Mechanism Design. Terence Johnson. December 7, University of Notre Dame. Terence Johnson (ND) Mechanism Design December 7, / 44

Mechanism Design Terence Johnson University of Notre Dame December 7, 2017 Terence Johnson (ND) Mechanism Design December 7, 2017 1 / 44

Market Design vs Mechanism Design In Market Design, we have a very specific market auctions to buy/sell, one-to-one matching, re-allocation of goods without money and analyze games or algorithms to determine what desirable properties they have, so that we develop a tool box that can be used in new markets In Mechanism Design, we go in the opposite direction: given a set of desirable properties and preferences, what outcomes can be achieved? Terence Johnson (ND) Mechanism Design December 7, 2017 2 / 44

Market Design vs Mechanism Design In Market Design, we have a very specific market auctions to buy/sell, one-to-one matching, re-allocation of goods without money and analyze games or algorithms to determine what desirable properties they have, so that we develop a tool box that can be used in new markets In Mechanism Design, we go in the opposite direction: given a set of desirable properties and preferences, what outcomes can be achieved? Application: we re going to look at how Facebook uses one approach to ad sales (the Vickrey-Clarke-Groves mechanism) while Google uses something else (Generalized Second Pricing), and why. Terence Johnson (ND) Mechanism Design December 7, 2017 2 / 44

The Revelation Principle The key tool in mechanism design is the Revelation Principle: every equilibrium in every game where the agents have private information can be thought of as a different game, in which the agents make a report (not necessarily truthfully) about their private information to the mechanism or the referee, and then the mechanism or referee plays the game for them as they would have played Terence Johnson (ND) Mechanism Design December 7, 2017 3 / 44

The Revelation Principle For example, in the second-price auction, we can reframe the game as follows: Make a report of your true value to me, about which you can obviously lie. I will give the good to the person reporting the highest value, and they will pay the second-highest report. It is its own direct mechanism. Terence Johnson (ND) Mechanism Design December 7, 2017 4 / 44

The Revelation Principle For example, in the second-price auction, we can reframe the game as follows: Make a report of your true value to me, about which you can obviously lie. I will give the good to the person reporting the highest value, and they will pay the second-highest report. It is its own direct mechanism. What about the First Price Auction? Terence Johnson (ND) Mechanism Design December 7, 2017 4 / 44

The Model There are i = 1,..., N agents Terence Johnson (ND) Mechanism Design December 7, 2017 5 / 44

The Model There are i = 1,..., N agents Each player i has a privately known type, θ i Terence Johnson (ND) Mechanism Design December 7, 2017 5 / 44

The Model There are i = 1,..., N agents Each player i has a privately known type, θ i There is an outcome, x = (x 1, x 2,..., x N ): this could be how many units of the good each agent i gets, whether i gets a kidney, etc., or it could be whether an airport is built (for all i, x i = 1) or not (for all i, x i = 0) Terence Johnson (ND) Mechanism Design December 7, 2017 5 / 44

The Model There are i = 1,..., N agents Each player i has a privately known type, θ i There is an outcome, x = (x 1, x 2,..., x N ): this could be how many units of the good each agent i gets, whether i gets a kidney, etc., or it could be whether an airport is built (for all i, x i = 1) or not (for all i, x i = 0) Each player s payoff is v i (x, θ i ) t i, and t i is a payment Terence Johnson (ND) Mechanism Design December 7, 2017 5 / 44

The Model This includes a bunch of examples: Auctions: v i (x, θ i ) = x i θ i, θ i is i s value for the good, x i is the probability that i wins, t i is i s payment Public goods: v i (x, θ i ) = x i θ i, θ i is i s value for the good, x i is whether a bridge/railroad/airport/climate change agreement happens, t i is i s payment Compensation: θ i < 0 is i s disutility of labor, x i is the level of effort requested by the firm, and t i is i s wage Matching: v i (x, θ i ) = j i x ijθ i θ j, x ij is the probability i and j are matched, θ i is i s marginal utility of partner quality, and t i is a costly signal of i s suitability as a mate Terence Johnson (ND) Mechanism Design December 7, 2017 6 / 44

Direct Mechanisms While θ i is agent i s true type, agent i can always lie and report any alternative type, ˆθ i Terence Johnson (ND) Mechanism Design December 7, 2017 7 / 44

Direct Mechanisms While θ i is agent i s true type, agent i can always lie and report any alternative type, ˆθ i A direct mechanism prescribes an allocation and a set of payments x(ˆθ) = (x 1 (ˆθ), x 2 (ˆθ),..., x N (ˆθ)) t(ˆθ) = (t 1 (ˆθ), t 2 (ˆθ),..., t N (ˆθ)). Terence Johnson (ND) Mechanism Design December 7, 2017 7 / 44

Direct Mechanisms While θ i is agent i s true type, agent i can always lie and report any alternative type, ˆθ i A direct mechanism prescribes an allocation and a set of payments Agent i s payoff, then is x(ˆθ) = (x 1 (ˆθ), x 2 (ˆθ),..., x N (ˆθ)) t(ˆθ) = (t 1 (ˆθ), t 2 (ˆθ),..., t N (ˆθ)). U i (ˆθ i, ˆθ i, θ i ) = v i (x(ˆθ i, ˆθ i ), θ i ) t i (ˆθ i, ˆθ i ) Terence Johnson (ND) Mechanism Design December 7, 2017 7 / 44

Direct Mechanisms While θ i is agent i s true type, agent i can always lie and report any alternative type, ˆθ i A direct mechanism prescribes an allocation and a set of payments Agent i s payoff, then is x(ˆθ) = (x 1 (ˆθ), x 2 (ˆθ),..., x N (ˆθ)) t(ˆθ) = (t 1 (ˆθ), t 2 (ˆθ),..., t N (ˆθ)). U i (ˆθ i, ˆθ i, θ i ) = v i (x(ˆθ i, ˆθ i ), θ i ) t i (ˆθ i, ˆθ i ) This lets us talk about how what outcomes (x, t) can be achieved, without worrying about the actual game (first or second price auction? TTC or SRD? etc) Terence Johnson (ND) Mechanism Design December 7, 2017 7 / 44

Direct Mechanisms A direct mechanism is individually rational if, for all i, θ i, θ i, U i (θ i, θ i, θ i ) 0, so no one would prefer to opt out of participating and get a payoff of zero Terence Johnson (ND) Mechanism Design December 7, 2017 8 / 44

Direct Mechanisms A direct mechanism is individually rational if, for all i, θ i, θ i, U i (θ i, θ i, θ i ) 0, so no one would prefer to opt out of participating and get a payoff of zero A direct mechanism is budget balanced if, for all θ, N t i (θ) = 0, i=1 so that whatever types are reported, the mechanism doesn t lose money Terence Johnson (ND) Mechanism Design December 7, 2017 8 / 44

Direct Mechanisms A direct mechanism is individually rational if, for all i, θ i, θ i, U i (θ i, θ i, θ i ) 0, so no one would prefer to opt out of participating and get a payoff of zero A direct mechanism is budget balanced if, for all θ, N t i (θ) = 0, i=1 so that whatever types are reported, the mechanism doesn t lose money A direct mechanism is efficient if, for all θ, N i=1 v i (x(θ), θ i ) = max x N v i (x, θ i ) i=1 so that the mechanism maximizes gross welfare Terence Johnson (ND) Mechanism Design December 7, 2017 8 / 44

Direct Mechanisms A direct mechanism is incentive compatible or implements truth-telling as a Nash equilibrium if, for all i, θ i, θ i, ˆθ i, U i (θ i, θ i, θ i ) U i (ˆθ i, θ i, θ i ), so no one wants to lie about their type if their opponents are honest A direct mechanism implements truth-telling as a weakly dominant strategy if, for all i, θ i, ˆθ i, ˆθ i, U i (θ i, ˆθ i, θ i ) U i (ˆθ i, ˆθ i, θ i ), so no one wants to lie about their type, regardless of what their opponents report Terence Johnson (ND) Mechanism Design December 7, 2017 9 / 44

The SPAR as a direct mechanism For the second price auction, the direct mechanism is { 1, ˆθ i > max j i ˆθ j x i (ˆθ i, ˆθ i ) = 0, otherwise and t i (ˆθ i, ˆθ i ) = { max j i ˆθj, ˆθi > max j i ˆθj 0, otherwise Is this mechanism individually rational? Efficient? Budget-balanced? Incentive compatible? Terence Johnson (ND) Mechanism Design December 7, 2017 10 / 44

Implementation Our goal is to implement different patterns of behavior, depending on the private information held by the agents. Terence Johnson (ND) Mechanism Design December 7, 2017 11 / 44

Implementation Our goal is to implement different patterns of behavior, depending on the private information held by the agents. Because of the Revelation Principle, if there exists a direct mechanism that implements some desirable pattern of behavior, we can find a game (like the SPA or the FPA) that implements the same outcome There are three commons kinds of implementation Dominant Strategy: Regardless of the types of the other agents, you want to participate honestly Incentive Compatible or Nash or Ex Post Implementation: Even if you knew the types of the other agents, you would still want to participate honestly Bayesian Incentive Compatible: Given your beliefs about your opponents, you want to participate honestly under the assumption that they do as well Terence Johnson (ND) Mechanism Design December 7, 2017 11 / 44

Implementation Our goal is to implement different patterns of behavior, depending on the private information held by the agents. Because of the Revelation Principle, if there exists a direct mechanism that implements some desirable pattern of behavior, we can find a game (like the SPA or the FPA) that implements the same outcome There are three commons kinds of implementation Dominant Strategy: Regardless of the types of the other agents, you want to participate honestly Incentive Compatible or Nash or Ex Post Implementation: Even if you knew the types of the other agents, you would still want to participate honestly Bayesian Incentive Compatible: Given your beliefs about your opponents, you want to participate honestly under the assumption that they do as well We focus on the first two because they don t require us to specify agents beliefs about one another, and the math is easier Terence Johnson (ND) Mechanism Design December 7, 2017 11 / 44

Dominant Strategy Implementation Let s look at the second price auction one more time, and recall that the SPA is fair or stable. Terence Johnson (ND) Mechanism Design December 7, 2017 12 / 44

Dominant Strategy Implementation Let s look at the second price auction one more time, and recall that the SPA is fair or stable. We can rewrite the auction as follows: x SPA i (v i, v i ) = { 1, v i > max j i v j 0, otherwise and { max j i v j, v i max j i v j t i (v i, v i ) = 0, otherwise Terence Johnson (ND) Mechanism Design December 7, 2017 12 / 44

Dominant Strategy Implementation Let s look at the second price auction one more time, and recall that the SPA is fair or stable. We can rewrite the auction as follows: and x SPA i (v i, v i ) = { 1, v i > max j i v j 0, otherwise { max j i v j, v i max j i v j t i (v i, v i ) = 0, otherwise Note that the payment here is equal to social welfare, if agent i refused to participate: we would give the good to the agent with the next highest value Terence Johnson (ND) Mechanism Design December 7, 2017 12 / 44

Dominant Strategy Implementation Let s look at the second price auction one more time, and recall that the SPA is fair or stable. We can rewrite the auction as follows: and x SPA i (v i, v i ) = { 1, v i > max j i v j 0, otherwise { max j i v j, v i max j i v j t i (v i, v i ) = 0, otherwise Note that the payment here is equal to social welfare, if agent i refused to participate: we would give the good to the agent with the next highest value This is the key idea of the Vickrey-Clarke-Groves mechanism Terence Johnson (ND) Mechanism Design December 7, 2017 12 / 44

Dominant Strategy Implementation Let s introduce a report, which means that agent i is simply opting out of participating Terence Johnson (ND) Mechanism Design December 7, 2017 13 / 44

Dominant Strategy Implementation Let s introduce a report, which means that agent i is simply opting out of participating Define the gross welfare of the other agents given i s report, W i (θ i, θ i ) = j i v j (x(θ i, θ i ), θ j ) and the gross welfare of the other agents without i s report, W \i (, θ i ) = max v j (x (, θ i ), θ j ) x j i Terence Johnson (ND) Mechanism Design December 7, 2017 13 / 44

Dominant Strategy Implementation Consider the following mechanism, called the Vickrey-Clarke-Groves mechanism or VCG mechanism: and the payment for each i, x (ˆθ) = max x I v i (x, ˆθ i ) i=1 t i (ˆθ) = W \i (, ˆθ i ) W i (ˆθ i, ˆθ i ) So, we ll take the efficient action given the reports, and charge each person who participates the difference between gross welfare of the other agents without their report and gross welfare of the other agents given their report Terence Johnson (ND) Mechanism Design December 7, 2017 14 / 44

Dominant Strategy Implementation Consider the following mechanism, called the Vickrey-Clarke-Groves mechanism or VCG mechanism: and the payment for each i, x (ˆθ) = max x I v i (x, ˆθ i ) i=1 t i (ˆθ) = W \i (, ˆθ i ) W i (ˆθ i, ˆθ i ) So, we ll take the efficient action given the reports, and charge each person who participates the difference between gross welfare of the other agents without their report and gross welfare of the other agents given their report We are internalizing the externality by charging each participant their impact on gross welfare Terence Johnson (ND) Mechanism Design December 7, 2017 14 / 44

Dominant Strategy Implementation Notice, it is now a dominant strategy to report honestly. Let ˆθ i be any reports by the other players. Then i is solving: v maxˆθi i (x (ˆθ i, ˆθ i ), θ i ) t i (ˆθ i, ˆθ i ) ( ) = maxˆθ i v i (x (ˆθ i, ˆθ i ), θ i ) W \i (, ˆθ i ) W i (ˆθ i, ˆθ i ) = maxˆθ i v i (x (ˆθ i, ˆθ i ), θ i ) + j i v j(x ( ˆθ i, ˆθ i ), ˆθ j ) max x j i v j(x (, ˆθ i ), ˆθ j ) N = max v i (x (ˆθ i, ˆθ i ), θ i ) + v j (x ( ˆθ i, ˆθ i ), ˆθ j ) ˆθ i j=1 }{{} Gross welfare max v j (x (, ˆθ i ), ˆθ j ) x j i }{{} Independent of ˆθ i Terence Johnson (ND) Mechanism Design December 7, 2017 15 / 44

Dominant Strategy Implementation U i (ˆθ i, ˆθ i, θ i ) = max ˆθ i v i (x (ˆθ i, ˆθ i ), θ i ) + N v j (x ( ˆθ i, ˆθ i ), ˆθ j ) j=1 }{{} Gross welfare max v j (x (, ˆθ i ), ˆθ j ) x j i }{{} Independent of ˆθ i Terence Johnson (ND) Mechanism Design December 7, 2017 16 / 44

Dominant Strategy Implementation U i (ˆθ i, ˆθ i, θ i ) = max ˆθ i v i (x (ˆθ i, ˆθ i ), θ i ) + N v j (x ( ˆθ i, ˆθ i ), ˆθ j ) j=1 }{{} Gross welfare max v j (x (, ˆθ i ), ˆθ j ) x j i }{{} Independent of ˆθ i Since (i) x (ˆθ) maximizes gross welfare, (ii) agent i s payoff is gross welfare less something he cannot control, (iii) agent i can now only harm himself by lying and skewing the outcome away from x (ˆθ), (iv) so i has a dominant strategy to report the truth Terence Johnson (ND) Mechanism Design December 7, 2017 16 / 44

Dominant Strategy Implementation Recall the SPA: x SPA i (v i, v i ) = { 1, v i > max j i v j 0, otherwise and t SPA i (v i, v i ) = { max j i v j, v i max j i v j 0, otherwise Terence Johnson (ND) Mechanism Design December 7, 2017 17 / 44

Dominant Strategy Implementation Recall the SPA: x SPA i (v i, v i ) = { 1, v i > max j i v j 0, otherwise and t SPA i (v i, v i ) = { max j i v j, v i max j i v j 0, otherwise The allocation rule x SPA maximizes gross welfare, and the payment is agent i s impact on the gross welfare of the other agents: the SPA is a VCG mechanism Terence Johnson (ND) Mechanism Design December 7, 2017 17 / 44

Example: Public Goods (Groves) Here s another example: each city i values the construction of a new regional airport at v i. The cost of the airport is c. The FAA wants to build the airport if it is efficient whenever but not otherwise. N v i c 0 i=1 Terence Johnson (ND) Mechanism Design December 7, 2017 18 / 44

Example: Public Goods (Groves) Here s another example: each city i values the construction of a new regional airport at v i. The cost of the airport is c. The FAA wants to build the airport if it is efficient whenever but not otherwise. N v i c 0 i=1 How can the FAA design a mechanism that implements the efficient outcome? Is it budget-balanced? Terence Johnson (ND) Mechanism Design December 7, 2017 18 / 44

Dominant Strategy Implementation Theorem The Vickrey-Clarke-Groves mechanism implements the efficient outcome and truth-telling as a weakly dominant strategy. Terence Johnson (ND) Mechanism Design December 7, 2017 19 / 44

VCG and Individual Rationality Suppose the values of all the agents are weakly positive, so v i (x, θ i ) 0, for all decisions x and types θ i This implies that the payoff to i is max x I j=1 v j (x, θ j ) max x so the mechanism is individually rational v j (x, θ j ) 0, Example: losers pay nothing in the SPA, and the winner s value is always greater than the highest losing value j i Terence Johnson (ND) Mechanism Design December 7, 2017 20 / 44

Example: Multi-Unit Auctions There are two spectrum licenses for sale, one in Seattle and one in New York. ATT has the value v ATT (S, NY ) for both, v ATT (S) for just Seattle, and v ATT (NY ) for just New York, and likewise for Verizon. What is the VCG mechanism for this setting? Suppose ATT and Verizon have the following values for the licenses: v ATT (S, NY ) = 10, v ATT (S) = 2, v ATT (NY ) = 6 v V (S, NY ) = 8, v V (S) = 5, v V (NY ) = 1 Who gets the licenses and what do they pay? Terence Johnson (ND) Mechanism Design December 7, 2017 21 / 44

Example: Bilateral negotiations There is a buyer with privately known value v for a good, and a seller with privately known cost c of providing it. If v c 0, we wish to arrange trade, but if v < c, we don t. What is the VCG mechanism for this setting? Is this mechanism budget balanced? Terence Johnson (ND) Mechanism Design December 7, 2017 22 / 44

Auctions for multiple goods with varying quality We ve covered the cases of one good, and a bunch of homogeneous goods Terence Johnson (ND) Mechanism Design December 7, 2017 23 / 44

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 Terence Johnson (ND) Mechanism Design December 7, 2017 23 / 44

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 Terence Johnson (ND) Mechanism Design December 7, 2017 23 / 44

Sponsored search auctions Whenever you run a search in Google, there s an auction: Terence Johnson (ND) Mechanism Design December 7, 2017 24 / 44

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 Terence Johnson (ND) Mechanism Design December 7, 2017 25 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 25 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 26 / 44

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. As market designers, we re interested in why the Adwords design is so much more stable and successful. Terence Johnson (ND) Mechanism Design December 7, 2017 26 / 44

Example Suppose there are two slots on a page and three advertisers. Terence Johnson (ND) Mechanism Design December 7, 2017 27 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 27 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 27 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 28 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 28 / 44

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 Terence Johnson (ND) Mechanism Design December 7, 2017 29 / 44

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 Terence Johnson (ND) Mechanism Design December 7, 2017 30 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 31 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 32 / 44

Generalized Second Pricing (GSP) We assumed buyers behave myopically: they only think about the current item under consideration If the buyers behave myopically, it seems the ones with the highest types get the best goods, so it would be efficient. Is this always true? Is the allocation efficient? How can this open format be turned into a closed one, since open formats aren t always convenient? What if buyers didn t know each other s values, so there was adverse selection? Would it be a dominant strategy to bid honesty? Terence Johnson (ND) Mechanism Design December 7, 2017 33 / 44

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 p }{{} k Price for the k-th slot Terence Johnson (ND) Mechanism Design December 7, 2017 34 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 35 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 36 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 36 / 44

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 satisfies α 2 r i p 2 = α 1 r i b 1, b 1 = p 2 + (α 1 α 2 )r i Terence Johnson (ND) Mechanism Design December 7, 2017 37 / 44

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 = p k+1 + (α k α k+1 )r i. 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. Terence Johnson (ND) Mechanism Design December 7, 2017 38 / 44

Adwords In the closed format, figuring out the right bid here is somewhat difficult. Terence Johnson (ND) Mechanism Design December 7, 2017 39 / 44

Adwords In the closed format, 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: Terence Johnson (ND) Mechanism Design December 7, 2017 39 / 44

Adwords Google s Adwords just asks you to submit your r i, and it figures out your bid for you: Terence Johnson (ND) Mechanism Design December 7, 2017 40 / 44

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. Terence Johnson (ND) Mechanism Design December 7, 2017 41 / 44

But, sealed-bid GSP is the VCG mechanism If we use this equation: and back-substitute, we get b k = p k+1 + (α k α k+1 )r i. b k = (α k α k+1 )r k + (α k+1 α k+2 )r k+1 +... + (α K 1 α K )r K 1 But VCG maximizes gross surplus by matching the highest-value type with the highest-quality good, second-highest-value type with second-highest-quality good, and so on. The payment is your impact on social welfare: t i (ˆr i, ˆr i ) = j i K k=1 P jk (, ˆr i )α k r j j i K P jk (ˆr i, ˆr i )α k r j k=1 Terence Johnson (ND) Mechanism Design December 7, 2017 42 / 44

But, sealed-bid GSP is the VCG mechanism But if you unpack t i (ˆr i, ˆr i ) = j i K k=1 P jk (, ˆr i )α k r j j i K P jk (ˆr i, ˆr i )α k r j, k=1 it has three features: (i) for goods/agents ranked higher than your allocation k, you don t affect the outcome, (ii) your participation only affects goods ranked below your allocation k, (iii) your externality is then equal to the impact on pushing everyone below you down a quality rung, q r j Terence Johnson (ND) Mechanism Design December 7, 2017 43 / 44

But, sealed-bid GSP is the VCG mechanism But if you unpack t i (ˆr i, ˆr i ) = j i K k=1 P jk (, ˆr i )α k r j j i K P jk (ˆr i, ˆr i )α k r j, k=1 it has three features: (i) for goods/agents ranked higher than your allocation k, you don t affect the outcome, (ii) your participation only affects goods ranked below your allocation k, (iii) your externality is then equal to the impact on pushing everyone below you down a quality rung, q r j If ˆr [k] is the value of the [k]-th highest report, t i (ˆr i, ˆr i ) = (α k α k+1 )ˆr [k] + (α k+1 α k+2 )ˆr [k+1] +... + (α K 1 α K )ˆr [K 1], which is the same as GSP. Terence Johnson (ND) Mechanism Design December 7, 2017 43 / 44

Facebook vs Google https://www.wired.com/2015/09/ facebook-doesnt-make-much-money-couldon-purpose/ They built a system based on what s called the Vickrey-Clarke-Groves auction, or VCG, an auction mechanism that dates back to the 60s and 70s. Named for a trio of academics, including Nobel Prize winner William Vickrey, VCG spent decades as little more than an academic exercise. But then Hegeman and Facebook came along and applied it to online ads. The VCG system provided a way of building an auction that advertisers couldn t game for their own monetary gain (at least in theory). And eventually, Hegeman and team modified this complex system so that it not only ranks ad against ad, but ad against all the other stuff on Facebook. Terence Johnson (ND) Mechanism Design December 7, 2017 44 / 44