Core-selecting package auctions. Han Dong, Hajir Roozbehani

Transcription

1 Core-selecting package auctions Han Dong, Hair Roozbehani

2 Direct Impression to the paper proposes QoS for mechanism design connects stable matching with mechanism analyses theorem deeply and theoretically the core concept - the "core" intuition for the theorem the proof

3 Outline Introduction Definition of Core Revenues vs. shills Truncation Minimizing incentives to misreport Monotonicity of revenues

4 Stable Matching no individual can do better by staying unmatched no pair can both do better by matching to one another pairs are the only significant coalitions a kind of core allocation

5 Intuition of Core no group of bidders: - strictly prefer an alternative deal - and also strictly better for seller advantage - no individual/group want to renege - non-core agreement vulnerable to defections

6 Marriage Problem man-optimal woman-optimal an interesting example m1: w1,w2,w3 w1:m2,m3,m1 m2: w2,w3,w1 w2:m3,m1,m2 m3: w3,w1,w2 w3:m1,m2,m3

7 Package Auction incentive compatibility revenues monotonicity of revenues to aviod shilling

8 Vickrey auction A B AB

9 A B AB = (0, 0,10,10) π VCG π π 2

10 Outline Introduction Definition of Core Revenues vs. shills Truncation Minimizing incentives to misreport Monotonicity of revenues

11 coalition value function a good assighment x is feasible (1) ˆ x X 0 (2) bidder and seller are both in the coalition coalition value function: w u ( S) x F ( S ) = max u ( x )

12 Cooperative game non-cooperative game: procedural game theory specifies various actions that are available to the players cooperative game: combinatorial game theory describes the outcomes that result when the players come together in different combinations

13 Core The core is defined as a set of payoffs or imputations, as described below. The core is the set of allocations and payment whose imputed payoffs are core imputations. = N S Core( N, w) π 0 π = w( N) and ( S N) π w( S)

14 Core-selecting mechanism a direct auction mechanism for every report profile π u Core( N, w ) û uˆ core vs core-selecting mechanism

15 b v w(0) = 0 w(0,1) =10 w(0,2) = 8 w(0,3) = 4 w(0,1,2) =10 w(0,1,3) =10 w(0,2,3) = 8 w(0,1,2,3) =10 π π π π π + π + π + π π 2 + π w(0,1) 3 w(0,2) w(0,3) = w(0,1,2,3) Assume that the price is k How to make it in the core? π 0 Θ π = 0 k + k k + π π 8 1 = 10 k w(0,2) = 10

16 A B AB = (0, 0,10,10) π VCG π π π = v + π = 2 1 p + π v = 20 ( p ( p 2 p 3 + π p ) p ) π 2

17 Lemma 1 For every core-selecting mechanism ( f, P) and every report profile f ( ˆ) u arg max ˆ u ( x ) x X Proof: assume that f ( ˆ) u arg max ˆ u ( x ) ˆ', u N s. t. ˆ u ( x ) < f ( ˆ') u arg max x X 0 N ˆ' u N x X 0 N ˆ' u contradict to the definition of core ( x ) 0 N then = w( N) ( x ) selecting mechanism

18 Outline Introduction Definition of Core Revenues vs. shills Truncation Minimizing incentives to misreport Monotonicity of revenues

19 Theorem 1 An efficient direct auction mechanism has the property that no bidder can ever earn more than its Vickrey payoff by disaggregating and bidding with shills if and only if it is a core-selecting auction mechanism.

20 Outline Introduction Definition of Core Revenues vs. shills Truncation Minimizing incentives to misreport Monotonicity of revenues

21 Truncation correctly rank all pairs consisting of a nonnull goods assignment but may falsely report that some of these are unacceptable

22 Lemma 2 û A report is a truncation report if and only if there exists some such that for all x α 0 X ˆ u ( x ) = u ( x ) α

23 Theorem 2 Suppose that ( f, P) is a core-selecting direct auction mechanism. and bidder is favored. Let û be any pro file of reports of bidders other than. Denote s actual value by and let π w N) w ( N ) be 's u = ˆ u, ( u ˆ u, u corresponding Vickrey payoff. Then, the π truncation of is among bidder s best replies in the u mechanism and earns a payoff for of. Moreover, this remains a best reply even in the expanded strategy space in which bidder is free to use shills. π

24 Theorem π π π is max 0 wins

25 Theorem 3 For every valuation profile u and corres ponding bidder optimal imputation π, the profile of π truncations of u is a full information equilibrium profile of every core selecting auction. The equilibrium goods assignment x* maximizes the true total value i N u i ( xi ) and the equilibrium payoff vector is π (including π 0 for the seller)

26 Theorem 3 1no goods assignment leads to a reported total value exceeding 2for any bidder, there is some coalition excluding for which the maximum reported value is at least π π 0 π 0 3 is a Nash Equilibrium as the precise payoff for each bidder

27 Outline Introduction Definition of Core Revenues vs. shills Truncation Minimizing incentives to misreport Monotonicity of revenues

28 Theorem 4 A core-selecting auction provides optimal incentives if and only if for every u it chooses a bidder optimal allocation.

29 Outline Introduction Definition of Core Revenues vs. shills Truncation Minimizing incentives to misreport Monotonicity of revenues

30 Theorem 5 The seller s minimum payoff in the core with bidder values û is non-decreasing in û. min π = w ˆ u ( N) π i N 0 i subect to i S π i w( S)

31 Connections to Marriage Problem Best reply: - the use of truncation strategies - the point of the truncation Equilibrium: similar to matching Incentives: no others improved for all bidders Monotonicity: adding men improves the utility of each woman if woman-pessimal, man-optimal is selected

32 Summary the definition of "core" "core-selecting mechanism" revenue and shills trunction reports incentives to misreport monotonicity of revenues

33 Open Questions budget constraint complexity problem - computational complexity - communication complexity