Ad Auctions: Game-Theoretic Perspectives

1. Ad Auctions: Game-Theoretic Perspectives Moshe Tennenholtz Technion—Israel Institute of Technology and Microsoft Research

2. Position (Ad) Auction

3. Ad Auctions k positions, n players (bidders) n ≥ k vi - player i’s valuation per-click, vi drawn from F~[0,1] (i.i.d) j- position j’s click-through rate 1 ≥ 2 ≥ ≥ k>0 Allocation rule – si(b1,…,bn) – player i’s position Payments – pi(b1,...,bn) - i’s payment per click Position payments - qj(b1,…,bn) payment per-click for player in position j Quasi-linear utilities: j(vi-pi) if player i is in position j and pays pi Reserve price: r ≥ 0

4. Special Ad Auctions • Allocation rule – jth highest bid to jth highest position • Tie breaks - fixed order priority rule • Payment scheme • Self price (first price): every player pays his bid qj(b1,…,bn)=b(j) • Next–price (GSP): a player who got slot j pays the bid of the player who submitted the j+1 highest bid qj(b1,…,bn)=b(j+1)

5. Generalized Second Price Auction (GSP) Every player pays the bid of the next player: ps=bs+1. Utility for player (in position) 2 is: 2(6-2)=8

6. Remarks 1. To move up one position – beat the bid of the player above 2. To move one position down – beat the price of the player below

7. Remarks 1. To move up one position – beat the bid of the player above 2. To move one position down – beat the price of the player below 3. GSP is not truthful 1=2 2=1.9 v1=5, v2=4, v3=1 Suppose 2 and 3 bid truthfully If 1 bids truthfully: u1=2(5-4)=2 If 1 bids 2: u1=1.9(5-1)=7.6>2

8. Nash Equilibrium Definition: A Nash Equilibrium (NE) is a profile of bids by the participants such that unilateral deviations are not beneficial. For GSP: Varian showed the existence of NE for GSP, and characterized explicitly a set of them – termed SNEs.

9. GSP and the Vickrey auction • In the case of only one position GSP coincides with the famous Vickrey auction: Each player submits his bid, the winner is the player who submitted the highest bid, and he pays the second highest bid. In the Vickrey auction it is a dominant strategy to submit the actual valuation as the bid: you can only gain by being truthful, regardless of the others do, This is an instance of the general VCG auction.

10. Vickrey-Clarke-Groves (VCG) Allocation rule: chooses an allocation that maximizes social welfare: assign players in decreasing order of their bids. A player pays for the difference between other players welfare when not participating to other players welfare when participating. Example:v=(10,5,2) payment for player 1: 5(1-2)+2(2-0)=7 payment for player 2: 10(1-1)+2(2-0)=2 vss

11. VCG and GSP VCG is truthful: it is a dominant strategy to bid your valuation as your bid. Basic Theorems about GSP (Varian): VCG payments coincide with the worst possible SNE payments of GSP. In Varian’s setting: GSP is preferable to VCG from the revenue perspective.

12. Some celebrated results • There exist pure strategy equilibria for the next-price position auction with complete information. • A set of equilibria of that kind has been characterized, and the best equilibrium in this set (from an agent’s utility perspective) coincides with the VCG outcome. Research highly influenced by: Position Auctions (Varian 2005) Internet Advertising and the Generalized Second Price Auction (Edelman, Ostrovsky and Schwarz 2005)

13. Quality Factors Google/Yahoo/MSFT assign advertisers according to quality × bid. Each advertiser pays the minimum price to retain her position. The click-rate an advertiser experiences in position s is ess where es is player’s s quality factor. qst – price for advertiser s in t qstes=bt+1et+1 qst=bt+1et+1/es The analysis remains as before when is replaced by

14. A game-theorist wish list • The system perspective: Deal with the general situation where more than one search engine conducts an auction for a given keyword. Who will make more money? • The agent (advertiser) perspective: can we provide an advice for agent bidding in a standard ad auction such as GSP; how much should I actually bid? • The user (surfer) perspective: re-visit the Varian’s model when explicit user models are taken into account: would this effect the basic results obtained about GSP/VCG? • The mediator perspective: can a mediator coordinate advertisers’ bidding, in order that they will be able to ignore uncertainty about each other bids, while yielding them high utility?

15. A game-theorist wish list Serious challenges: • We wish to deal with the (more realistic but more challenging) incomplete information setting. • The game-theoretic analysis of the system perspective, addressing competition among auctions, has been tackled without success in the past. • The user perspective introduces the need to consider dependence between ads. • Can positive results be obtained for the agent perspective? • Can positive results be obtained for the mediator perspective?

16. The System Perspective: Competing Ad Auctions

17. A basic story

18. It is not a monopolist market after all: the system perspective

19. Which auction will gain more revenue?

20. Two Ad Auctions – The Setting Auction A Auction B No. of positions kA kB Click-through rates 1¸ 2¸¸kA1¸2¸¸kB Reserve prices rA rB A and B are regular 1 1 Definition: A is stronger than B if jj for every position j

21. Incomplete Information:Symmetric Bayesian Equilibrium Equilibrium analysis in the case of incomplete information is rather detailed. Roughly speaking, given an auction, an equilibrium is a profile of bidding strategies (mapping from valuations to bids), such that each player strategy is the best response to the others, given the probabilistic assumptions. This is termed Bayesian Equilibrium. In a symmetric Bayesian equlibrium all players use the same bidding strategies.

22. Regular ad auctions In a regular ad auction symmetric Bayesian equilibrium exists under some assumptions. Our results do hold when the individual auctions are regular. VCG ad auctions are regular, and for ease of exposition we will refer to them. This is done in order to concentrate on the complexity arising from the fact we have two ad auctions rather than only one, and not in open questions regarding a single ad auction.

23. Symmetric Bayesian Equilibrium Valuations are assumed to be selected from [0,1] according to distribution F with density f. Strategies - bi:[0,1]→[0,1], bi(vi) – player i’s bid when her value is vi Utilities - Ui(vi,bi,b-i) – expected utility for player i in the strategy profile (b1,…,bn) given that her valuation is vi. Bayesian equilibrium – a profile (b1,…,bn) such that for every player i, every vi Ui(vi,bi(vi),b-i) ≥Ui(vi,bi’(vi),b-i) for every strategy bi’. Symmetric Bayesian equilibrium – if in addition b1=b2=…=bn

24. Regular Ad Auctions An ad auction is regular if there exists a symmetric equilibrium which is increasing in types above the reservation price for any identical independent distribution F with the following structure: let F(0) can be positive for every for every

25. Regular Ad Auctions • All the results presented for competing ad auctions are applicable for the case where A and B are regular ad auctions. • The VCG ad auction is regular. • For simplicity of exposition, we assume both A and B are VCG ad auctions, where agents use the truth-revealing dominant strategy equilibrium.

26. Two Ad Auctions Non-Competingeverybidder participates in both auctions. Competing everybidder chooses to participate only in a single auction.

27. Non-Competing Auctions Claim: There exist a non-competing setting and a distribution F, such that (i) rA=rB=0, (ii) A is stronger than B, but Rev(A) < Rev(B). (Rev(A)–expected revenue in A) The claim follows from the following…

28. Non-Competing Auctions Proposition: Let A and B be VCG ad auctions and suppose A is stronger than B where j>j for only a single slot j. If j=1 Rev(A)>Rev(B). If j > 1 Rev(A) > ( ) Rev(B) if and only if jv(j+1) > ( ) (j-1)v(j) where v(j) is the expected jth highest bid.

29. Non-Competing Auctions - Intuition 2 players and 2 positions: A B 1=10 1=10 2=9.98 2=5 Position 1 is less attractive in auction A More generally: Increasing number of click-rates may yield less competition for attractive click-rates, which may result in lower revenues.

30. Two Ad Auctions Non-Competing everybidder participates in both auctions. Competingeverybidder chooses to participate only in a single auction.

31. Competing Auctions The gameH=H(A,B): • Strategy for i, i=(qiA,qiB,biA, biB), consists of: • qiA(vi), qiB(vi), - probabilities attending A and B given vi (participation function). • qiB(vi)=1- qiA(vi) ; • biA(vi) – the bid in A • biB(vi) – the bid in B • Utility for i - UiH(vi,i,-i). To compare revenues we need to do some equilibrium analysis…

32. Equilibrium in H(A,B) A profile of strategies (1,…,n) such that: In both auctions, A and B, the strategies induce a Bayesian equilibrium (according to the induced value distributions in each auction). Given -i player i does not want to change her participation function. Does there exist an equilibrium?

33. Equilibrium - Result Theorem: There exists a unique symmetric equilibrium in H(A,B). (for any number of positions, any click-through rates, and any reserve prices). Uniqueness is with regard to participation function. Two things to prove: Existence Uniqueness Elaborated proof employing the implicit function theorem…

34. Equilibrium - Examples kA=kB=1 - single position general structure

35. Revenues in Competing Auctions: Main Result Theorem: Suppose A is stronger than B, and rA = rB. Then Rev(A) Rev(B).

36. The Sellers Let the sellers choose reserve prices: There is no pure strategy equilibrium in the 2-stage “sellers game”! Let Rev(A, rA, rB) (resp. Rev(B, rA, rB)) be the expected revenue in A (resp. B) given reserve prices rA and rB in A and B respectively. Proposition: Let A and B be VCG auctions. Suppose A is stronger than B and rA=rB. Then for every rB’>rB such that Rev(B,rA,rB) < Rev(B,rA,rB’) we have that Rev(A,rA,rB’)-Rev(A,rA,rB) > Rev(B,rA, rB’)-Rev(B,rA,rB)

37. Intermediate conclusions: the effects of competition The revenue in a “more visited” auction can be lower than the revenue in a “less visited” auction. In the competing setting the revenue of the stronger auction is always higher. Mechanisms can have different properties in a competing and non-competing settings: Ad revenue is not a direct implication of search engine success; advertisers’ acquisition / mechanism selection may be a winning strategy!

38. The Agent Perspective: Competitive Safety Analysis in Ad Auctions

39. But what should I do? The agent perspective $7.6?

40. An Agent Centric Approach to Games • Given a game, what is a useful strategy for an agent in this game? • The criteria for the evaluation of the strategy are: The utility of the agent. The assumptions about other agents. Safety-level strategy guarantees a payoff regardless of what others do, but is it good enough?

41. An Agent Centric Approach to Games • Example (Aumann 1985): • Pr(U)=3/4 is the safety-level strategy for the row player. • Pr(U)=Pr(D)=1/2 ; Pr(L)=Pr(R)=1/2 is the unique NE • The row player expected payoff in the safety-level strategy is as in the Nash equilibrium!

42. Competitive Safety Analysis • A c-competitive safety level strategy guarantees a payoff which is at least 1/c of what is obtained in a (best) Nash equilibrium. • C-competitive strategies extend upon Aumann observation, and enables it to be applicable as a design paradigm. • C-competitive strategies for small C exist for interesting classes of congestion games.

43. Competitive Safety Analysis in Position Auctions • In Varian’s (complete information) model we prove that for GSP: With exponentially decreasing click-rate functions, assuming N bidders, the competitive safety ratio can be arbitrary close to N. With linearly decreasing click-rate functions the ratio can not be greater than 1+ln(n); this bound is tight.

44. Click-Rates Exponential click rate: for slot i Linear click rate: for slot i

45. An incomplete information setting • Each advertiser knows his valuation per-click, but not the ones of others. • The agents’ valuations are assumed to be taken from a known distribution. • We consider an agent’s expected payoff under the VCG position auction, for a given valuation, when taking expectation over other agents’ valuations. • This coincides with the expectation of the agent’s payoff assuming the system converges to the best SNE for the agents in the GSP auction for the actual valuations.

46. An incomplete information setting • We consider safety-level strategies, where the other agents can see the agent’s strategy and valuation, and minimize its payoff in purely adversarial manner, as long they do not overbid. • This gives a highly demanding setting. • We present our results for the uniform distribution over the agents’ valuation

47. Competitive Safety Analysis in Position Auctions (revisited for incomplete information) Results: • The competitive safety ratio for exponentially decreasing click-rates is e. • The competitive safety ratio for linearly decreasing click-rates is 2. • Close to 1 competitive safety ratio if the agent’s valuation is “low”.

48. Intermediate Conclusions: providing advice to the bidder • Given the GSP auction, competitive safety strategies exist for the (more realistic) incomplete information setting!

49. The User Perspective: Revisiting the basic (complete information) model

50. We have users and not only advertisers! I will try the first Is ad and see