Revenue Maximization in Multi-Object Auctions

Transcription

1 Revenue Maximization in Multi-Object Auctions Benny Moldovanu April 25, 2017 Revenue Maximization in Multi-Object Auctions April 25, / 1

2 Literature Myerson R. (1981): Optimal Auction Design, Mathematics of Operations Research 6, Manelli, A and Vincent, D. (2007): "Multidimensional Mechanism Design: Revenue Maximization and the Multiple-Good Monopoly", Journal of Economic Theory 137, Revenue Maximization in Multi-Object Auctions April 25, / 1

3 Multi-Object Auction Model n agents and m objects. Value of object j to agent i is v j i R +. Value of a bundle of objects is the sum of the values of the objects in the bundle. The utility of agent i who obtains set of objects S and makes monetary transfer t is v j i t. j S Auctioneer maximize revenues = maximize the sum of monetary transfers made by agents. Revenue Maximization in Multi-Object Auctions April 25, / 1

4 Information Each agent i knows v i = (v 1 i,.., vm i ) but regards v k, k i, as being a random draw from a compact and convex set V k R m +. v k independent of v l, l k. Distribution F k with density f k > 0 is common knowledge Auctioneer regards all values as random variables. Revenue Maximization in Multi-Object Auctions April 25, / 1

5 Mechanisms In a direct revelation mechanism (DRM) each agent i reports v i (not necessarily truthfully!), and the mechanism determines a physical and monetary allocation. Formally, a mechanism consists of: 1 i, j, p j i (v 1,.., v n ) [0, 1] is the probability that agent i gets object j when the reports are (v 1,..v n ). Feasibility requires that j, v 1,.., v n, n p j i (v 1,.., v n ) 1 i=1 2 i, t i (v 1,.., v n ) R is the monetary transfer made by agent i to the auctioneer when reports are (v 1,..v n ). Revenue Maximization in Multi-Object Auctions April 25, / 1

6 The Revelation Principle In a Bayes-Nash equilibrium, each agent optimizes given his expectation about the actions of others and expectations are correct. The Bayes-Nash equilibrium of any possible selling scheme can be replicated by a DRM where it is a Bayes-Nash equilibrium for each agent to truthfully reveal his values. Thus, in order to find the revenue maximizing selling scheme it is enough to restrict attention to truthful (or incentive compatible) DRM s Revenue Maximization in Multi-Object Auctions April 25, / 1

7 Incentive Compatibility Fix a DRM (p, t) and define: q j i ( v i) = p j i (v i, v i )f i (v i )dv i V i T i ( v i ) = t i (v i, v i )f i (v i )dv i V i Agent i s expected utility is given by : U i (v i, v i ) = q i ( v i ) v i T i ( v i ) Truthful telling requires then that for all i,and for all v i, v i V i we have: S i (v i ) := U i (v i, v i ) = sup v i V i U i (v i, v i ) Revenue Maximization in Multi-Object Auctions April 25, / 1

8 Consequences of IC The function S i (v i ) = U i (v i, v i ) is the supremum of linear functions, hence it is continuous and convex, and S i = q i a.e. q i is cyclically monotone and S i (v i ) = S i (v i ) + vi v i q i (w i ) dw i where v i is an arbitrary vector in V i, and where the integral does not depend on the path of integration. We can write: T i (v i ) = S i (v i ) + q i (v i ) v i = T i (v i ) q i (v i ) v i + q i (v i ) v i vi v i q i (w i ) dw i Revenue Maximization in Multi-Object Auctions April 25, / 1

9 Payoff Equivalence In other words, in a truthful DRM, the expected monetary transfer T i is completely determined, up to a constant, by the expected probabilities of getting the various objects q i (and hence by the primitive probabilities p j i ). This important result is called the payoff-equivalence theorem. Conversely, any mechanism (p, t) that satisfies the above conditions is a truthful DRM. Revenue Maximization in Multi-Object Auctions April 25, / 1

10 Revenue Maximization: Individual Rationality Since S i (v i ) = S i (v i ) + v i v q i (w i ) dw i and q i is non-negative we i obtain that S i (v i ) 0 S i (v i ) 0, v i. Let assume now for simplicity that V i = [0, 1] m, and let us take v i = (0, 0,..0). Then, S i (v i ) = T i (v i ), so that S i (v i ) 0 T i (v i ) 0. In other words, the agent with zero values for all objects cannot be asked to make a positive transfer to the auctioneer. Thus, in a revenue maximizing mechanism we have T i (v i ) = 0. Revenue Maximization in Multi-Object Auctions April 25, / 1

11 Expected Revenue We have: T i (v i ) = S i (v i ) v i S i (v i ) = q i (v i ) v i The expected revenue is then given by R = = n [ ] T i (v i )f i (v i )dv i V i [ n [ q i (v i ) v i i=1 i=1 V i vi v i vi q i (w i ) dw i v i ] ] q i (w i ) dw i f i (v i )dv i Revenue Maximization in Multi-Object Auctions April 25, / 1

12 Constraints The above revenue can be written directly in terms of the mechanism s primitive probabilities p j i. Thus the auctioneer maximizes R over all functions p j i (v 1,..v n) such that: 1 i, j, v 1,.., v n, p j i (v 1,..v n ) [0, 1]. n 2 j, v 1,.., v n, p j i (v 1,.., v n ) 1. i=1 3 i, the vector field q i defined by q j i (v i) = p j i (v i, v i )f i (v i )dv i V i is the gradient of a convex function. Revenue Maximization in Multi-Object Auctions April 25, / 1

13 Remarks Myerson (1981) completely solves the 1-object case. Heuristic: ignore constraint 3; solve pointwise; find conditions on the distribution under which constraint 3 is satisfied. Solution is a deterministic allocation. Manelli and Vincent (2007) have shown that the extreme points in the multidim. case need not be deterministic Revenue Maximization in Multi-Object Auctions April 25, / 1