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2 2 Lecture 11: Sponsored search α ij - Probability that a user will click on jth slot when it is occupied by advertiser i, also known as the Click Through Rate (CTR). We assume that j : i, i : i i α ij = α i j. Meaning CTR is independent on which advertiser occupies it. We ll denote the jth slot s CTR as α j. β i - The click through rate for advertiser i (independant from slot) We denote v ij = v i β i α j Matching In order to maximize the social welfare we need to match advertisers with spots: We denote v i = v i β i α 1 α 2... α k We sort v i and match them with the corresponding α i VCG Prices In general, the price of the ith slot to the ith bidder is the total price if the ith bidder didn t exist (hence didn t win any slot), minus the total price subtracting the ith bidder part (v iα i ). Formally: Example v jα j + i 1 ( j=1 k i 1 v jα j 1 ) ( v jα j + j=i+1 j=1 k j=i+1 v jα j ) = k j=i+1 v j(α j 1 α j ) We ll calculate the prices for advertisers x, y, z in figure We ll first notice that α a > α b > α c and v x > v y > v z hence we match x, y, z with a, b, c respectively. Prices Calculation: We ll denote P i the price of advertiser i P x = (0 + v yα a + v zα b ) (0 + v yα b + v zα c ) = (20 + 5) (10 + 2) = = 13 P y = (v xα a + v zα b ) (v xα a + v zα c ) = (30 + 5) (30 + 2) = = 3 P z = (v xα a + v yα b ) (v xα a + v yα b ) = ( ) ( ) = = 0

3 Sponsored Search 3 Figure 11.1: 3 advertisers x, y, z and 3 slots a, b, c Generated Second Price Auction (GSP) The search engine assigns a weight β j to each advertiser. The weight can be thought of as a relevance or quality metric. Through the lecture we ve assume for simplicity j : β j = 1. Historically, the sponsored mechanism was introduced by Overture called rank by bid. Each bidder i bids b i Score for the ith bidder is b i = b i β i We sort the bidders by their scores: b 1 b 2... b n Now the bidders are sorted so that bidder i obtains slot i Payment for a click in slot i is P i = b i+1β i+1 β i b i GSP Strategy-Proofness Claim 11.1 GSP is not Strategy-Proof

4 4 Lecture 11: Sponsored search Figure 11.2: 3 advertisers x, y, z and 3 slots a, b, c where advertiser s x value is higher when he bids a number other than his real click-value Definition Let g i to denote the total gain of advertiser i, i.e., g i = clicks (v i b i) Proof: Consider figure We ll calculate the advertisers gain in the case where i : b i = v i, that is, advertisers bid their actual click value: Now we ll change x s bid to be \$5: b x = b y = 6 g x = 10 (7 6) = 10 b y = b z = 1 g x = 4 (6 1) = 20 b z = 0 g z = 0 b x = b z = 1 g x = 4 (7 1) = 24 We ve seen that if x lies about it s click-value (that is, bids differently than its value) his gain increases. Thus, x gains from lying, and therefore GSP mechanism is not strategy-proof. Example - Existance of multiple equilibria under GSP

5 Sponsored Search 5 Figure 11.3: 3 advertisers x, y, z and 3 slots a, b, c, the orange bids and the green bids form 2 different equilbria under GSP We ll now see that there are sets of bids {b i } s forming an equilbrium under GSP. Consider the green and orange bid sets in figure We ll see that both are an equilbrium under GSP. Claim 11.2 The orange bid set is an equilbrium under GSP Proof: We ll calculate the gains: Changing x s bid to \$3 isn t worthy: g x = 10 (7 4) = 30 g y = 4 (6 2) = 16 g z = 0 g x = 4 (7 2) = 20 < 30 Changing y s bid to so that b y > \$5 isn t worthy: g y = 10 (6 5) = 10 < 16 Thus, the orange bid set is an equilbrium under GSP. Claim 11.3 The green bid set is an equilbrium under GSP

6 6 Lecture 11: Sponsored search Proof: We ll calculate the gains: g x = 4 (7 1) = 24 g y = 10 (6 3) = 30 g z = 0 Changing x s bid so that b x > \$5 isn t worthy: g x = 10 (7 5) = 20 < 24 Changing y s bid to \$2 isn t worthy g y = 4 (6 1) = 20 < 30 Thus, the green bid set is an equilbrium under GSP. Payment Payment is the sum of prices for each advertiser: i P i GSP payment in the orange equilbrium: GSP payment in the green equilbrium: VCG Prices: = = 34 P x = v y(α a α b ) + v z(α b α c ) = 6 (10 4) + 1 (4 0) = = 40 Total VCG Payment is = 44. Summary GSP Payment in green equilbrium We found multi equilbria under GSP P y = v z(α b α c ) = 1 (4 0) = 4 P z = 0 < VCG Payment < GSP Payment in orange equilbrium

7 Sponsored Search Matching Market Model n - buyers m - products Each buyer wants to buy exactly 1 product Social Optimum Maximum Matching Each buyer i has a value vj i to item j Product j has price p j Buyer i will choose a product out of arg max j {vj i p j } Goal Set prices that clear the market, i.e. sell all the products to the buyers. Example Figure 11.4: 3 buyers x, y, z and 3 products a, b, c, and values of each product for each buyer. Best response for prices 1 marked with solid lines, and for prices 2 with dashed lines. There s a perfect matching in BR graph for prices 1, i.e. those prices clear the market, while prices 2 don t clear the market.

8 8 Lecture 11: Sponsored search For a set of prices for the products, we define preferred-product graph, or best response graph on buyers and products by constructing an edge between each buyer and his preferred product. There can be different edges from a buyer to the products, if he s indifferent for their prices. Now it s simple to see that the prices are market-clearing iff the resulting preferredproduct graph has a perfect matching. We shall show an algorithm which constructs a set of market-clearing prices. Algorithm for constructing a set of market-clearing prices 1. Set p j =0 2. Start of the round: at this point we have a set of prices, with smallest price being 0 3. Build BR graph, given the prices 4. Find perfect matching in the graph 5. If there is, we re done: the current prices are market clearing 6. Else we find a set of buyers S with BR set N(S) which satisfies S > N(S) 7. Increase prices in N(S) by 1 8. If the minimal price p is non-zero, decrease each price p j by p 9. Goto 2 Example run of the algorithm Figure 11.5: Round 1: 3 buyers x, y, z and 3 products a, b, c, and values of each product for each buyer. Starting prices are 0. Best response is graphed. There s no perfect matching, so S is chosen to be {x, y, z} with N(S) = {a}

9 Sponsored Search 9 Figure 11.6: Round 2: There s no perfect matching, so S is chosen to be {x, z} with N(S) = {a} Figure 11.7: Round 3: There s no perfect matching, so S is chosen to be {x, y, z} with N(S) = {a, b}

10 10 Lecture 11: Sponsored search Figure 11.8: Round 4: There is perfect matching, marked by bold lines, so we have found market-cleaning prices. Hall s theorem claims that if we do not have a perfect matching, then there is a set S, such that S > N(S). This implies we will not fail in step 6. Theorem 11.4 Hall s theorem: Given a finite bipartite graph G := (S T, E) with two equally sized partitions S and T, a perfect matching exists if and only if for every subset X of S X N(X), where for a set X of vertices of G, N(X) is the neighborhood of X in G. Claim 11.5 Step 6 in the algorithm is viable Proof: Follows directly from Theorem 11.4, since there s such set S with S > N(S) if and only if there s a perfect matching in G Claim 11.6 The algorithm eventually stops, therefore achieving market-clearing prices. Proof: We prove the claim for integer prices and values for simplicity, the proof can be easily extended to the general case. Define potential function φ such that φ(p roduct i ) = p i, the product price, and φ(buyer i ) = max k {v j k p k}, the buyer s gain. The cummulative potential function Φ is the sum of φ s values over all buyers and all products. The starting prices are 0, so φ over all products is 0. Initially, φ(buyer i ) = max k {v j k } since p j = 0 in the beginning, therefore when the algorithm starts Φ = i max k{v j k }, which is some finite non-negative integer value. The prices are always non-negative, since we decrease them only by the smallest price The gain of any buyer is non-negative in step 2, since there s a product with 0 price

11 Sponsored Search 11 The potential Φ changes only when the prices change Let p be the change in the potential of the prices, B the change in the potential of the products, and Φ = p + B the change in the potential. We consider the two cases where prices change: Decrease all prices by p: p = n i=1 φ(p roduct i) = np, while B = n i=1 φ(buyer i) = np, so overall Φ = 0. Increase prices in N(S) by 1: p = n i=1 φ(p roduct i) = N(S), while B = n i=1 φ(buyer i) = S, so overall Φ = N(S) S < 0. Since the initial value of Φ is bounded, Φ 0, Φ < 0 and all the values are integers, the algorithm stops in finite time, reaching a perfect matching in preferred-product graph, and thus finds market clearing prices. Theorem 11.7 Market clearing prices maximize the social gain. Proof: Suppose there s a perfect matching in the preferred-product graph, i.e. prices p i that clear the market. A perfect matching matches buyer i to some product j i, thus having buyer i recieves payoff vj i i p ji. Given a perfect matching M, summing over all buyers payoff yields n i=1 P ayoff(m i) = n i=1 V aluation(m i) n i=1 p i so V aluation(m) = P ayoff(m) + n i=1 p i, so choosing M among the perfect matchings that maximizes buyer payoff will maximize the social gain, since the prices are constant Back to GSP Now we show a way to convert a sponsored search instance into a matching market instance in such way that finding market clearing prices will be equivalent to finding a GSP equilibrium. The sponsored search slots are the matching market products, and the advertisers are the buyers. An advertiser s valuation for a slot is simply the product of it s own revenue per click and the clickthrough rate of the slot. Figure 11.9 shows a sponsored search instance translated into a matching market instance: advertiser x has click value 7, thus his item values are his click value multiplied by slot click rate: 7 10, 7 4, 7 0, and the item values for y and z are computed similarly. We chose market clearing prices 40, 40 and 0, which impose matching between advertisers and slots. The costs per click are the slot price divided by click rate - 40 = 4 for x, 4 = 1 for y and 0 for z. Player y bids x s cost per click b 10 4 y = 4, player z bids y s cost per click b z = 1, and x bids some bid larger than y s, i.e., b x > b y. This is a GSP equilibrium. General claim on GSP equilibrium construction: Given advertiser s click values and slot click rates, we find market clearing prices p 1,..., p n. The matching maps j-th advertiser to slot j with price p j. The price is for all the clicks

12 12 Lecture 11: Sponsored search Figure 11.9: 3 advertisers x, y, z and 3 slots a, b, c, with click values for advertisers and click rates for slots. Constructing a matching market instance with market clearing prices imposes matching between advertisers and slots. Bids definition: for each slot j define cost per click cost p j = p j α j Claim 11.8 p 1 p 2... p n Proof: Prices p j clear the market. Consider buyer i and compare to place l < i. In i-th place the gain is v i p i per click, α i (v i p i ) for all clicks. In l-th place - α l α j, since l-th place has higher clickthrough rate. Since buyer i preferrs i-th place, his gain is larger than in l-th place - v i p l v i p i p i p l Now, given prices p 1,..., p n we define the GSP bids. i + 1-th buyer has bid i+1 = p i, and 1-st buyer has bid 1 = p 1. Claim 11.9 The bids are a GSP equilibrium. Proof: Consider any change of bids: Lowering a bid If player i lowers his bid he will be between some player j and j + 1 with i < j and will pay j-th player price. It won t increase i-th player gain, since the prices clear the market. Raising a bid Player i raises his bid and get between some player j and j 1 with i > j and pays p j 1 p j, which is higher than i-th player original price GSP and VCG prices VCG price: p i = k j=i+1 v j(α j 1 α j ) GSP bid: bid i+1 = p i, bid 1 = p 1

13 Sponsored Search 13 The VCG prices can be rewritten as p i 1 = p i + v i (α i 1 α i ), so the prices are ascending and v i (α i 1 α i ) is the marginal price of moving from price p i to p i 1. Every player can choose the highest price as long as the marginal cost is smaller than his value, and a player can lower his price until the gain is smaller than the marginal cost. This is exactly the VCG prices, therefore these bids are a GSP equilibrium.

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