Auctions. Market Design. University of Notre Dame. Market Design (ND) Auctions 1 / 61

Auctions Market Design University of Notre Dame Market Design (ND) Auctions 1 / 61

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 Market Design (ND) Auctions 2 / 61

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 si is dominant for player i if, given any selection of strategies a i by i s opponents, si maximizes i s payoff If all the players are playing dominant strategies, they are playing a pure-strategy Nash equilibrium of the game Market Design (ND) Auctions 2 / 61

What are auctions? Market Design (ND) Auctions 3 / 61

What are auctions? They are games. Market Design (ND) Auctions 3 / 61

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 Market Design (ND) Auctions 3 / 61

The English Auction This is the most important game in all of economics. Market Design (ND) Auctions 4 / 61

The English Auction This is the most important game in all of economics. The price clock starts at zero Market Design (ND) Auctions 4 / 61

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. Market Design (ND) Auctions 4 / 61

The English Auction How should bidders behave in the English auction? Market Design (ND) Auctions 5 / 61

The English Auction Theorem It is a dominant strategy a to drop out of the English auction at b i = v i. a Remember, b i is a dominant strategy if, regardless of what your opponents do, you prefer one of your strategies to all others. Market Design (ND) Auctions 6 / 61

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. Market Design (ND) Auctions 7 / 61

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. Market Design (ND) Auctions 8 / 61

The English Auction Why is the English auction so important? Market Design (ND) Auctions 9 / 61

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 second-highest 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 Market Design (ND) Auctions 9 / 61

The Second-price Auction The English auction is an open format: the buyers indicate their interest in participating over time Market Design (ND) Auctions 10 / 61

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 Market Design (ND) Auctions 10 / 61

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 Market Design (ND) Auctions 11 / 61

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. Market Design (ND) Auctions 11 / 61

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 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. Market Design (ND) Auctions 11 / 61

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. Market Design (ND) Auctions 12 / 61

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) Market Design (ND) Auctions 13 / 61

The First-Price Auction: Complete Information Suppose each buyer i knows each other buyer j s value v j Market Design (ND) Auctions 14 / 61

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 Market Design (ND) Auctions 14 / 61

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? Market Design (ND) Auctions 14 / 61

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. Market Design (ND) Auctions 15 / 61

The First-Price Auction: Complete Information Suppose values and bids are on a grid, like {0,, 2,..., }, like pennies Market Design (ND) Auctions 16 / 61

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 1 v 1 (v 2 + ) > }{{} 2 (v 1 v 2 ) + 1 2 0, }{{} Win for sure, pay v 2 + Win with probability 1/2, pay v 2 or v 1 v 2 > 2 So if is sufficiently small, this is a pure-strategy Nash equilibrium of the game Market Design (ND) Auctions 16 / 61

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 Market Design (ND) Auctions 17 / 61

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 second-highest bid, v 2 Theorem (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 Market Design (ND) Auctions 17 / 61

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 Market Design (ND) Auctions 18 / 61

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 Market Design (ND) Auctions 19 / 61

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 Market Design (ND) Auctions 19 / 61

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 Market Design (ND) Auctions 20 / 61

Solving the FPA with incomplete information Each buyer solves Call Notice that max p(b)(v b) b U(v) = max p(b)(v b) b U (v) = [ p (b(v))(v b(v)) p(b(v)) ] b (v) + p(b(v)) = p(b(v)). Then This implies that p(b(v))(v b(v)) = U(v) = v 0 v 0 p(b(x))dx + U(0) p(b(x))dx b(v) = v so if we can determine p(b(x)), we ve solved for b(v). v 0 (p(b(x))dx, p(b(v)) Market Design (ND) Auctions 21 / 61

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 b (v) = v }{{} True value v 0 F (x)n 1 dx } F (v) {{ N 1 } Profit Market Design (ND) Auctions 22 / 61

Solving the FPA with incomplete information What is this expression? Well, we can use an integration by parts to re-write the bid as b (v) = v 0 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 Market Design (ND) Auctions 23 / 61

Simulations (Revenue Equivalence) https://marketdesign.shinyapps.io/simulation1/ What happens to revenue as the number of bidders increases, in particular? Market Design (ND) Auctions 24 / 61

Revenue maximization What s the worst thing that can happen to a seller in the SPA? Market Design (ND) Auctions 25 / 61

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. Market Design (ND) Auctions 25 / 61

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. Market Design (ND) Auctions 26 / 61

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. Market Design (ND) Auctions 27 / 61

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 Market Design (ND) Auctions 28 / 61

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. Market Design (ND) Auctions 28 / 61

Simulations (Optimal Reserve Prices) https://marketdesign.shinyapps.io/simulation2/ What happens to revenue as the number of bidders increases, in particular? Market Design (ND) Auctions 29 / 61

Auction-like markets ebay looks like a second price auction: http://www.ebay.com/ Market Design (ND) Auctions 30 / 61

Auction-like markets ebay looks like a second price auction: http://www.ebay.com/ 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 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) Market Design (ND) Auctions 30 / 61

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 Market Design (ND) Auctions 31 / 61

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. Market Design (ND) Auctions 32 / 61

Procurement auctions As a tool of public policy, auctions are incredibly popular. Market Design (ND) Auctions 33 / 61

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. Market Design (ND) Auctions 33 / 61

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? Market Design (ND) Auctions 34 / 61

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 Market Design (ND) Auctions 35 / 61

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: https://www.iso-ne.com/ http://wireless.fcc.gov/auctions/default.htm?job= auctions_home http://www.ppaghana.org/ Market Design (ND) Auctions 36 / 61

Example: Clean Power Plan Here s a case study that s in court right now: https://www.epa.gov/ cleanpowerplan/clean-power-plan-existing-power-plants Market Design (ND) Auctions 37 / 61

Example: Clean Power Plan Here s a case study that s in court right now: https://www.epa.gov/ cleanpowerplan/clean-power-plan-existing-power-plants 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 market-based mechanisms Market Design (ND) Auctions 37 / 61

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 Market Design (ND) Auctions 38 / 61

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 Market Design (ND) Auctions 39 / 61

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. Market Design (ND) Auctions 40 / 61

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. Market Design (ND) Auctions 41 / 61

The Clean Power Plan How do we adapt the LRB to the CPP problem? Market Design (ND) Auctions 42 / 61

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. Market Design (ND) Auctions 42 / 61

Auctions for multiple goods with varying quality We ve covered the cases of one good, and a bunch of homogeneous goods Market Design (ND) Auctions 43 / 61

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 Market Design (ND) Auctions 43 / 61

Sponsored search auctions Whenever you run a search in Google, there s an auction: Market Design (ND) Auctions 44 / 61

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 Market Design (ND) Auctions 45 / 61

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. Market Design (ND) Auctions 45 / 61

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-as-bid 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. Market Design (ND) Auctions 46 / 61

Example Suppose there are two slots on a page and three advertisers. Market Design (ND) Auctions 47 / 61

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. Market Design (ND) Auctions 47 / 61

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. Market Design (ND) Auctions 47 / 61

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. Market Design (ND) Auctions 48 / 61

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. Market Design (ND) Auctions 48 / 61

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 $10 200 -$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 $10 100 - $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 Market Design (ND) Auctions 49 / 61

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. Once a buyer exits, he cannot re-enter 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 Market Design (ND) Auctions 50 / 61

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? 10 200 p 1 = 10 100 2 100, or p 1 = 2000 1000 + 200 = 1200. When is Buyer 2 indifferent between the first and second slots? 8 200 p 2 = 8 100 2 100, or p 2 = 1600 800 + 200 = 1000. So 2 drops out first at a price of $1000. Market Design (ND) Auctions 51 / 61

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: 10 200 1000 = $1000 > 10 100 200 = $800. Market Design (ND) Auctions 52 / 61

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 Market Design (ND) Auctions 53 / 61

Local envy-freeness Definition 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. Market Design (ND) Auctions 54 / 61

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. Market Design (ND) Auctions 55 / 61

Solving the GSP Suppose we use a system like the English auction, rather than the second-price 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 ). Market Design (ND) Auctions 56 / 61

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. Market Design (ND) Auctions 57 / 61

Adwords Figuring out the right bid here is somewhat difficult. Market Design (ND) Auctions 58 / 61

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: Market Design (ND) Auctions 58 / 61

Adwords Google s Adwords just asks you to submit your r i, and it figures out your bid for you: Market Design (ND) Auctions 59 / 61