All-Pay Auctions and Contests

1 Plan

• Today: contests

• Thursday: recap

• Thursday night: HW5 due

• Online course evaluations are open till Friday

• Final exam next Thursday, 7:45 am, Soc Sci 6102

• Any questions before we get started?

398 2 Before we start: a common mistake on HW4

• Before we start, I wanted to mention a common mistake that the grader ﬂagged on HW4

• On question 1.e, you’ve already veriﬁed that in the two-bidder all-pay auction with valuations 2 vi uniform on [0, 30], it’s an equilibrium for both bidders to bid bi = 60

• Now you’re asked to calculate expected revenue, which is the expected sum of the payments from both bidders, or

v2 v2 E 1 + 2 60 60

• We can think of these as bidder 1 and bidder 2, or we can think of them as the higher-value and the lower-value bidder, and rewrite this as

v2 v2 1 1 E max + min = E v2 + E v2 60 60 60 max 60 min

where vmax and vmin are max{v1, v2} and min{v1, v2}, or whichever valuation turns out to be higher and whichever turns out to be lower, respectively

• Up to here, everything’s ﬁne;

but at this point, several people noted that Evmax = 20 and Evmin = 10, 202+102 plugged those in, and mis-calculated revenue as 60 , which gave the wrong answer

• Why can’t you do that? Because if x is a random variable, E(x2) 6= (E(x))2

2 2 • (In fact, E(x ) − (E(x)) is the variance of x, so unless vmax and vmin have zero variance, this is guaranteed to give the wrong answer)

• More generally, for a general function g, the only time that E(g(x)) = g(E(x)) – the only time you can plug in the mean of x to evaluate the mean of g(x)– is when g is linear

• In all other cases, you have to calculate the expected value of a function by integrating

399 3 All-Pay Auctions

• This segues nicely into today’s starting point: all-pay auctions

• When I ﬁrst mentioned the all-pay auction in class, I mentioned that these are rarely if ever literally used to sell things, but that they’re sometimes used as an analogy for certain types of winner-take-all competition

• A paper from 19931 uses an all-pay auction as a model for politicians and lobbying

• Suppose a corrupt politician gets to make a policy choice – say, the location of a military base – and would happily sell the outcome to whichever side is willing to pay more

• However, that’s illegal, so instead, the diﬀerent sides “wine and dine” the politician with campaign contributions, fancy meals, and so on, and then the politician makes a decision, but all the lobbyists have “paid” their bribes – nobody gets their campaign contribution back because they’re unhappy with the policy the politician chose

• The structure of payoﬀs is exactly the same as in an all-pay auction – each lobbyist gets a certain value from winning, minus the bribe they pay, and the politician awards the outcome based on the biggest bribe, but keeps all the bribes, not just the winner’s

1Baye, Kovenock and de Vries (1993), “Rigging the Lobbying Process: An Application of the All-Pay Auction,” American Economic Review 83.1

400 • So, an all-pay auction might be a pretty reasonable, natural way to think about certain legislative/lobbying settings

• However, when all-pay auctions are used to model situations like this, there’s one key diﬀerence from the way we’ve been thinking about auctions

• We’ve been thinking of settings where bidders’ valuations are private information; in lobbying models, it’s assumed that while diﬀerent lobbyists may value winning diﬀerently, everyone knows everyone else’s valuation

• That is, if the politician is considering putting a military base in Bethesda, or Portland, or Newport News, everyone knows how much value Bethesda would get from being chosen, and how much value Portland would get, and so on

• Modeling auctions with complete information is messier than with private information

• When bidders’ valuations are private information, each “type” of bidder can have a unique equilibrium bid, and uncertainty about which valuations your opponent has creates a tradeoﬀ between price and likelihood of winning

• That is, perhaps in equilibrium, if I have valuation 70, you know I’ll bid 35, and if I have valuation 80, you know I’ll bid 40, but since you don’t know what my valuation turned out to be, you don’t know how I’ll bid, which creates a trade-oﬀ for you between how much you pay and how likely you are to win, which is what makes the equilibrium “work”

• But when bidders know each others’ valuations, this no longer works

• There can’t be an equilibrium in pure strategies – an equilibrium where each bidder makes a predictable bid – or someone would always have an incentive to change their bid

401 • For example, consider an all-pay setting with two bidders,

where their valuations for winning are v1 = 1 and v2 = 2

• Of course, bidder 1 can’t bid more than 1 in equilibrium, or he’d get a negative payoﬀ even if he won; so bidder 2 would have to win in any equilibrium

• If bidder 1 was bidding more than 0 and losing, he’d want to lower his bid; but if bidder 1 was bidding 0, bidder 2 would want to win as cheaply as possible; and then bidder 1 would want to outbid him

• So in an all-pay auction with complete information, there’s no equilibrium in the sense we’ve been considering so far – for any combination of bids, some bidder will always have an incentive to change their bid

• So how do we proceed?

• Let’s put aside the all-pay auction for a few minutes, and think about a game you might have more intuition for: Scissors-Paper-Rock

402 3.1 Scissors-Paper-Rock and Mixed Strategies

• Suppose you and I decide to bet a dollar on a single game of scissors-paper-rock

• If you play scissors and I play rock, rock smashes scissors and I win your dollar

• If you play scissors and I play paper, scissors cuts paper and you win my dollar

• If we both play paper, we tie, and nobody wins any money

• There’s clearly no equilibrium where we both pick a single action to play

• Remember, the logic of equilibrium is that in equilibrium, I correctly anticipate how you’re going to play, and best-respond to that, and you correctly anticipate how I’m going to play, and best-respond to that

• But if I’m going to play rock, your best-response is easy – play paper – but then if you’re going to play paper, I don’t want to play rock, I want to play scissors

• So how do we deal with this?

• Instead of thinking of each of us choosing a single action to play, think of each of us choosing a probability distribution over actions – a probability of playing rock, a probability of playing paper, and a probability of playing scissors – and then at the very last second before we play, choosing randomly according to those probabilities

• So, maybe I’ll play rock half the time, scissors a quarter of the time, and paper a quarter of the time – and maybe you know that in equilibrium, but you don’t know which one I’ll choose this particular time

403 • Thinking about strategies in this way, scissors-paper-rock has a unique equilibrium: I play each action with probability exactly one-third, and you play each action with probability one-third

• If I’m playing each action with probability one-third, then no matter what you do, your expected outcome is the same: you’ll win a third of the time, lose a third of the time, and tie a third of the time

• So any action, or any combination of actions, is a best-response, because each one is just as good as the others

• And if you’re playing each action one-third of the time, then each action is just as good for me, so any combination is a best-response

• And that’s an equilibrium

• A little more formally, a mixed-strategy equilibrium is when...

– each player chooses a probability distribution over actions, and – given the other players’ strategies, each action I play in equilibrium gives me the same expected payoﬀ, which is at least as high as anything else I could do

404 3.2 Back to the complete-information all-pay auction...

• So let’s go back to the all-pay auction,

where there’s no private information, v1 = 1, and v2 = 2

• The mixed-strategy equilibrium turns out to look like this: bidder 2 always bids, and mixes uniformly over the interval (0, 1]; and bidder 1 only bids half the time, and mixes uniformly over (0, 1] when he bids

• Why is this an equilibrium?

• First, consider bidder 1’s problem, if bidder 2 is following this strategy

• If bidder 1 bids b1, her expected payoﬀ is

v1 Pr(win|b1) − b1 = Pr(b2 < b1) − b1 = b1 − b1 = 0

which is the same for every b1 ∈ [0, 1], and she can’t do better than that by bidding more than 1, so any bid in [0, 1] – or any mixture of those bids – is a best-response

• What about bidder 2?

• If bidder 2 bids b2 > 0, his expected payoﬀ is

1 1 v Pr(win|b ) − b = 2 + b − b = 1 + b − b = 1 2 2 2 2 2 2 2 2 2

for any b2 ∈ (0, 1]

• (If bidder 2 bids exactly 0, then when b1 = 0, they tie, and he only wins half the time, 1 1 which would make his expected payoﬀ 4 2 − 0 = 2 )

• So any bid in (0, 1] is equally good, giving payoﬀ 1, and is beter than bidding 0 or strictly above 1; so any bid in (0, 1], or any mix of such bids, is a best-response

• so given these strategies, both bidders are best-responding to each other, and so this is the mixed-strategy equilibrium

405 • So, many papers consider the complete-information all-pay auction, as an analogy for lobbying

• Baye, Kovenock and de Vries show, interestingly, that if there are more than two lobbyists, the politician (seler) may increase revenue by excluding some competitors – and may want to exclude the one who values the outcome the most!

• (They note that one example of excluding some competitors is announcing a small ﬁeld of “ﬁnalists”)

• If one competitor is much stronger than the others, the others may not want to compete much knowing they’ll probably lose, and so the strong one may not have to compete much either;

• If the politician excludes the strong competitor, the weaker ones have a better shot at winning, and will be willing to expend more resources)

• In the example above, suppose we started with three lobbyists, with vi = 1, 1, and 2

• If you exclude one of the weak bidders...

– then we’re in the case we just considered, say, with v2 = 1 and v3 = 2, and we know that bidder 2 only bids half the time, and bids uniformly on (0, 1] when she does bid, 1 so her expected payment is 4 ; 1 and bidder 3 bids uniformly on (0, 1], so his expected payment is 2 ; 3 so expected revenue is 4

• If instead you exclude the strong bidder...

– we need to ﬁgure out the new equilibrium, but it turns out to be both weak bidders bidding uniformly on [0, 1] (you can verify this), 1 so each one’s expected payment is 2 , so expected revenue is 1

406 • Is this better than just letting all three bidders compete?

• If you let all three bidders compete, there are multiple diﬀerent equilibria, but if let’s look at the one where the two weak bidders play the same mixed strategy

• In that case, strategies are such that for bidders 1 and 2, r 1 + b Pr(b < b) = i 2

and for bidder 3, r 2b2 Pr(b < b) = 3 1 + b

• To verify that: suppose bidders 2 and 3 are following these strategies,

and consider bidder 1’s problem; a bid of b1 earns s r 2 1 + b1 2b1 v1 Pr(win|b1) − b1 = 1 · · − b1 = b1 − b1 = 0 2 1 + b1

for any bid in [0, 1], so any mix over this range is a best-response; by symmetry, the same is true for bidder 2;

and for bidder 3, a bid of b3 earns r r 1 + b 1 + b v Pr(win|b ) − b = 2 · 3 · 3 − b = 1 + b − b = 1 3 3 3 2 2 3 3 3

for any bid in (0, 1], so any mix over this range is a best-response

• Given these strategies, we can work out each bidder’s expected payment, and therefore the seller’s expected revenue; I’ll spare you the algebra, but it comes out to about 72.4 cents

• So indeed, the seller gains by excluding a bidder; and gains more by excluding the strong bidder

• (Basically, the two weak bidders bid more aggressively when they’re only against each other, and this more than compensates for losing the strong bidder’s revenue)

• But of course, excluding the strong bidder leads to an ineﬃcient allocation – the policy chosen is not the most socially valuable one

407 3.3 Other things the all-pay auction is a decent model for

• There’s also an old working paper – never published – with the suggestive title, “Games of Redistributive Politics Are Equivalent to All-Pay Auctions with Consolation Prizes”2

• Persico thinks of elections as all-pay auctions – the different candidates compete by promising different things to voters – and shows that different electoral systems can be represented by different versions of an all-pay auction, modified to give prizes to losers as well

• Another setting often thought of as analogous to an all-pay auction is a patent race – multiple ﬁrms doing R&D in the same area, racing to develop competing products, knowing that whoever succeeds ﬁrst will have a patent granted and get to be a monopolist

• If we think of each ﬁrm choosing up front how much to spend on R&D, and the “prize” being the value of serving the market as a monopolist, and assume the ﬁrm that commits the most resources will win the race, this is a reasonable way to model them

• However, it might be more natural to think ﬁrms commit resources over time, and get some information over time about how things are going, and can start out working on a project and decide to keep going or quit

• This is called a war of attrition – there’s a nice paper that models them3 and ﬁnds some nice results

2by Nicola Persico, available here: http://nicolapersico.com/ﬁles/klemperer3.pdf 3Bulow and Klemperer (1999), “The Generalized War of Attrition,” American Economic Review 89.1

408 • Aside from racing for patents, there are also settings where someone – a government, or a private foundation – oﬀers a large cash prize to encourage research in a particular area,

• A few examples: in 1919, a New York hotel owner, Raymond Orteig, offered a $25,000 prize (equivalent to about $360,000 today) to the first people to fly non-stop between New York and Paris; the prize was eventually won by Charles Lindbergh in 1927, although six people died in earlier attempts

• This would later inspire an engineer named Peter Diamandis to raise funds for the X Prize – renamed the Ansari X Prize – which was created in 1996,

and oﬀered $10,000,000 to the ﬁrst non-government organization to launch a reusable manned spacecreaft into space twice within two weeks

• The prize was claimed in 2004 by a group ﬁnanced by Microsoft co-founder Paul Allen; the prize led to more than $100,000,000 being invested in new technologies

• In 2007, Google announced a similar prize for the ﬁrst group to land a robotic spacecraft on the moon, travel around a bit, and send back video;

a total of $30,000,000 in prize money was oﬀered; nobody won the main prize, although just a few weeks ago, $1 million was awarded to a group whose entry made it to the moon but crashed

• (Sources: Wikipedia pages on Orteig Prize, Ansari X Prize, and Google Lunar X Prize, although I’m having trouble getting links to work)

• In all these cases, the contest functions like an all-pay auction – the prize gives people an incentive to invest time and money, knowing they won’t get it back in the event they don’t win

409 4 Procurement Contests

• So, those are some settings where all-pay auctions seem like a reasonable modeling analogy

• There are also some settings where a market is deliberately designed to have the features of a contest

• One is in government procurement

• Government agencies hold lots of auctions where the “prize” is a contract to provide a good or a service, and bids are the prices that will be paid by the government for the service

• In some settings, the auctions are quite simple – procurement auctions to repave a certain length of highway, or to provide road salt to a city government, can specify the exact details of what’s required, and allow ﬁrms to just compete on price

• However, some procurement problems are for more complicated things, which need to be custom designed and then built

• When the Air Force or Navy needs a new ﬁghter jet system, they don’t simply ask each possible supplier to name a price per jet and go with the cheapest; the process is much more complicated

• As one example, the Department of Defense spends about $1 Billion a year through the DOD Small Business Innovation Research program

• This isn’t for ﬁghter jets – the SBIR is open to ﬁrms with 500 employees or less

• This is for smaller projects, or small pieces of larger ones, where the DOD needs a new technology developed

410 • All federal agencies with at least $100 million annual R&D budget are required to allocate some of it to small businesses through the SBIR; the DOD uses it extensively; the paper I’m borrowing from focuses on the Navy

• The SBIR competitions have three stages4

• Each year, the DOD posts 150 to 250 research projects, giving the technical requirements in detail, including goals for each stage

• Firms that want to compete for a Phase I contract submit a proposal explaining how they’ll meet the Phase I goals

• Based on those proposals, the DOD awards several Phase I contracts

• Phase I is “a feasibility study to determine the scientiﬁc or technica merit of an idea or technology that may provide a solution to the [Navy]’s need or requirement”

• “It involves preliminary prototyping, benchtop testing, computer simulations, and other low- cost preliminary research. The Navy currently awards approximately $80,000 for the base Phase I contract, and there is little variation across competitors and projects in this amount.”

• After six months or so, the ﬁrms submit reports, along with Phase II proposals to implement or manufacture the product and a detailed cost proposal

• The Navy typically awards Phase II contracts to about 40% of the Phase I ﬁrms; these contracts are between $200,000 and $2 million, and include much more detailed proto- type testing

• Phase II is set up to “allow for increased funding levels ased on the project’s transition potential”

• After two years or so, reports are due; unlike many federal agencies, the DOD process includes Phase III, where the ﬁrm actually implements or produces the technology,

with contracts up to and above $25 million, although many contests don’t lead to a Phase III contract at all

4There’s lots of background, and more, in Bhattacharya (2018), “An Empirical Model of R&D Procurement Contests: An Analysis of the DOD SBIR Program,” working paper.

411 • How do we model innovation contests like this?

• Putting aside multiple rounds, take the simpler view, where each ﬁrm develops a product, choosing both the quality of the product and the price they will ask for it

• In the simple model,5 there’s no uncertainty in R&D – the ﬁrms know exactly what they’re doing, and just decide how much to spend on quality

• For ﬁrm i, developing a product of quality level xi costs ci(xi);

each ﬁrm chooses both a quality level xi and a price pi to ask,

giving surplus of xi − pi to the government,

incurring a development cost of ci(xi) and then receiving payment of pi if they win

• It’s assumed the procurer will buy from whichever firm offers the highest surplus xi − pi; firms may be asymmetric, in the sense of having different cost functions, but there’s complete information; and as before, the equilibrium involves mixed strategies

• The government can design the contest, in the sense that it decides which bidders to invite, and whether to restrict the prices that the ﬁrms can ask for

• One of the main ﬁndings of this literature is that in many cases, it’s optimal for the government to invite just two ﬁrms to compete

• Basically, if you invite too many, each one has an incentive to invest very little, since they’re unlikely to win; but if you only invite one, they can charge a really high price; so under many circumstances (though not always), the optimal number to invite is exactly two

5See Che and Gale (2003), “Optimal Design of Research Contests,” American Economic Review 93.3

412 5 Grant Competitions

• The traditional view of contest design is that the designer wants to maximize eﬀort – the Navy wants ﬁrms to spend a lot developing a high-quality product, innovation prizes are supposed to spur investment in developing particular technologies, the politician wants to maximize the bribes he receives

• However, there are some settings where contests are used, where the eﬀort devoted to the contest isn’t the actual goal

• A recent paper6 discusses the scientiﬁc grant process

• Over the last 50 years, research grants have become more competitive; it’s estimated that faculty devote about 8% of their total time – and 19% of the time available for research – to grant proposals

• But generating grant proposals isn’t the point of the grant process; the proposals are just used to decide who should receive money to do the actual research

• As grants have gotten more competitive, faculty have needed to spend more and more time on grant-writing, and therefore less and less time on actual research

• This looks a lot like... rent dissipation! (except in this case, the thing being “dissipated” isn’t just the payoﬀ to the individual competing, but a public good)

• The article claims that the resources being wasted on grant competition could be on the same order of magnitude as the social value of the science actually getting done;

• they propose switching to a partly-random system – rather than awarding grants based on the best proposals, a larger fraction of them could be judged “good enough to fund,” and then the actual grants awarded randomly among this group

• This way, scientists could spend less time on grant-writing, and more time doing the actual science

6Gross and Bergstrom (2019), “Contest Models Highlight Inherent Ineﬃciencies of Scientiﬁc Funding Competi- tions,” PLOS Biology 17.1

413 6 Tennis

• Going back to contests where the eﬀort is actually the point...

• A recent paper7 considers a classic example of contests: actual sporting events

• She considers tennis tournaments, which she thinks of as being designed to get maximum eﬀort from players, as this leads to higher-quality matches

• She points out that tennis tournaments tend to give prizes to everyone who plays, even those beaten quickly in the first round, which goes against usual understandings of optimal contest design: you generally want to maximize the difference between winning and losing, to maximize the incentives for effort

• She points out, though, that many contests award prizes in multiple dimensions: a manager might give her best employee a raise and a promotion; tennis matches oﬀer players both prize money and ranking points, which get players higher rankings and invitations to more prestigious tournaments

• Her point is that diﬀerent players might value the two prizes asymmetrically: some players might care more about the prize money, others more about the ranking points

• We know from before that when one participant in a contest is much stronger than the other, the weaker competitor has little incentive to try hard, since they’ll probably lose

• Antsygina shows that if the two players value the prizes differently, giving some of the prize money to the loser will reduce the strong bidder’s effort level, but this increases the weak bidder’s incentive to try, since she now has a better chance to win; depending on the details, the net effect can be an increase in overall effort

• (She then uses her model to evaluate recent changes made to the allocation of prizes at the Australian Open, and ﬁnds that it did indeed lead to an increase in overall eﬀort, but only very marginaly) 7Anastasia Antsygina (2019), “Optimal Allocation of Multi-Dimensional Prizes in Contests with Heterogeneous Agents: Theory and an Empirical Application,” working paper

414 7 Some dumb auctions that are sort of all-pay

• I said at the outset that all-pay auctions are never really used to sell things, but are more of a way to model certain real-world competitions

• However, that’s not completely true

• There are a few websites that have set up auctions that have some all-pay characteristics, and use them to actually sell things; although I think they’re really pretty dumb, and are mostly meant to be lotteries that are called auctions to evade gambling laws

• First is the “highest unique bid auction”

• These were popular for a few years, around 2007-2009

• The idea is, you take a valuable prize – a new iPad, or a laptop, or a ﬂatscreen TV, worth several hundred dollars –

and sell it via an auction where the maximum bid is, say, $30

• The twist: bidders have to pay for each bid they submit; bidding is in one-cent increments; and the winner is the highest bid that doesn’t tie with any other bidders

• So, lots of bidders bid $30; lots of bidders bid 29.99; lots bid 29.98; a few bid 29.97; a few bid 29.96; and so on, all the way down to, say, 28.43, which only one bidder bids

• So, the guy who bid 28.43 wins the TV, and pays just 28.43 – plus a dollar for each bid he submitted, and all the losers still have to pay a dollar for each bid they made

• Like I said, these were popular for a few years – on sites called uBid, QuiBids, and several others

• Since even the losers had to pay, they have some similarity to all-pay auctions, but like I said, I think of them more as a way to gamble legally – and they seem to have completely vanished8

8For more, see Samorodnitzky, Troyer and Wool (2013), “Analyzing Unique-Bid Auction Sites for Fun And Proﬁt,” Proc. Network & Distributed System Security Symposium, and https://en.wikipedia.org/wiki/Unique bid auction

415 • What’s replaced these seem to be “penny auctions”

• Like with unique-bid auctions, you have to pay for each bid you submit; these are run as actual ascending-price auctions, with the highest bid winning, but bidding is again in one-cent increments, and there’s no ﬁxed end time: and each time someone bids, the auction is extended for another ten seconds

• These are still being used – see DealDash, for example – but the FTC has a consumer information page up about auctions like this that seems to be saying, more or less, “we’re not saying these are a scam, but, well, maybe they’re a little bit kind of a scam”9

9See https://www.consumer.ftc.gov/articles/0037-online-penny-auctions and also https://www.forbes.com/sites/nextavenue/2014/05/13/are-penny-auction-sites-seen-on-tv-for-real/

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