Robust Trading Mechanisms with Budget Surplus and Partial Trade Jesse A. Schwartz Kennesaw State University Quan Wen Vanderbilt University May 2012 Abstract In a bilateral bargaining problem with private values, Hagerty and Rogerson (1987) showed that essentially all dominant strategy incentive compatible, ex post individually rational, and budget balanced mechanisms are posted-price mechanisms, where a price is drawn from a distribution, and trade occurs if both players benefit from trading at this price. We relax budget balancedness to feasibility, and demonstrate that allowing a budget surplus can increase expected gains from trade. Our example demonstrates that when searching for the optimal bargaining mechanism, we cannot generally limit our attention to budget-balanced mechanisms. Keywords: Dominant strategy implementation; Vickrey-Clarke-Groves mechanisms; bilateral bargaining; budget balancedness JEL Classification Numbers: C72 (Noncooperative Games); C78 (Bargaining Theory); D82 (Asymmetric and Private Information) Department of Economics, Finance, and Quantitative Analysis, Kennesaw State University, 1000 Chastain Road, Box 0403, Kennesaw, GA 30144, U.S.A. Email: jschwar7@kennesaw.edu Department of Economics, Vanderbilt University, VU Station B #351819, 2301 Vanderbilt Place, Nashville, TN 37235-1819, U.S.A. Email: quan.wen@vanderbilt.edu
1 Introduction It is well-known that dominant strategy incentive compatibility (DIC), ex post individual rationality (EIR), budget balancedness (BB), and allocative efficiency are not compatible in the bilateral bargaining model of Myerson and Satterthwaite (1983). Trade-offs must be made among these properties. Hagerty and Rogerson (1987) characterize essentially all DIC, EIR, and BB mechanisms as posted-price mechanisms, where a price is drawn from a probability distribution, and trade occurs at this price if both players agree. We address whether the players can obtain more expected gain from trade by relaxing BB to feasibility, allowing the buyer to pay more than the seller receives. It may seem counter-intuitive that raising surplus revenue only to destroy it can be beneficial. Yet, Čopič and Ponsatí (2008, pp 16) recognize the possibility that a mechanism which maximizes expected gain from trade could lie outside of the class of posted-price mechanisms. We provide such an example of a feasible but not BB mechanism that does indeed generate larger expected gain from trade than any posted-price mechanism. We first show that any posted-price mechanism does no better than choosing a single price. However, such a mechanism never admits trade when the seller s and buyer s values are both below or both above the price. We then devise an example that places enough probability on the possibilities that players values are both low and both high, and show that a feasible mechanism that allows for trade in these realizations generates a higher expected gain from trade than any single-price mechanism. Our paper fits into the emerging literature on trading off allocative efficiency and BB among DIC mechanisms. An early example is McAfee s (1992) double auction that sometimes sacrifices the least efficient trade to set the price for the other traders, where the efficiency loss becomes negligible as the number of traders grows. For cost-sharing problems, Moulin and Shenker (2001), Roughgarden and Sundararajan (2006), and Mehta, Roughgarden and Sundararajan (2009) strengthen DIC to coalition strategy-proofness, allowing for 1
either budget imbalance, or inefficiency, or both. For rationing problems, Moulin (2009) considers designing the lump-sum payments in an allocatively efficient mechanism to alleviate budget imbalance. De Clippel et al. (2009) show that destroying the objects in rationing can reduce the surplus revenue and improve the performance. Observe that there are no sellers in the rationing and cost-sharing problems, only buyers. We study a bargaining problem between a seller and buyer, where there can be no lump-sum payments without violating feasibility and individual rationality. We find that partial trade can improve the overall performance of the mechanism, even if it leads to more surplus revenue, not less. 2 Preliminaries Consider the bargaining problem with private values, analyzed for Bayesian incentive compatibility (BIC) by Myerson and Satterthwaite (1983) and DIC by Hagerty and Rogerson (1987). Player 1 is a seller who has one object worth 1 to herself and player 2 is a buyer whovaluesthisobjectat 2.Values 1 and 2 are private information and are drawn from distribution function ( ) with density function ( ) that is positive on support [ ] 2. Both players are risk-neutral, with linear utility functions. A mechanism consists of quantity and payment rules. For reported values ( 1 2 ),the seller supplies quantity ( 1 2 ) 0 and receives payment 1 ( 1 2 ) 0, and the buyer obtains quantity ( 1 2 ) 0 and pays 2 ( 1 2 ) 0. The seller s ex post utility is 1 ( 1 2 ) 1 ( 1 2 ) and the buyer s ex post utility is 2 ( 1 2 ) 2 ( 1 2 ).Amechanismis:dominant strategy incentive compatible (DIC) if it is always optimal for each player to report her value truthfully; ex post individually rational (EIR) if each player always earns non-negative ex post utility; budget balanced (BB) if the buyer always pays exactly what the seller receives; and feasible if the buyer always pays no less than the seller receives. Myerson and Satterthwaite (1983) and Hagerty and Rogerson (1987) implicitly limit their attention to BB mechanisms by a single payment rule ( 1 2 ) from the buyer to the seller. As we will show in this paper, however, limiting attention to BB mechanisms can come at 2
a cost, sometimes leading to lower expected gain from trade than some feasible mechanism. We see no reason to favor BB over feasibility if BB reduces gain from trade. In contrast, although DIC leads to lower gain than BIC, there are normative reasons for preferring DIC, as discussed by Mookherjee and Reichelstein (1992). Hagerty and Rogerson (1987) posted-price mechanism gives a probability distribution function ( ) on [ ] from which a posted-price is drawn, and then the trade occurs at price if and only if 1 2. Recall that a distribution function may not be left continuous: denote the left limit of ( ) at by (). The quantity and payment rules from a posted-price mechanism are ( 1 2 ) = ( 1 2 ) = ( (2 ) ( 1 ) if 1 2 0 otherwise, ( { [1 2 ]} if 1 2 0 otherwise, for =1 2. Because the posted price is chosen from an exogenous distribution, it does not depend on the players reports. In this model with linear utility, we may synonymously refer to ( 1 2 ) as either the probability of trading one indivisible unit or the partial quantity traded of a divisible unit. By construction, a posted-price mechanism is DIC, EIR, and BB. Within the class of BB mechanisms that Hagerty and Rogerson (1987) restrict attention to, they show that posted-price mechanisms are the only mechanisms that are DIC and EIR as long as the quantity rule is: twice continuously differentiable, into {0 1}, or a block function (similar to a step function). They conjecture that any DIC, EIR, and BB mechanism is a posted-price mechanism. If true, then we can simply focus on posted-price mechanisms when searching for the optimal DIC, EIR, and BB mechanism. We will show that it may be worthwhile to explore outside the class of BB mechanisms. We denote as a single-price mechanism the special case of the posted-price mechanism when ( ) is degenerate, placing probability 1 on some price [ ], and now show the 3
single-price mechanism performs at least as well. Proposition 1 For any posted-price mechanism, there exists a single-price mechanism that yields at least as much expected gain from trade. Proof. Consider any posted-price mechanism with price drawn from ( ). Given [ ], trade occurs whenever ( 1 2 ) () {( 1 2 ) [ ] 2 : 1 2 }. The expected gain from trade is R () (),where Z () = ( 2 1 ) ( 1 2 ) ( 1 2 ) () Since () is continuous for [ ], it contains a maximum, say at. Then the single-price mechanism with this yields at least as much expected gain from trade. Although a single-price mechanism generates the largest gain from trade among all posted-price mechanisms, it foregoes any trade on a large set of values where trade is beneficial: 1 2 and 1 2. To further improve performance, trade must be extended to these regions. 3 A Superior Feasible Mechanism We now provide a DIC, EIR, and feasible mechanism that has larger expected gain from trade than any posted-price mechanism. We first present a Vickrey-like (1961) framework for a mechanism that is sufficient to induce DIC and EIR, but unlike Vickrey in that efficiency is not imposed. Let each player face some price schedule that is independent of her own report, though possibly dependent on the other player s report. Denote the seller s and buyer s price schedules by 1 ( 2 ) and 2 ( 1 ). The mechanism chooses the quantity that maximizes 4
each player s utility based on her price schedule, and each player is paid or pays accordingly: Z 1 ( 1 2 ) arg max 1 ( 2 ) 1 1 (1) 1 [01] Z (1 2 ) 0 1 ( 1 2 ) = 1 ( 2 ) (2) 0 Z 2 ( 1 2 ) arg max 2 2 2 ( 1 ) (3) 2 ( 1 2 ) = 2 [01] Z (1 2 ) 0 0 2 ( 1 ) (4) It is straight-forward to show that it is always optimal for each player to report her true value in a mechanism satisfying (1) and (3), and thus such a mechanism is DIC. As an aside, the DIC of this mechanism with exogenous price schedules has been earlier established as part of the taxation principle (Rochet, 1985). Because the maximizations in (1) and (3) allow for =0and a zero payoff, payoffs can never be negative, thereby also establishing EIR. Consider a bargaining problem where =0and =1. The distribution of the players values will be specified later. In what follows, we first specify the quantity rule and then construct each player s price schedule to be used in (1)-(4) to implement this prescribed quantity rule. In order for the mechanism to outperform any single-price mechanism, we allow for partial trade so that we can enlarge the set of players values where positive trade occurs, getting trade when both players have low values and high values. More specifically, for some 1, consider the following quantity rule: 2 1 if ( 1 2 ) ( 1 2 )= ˆ if ( 1 2 ) 0 otherwise. (5) where ˆ = 1 2 1 2 1 1 2 1 due to 1 2 and where, as shown in Figure 1, =[0] [1 1] =[0] [1 ) and =( 1 ] [1 1] 5
2 6 1 1 0 1 1-1 Figure 1: Trade regions For each 2 [0 1], the seller s price schedule at quantity is the highest value she can have and still supply at least : 2 [0) 2 [1 ) 2 [1 1] ˆ 1( 2 )=0 1( 2 )= 1( 2 )=1 ˆ 1( 2 )=0 1( 2 )=0 1( 2 )= Likewise, for each 1 [0 1], the buyer s price schedule at quantity is the lowest value she can have and still obtain at least : 1 [0] 1 ( 1 ] 1 (1 1] ˆ 2( 1 )= 2( 1 )=1 2( 1 )=1 ˆ 2( 1 )=1 2( 1 )=1 2( 1 )=1 It is straight-forward to verify that these price schedules used in (1) and (3) do indeed induce ( 1 2 ) of (5). For example, for ( 1 2 ), 2 [1 ), the seller faces price up to quantity ˆ and price 0 thereafter. At these prices, for 1 [0], the payoff-maximizing quantity for the seller is indeed ˆ, asspecified by ( 1 2 )=ˆ for all ( 1 2 ). Consider the mechanism with quantity rule ( 1 2 ) and the following payment rules 6
derived from (2) and (4): 1( 1 2 )= 2( 1 2 )= for ( 1 2 ) +(1 )ˆ +(1 )(1 ˆ) for ( 1 2 ) ˆ ˆ for ( 1 2 ) (1 )ˆ (1 )ˆ otherwise 0 0 This mechanism is DIC and EIR because of (1)-(4). We now verify its feasibility. For ( 1 2 ), wehave 1( 1 2 )= 2( 1 2 ) due to ˆ = 1 2 1 2 1. For ( 1 2 ), wehave 1( 1 2 )=ˆ ˆ = 2( 1 2 ) due to. For ( 1 2 ), wehave 1( 1 2 )=(1 )ˆ(1 )ˆ = 2( 1 2 ) due to. For ( 1 2 ), there is no trade and no payment. Suppose that players values are more concentrated at (0 1 2 ) and ( 1 2 + 1). Assume ( 1 2 )=(1 )(1 1 2)+( 1 1 2 2) 2 where (1 1 2) 1 is distributed at (0 1 ) and ( 1 + 1) with 2 2 equal probability, and (1 2 2) 2 is uniformly distributed on [0 1] 2. Notethatnosingle-price mechanism can admit trade at both (0 1 ) and ( 1 + 1). The expected gain from trade 2 2 of the optimal (single) posted-price mechanism (with price at 1 ) is 2 = (1 ) 1 µ 2 1 1 Z 1 2 + 2 Z 1 + µ 1 = (1 ) 4 + 1 µ 1 2 + 8 1 2 2 0 1 4 ( 2 1 ) 2 1 1 2 as 0 and 0 On the other hand, our feasible mechanism ensures trade with quantity ˆ as long as 1 2 + [ 1 ]. Inparticular,choose = 1 2 and = 1 2 1 2.For( 1 2 ), the seller and buyer will always trade at quantity ˆ = 1 1 2 = 2. The expected gain from trade 2 1 3 of our feasible mechanism is µ 1 1 = (1 ) 2 ˆ 2 Z Z + ( 2 1 ) 2 1 +ˆ + ˆ( ) + 12 µ 12 ˆ + ˆ( ) Z ( 2 1 ) 2 1 +ˆ 7 ( 2 1 ) 2 1
As 0 and 0, wehave 1 2 2 3 1 2 + 1 2 2 3 1 2 = 1 3 1 4 Because the limit of exceeds of as 0 and 0, our feasible mechanism yields a higher expected gain from trade than the optimal posted-price mechanism when 0and 0aresmallenough. Asanaside,itispossibletofurtherimprovethegainfromtrade by adding positive trade to ( 1 2 ) for some ( 1 2 ) such that 1 2 1. This, however, would increase the complexity of the mechanism. To summarize, in the class of DIC mechanisms, we demonstrated that it is possible improve the players payoffs by allowing for budget surplus, using the weaker requirement of feasibility rather than BB. Our counter example demonstrates that when searching for the optimal bargaining mechanism, we cannot generally limit our attention to budget-balanced mechanisms. This paper raises two fundamental questions. Can the optimal DIC, EIR, and feasible, bilateral-bargaining mechanism be characterized? Does the characterization in Myerson and Satterthwaite (1983) of optimal BIC, EIR, and BB bargaining continue to hold if we relax BB to feasibility? References [1] Čopič, J. and C. Ponsatí (2008): Ex-Post Constrained-Efficient Bilateral Trade with Risk-Averse Traders, manuscript, UCLA. [2] De Clippel, G., V. Naroditskiy, M. Polukarov, A. Greenwald, and N.R. Jennings (2009): Destroy to Save, in the: Proceedings of the 10th ACM conference on Electronic Commerce. [3] Hagerty, K.M., and W.P. Rogerson (1987): Robust Trading Mechanisms, Journal of Economic Theory, 42, 94 107. [4] McAfee, R.P. (1992): A Dominant Strategy Double Auction, Journal of Economic Theory, 56, 434 450. 8
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