Simplified Mechanisms

Transcription

1 Simplified Mechanisms A study on the applications to sponsored search by: Hajir Roozbehani, Dong Han

2 Outline Properties of the Core Why Simplification? Simplification Theorems GSP vs VCG Package auctions Conclusion

3 The Core The core provides stability with respect to group deviations: No coalition can block the payoff allocation (imputation) Feasibility is guaranteed No individual ends up with negative payoff Coalitional Value Function: max vl( xl) if 0 S l S ws ( ) x X = 0 otherwise

4 The core of a cooperative game with player set L and coalitional value function w(.) is defined as follows: Core ( L, W ) = { π 0: w ( L ) = π, w ( S ) π for S L } l L The above definition implies a full information game. what about individual deviations? Does the core contain a NE? what about bidder-optimality? l l S l

5 Simplification Example: Consider the following Vickrey auction game with two slots and three bidders. The first and second slots receive 20 and 10 clicks per hour respectively. 10$ 0 0 5$ 0 0 The Vickrey payoff profile is given by: Zero revenue! π = (200,50,0)

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7 The Vickrey payment of the preceding example in the core Vickrey mechanism is DSIC Does there exist any other equilibrium in the core that can be implemented in dominant strategies leading to a higher revenue? Theorem: If the Vickrey payoff vector is in the core, then it is the bidder-dominant point in the core. If not, then there is no bidderdominant point in the core. (see pp in The Lovely but Lonely Vickrey auction)

8 Why one dimensional bidding?

9 Consider the same game with a one dimensional report space. The value per clicks are (10$,5$,2$). This gives a revenue of 90$ to the seller.

10 Simplification Theorems A product set of strategy profiles X ˆ = X ˆ 1 X ˆ 2 X ˆ ˆ 3... XL has the best-reply closure property in ( LXπ,, ) if: x X x x x x for all x X ' ' l l: πl( l, l) πl( l, l) l l Simplification Theorem: Suppose x X has the best-reply closure property in ( LXπ,, ). Then, a pure strategy profile is a Nash equilibrium of the restricted game iff it is also a Nash equilibrium of the unrestricted game. Restriction under the best-response closure property does not extend the set of pure equilibriums. New mixed equilibriums may be introduced.

11 Definition: An auction mechanism is said to be standard if: It is a direct mechanism The payment required from any bidder is a function of the reports of the other participants and of its own winning bid Definition: A simplification of an auction mechanism is complete when for any v that wins x at price p j j j there exists a v j that wins at p j x j Theorem: A complete, standard auction mechanism satisfies the best reply closure property Remark: the standard condition is not required!

12 One dimensional-bidding Theorem: In any pure strategy equilibrium of VCG/GSP auction with costly bidding, the seller s revenue is zero. If the equilibrium bids are undominated, then the winner i of position n bids bin = vin for that position. This efficient zero-revenue equilibrium is also a pure strategy NE when the bidding cost is zero. Note: restricting the strategy sets to one dimension leads to higher revenue for both Vickrey and GSP auctions. Theorem: There is no zero revenue equilibrium of the simplified VCG/GSP auction for more than N > 1 slots with at least N bidders and zero or small positive bidding costs. At any pure strategy equilibrium, the price paid for position N is zero.

13 Why prefer GSP to VCG? The well known problems with VCG pricing: Low (or zero) seller revenues Non-monotonicity of the seller s revenues in the set of bidders and amount of bids Vulnerability to collusion Vulnerability to use of shill bids These problem do not occur in the sponsored search. (the bidders have substitutes preferences). What about the required information?

14 Assume there are two kinds of searchers: Some are looking for a product Others are just curious Allow heterogeneity among the searchers, i.e., each group has a different click rate for different positions on the page: set α1 > α2 >... > α N > 0 for potential customers and β1 > β2 >... > β N > 0 for curious searchers. The total click rate for position n is αn + λβn

15 Define the relative click through rate for position αn + λβn n as γ n = α + λβ 1 1 If a bidder i names a bid b i1 for position 1 in a simplified auction, its bid for position n can be imputed as b = b γ in i1 n The auctioneer can not observe the actual value of the slots from its own observation, i.e., it does not know enough to compute Vickrey payoffs

16 Theorem: For the presented two searcher-type model, there is pure NE under GSP in which the assignment is efficient and the payoff profile is pn that of Vickrey iff { } N n= 1 is decreasing γ n This implies that a structure on the price profile is sufficient to obtain an equilibrium with the same payoff profile as that of Vickrey The prices must be dropping faster than relative click rates Weakening both VCG and GSP!

17 Theorem: GSP is a complete, standard auction mechanism. The price imposed on each position depends on the next highest bid. Hence, the best reply in the restricted set is also the best reply in a richer strategy space. Besides, simplified GSP entails no zero-revenue equilibria.

18 Package Auction Definition: The set of values V is closed under fixed costs if for all k > 0, v V v k V Closure under fixed cost can be used for simplifying package bidding mechanisms. Each agent is restricted to express its value for any nonempty package: v ( S) = α k i j S ij i This can be done by restricting the reports to a tuple: v = ( α, k ) with α n +, k + i i i i i

19 Example: item 1 item 2 item 3 k bidder bidder bidder

20 Inefficient NE: each bidder bids its value for the whole set. Items 1,2,3 will be allocated to bidder 3 with a total revenue of 15. This equilibrium will be eliminated in the simplified mechanism. Bidders 1,2, and 3 are restricted to submit (10,0,0,0), (0,10,10,10), and (5,5,5,0) respectively.

21 Fixed cost package auction {1} {2} {3} {12} {23} {123} bidder bidder bidder The items are allocated efficiently No coordination failure Low dimensional bids Linear complexity in the number of items

22 Theorem: Let Γ V be a simplified core-selecting auction with reports restricted to lie in the set V. Suppose that V is closed under fixed costs and that actual bidder values lie in the set V. Then, Γ V has the best-reply closure property. Suggestion: fixed-cost closure is sufficient for simplification.

23 Affine Approximation Mechanisms Affine approximation mechanisms are simplified coreselecting package auctions in which the bidders are interested in a particular bundle. Each bid is a tuple ( T, β, α, r) comprising a package T, a n price β, individual item price α +, and a radius of approximation. r 1 Theorem: An affine approximation mechanism has a best-reply closure property regardless of the bidder s actual valuations.

24 Proof: For each bidder with a best reply winning report: α α jn jn > α j ' n for all j ' j if n T = 0 if n T j must be the winner with the proposed bid in any efficient mechanism. Since the mechanism is standard, the winning bid is unchanged. Hence, the winner always wins the same package at the same price. The standard property allows implementing a best-reply closure property with a simplified bidding language

25 Affine approximation mechanisms: Requires full information As the number of items grows large, may run at many coordination failures The coordination problem can be circumvented by restricting a wide radius of r.

26 Conclusion Simplified mechanisms can be used to eliminate undesirable equilibriums. Package auctions are implementable through coarse bidding in simplified languages Offers attractive efficiency and tractability properties Puts a high emphasize on full information game Can we define similar mechanisms for incomplete information games?