Game theory review. The English Auction How should bidders behave in the English auction?

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1 Game theory review A game is a collection of players, the actions those players can take, and their preferences over the selection of actions taken by all the players A strategy s i is dominant for player i if, given any selection of strategies a i by i s opponents, s i maximizes i s payoff If all the players are playing dominant strategies, they are playing a purestrategy Nash equilibrium of the game We d like to design games for which the players dominant strategies are simple, and the outcomes are desirable. Today we ll look at the quintessential example of such a game. What are auctions? They are games. Players: the bidders, denoted i = 1, 2,..., N Actions: their bids, b 1, b 2,..., b N Payoffs: if buyer i wins, he gets a payoff v i t i, where v i is bidder i s value and t i (b 1, b 2,..., b N ) is a payment (not necessarily his bid), and if buyer i loses, he gets a payoff of zero The English Auction This is the most important game in all of economics. The price clock starts at zero The auctioneer raises the price clock slowly, allowing agents to indicate whether they want to continue or withdraw When the second-to-last buyer withdraws from the auction, the winner s payment is set equal to the current price on the clock (call it b (2) ), and the winner is the buyer who failed to withdraw Let s play a few rounds. The English Auction How should bidders behave in the English auction? The English Auction Theorem 1. It is a dominant strategy 1 to drop out of the English auction at b i = v i. If a buyer bids b i = v i, we say he bids sincerely or honestly. 1 Remember, b i is a dominant strategy if, regardless of what your opponents do, you prefer one of your strategies to all others. 1

2 The English Auction The key to the auction is that you control the likelihood you win, but not the price you pay, which will always be the lowest price you could pay and still be a winner (meditate on this). What if i was to stay in until b i > v i? There s two cases: (1) i would have won when bidding b i = v i, and (2) i would have lost when bidding b i = v i. Case 1: if i wins by bidding b i = v i, then the next-highest bid is less than v i, so i makes the same payment when bidding b i > v i and i s payoff doesn t change. Case 2: if i loses by bidding b i = v i, then the next-highest bid is greater than v i, so overbidding to win leads i making a loss, since b i > v i, or i still loses, and his payoff doesn t change. In either case, overbidding is weakly dominated by bidding sincerely. The English Auction What if i was to bid b i < v i? There s two cases: (1) i would have won when bidding b i = v i, and (2) i would have lost when bidding b i = v i. Case 1: if i wins by bidding b i = v i, then the next-highest bid is less than b i, so i makes the same payment when bidding b i < v i if i still wins, but risks losing by cutting his bid too far. Case 2: if i loses by bidding b i = v i, then the next-highest bid is greater than b i, by cutting his bid, i still doesn t win. In either case, underbidding is weakly dominated by bidding sincerely. Since b i > v i and b i < v i are both dominated by bidding b i = v i, b i = v i is a dominant strategy. The English Auction Why is the English auction so important? Dominant strategies: the buyers have a dominant strategy to bid honestly Efficiency: the buyer with the highest value wins Stability: if the price paid by the winner were any lower, some other buyer could rightfully object Individual rationality: since b i b (2), any winner s payoff is v i b (2) v i b i 0, so no buyer will regret winning Privacy preserving: if the auction ends then the buyer with the secondhighest value drops out, no one ever learns the winner s true value Robust: players optimal strategies do not depend on their beliefs about their opponents It is easy to participate, and the outcome satisfies many desirable properties. 2

3 The Second-price Auction The English auction is an open format: the buyers indicate their interest in participating over time Sometimes, this is infeasible or undesirable, and a closed or silent format is adopted It retains most of the positive features of the English auction, except, potentially, for privacy preservation of the winner s value The Second-price Auction In the second price auction (SPA), Each buyer i submits a bid b i The highest bidder wins, but pays the second-highest bid A buyer bids honestly if b i = v i. Theorem 2. It is a dominant strategy to bid honestly in the second-price auction, and it is outcome equivalent to the English auction: the winner and payment to the seller are the same. The First-price Auction Consider the first-price auction (FPA): Each buyer i submits a bid b i A winner is chosen randomly from the set of buyers placing the highest bid. The winner pays his bid. How do people play this game? Does this raise more or less revenue than the second-price auction? The First-price Auction The Dutch or first-price auction is much harder to analyze Optimal play depends on your beliefs about your opponents types, and what you need to bid to win: honesty isn t a dominant strategy If information about your opponents is complete, it s difficult to find a statisfying pure-strategy Nash equilibrium because of the possibility of ties (for reasons we ll discuss) 3

4 The First-Price Auction: Complete Information Suppose each buyer i knows each other buyer j s value v j For simplicity, order the values v 1 > v 2 >... > v N Bidding more than one s value is a dominated strategy, since if b i > v i, then i s payoff is v i b i < 0 If buyer 2 bids b 2 < v 2, then buyer 1 could safely bid b 1 = v 2 and win but then what does 2 bid? Problem: if 2 bids b 2 = v 2, then 1 wants to bid the smallest amount less than v 2, so we need to discretize the set of bids that players can submit The First-Price Auction: Complete Information Here s our strategy: If v 1 and v 2 both bid v 2, then 1 wins with probability 1/2, but to win for sure, 1 needs to outbid 2 strictly Put bids on a very fine grid. Then if there is a bid v 1 > b > v 2 and b 2 = v 2, we can determine if buyer 1 prefers b over tying with 2 at a bid of v 2 and winning with probability 1/2. The First-Price Auction: Complete Information Suppose values and bids are on a grid, like {0,, 2,..., }, like pennies Suppose buyer 1 bids b 1 = v 2 +, buyer 2 bids b 2 = v 2, and all other buyers bid something weakly less than their true value Then buyer 1 prefers to bid v 2 + rather than v 2 if or 1 v 1 (v 2 + ) > }{{} 2 (v 1 v 2 ) , }{{} Win for sure, pay v 2 + Win with probability 1/2, pay v 2 v 1 v 2 2 > So if is sufficiently small, this is a pure-strategy Nash equilibrium of the game (Why doesn t 2 bid b 2 < v 2? Then 1 would bid b 1 < v 2, and then 2 could raise his bid to b 1 + and win instead of 1.) 4

5 Revenue Equivalence Note that in the FPA, the winner bids (approximately) b 1 = v 2, so the seller s revenue in the FPA is v 2 Note that in the SPA, everyone bids b i = v i, and the revenue is the secondhighest bid, v 2 Theorem 3 (Revenue Equivalence). The FPA and SPA raise the same amount of revenue, equal to the second-highest value. This is a fundamental idea in mechanism design: you can change the rules all you want, but players will respond to your design and can potentially unravel the effects of your decisions Incomplete information It s unrealistic that buyers know each others values If they don t, we re facing a situation with incomplete information We imagine each buyer draws a value v i, which has a cumulative distribution function F (v i ) with a probability density function f(v i ) The probability I win is then the probability that I drew the highest type The First-price Auction: Incomplete Information In the FPA, buyers solve max b i p(b i )(v i b i ) where p(b i ) is the probability of winning, given a bid of b i Is honest bidding a good strategy? The FONC is dp(b i ) db i (v i b i ) }{{} Benefit of a higher likelihood of winning and getting a payoff just like a monopolist. p(b i ) }{{} = 0, Cost of paying a higher bid, conditional on winning 5

6 Solving the FPA with incomplete information There are two steps: Determine the buyers payoffs, given their values Determine the probability they win, given their values Solving the FPA with incomplete information Each buyer solves Call max p(b)(v b) b U(v) = max p(b)(v b) b Notice that U (v) = [p (b(v))(v b(v)) p(b(v))] b (v) + p(b(v)) = p(b(v)). Then This implies that U(v) = v 0 p(b(x))dx + U(0) p(b(v))(v b(v)) = v 0 v 0 p(b(x))dx b(v) = v (p(b(x))dx, p(b(v)) so if we can determine p(b(x)), we ve solved for b(v). Solving the FPA with incomplete information Recall that F (v) = pr[buyer s value v]. The probability that buyer i outbids buyer j is pr[b(v i ) > b(v j )] = pr[b 1 (b(v i )) > v j ] = pr[v i > v j ] = F (v i ) Then the probability that buyer i outbids every other buyer is p(b(v i )) = F (v i ) N 1 And v b 0 (v) = }{{} v F (x)n 1 dx F (v) N 1 True value }{{} Profit 6

7 Solving the FPA with incomplete information What is this expression? Well, we can use an integration by parts to re-write the bid as v b 0 (v) = xnf (x)n 1 f(x)dx F (v) N 1 This is a conditional expectation: it is the expected value of the highest of the N 1 other draws, given that x v This is an expression for the second-highest value, given that v is the highest value So revenue equivalence holds even with incomplete information: the winning bid is the expectation of the second highest-bid, given the winner s value Simulations (Revenue Equivalence) What happens to revenue as the number of bidders increases, in particular? Revenue maximization What s the worst thing that can happen to a seller in the SPA? Some buyer i submits a huge bid, but......no one else does, so the good is sold for a very low price despite the buyer having a really high value for it. Basically, the seller inadvertently faces a monopsonist, and would be committed to trading at a price of zero. To protect sellers from bad outcomes like this, we add reserve prices: a price r below which bids are disqualified. Revenue maximization In the second-price auction with a reserve price (SPAR), The seller sets a reserve price r Each buyer i submits a bid b i The highest bid greater than the reserve price wins. The winner pays the maximum of the second-highest bid and the reserve price. A buyer bids honestly if b i = v i. Theorem 4. It is a dominant strategy to bid honestly in the SPAR. 7

8 Revenue maximization In the first-price auction with a reserve price (FPAR), The seller sets a reserve price r Each buyer i submits a bid b i A winner is chosen randomly from the set of buyers making the highest bid above the reserve price. The winner pays his bid. What is the optimal reserve price? Imagine you knew there was only one sincere buyer: everyone else has values below the reserve price, so they are disqualified Then you are simply posting a price, r, at which the buyer purchases or not: if v > r, the buyer makes the purchase, and fails to do so otherwise Then the seller s profits are pr[v > r]r = (1 F (r))r, and the optimal reserve price satisfies or (1 F (r)) f(r)r = 0, r 1 F (r) f(r) = 0 For the uniform case, F (v) = v, so r = 1/2. Simulations (Optimal Reserve Prices) What happens to revenue as the number of bidders increases, in particular? Auction-like markets ebay looks like a second price auction: The seller can set a secret reserve price, buyers can make open bids or use a proxy bid (a robot that bids for them up to a certain limit), and the auction ends at a pre-specified time Bid-sniping: enter bids right at the end of the auction in order to steal the good from the current standing high bidder 8

9 40 percent of all ebay-computers auctions and 59 percent of all ebay- Antiques auctions as compared to about 3 percent of both Amazon-Computers and Amazon-Antiques auctions, respectively, have last bids in the last 5 minutes. The pattern repeats in the last minute and even in the last ten seconds. In the 240 ebay-auctions, 89 have bids in the last minute and 29 in the last ten seconds. In Amazon, on the other hand, only one bid arrived in the last minute. (Roth and Ockenfels, 2002) Auction-like markets In a penny auction or all-pay auction, we start at a price of zero. Each bidder pays a bid increment to stay in. Once all a bidder s competitors have dropped out, he is declared the winner and given the good This is a popular ecommerce business model It is totally evil. It is not an auction, it is a war of attrition. Auction-like markets The Better Business Bureau warns consumers, although not all penny auction sites are scams, some are being investigated as online gambling. BBB recommends you... know exactly how the bidding works, set a limit for yourself, and be prepared to walk away before you go over that limit. The idea is that the goods might sell for low prices: $30 for a tv, say. But hundreds or thousands of people each bid a few bucks on it, so the company is making a ton of money in the background. These losses add up for the poor people involved. It s really a lottery. Procurement auctions As a tool of public policy, auctions are incredibly popular. Suppose the government is trying to procure some good (like a bridge or computer or fighter plane) that it values at v. It wants to buy at the lowest price it can. The reverse auction or procurement auction is the game where i = 1, 2,..., N sellers each submit a bid b i The lowest bidder wins, and is paid the second-highest bid We can also impose a reserve price: if all the losing bids are above r, the winner is paid only r. The efficient reserve price would be that any winning bid must be less than v, so the government doesn t pay more than the good is worth. 9

10 The FPAR and SPAR Now you know everything about how to sell one unit of a good (revenue equivalence, dominant strategy implementation, reserve prices, etc) What about multiple units of a homogeneous good? What about single units of varying quality? What about multiple, heterogeneous goods? Multi-unit auctions Let s start with a simple multi-unit problem: there are N buyers, but each buyer only wants one unit, and the seller has K units available Examples: tickets to events, licenses/permits, over-booked seats on airplanes Multi-unit auctions There are a lot of important multiple unit auctions, where the market designer is looking to sell or buy more than one unit at a time: h omehttp : // But in the SPAR, only one unit is sold or purchased at a time. We want to sell or buy more than one unit at a time. Example: Clean Power Plan Here s a case study that s in court right now: Using the Clean Air Act, the Federal government has set emissions targets for power generation for each state in the country that need to be met by 2022, 2029, and 2030 The reality of the law is that it requires each state to close a certain number of coal-burning power plants The Federal Government has advocated using cap-and-trade or other marketbased mechanisms Many actors have sued the Federal Government over this; regardless of how those lawsuits are decided, many actors will then sue the states when they try to implement their plans 10

11 Example: Clean Power Plan The expected discounted profit of a plant to owner i is π i, i = 1, 2,..., N; suppose they are all equally dirty, to keep the problem simple, but they have varying levels of investment that make some more profitable/efficient than others The market is individually rational only if the owner receives at least π i for agreeing to close his plant down; otherwise there s a lawsuit The government has told us we need to close down at least K plants, K < N We want to find the cheapest way to induce the least profitable plants to exit the market, such that we satisfy the Federal Government s emissions constraint Multiple units The second price auction gives us an important principle: if you control the likelihood you win, but not the price you pay, it s possible to give you a dominant strategy to bid honestly We can only implement the efficient outcome (closing down the least profitable plants) if we know the agents true valuations, which are only known to the agents themselves But we can extend the basic idea of the SPAR pretty easily here: all of the winners pay (receive) the bid of one of the losers Highest-Rejected Bid (HRB) Auctions Suppose we want to sell K units to N buyers, N > K: All the buyers i = 1, 2,..., N submit a bid, b i. Order the bids from highest to lowest, b (1) b (2)... b (N). The, at most, K buyers with bids above a reserve price r win, but they all pay the maximum of the highest losing bid or the reserve price. A winner then gets a payoff of v i max{b (K+1), r}, where v i is i s value of winning the good. Notice, it is a dominant strategy to bid honestly in the HRB auction, just like the SPAR. 11

12 Lowest-Rejected Bid (LRB) Auctions Suppose we want to buy K units from N sellers, N > K: All the sellers i = 1, 2,..., N submit a bid, b i. Order the bids from lowest to highest, b (1) b (2)... b (N). Then, at most, K sellers with bids below a reserve price r win, but they all receive the minimum of the lowest losing bid or the reserve price. A winner then gets a payoff of min{b (K+1), r} c i, where c i is i s cost of providing the good. Again, it is a dominant strategy to bid honestly in the LRB auction. The Clean Power Plan How do we adapt the LRB to the CPP problem? So we can induce the least profitable/most inefficient plants to leave the market, and we implement the efficient outcome in dominant strategies. Auctions for multiple goods with varying quality We ve covered the cases of one good, and a bunch of homogeneous goods Heterogeneous goods when there are multiple goods of varying quality or type, like a lot of different Van Gogh sketches are generally much harder to understand and design We ll focus on a special case you use everyday Sponsored search auctions 12

13 Whenever you run a search in Google, there s an auction: Sponsored search auctions Google and Yahoo! both use a similar mechanism to sell sponsored search results, called the Generalized Second-Price Auction, or GSP auction Google s total revenue in 2005 was $6.14 billion, over 98 percent of which came from sponsored search auctions. Yahoo! s total revenue was $5.26 billion, and over half is estimated to come from sponsored search. History of Internet Advertising From 1994 to 1997, Internet ads were typically sold by fixed prices for the number of times the ad would be shown, and prices were determined by negotiation In 1997, Overture (now part of Yahoo!) started selling ads on a per-click basis for a particular keyword, mainly through banner advertisements. Bidding determined the order in which ads were shown, but used paid-asbid pricing. This led to price and rank fluctuations and instability in the market. In 2002, Google introduced Adwords, which used second-pricing instead of paid-as-bid pricing, leading to much more stable prices and rankings over time. As market designers, we re interested in why the Adwords design is so much more stable and successful. 13

14 Example Suppose there are two slots on a page and three advertisers. An ad in the first slot receives 200 clicks per hour, while the second slot gets 100. Advertisers 1, 2, and 3 have values per click of $10, $8, and $2, respectively. Example: Overture design (pay-as-bid) Suppose advertiser 2 bids $2.01, to guarantee that he gets a slot. Then advertiser 1 will not want to bid more than $2.02 to get the top spot. But then advertiser 2 will want to revise his bid to $2.03 to get the top spot, advertiser 1 will in turn raise his bid to $2.04, and so on. This over-cutting will not lead to an equilibrium of any kind: someone can always deviate and improve their payoff. This is why Overture had so much instability over time. Example: Next-pricing (An ad in the first slot receives 200 clicks per hour, while the second slot gets 100. Advertisers 1, 2, and 3 have values per click of $10, $8, and $2, respectively.) Suppose we have the agents bid in terms of revenue-per-click, the top bidder gets the top slot and pays what the next highest bidder would have been worth in the top slot, and so on. Does this induce honest bidding? The second person in the top slot would have gotten $8 200 = $1600, so the top person s payoff would be $ $8 200 = $400. But if the top person submitted a bid instead of, say, $3, he would have gotten the second slot at a price of $200, and made a profit of $ $2 100 = $800. So the top advertiser doesn t want the top spot if everyone bids honestly. So honesty isn t a dominant strategy in next-price auctions Generalized Second Pricing (GSP) What does Google do? We ll first think of it like an open format, like the English auction: Buyers indicate whether they are in or out at the current price, starting from a price of zero The auctioneer raises the clock slowly. re-enter Once a buyer exits, he cannot 14

15 Once the number of bidders equals the number of items, we begin awarding items: then the k-th person drops out, he gets the k-th item at the price at which the k + 1-st person dropped out Generalized Second Pricing (GSP) (An ad in the first slot receives 200 clicks per hour, while the second slot gets 100. Advertisers 1, 2, and 3 have values per click of $10, $8, and $2, respectively.) The first drop-out occurs when the clock-price reaches $2 100, and the third buyer exits. This sets the price of the second good. The remaining two buyers have to decide at what point to give up. At what price is Buyer 1 indifferent between the first and second slots? p 1 = , or p 1 = = When is Buyer 2 indifferent between the first and second slots? p 2 = , or p 2 = = So 2 drops out first at a price of $1000. Generalized Second Pricing (GSP) So 1 gets the first slot at a price of $1000, 2 gets the second slot at a price of $200. Notice that at those prices, advertiser 1 prefers his slot to the second one, unlike the next-price auction: = $1000 > = $800. A more formal model There are i = 1, 2,..., N advertisers The click-through-rates (CTR) for the slots are α 1 > α 2 > α N Each advertiser has a value r i for each click. Advertiser i s payoff from the k-th slot at a bid of b k is α k }{{} Click-through rate r }{{} i Revenue per click This is called pay-per-click pricing. p }{{} k Price at the k-th slot 15

16 Local envy-freeness Definition 5. A set of prices (p 1, p 2,..., p N ) is (locally) envy-free if the advertiser in each slot k prefers his CTR and price to the CTR and price of the k 1-st and k + 1-st advertisers, or for i N, α k 1 (r k p k 1 ) α k (r k p k ) α k+1 (r k p k+1 ). This is a stability or fairness concept: no one wants to trade their slot with the person above or below them, given what everyone else is getting. Local envy-freeness Global envy-freeness If the assignments of slots and prices are locally envy-free, then for k and m we have the set of inequalities α k (r k p k ) α k 1 (r k p k 1 ) α k 1 (r k 1 p k 1 ) α k 2 (r k 1 p k 2 ). α m+1 (r m+1 p m+1 ) α m (r m+1 p m ). Raise all the r j, j k, terms to r k, and add the inequalities to get: α k (r k p k ) α m (r k p m ), so that k doesn t want to deviate to m. Reversing the inequalities and lowering the r j s proves the same for m > k. So if we can find locally envy-free prices, they are globally envy-free: myopic bidding will be optimal. Solving the GSP Suppose we use a system like the English auction, rather than the secondprice auction, to solve for the bidding: players stay in as long as they like, and indicate when they want to drop out. The advertiser who drops out k-th receives the k-th slot at the price at which the k 1-st person dropped out. Suppose there are only two agents left, who both know they face a price p 2 for the second slot and the time at which one or the other drops out determines the price for the first slot, b 1. Then they are indifferent between dropping out and continuing if implying the optimal bid is α 2 (r i p 2 ) = α 1 (r i b 1 ), b 1 = r i α 2 α 1 (r i p 2 ). 16

17 Solving the GSP At the k-th stage, there are N k advertisers left vying for at least the k-th slot. The price for the k + 1-st slot, p k+1 is known. Then the advertisers are indifferent between dropping out and continuing if implying the optimal bid is α k+1 (r i p k+1 ) = α k (r i b k ), b k = r i α k+1 α k (r i p k+1 ). Working backwards in this way, we construct a set of bids b k (r i, α, p) that is locally envy free, and therefore globally envy free. Therefore, there are no profitable deviations, and these strategies are a Nash equilibrium. Adwords Figuring out the right bid here is somewhat difficult. Google s Adwords just asks you to submit your r i, and it figures out your bid for you: Adwords Google s Adwords just asks you to submit your r i, and it figures out your bid for you: 17

18 GSP There is an easy and relatively intuitive way to auction off goods differentiated in quality when agents have unit demand. We talked about sponsored search, but it would work with anything. We used a new kind of condition envy-freeness to show that being honest was an equilibrium The more general definition is stability: an outcome is stable if no subset of agents can get together and jointly deviate to improve their payoffs. This will be very important in a matching setting. Conclusion When the issue is incomplete information the buyers values or sellers costs are unknown we can implement efficient or profit-maximizing allocations using auctions So we have a lot to say about the one-unit case... we have some things to say about the case where we have multiple identical units to sell but buyers only want one unit... we have some things to say about the case where we have multiple goods of varying quality but buyers only want one unit... How do we go beyond this cases to situations where buyers or sellers can have complex demand curves over many similar or different goods? 18