation but I will talk though the intuition This theorem states that a risk averse bidder will bid higher than a risk neutral bidder To gure that out let us look rst about the rst price auction Notice that what he cares about is My 7 bPuinning Consider that our standard RA function looks like m5 as opposed to RN of 33 What is happening is that you are now caring less about adding more surplus so adding marginal surplus doesn t help you Adding more probability of winning however does help you So you raise your bid lower your surplus if you do win but raise your probability of winning What risk aversion means in this context is that you are averse to the risk of losing the auction Another way of stating this is that a bidder being risk averse means that he is willing to accept a lower surplus so long as he has a higher probability of getting it This means explains exactly why a risk averse bidder would be willing to bid more than a risk neutral bidder as he is perfectly happy making the implied tradeoff between declining surplus yet increasing probability of winning Now let us consider the second price auction What happens to our bidder here Previously his bidding strategy was b U U Does that change here No If you go back through the previous proof of the optimal bidding strategy there was never any risk or probabilities involved in evaluating strategies so risk aversion does not enter into consideration Result is that we know that in the sealed bid case a risk averse bidder bids higher than a RA bidder Therefore revenue will be higher with risk averse bidders We also know that the second price auction will generate the same amount of revenue with RA or RN bidders Taken together this implies that with RA bidders the revenue generated by the rst price or dutch auction will be higher than that of second price or English So if you are a seller and think that you potential bidders are RA then you might want to hold a sealed bid auction as you expect to make more money 412 Risk Averse Sellers What if our bidders are RN but our seller is RA What should our seller prefer We won t do a proof but it can be shown that in this case sellers will also prefer the sealed bid rst price auction To show this you need to examine the variances of the expected selling prices under the different formats The base issue is that in a second price or English auction prices are very steady In the rst or dutch prices vary much more even though on average they come out the same The key though is that a high price will emerge with higher probability under the dutch or sealed bid rst price For example consider the case in which someone draws a value of 8 and bids 4 expecting that to be the second highest value It may be that the other player drew 05 In this case the ascending or second price auction would have only raise 05 while the rst price 4 ln expectation this all works out to be equal but the tails are what matter for this case 42 Common Values Another possibility is that the values between bidders may not be independent but related In the extreme let us assume that the value of the object is the same for all bidders With this assumption though it would be a pretty boring auction if everyone actually knew what that common value was so let us assume further that instead of each bidder knowing the value of the object each bidder receives a signal t of the value of the item These signals are drawn from some common distribution F and there is some U that is the true value of the object and common to all bidders Consider a sample auction in which there are 4 people who have drawn estimates of t1 15 t2 11 t3 8 t4 5 and the true value is U 9 Who is likely to win the auction Probably not 4 but it is much more likely to be bidder 1 This will be the bidder who has drawn the highest signal or who has most severely overestimated the value of the item If the bidder bids in such a way as to not account for this overestimation he can end up overpaying for the object This phenomenon is known as the Winner s Curse What happens is that the bidder who just found out he won also just found out that he necessarily overestimated the value of the item and probably just lost money if he didn t take these effects into account Experimental results show that some people can learn to deal with this effect but it isn t easy So how do you bid to get over it Well it isn t easy I m going to avoid doing the full solution as it is rather messy Instead I will explain the bidding strategy in the English auction and what this tells us about bidding strategy in other auctions 43 English Auction Remember in an English auction the price starts low and increases until only one person is left So what every bidder needs to do is gure out where to drop out Should a bidder drop out at their signal Well if they wait that high then they may win and have overestimated the value which means they lose money Perhaps they should go higher than their signal Why Well if they estimated low then they would be dropping out well below the value of the item Look at player 4 above Sounds plausible but there seems to be a problem with it To determine this we rst need to clarify the environment We will assume that the actual ti tj N Z N 19 1 value of the object is We will further assume that t is uniformly distributed on 05 We will also denote the jth highest signal by 7 This then means that the actual value of the object is just the average of all of the signals received Yes this is arbitrary and cooked but it makes the math easier and changes nothing about the substance of the approach What 1 am going to do is go through and explain the full equilibrium bidding strategy What it involves is each of the bidders taking the best information they have available to estimate the true value of the object and being willing to bid up to that point As the auction goes on their information gets better and better So let us look at bidder N who has signal of tN or the lowest signal How does he decide a dropout point He estimates the value of the object based upon all of the information he has available which is his own signal He know that the other bidders have signals as well so he needs to come up with some belief about what those are The one he is going to use is that he is going to believe that every one else has seen the same signal he has If he does then this means his estimate of the value of the object is lt N 7 rm 20 N NQN 21 There are two issues with this First when auction starts no one knows that they are bidder N so no one knows to use this The way this will work is that everyone assumes they are bidder N until proven otherwise by seeing another person drop out earlier Another way of viewing this gets at the second point which is why would the person use their own signal as the estimate of the signals of the other players As the price starts low and goes up the bidder knows who else is in and can assume that the signals of the other bidders are at least as high as the current price If their signals are higher than his he is going to lose the auction so he doesn t care about what happens in that case so he assumes that the signals of other people are below his and above the current price If he doesn t see anyone else drop out below his signal then his effective estimate of their signals at that point is that they must all be equal to his What will happen then is that everyone uses this to estimate the value of the object and prepares to drop out at their signal of the value As the price rises it eventually hits the actual tN and the real bidder N drops out So the rst bidder has dropped Where does bidder N 7 1 dropout at All of the bidders now update their estimation of the value of the object They still know their own signal but now they can also infer what bidder N s signal was This gives them 2 out of N data points they need to estimate the value accurately Since they don t know the others again they make their best estimate which is that they are equal to their own value Bidder N 7 1 3 estimate and drop out point then is ttm RN71gt lt 75ltN71gt 22 The point to realize is that he is dropping out at a point below his signal Again this bidder doesn t know he is bidder N 7 1 but everyone constructs their next possible dropout point like this and as it turns out bidder N 7 1 hits his rst This process continues with the bidder updating their value estimates based on backing out what signal bidder N 7 1 must have had to drop out there and then bidder N 7 2 drops out at some point The others update their estimate again and this continues on until we get down to the last two bidders The drop out point for bidder 2 is N 1 2 N E 750 NR2 lt U ltlt 751 23 j3 10 This is also the nal price paid by bidder 1 The important point to see is that this is less than the value of the item and in most cases this will be well below the value for ta and t2 as both are likely to be greater than U So the strategy in this case is to use the information conveyed by where other bidders have dropped out to improve your estimation of the value and to drop out where you estimate the value to be Why though should you assume that everyone else who hasn t dropped has the same value as yourself I tried one attempt at explaining this below but this ultimately relies on showing that this is an equilibrium strategy by verifying that no one has an incentive to deviate This is easiest to verify for bidder 2 and it should be obvious that if it works for him it will work for all others The only way 2 can deviate and have it matter to him is if he decides to drop out at a higher price and wins lf bidder 2 drops out at a price under tm t1 where bidder 1 is intending to drop out after the other N 7 2 bidders have dropped then he has no change in his outcome as he still loses If he drops out at a point above this he knows that he is going to pay more than the value of the object since 1 is following the equilibrium strategy 1N 2 N Ztm NR1 gt v 24 j3 thus the only way he can win the auction is by bidding above the value of the item So he won t do it The intuition from the private case still holds you drop out where you think the value of the item is given your information The difference is that in the lPV case you have full info on that matter in this case you have partial info The moral of the story is that in this case where you only have partial information you use what you can to your best ability or you will end up bankrupt I won t go through and prove it but it turns out that revenue equivalence is also going to hold in the common value environment I won t derive these for you either but in case you are curious the bid function in the second price auction is now 7 2 N b t 2N1 and for the rst price sealed bid the bid function is t72NNigv 1 2N iv In both of these cases what you have to do is come up with your best estimate of what the signals are that the other people must have had if yours was the highest Based on what those signals in the second price case you place a bid equal to that estimate and in the rst price you try to place a bid just above where the person with the second highest signal would be willing to bid This is essentially the same approach as in the SlPV environment the only problem is that estimating that point is more dif cult 44 Optimal Auction Design To this point we still have not solved the problem we started with which was what is the actual optimal auction design The real answer is that the optimal auction design depends on the speci cs 11 of the environment I will discuss the classic solution for the SlPV environment but for other environments the answer changes In getting to the point of designing an optimal auction we have derived several intermediate steps 1 We know how to model what bidders are going to do in a number of different institutions 2 Even better we know that at least in certain cases expected revenue is going to be the same across a broad range of auction forms ltRevenue EquivalencegtThis means I don t have to check 50 different auction formats for which one makes me the most money 9 Another fundamental and important result that we haven t talked about but which I will mention is something called the Revelation Principle What it tells me is that the outcome of any dominant strategy mechanism can be achieved in a direct revelation mechanism for which truthitelling and participation is optimal So what does this do It means that for any dominant strategy equilibrium of any auction game there is an incentive compatible direct auction that gives us the same outcome that is an auction where the bid a bidder send in is equal to their true value So I can set up the game such that I can ask people for bids and get one result lfl can do that then I know that I could also set up a different game and ask them to submit their values they will and get the exact same result How The basic idea is that you submit those values to an independent third party who then makes optimal choices from the point of view of the bidders and does what they should have done in the original game For example Let us go back to our rst price auction in the SlPV RN case We showed in the case in which values were uniformly distributed on 0 1 that the optimal strategy was I quot7711 So we set up the mechanism as follows All bidders send in their bids bi The bidder with the highest bid will win the item and will pay I quot771b It is easy to show that in this case the mechanism elicits all participants to submit bf vi What points 2 and 3 do for me is make it to where I do not have to spend an eternity checking out every bizarre mechanism someone dreams up to see if it can do better than what I m about to show you Since I have 2 and 3 my problem is now very easy in relative terms Deriving this result in full generality is dif cult but again the intuition should be clear The basic point is that the mechanism ends up looking like the following 1 Ask bidders to submit a bid as it will turn out they should submit their true value 2 Compute what is essentially the marginal revenue sometimes called priority level of each bid or 1 7 U U 7 25 M l 1 MW l for the uniform distribution case we have been using this is just vi 7 1 7 vi 2W 7 1 9 Keep the item if all MR s are below U0 value of item to seller 4 Give the item to the bidder with the highest MR works out to be to give the object to the bidder with the highest bid in most cases 5 Winning bidder pays a price equal to the minimum he could have reported in the place of his true value and still won the auction In the case of for all 239 and j then this is just a second price auction with a reserve price but the reserve price isn t U0 it is 771U0 The idea that this is suggesting is that if the true value of the seller is 2 then they should set a reserve price equal to 2r 7 1 6 2 7quot and hold a second price auction If no bids are received above 6 then no sale is made If only a single bid is above 6 that bidder will pay a price of 6 If two bidders submit bids above 6 then standard second price auction rules hold This is a curious idea though that the seller should actually refuse Pareto improving offers If a bidder bid 4 and another 3 this would imply a sale price of 3 gt 2 or the value to the seller I am arguing he should forego that trade Why This is derived from the seller looking at the expected pro t or 7 7 U0 Pr 0M7 If the seller gives up a few transactions between their value and some reserve price fact is they are not giving up much in expectation They only lose in those relatively few cases in which there are not two values greater than 771U0 but greater than U0 On the other hand for those cases in which there is only a single value above the reservation price the seller makes more money In expectation it works out that this is a good tradeioff 5 Legal Systems I want to take a bit of a diversion from auction theory speci cally to show you one of the cooler applications of the RET What I want to do is look at legal systems and in particular how lawyers are compensated by their clients In 1991 Vice President Dan Quayle decided that the American legal system needed to be reformed to reduce expenditures on legal costs Many people have been concerned over how much money is being spent on legal bills and part of that concern is over the fact that it clogs up court systems and makes guaranteeing the right to a speedy trial di cult to do One of his proposals was to change who bears the legal expenses in a lawsuit He proposed that the losing party pay to the winner an amount equal to the loser s own legal expenses The result is that the loser s legal expenses would be effectively doubled He based the justi cation of this argument on a marginal cost basis that if the costs of pressing a lawsuit might cost you double then less would be spent pursuing lawsuits What we want to do is determine if his claim was correct Analyzing this in full generality gets quite dif cult so we will analyze it in the context of a simple model Assume that each party in the lawsuit has a privatelyiknown value of winning the lawsuit relative to losing that is independently drawn from a common distribution say from the range 2 What this represents for the plaintiff is what he expects to win and for the defendant this is more or less the negative of what he expects to lose Further assume that both parties simultaneously and independently decide how much to spend on legal expenses and then whoever spends the most will win the lawsuit This is what is known as an allipay auction That is bidders place their bids the highest bidder wins but everyone pays their bid It is by no means an exact model of the court system but it really isn t such a bad one What we want to look at is if the regime proposed by Quayle reduces legal expenditures as compared to the standard US system To do that we need to compare the revenue generated by the 13 two mechanisms as their bids are essentially what they are paying to lawyers So if one generates higher revenue this means it is generating higher legal fees We have two systems to analyze one we will call the American system in which what people pay is what they bid regardless of whether they won or lost The Quayle system would have people pay twice their bid in the event that they lose and the winner would pay their bid but get back the amount of the bid by the loser In order to compare these two systems we could go through and nd the equilibrium strategy for our players and then use that to compute expected revenue As you might imagine actually doing that would be very dif cult The beauty of the RET is that it gives us another option At least for starting the analysis We can ask the simple question will they generate the same revenue in terms of payments to lawyers At this point all we need to do is see if the RET holds That means verifying two conditions 1 the winner will be the player with the highest value and 2 any bidder with value 2 expects 0 surplus In this case the guy who draws Q can guarantee a surplus of 0 by bidding 0 in either mechanism He knows he has no probability of winning so why spend anything on lawyers Further there is no reason that the bidder with the lower value would ever be able to outbid the guy with the higher value in eq for either mechanism The one with the higher value will always be willing to bid more Consequently the RET holds and in both institutions the lawyers expect to take the same exact amount of money from people with no real difference and the expected surplus to the actual participants is the same as well The implication is that Dan Quayle s proposal should not be expected to make any difference The key insight as to why is that he left out one point in his analysis He looked at the losing side and said If court is more expensive you will go less often The problem is that he ignored what happened when you win If you do win you make even more money than you would under the base American system as the other guy pays part of your bill As it turns out these two effects will off set each other resulting in no change in revenue or even in the expected surplus to the participants That doesn t mean however that some form of his idea would not have a change on the system Others who believe legal expenses and lawsuits have gotten out of hand in this country have proposed adopting a European or British style system In this system the loser would pay a fraction perhaps 25 of the winners legal expenses What effect should this be expected to have on legal expenditures Well lets try to compare the revenue generated in the two systems and try to gure it out Again it would be really cool if we could use the RET One thing we also want to try to get an idea about is which system people would be happiest in To apply the RET we would need to show that those two conditions hold In the British system however it is no longer true that the person with the lowest value can guarantee themselves a payment of 0 as when they lose they will have to pay part of the winners legal expenses They have effectively lost control of their legal payments when they lose in the UK system This means that the RET will not hold and revenue will be different between the two institutions We can however still get something out of this We already showed that people are indifferent between the current American system and the Quayle system In the proof of the RET we relied on starting from the lowest value bidder s expected utility and then showing that everyone else s depended only on that and their probability of winning Well the probability of winning should be the same across all three systems but the expected value to the lowest value person is lower in the 14 British system So what happens is that the expected value graph will start at a lower intercept but then go up in the same slope This demonstrates quite clearly that everyone is worse off in the British system People would prefer either the American or Quayle systems What is accounting for this preference It would take a while to prove it formally but it should be relatively clear What is going on is that in the British system legal expenses are raised so lawyers get more money As in moving to the Quayle system in the British system winning becomes more valuable and losing does become more costly but these effects no longer exactly offset The British system causes people to pay more to lawyers in order to avoid losing and because when you win the other guy subsidizes your legal fees The idea is that if you win you only have to pay 75 of your legal bills so you are willing to pay more In fact some proposals for systems of this sort want the loser to pay 100 of the legal bills of the winner The equilibrium of this game is that both bidders spend an in nite amount on legal bills as no matter what your opponent pays you want to be 1 higher What this tells us is that if a lawsuit is led then legal expenses are higher in the British system but in reality that doesn t necessarily mean that total receipts should be higher Here we get to some of the limitations of the simple model I have set up It ignores the fact that once a plaintiff sees the rules of the system and their expected value from pressing a lawsuit they might decide not to go to court If we looked at this issue again there should be no difference between the US and Quayle systems Expected value is the same probability of winning is the same so if you want to press a lawsuit in one you do in the other In the UK system though plaintiffs with lower values might not think it is worthwhile to push a lawsuit since their expected value is lower So the prediction on total legal bills is uncertain This does give us other empirically testable predictions that we should see more cases in the US but higher legal bills per case in Britain This prediction can be tested both experimentally and with eld data from US and UK trials comparing size of legal bills and number of cases led corrected for population and different legal systems It works out in both cases that the data support the prediction Fewer cases are led in the UK but the per case legal expenses are huge So should we adopt the British system Depends Since this model isn t allowing the dropout decision we can t say for sure which one will really do what in terms of total revenue We can say without that the British system unambiguously reduces welfare unless the fewer cases allowed cases to go to trial more quickly and that made up for the increased cost It is quite likely therefore that of the three systems we are ne with the one we have There may be better systems out there but the Quayle system does nothing and the UK system actually increases legal fees and reduces welfare except for the lawyers So this is a very nice clean argument and result that while based on a simple model actually does capture what is going on in these various legal systems fairly well Doing this analysis without the BET would have taken us days but we could do it all very quickly using this theorem

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