Multidimensional Mechanism Design for Auctions with Externalities

In our framework, when a buyer does not obtain the auctioned object, he is no longer indifferent about the identity of the winner (i.e., eyternal effects are present). Buyer i's preferences are characterized by an N-dimensional vector t^i = (t1^i, t2^i,..,tN^i). The coordinate ti^i can be interpreted as the usual "private value" of player i, while each other coordinate tj^i represents i's total payoff should j get the object. In this framework, we characterize incentive-compatible and individually-rational mechanisms, and look at second price auctions (which, under some conditions, maximize the seller's revenue). Any incentive combatible mechanism induces a conditional probability assignement vector field which is conservative. A useful geometric property of conservative vector fields is used for the derivation of a differential equation which determines equilibrium bids. Finally, we show that exclusion (i.e., the announcement of a reservation price such that a measure can never get the object) is not necessarilly optimal for the seller. This contrasts with Armstrong's (Econometrica, 1995) insight about the optimality of exclusion in another multidimensional setting.

[1]  Leonard J. Mirman,et al.  Optimal Nonlinear Prices for Multiproduct Monopolies , 1980 .

[2]  M. Spence.  Multi-product quantity-dependent prices and profitability constraints , 1980 .

[3]  Roger B. Myerson,et al.  Optimal Auction Design , 1981, Math. Oper. Res..

[4]  Paul R. Milgrom,et al.  A theory of auctions and competitive bidding , 1982 .

[5]  Thomas R. Palfrey,et al.  Bundling Decisions by a Multiproduct Monopolist with Incomplete Information , 1983 .

[6]  J. Rochet.  The taxation principle and multi-time Hamilton-Jacobi equations☆ , 1985 .

[7]  Steven A. Matthews,et al.  MONOPOLY PROVISION OF QUALITY AND WARRANTIES: AN EXPLORATION IN THE THEORY OF MULTIDIMENSIONAL SCREENING , 1987 .

[8]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[9]  R. Radner,et al.  Optimal Nonlinear Pricing with Two-Dimensional Characteristics , 1987 .

[10]  R. McAfee,et al.  Multidimensional incentive compatibility and mechanism design , 1988 .

[11]  D. Sappington,et al.  Regulating a Monopolist with Unknown Demand and Cost Functions , 1988 .

[12]  M. Whinston,et al.  Multiproduct Monopoly, Commodity Bundling, and Correlation of Values , 1989 .

[13]  Jean Tirole,et al.  The regulation of multiproduct firms: Part I: Theory , 1990 .

[14]  Robert Wilson.  Multiproduct tariffs , 1991 .

[15]  E. Stacchetti,et al.  How (not) to sell nuclear weapons , 1996 .

[16]  M. Armstrong.  Multiproduct Nonlinear Pricing , 1996 .

[17]  J. Rochet,et al.  Ironing, Sweeping, and Multidimensional Screening , 1998 .

[18]  Raghuram Iyengar,et al.  Nonlinear pricing , 2022 .