The cascade auction – a mechanism for deterring collusion in auctions

1. The cascade auction – a mechanism for deterring collusion in auctions Uriel Feige Weizmann Institute Work done at Microsoft Herzeliya Joint work withGil Kalai andMoshe Tennenholtz

2. Ad auctions – market for impressions

3. Market for impressions • End users (“impressions”) – items • Advertisers / networks– bidders (wish to target ads at the right end users) • Publisher/Exchange – auctioneer • Different advertisers have different valuations for different impressions

4. Motivating Example: classical failure in ad exchanges • In current ad exchanges each ad impression is sold using second price auction. • Ad networks serve as mediators on behalf of the bidders. • Advertisers submit bids through their ad networks. • Say network has two advertisers and with bids 10 and 8 resp., while network has a single advertiser with bid 5. • forwards 10to the exchange, and forwards 5. • Second price auction at the exchange will declare the winner with price 5. The publisher gets less than the second price.

5. Our goal Design an auction mechanism that strikes a good balance between: • Bidding truthfullyis a dominant strategy for the advertisers (as in second price auctions). • The collusion implied by using mediators does not lower the revenue of the seller significantly below the “fair price”: that of second price auction without collusion.

6. Our Suggested Mechanism:The Cascade Auction • Parameter • Probabilities ,,…., 1 • Floor price (equals in this talk). • Let ,……, be the highest bids (above floor price) in decreasing order. • For 1 , the bidder of bid wins the auction with probability (If corresponds to a floor price rather than an actual bid, the item is not allocated.)

7. The Cascade Auction (cont.) • When the winner is the-th highest bid the payment made by winner is the Vickrey-Clarke-Groves (VCG) payment -) • In other words, the expected payment associated with winning by the-th highest bid is the expected loss it causes to the rest of (lower) bids: -)

8. Common sense • Bidders can bid truthfully, because of VCG prices. • Advertisers will expect ad networks to forward their bids to the exchange network, even if the ad network sees a higher bid. • If is sufficiently high, revenue will be similar to that of second price auction without collusion.

9. Rest of the talk • An exercise in mathematical modelling: provide a formal model and formal theorems in support of what our common sense already tells us. There are some subtleties that need to be handled: • A bidder may want to provide more than one bid, and then there is no dominant strategy. • What are the strategic goals of mediators?

10. Multiple Bids • A seller has one item for sale. There are multiple buyers. • The cascade auction mechanism allows a single bidder to submit multiple bids. • Bids specify which buyer made the bid and a positive bid value. • Tie-breaking among different buyers is random.

11. Collusion and agents • A collusion is a set of buyers that coordinate their bids. • We model collusion by a notion of a mediator (e.g. an ad network). • An mediator is a bidding algorithm, and the algorithm of the mediator is effectively a contractthat the mediator offers to buyers. • The contract says that the buyers may provide their inputs to the bidding algorithm, and the mediator will bid on their behalf the output of the algorithm.

12. Collusion and agents (cont.) • Each mediator is a function of its incoming bids to at most bids on behalf of the buyers. • Buyers may submit bids directly to the seller, and also through one or more mediators. We refer to a buyer submitting bids only directly to the seller as being independent. • If a mediator submits a bid and the bid wins, the item goes to buyer . The mediator is not allowed to instead give the item to a sibling of who is using its services.

13. The cascade auction as a multi-player game • Given the set of mediators, we get a game in strategic form, where the players are the buyers • An action of a player is a pair , where is a set of bids it submits directly to the seller, and =( ,…., ) is the sets of bids it submits to the mediators.

14. Are mediators also players? Yes, but our analysis will circumvent the need to model their strategic behavior (the type of contracts that they offer).

15. Naive buyers A buyer is naive if he is independent and he submits a single bid. Proposition: There is a strictly dominant action for a naïve buyers, and this action is to bid his value. Proof: follows from VCG prices.

16. Non-Naïve Buyers Proposition: In a cascade auction with = 2, if a buyer has value and the two top bids by other buyers are and , then submitting two independent bids is preferable over one independent bid if and only )/2 > . The two bids will then be and .

17. Naive mediator • The naive mediator asks each buyer for his bids. • The naive mediator sorts all bids that he receives in order of decreasing value, determines the value of the th highest bid, and passes to the seller those bids having at least this value.

18. Naive mediators Proposition: If in the cascade auction all mediators are naive, then for every buyer, in every undominated action, 1. The buyer does not submit an independent bid. 2. If the buyer submits multiple bids, all these bids are submitted through the same naive mediator. 3. At least one bid that the buyer submits to the naïve mediator is the true value for the buyer, and the other bids (if any) are not higher.

19. Notation for rest of talk • Consider the realistic case and let =. • Denote the three highest values that buyers have by (for simplicity we assume that there are no ties). Without loss of generality, these values are held by buyers 1, 2and 3 respectively.

20. Naïve mediators Theorem: In the cascade auction with , if all mediatorsare naive and if every buyer uses an undominatedaction, then the expected revenue of the seller is at least

21. Arbitrary mediators The need to model the strategic behavior of mediators (the type of contracts that they offer) is circumvented by considering pure Nash equilibria, modeling the possibility of a player to leave a mediator and become independent. It can be shown that the cascade auction always has pure Nash equilibria(regardless of the mediators).

22. Nash Profiles (best response) Theorem: In every pure Nash profile, if the expected revenue of the seller is at least , and if the expected revenue of the seller is at least The expectation is over randomness of the auction, not of the players!

23. Semi best response A weaker (and hence more general notion) than Nash equilibrium. Applies even when players do not know the input of other players to the mediators. Given the set of bids received by the seller, is a semi-best response if no independent action results in an action that offers the buyerhigher expected payoff than does. A profile in which each buyer plays a semi-best response to the others’ actions is a Semi-Nash profile.

24. Arbitrary mediators Theorem: In every semi-Nash profile the expected revenue for the seller is at least

25. Conclusion • Introduced a sealed bid auction of a single item in which the winner is chosen at random among the highest bids according to a fixed probability distribution, and the price for the chosen winning bid is the VCG price. • Our analysis suggests that this type of auction gives higher revenues compared to second price auction in cases of collusion/mediation, as common in ad exchanges.

26. Principles of analysis • Informative special cases: naïve buyer, naïve mediator. • Multiple bids by the same buyer. • Independent bids allow reasoning about outcome without specifying mediators. • Undominated actions circumvent difficult (and uninformative) equilibrium analysis. • Nash and semi-Nash profiles as a way of quantifying over all reasonable contracts offered by mediators.