Optimal Allocation of an Indivisible Good

Transcription

1 Optimal Allocation of an Indivisible Good Ran Shao y Yeshiva University Lin hou z Shanghai Jiao Tong University This Version: January Abstract In this paper we consider the problem of allocating an indivisible good between two agents under incomplete information. We provide a characterization of mechanisms that maximize the sum of the expected utilities of the agents among all deterministic feasible strategy-proof mechanisms: Any optimal mechanism must belong to either one of the two x price mechanisms or one of the two option mechanisms. Furthermore, when stochastic mechanisms are allowed and agents types are distributed uniformly, we show that any optimal mechanism must be a convex combination of two xed price mechanisms and two option mechanisms. Introduction In this paper we consider the problem of allocating an indivisible good between two agents when agents valuations of the good are private information. A typical problem of such is the bilateral bargaining model in which a seller and a buyer negotiate with each other as to if and how to trade a particular good. This problem has generated a large literature since the pioneering work of Myerson and Satterthwaite (983). Another more practical example is how to allocate a more desirable o ce between two interested employees. The The authors want to thank the the following scholars for their comments on the work reported in this paper: Hector Chade, Kim-Sau Chung, Jerry Green, Alejandro Manelli, Eric Maskin, David Miller, Benny Moldovanu, Stephen Morris, Ed Schlee, and participants at the 7 Far Eastern Meeting of the Econometric Society in Taipei and the 8 Southwest Economic Theory Conference at UC Santa Barbara. y Department of Economics, Yeshiva University, 5 Lexington Ave. Room 75, New York, NY 6. rshao@yu.edu. z Antai College of Economics and Management, Shanghai Jiao Tong University, The People s Republic of China. zhoulin@sjtu.edu.cn

2 intense research interest in this type of problem is derived from a fundamental dilemma due to Green and La ont (977): when agents valuations are private information, it is impossible to nd a costless method that always gives the good to the agent with the higher valuation. Several methods are commonly used in practice to solve such allocation problems: Lotteries, seniority ranking or other type of queuing, or even auctions, are just few of the well-known examples. Many of such methods are quite e ective in soliciting agents revelation of true valuations. Yet some methods may often assign the good to the agent with the lower valuation (lotteries), others may incur negative cash out ows from the agents (auctions). At the more theoretical level, two particular classes of methods have received researchers attentions. The rst class consists of all Vickery-Clark-Groves pivotal mechanisms (Vickery, 96, Clarke, 97, Groves, 973) that extend the conventional English auction scheme. The second class consists of all xed-price mechanisms (Hagerty and Rogerson, 987), in which the good is assigned to one agent (the seller) unless both agents are willing to trade the good at a predetermined price. It is well-known that these two classes of methods have their own strength and weakness, yet no one has ever carried out any formal comparisons of these two methods, not to mention comparisons of more general methods. We shall conduct a systemic investigation of various allocation methods from the optimal mechanism design perspective. We focus on methods that are both robust and practical. Speci cally, we require that all methods be immune to individual manipulation so that it must be a dominant strategy for each agent to reveal his/her true valuation. We also require that all methods be feasible so that there would be no need of injection of money from outside. Our task is to nd among all such methods those that maximize the sum of the utilities of both agents. We show that xed price mechanisms are indeed optimal when primitives are well behaved. In addition, this exercise leads us to another class of mechanisms, called the option mechanisms. In an option mechanism, one agent is the temporary holder of the good and the other agent is the recipient of a call option that allows him to purchase the good from the rst agent at a predetermined price. The good changes hands as long as the option recipient is willing to buy the good from the temporary holder at the predetermined price. In comparison, under the xed price mechanism, the good changes hands only when both agents agree to the trade at a predetermined price. We can show that option mechanisms are also optimal. In fact, one of the main results of our paper is that any optimal mechanisms must be a lottery of xed price mechanisms and

3 option mechanisms. It is also interesting that we have found a new role for options. In the traditional nance literature options are either used as instruments for risk management for investors or as means to provide incentives for managers. In our model, however, they are used as a part of a mechanism to maximize the sum of agents utilities. Our results make a signi cant contribution to the literature of mechanism design. While the optimal mechanism design approach has been standard for the study of Bayesian mechanisms, it has rarely been applied by anyone to study strategy-proof mechanisms. Our paper is one of the very few that have identi ed the structure of optimal strategy-proof mechanisms in a canonical allocation model. The most closed related paper is Miller (). Miller considers a model that is similar to ours in a concurrently developed paper (Miller, ). The two- rm cartel with private information case subsumes the same math structure as our model. He shows that in general the mechanisms (V-C-G mechanisms) that guarantee the allocation e ciency cannot be optimal (Theorem of Miller ()). However, the paper s major focus is to identify the conditions under which the money burning (price war) is unavoidable. Moreover, there is no mentioning of the option mechanism and the characterization of all optimal mechanisms. The fundamental di culties of analyzing such kind of problem are underlying the problem of in nite dimension linear programming. In general, solving the optimal solution explicitly is extremely di cult. It is much easier to compute it numerically other than characterizing the properties the solution in general even after discretizing the problem into a nite problem. The Main Results Consider a model with two agents and one indivisible good. The good is a private good so that it can be consumed by one agent only. Each agent has a quasi-linear utility function The fact that there few known results with strategy-proof mechanisms is not the only motivation behind our work. The predicted outcomes of a strategy-proof mechanism are deemed reliable since all agents have unambiguous optimal actions regardless others actions. The predicted outcomes of a Bayesian mechanism are accepted only under strong informational and behavioral assumptions. For example, one must assume that the distribution of agents types is common knowledge among all agents and is also known to the designer of the mechanism. We will not debate on the relative merits of Bayesian mechanisms, strategyproof mechanisms and other alternatives here. Interested readers can nd them in other papers on this issue (Chung and Ely, 4, d Aspremont and Gerard-Varet (979), Bergemann and Morris, 5, Jehiel et al, 6). 3

4 for the good and the money transfer, v i (x i ; t i ; i ) = i x i t i : Here the parameter i [; ] is agent i s valuation of the good, or i s type and x i [; ] is the probability that agent i receives the good. The value of i is known to agent i only. When the values of and are commonly known, the e cient allocation is to give the good to the agent with the higher i. However, when and are privately known only, it is not always possible to identify and execute the e cient allocation. We consider here direct mechanisms that ask agents to report their types and use their reported types to determine the allocation as well as transfers to the agents. This attention on direct mechanisms is not excessively restrictive since, by the revelation principle, our analysis immediately extends to all direct mechanisms in which both agents have dominant strategies at every type pro le. Otherwise, we allow nearly all direct mechanisms. For the general setup, we allow all mechanisms that allocate the good to agents randomly. In fact, one of our main results is derived only allowing deterministic mechanisms without withholding by the designer. Formally, a direct mechanism M consists of four integrable functions on [; ] [; ]: x ( ; ); x ( ; ); t ( ; ); t ( ; ). Since x ( ; ) and x ( ; ) are respectively the probabilities agent and agent receive the good, they must satisfy x i ( ; ) [; ] ; and x ( ; ) x ( ; ) ; 8 ; : The functions t ( ; ) and t ( ; ) are transfers to the agents. To ensure that agents have incentives to report their true types, we require that all mechanisms under consideration be strategy-proof, i.e., (IC) x ( ; ) t ( ; ) x ( ~ ; ) t ( ~ ; ); 8 ; ~ ; x ( ; ) t ( ; ) x ( ; ~ ) t ( ; ~ ); 8 ; ; ~ : We also require that all mechanisms be feasible so that it does not need outside money, (F) t ( ; ) t ( ; ) ; 8 ; : We are still working on the more general result which uni es both Theorem and by allowing all stochastic mechanisms. 4

5 The majority of work on bilateral bargaining further assumes budget-balanced-ness, i.e., (BB) t ( ; ) t ( ; ) = ; 8 ; : We don t want to impose budget-balanced-ness in our work. Although money burning seems ine cient ex post, it might conceivably increase the ex ante e ciency of a mechanism since it could provide incentives more e ectively. In fact, all Vickery-Groves-Clarke mechanisms that implement the e cient allocation of the good do not satisfy budgetbalanced-ness. If we were to impose budget-balanced-ness, we would have excluded a large class of mechanisms that are popular in the literature of mechanism design from consideration and the end result would be much weaker. So far the basic structure of our model looks virtually the same as the auction model with private values (Myerson, 98) and the bilateral bargaining model (Myerson and Satterthwaite, 983, Hagerty and Rogerson, 987). The main distinction between our model and the others is the di erence in objectives. In the auction literature the objective is maximal revenue extraction by an outsider from the agents, whereas in the bilateral bargaining model the objective is maximal revenue extraction by one agent (the seller) from the other (the buyer). In contrast, we adopt the utilitarian viewpoint here and our objective is to nd mechanisms that maximize the sum of ex-ante utilities of both agents. Given any feasible strategy-proof mechanism M = fx i ( ; ); t i ( )ji = ; g, the sum of agents utilities is at each ( ; ) is U ( ; ) U ( ; ) = x ( ; ) x ( ; ) t ( ; ) t ( ; ): Since the mechanism M satis es feasibility, U ( ; ) U ( ; ) max f ; g ; 8 ; : If we could nd a feasible strategy-proof mechanism M for which U ( ; ) U ( ; ) = max f ; g ; 8 ; ; then M would be the optimal mechanism. However, this is impossible the classical result by Green and La ont (977) shows. On the other hand, for every ( ~ ; ~ ), it is easy to nd 5

6 a feasible strategy-proof mechanism M ( ~ ; ~ ) for which U ( ~ ; ~ ) U ( ~ ; ~ n ) = max ~ ; ~ o ; (just the trivial mechanism that always give the good to the agent with the higher ~ i ). Hence, we cannot nd a rst-best mechanism that (weakly) dominates all others at all type pro les. As a result, a meaningful criterion of optimality should be based on some average measurement. Throughout the paper, we assume that agents types are distributed symmetrically according to a nonatomic distribution function F with density f. Let F denote the class of all feasible strategy-proof mechanisms. For each M F, the total utilities of M is given by: T U(M) = ( x ( ; ) x ( ; ) t ( ; ) t ( ; ))df ( ) df ( ) : Our goal is to characterize the mechanisms M F that yield the highest ex ante total utilities, that is, T U(M ) = max MF T U(M): To convey the intuition of our main theorems, let us calculate T U(M) for some wellknown mechanisms under uniform distribution. First, the canonical pivotal mechanism (or the second price auction mechanism) M SP has the total utility T U(M SP ) = 3, which is not very high. It is not even the best one among all V-C-G mechanisms. In a separate paper we nd the best V-C-G mechanism M BV CG with T U(M BV CG ) = 7 (Shao and hou, 7). Hagerty and Rogerson (987) consider xed price mechanisms: assuming that agent one is the seller and agent two the buyer, a trade will take place at some xed price p if and only both the seller and the buyer agree. Formally, the xed price mechanism with price p is de ned as follows (see Figure ): 8 >< >: x ( ; ) = ; t ( ; ) = p; x ( ; ) = ; t ( ; ) = p; when p and p; and 8 >< >: x ( ; ) = ; t ( ; ) = ; x ( ; ) = ; t ( ; ) = ; otherwise. 6

7 Figure Among all xed price mechanism, the mechanism with the price p = yields the highest total utility T U(M F ) = 5 8. (The same holds for the xed price mechanism in which agent two is the designated seller.) In this paper we also consider another mechanism, called the option mechanism, which is related to, but di erent from, the xed price mechanism. It gives the good to agent conditionally and, at the same time, issues a call option to agent that allows him to buy the good from agent one at a xed exercise price of p. Obviously, agent will exercise the option if and only if > p. Formally, it is de ned as follows (see Figure ): 8 >< >: x ( ; ) = ; t ( ; ) = p; x ( ) = ; t ( ; ) = p; when p; and 8 >< >: x ( ; ) = ; t ( ; ) = ; x ( ; ) = ; t ( ; ) = ; otherwise Among all option mechanisms, the mechanism with the option price p = yields the highest total utility T U(M O ) = 5 8. (The same holds for the option mechanism in which agent two is the conditional owner of the good and agent one is awarded the option.) It is interesting that the best xed price mechanism and the best option mechanism yield 7

8 Figure the same level of total utilities. These two mechanisms di er only in the region and where both agents types are greater than or equal to. The xed price mechanism favors agent by giving the whole region to agent, whereas the option mechanism favors agent. The total utilities are the same since agents types are distributed symmetrically. Our main ndings consists of two theorems. The rst theorem con rms that the aforementioned two types of budget-balance mechanisms x price mechanisms and option mechanisms are indeed the optimal mechanisms among all deterministic ones under a class of well-behaved distribution functions. By allowing stochastic mechanisms, the second theorem shows that both the xed price mechanisms and the option mechanisms are optimal when agents types distributed uniformly. Furthermore, all optimal mechanisms are convex combinations of these four mechanisms. Optimal Mechanisms under General Distributions As shown numerically by Example 4 of Miller (), under some distribution functions the optimal mechanisms could display both allocation ine ciency and money burning which is a compromise of both V-C-G mechanisms and budget-balance mechanisms. This implies that in general it is very di cult to solve for the optimal mechanisms analytically under arbitrary distributions. Hence, we restrict our attention to characterizing the optimal mechanisms 8

9 under a class of well-behaved distribution functions of agents types. In particular, we restrict the solution set to all the deterministic feasible mechanisms without withholding by the designer. Precisely, we requires that x ( ; ) ; x ( ; ) f; g and x ( ; ) x ( ; ) =. We rst derive some lemmas to help to prove the rst theorem. In fact, unlike the rst theorem, these lemmas also hold when stochastic mechanisms allowed. We begin with the structure of a generic mechanism M F. First, since M is strategyproof, x ( ; ) is non-decreasing in and x ( ; ) is non-decreasing in. For every pair of such x ( ; ) and x ( ; ), we can nd a continuum of pairs of t ( ; ) and t ( ; ) such that these four functions de ne a feasible strategy-proof mechanism. The canonical transfers are the generalized pivotal-taxes: t p ( ; ) = x ( ; ) t p ( ; ) = x ( ; ) x (; )d; and x ( ; )d: By de nition, the pivotal-taxes are non-positive so that they de ne a feasible mechanism. However, they represent out ows of money from agents so the resulting mechanism is not e cient. To improve the e ciency of the mechanism, we consider redistribution of the pivotal transfers between two agents while keeping the incentive property of the mechanism intact. To achieve this goal, we add some function of only h ( ) to t p ( ; ) and some function of only h ( ) to t p ( ; ): t ( ; ) = x ( ; ) t ( ; ) = x ( ; ) x (; )d h ( ) ; and x ( ; )d h ( ) : Obviously, the new mechanism is still strategy-proof. In fact, this is actually the only way to maintain the incentives. We may consider these functions h ( ) and h ( ) as rebates to the agents h ( ) is the amount of money agent receives when agent s type is and h ( ) is the amount of money agent receives when agent s type is. The total amounts of money that can be redistributed between the agents are limited from above by 9

10 the feasibility condition: (F) h ( ) h ( ) x ( ; ) x (; )d x ( ; )d: Based on the above discussion, the optimal mechanism problem can be written as the following programming problem: T U (M) = subject to max fx i ;t i ji=;g x ( ; ) x ( ; )t ( ; )t ( ; ) df ( ) df ( ) : () () (F) h ( ) = x ( ; ) h ( ) = x ( ; ) x (; ) d t ( ; ) x ( ; ) d t ( ; ) t ( ; ) t ( ; ) (a:e:) x ( ; ) is weakly increasing in x ( ; ) is weakly increasing in : (a:e:) (a:e:) A mechanism M is feasible if it satis es all the constraints above. To avoid unnecessary repetitions, we will drop the quali er (a.e.) from the proof whenever we invoke the (), () and (F) 3. Since x ( ; ) t ( ; ) = R x (a; )da h ( ), we have = = = ( x ( ; ) t ( ; ))df ( ) df ( ) x (; )d df ( ) df ( ) x (; )df ( ) ddf ( ) ( F ())x (; )ddf ( ) h ( )df ( ) h ( )df ( ) h ( )df ( ) 3 The proof remains valid if we have to be more rigorous. At some steps we need to use the Fubini theorem to justify our argument.

11 = F ( ) x ( ; )df ( ) df ( ) f ( ) Thus, we can rewrite the objective function T U (M) as T U (M) = h ( ) df ( ) F ( ) f ( ) h ( )df ( ) : x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) h ( ) df ( ) : Claim: x ( ; ) R x (; ) d is increasing in ; and x ( ; ) R x ( ; ) d is increasing in. To see this, x, assume <, x ; = = x ; x ; x ( ; ) d x ( ; ) d x ( ; ) x ( ; ) d x ; x ( ; ) d (x ( ; ) x ( ; )) d x ( ; ) d The last step holds because x ( ; ) is increasing in : Hence, t ( ; ) is decreasing. R Similarly, x ( ; ) x (; ) d is increasing in and t ( ; ) is decreasing in. Lemma Given any M ~ F, there exists a symmetric M = fx i ( ; ) ; h i () ji = ; g F with T U ~M = T U (M), such that x ( ; ) = x ( ; ) ; t ( ; ) = t ( ; ) and h () = h (). The idea of this lemma is that the two agents are ex-ante the same. If any optimal mechanism is in favor of one of the two agents, there must exist another optimal mechanism which is in favor of the other agent. Then, based on these two mechanisms, one can derive a "fair" mechanism which treat them equally. Hence, given a arbitrary mechanism, one can always nd a symmetric mechanism which is feasible and yields the same total utility as the given one. Technically, such symmetric mechanisms have more structure and therefore

12 easy to work with. Without loss of generality, we only need to focus on such symmetric mechanisms to nd the optimal mechanisms. Proof: Given ~ M = Obviously, the mechanism utility. Now, let n ~x ( ; ) ; ~x ( ; ) ; h ~ ( ) ; h ~ o ( ), if it is not symmetric, let ~x ( ; ) = ~x ( ; ) ; ~x ( ; ) = ~x ( ; ) and ~h () = ~ h () ; ~ h () = ~ h () : n ~x ( ; ) ; ~x ( ; ) ; h ~ ( ) ; h ~ o ( ) gives the same total x ( ; ) = ~x ( ; ) ~x ( ; ) x ( ; ) = ~x ( ; ) ~x ( ; ) h () = h ~ () h ~ () h () = h ~ () h ~ () : It is immediate to see that the mechanism fx ( ; ) ; x ( ; ) ; h ( ) ; h ( )g is symmetric and also yields the same total utility. We only need to verify it is feasible. By letting t ( ; ) = ~ t ( ; ) ~ t ( ; ) and t ( ; ) = ~ t ( ; ) ~ t ( ; ), it is straightforward to verify the feasibility. Q.E.D. Note that under the symmetric mechanism along the diagonal = of the type space, x (; ) = x (; ) = and h () = R x (; ) d t (; ). Furthermore, t (; ). When <, t ( ; ). Otherwise, t ( ; ) >, x and decrease, since t ( ; ) is decreasing in we have t ( ; ) > contradiction. Similarly, h () = R x (; ) d t (; ), t (; ) and t ( ; ) when >. Due to x ( ; ) is weakly increasing in and x (; ) =, x ( ; ) when <. Similarly, x ( ; ) when >. Lemma If A () is decreasing, R b R b a a A ( ) x ( ; )A ( ) x ( ; ) df ( ) df ( ) is maximized by letting x ( ; ) and x ( ; ) be constant with x ( ; )x ( ; ) =. Proof: Since x ( ; ) is increasing in and A ( ) is decreasing in, x, by

13 Chebyshev inequality 4, b a Similarly, b a A ( ) x ( ; ) df ( ) A ( ) x ( ; ) df ( ) Hence, F (b) F (b) F (a) F (a) b a b a A ( ) df ( ) A ( ) df ( ) b a b a x ( ; ) df ( ) : x ( ; ) df ( ) : b b a a F (b) F (a) A ( ) x ( ; ) A ( ) x ( ; ) df ( ) df ( ) b F (b) F (a) = F (b) F (a) = (F (b) F (a)) a b a b a b a A ( ) df ( ) A ( ) df ( ) A ( ) df ( ) A ( ) df ( ). b b a a b b a a b b a a x ( ; ) df ( ) df ( ) x ( ; ) df ( ) df ( ) x ( ; ) x ( ; ) df ( ) df ( ) It is easy to verify that any constant x and x with x x = can yield the derived upper bound. Q.E.D. REMARK: This lemma indicates that how to allocate the indivisible good with agents types belong to any block along the diagonal doesn t change the overall welfare as long as the probability remain constant within such blocks. Later, when dealing with symmetric mechanisms, we only need to set x = x = for such blocks. REMARK: The above lemmas hold for not just deterministic mechanisms but also stochastic ones. Theorem If ( F ) =f is decreasing and F=f is increasing, the optimal deterministic mechanisms must be x price mechanisms or option mechanisms with p = where = E (). 4 The conditions required by Chebyshev inequality can be further relaxed by using Ste ensen inequality (see also Shao ) 3

14 To prove this theorem: ) we restrict the solution set to these symmetric mechanisms corresponding to all deterministic mechanisms; ) we show that there is a unique optimal mechanism among the symmetric mechanisms; 3) we then derive the corresponding deterministic mechanisms which are optimal among all deterministic mechanisms automatically. Proof: Since M is strategy-proof, x ( ; ) is (weakly) increasing in and x ( ; ) is (weakly) increasing in. As we de ne ' ( ) = inf fbjx (b; ) = g ; and ' ( ) = inf fajx ( ; a) = g ; both ' ( ) and ' ( ) are (weakly) increasing functions from [; ] to [; ]. Since they are increasing, they are almost continuous everywhere on [; ]. They are virtually inverse functions to each other since = ' ( ) if and only if = ' ( ) at every pro le ( ; ) where both ' ( ) and ' ( ) are continuous. For simplicity, WLOG, we denote ' () as ' () and denote ' () as ' (). That is x ( ; ) = if ' ( ) and x ( ; ) = if < ' ( ). When we plot the graphs of these two functions on the unit square [; ] [; ], they separate the region in which agent gets the good from the region in which agent gets the good (see Figure 3). STEP : By the rst lemma, given any deterministic mechanism, there exists a symmetric mechanism which provides the same total utility. The corresponding symmetric allocation rule has x i ; ;. Moreover, x i = happens only within the area bounded by ' ( ) and ' ( ) as illustrated in the Figure 3. Next, we show that each corresponding symmetric mechanism is dominated by a unique symmetric mechanism meanwhile it is budget-balance. STEP : Denote ; as the point where h i achieves the maximum value. Also h () = R x (; ) d t (; ) together with t i (; ) imply that. By () and (); h ( ) h = x ; x ; x (; ) d x ( ; ) d t ; t ; 4

15 Figure 3 and h ( ) h = x ( ; ) x ; Due to (F), we have x (; ) d x ; d t ; t ; : h ( ) x ; x ; x ; d x ( ; ) d and h ( ) x ; x ; x (; ) d x ; d : As shown in Figure 3, ' (a) =. When i a, we know that, x ( ; ) = over 5

16 [; a] ; and x ( ; ) = over ; [; a]. Hence, h ( ) x ; x ; x x ( ; ) d for [; a] ; d x ( ; ) d and h ( ) For [a; ] and =, x ( ; ) =. Hence, x (; ) d for [; a] : h ( ) = x ( ; ) x ( ; ) d t ( ; ) x ( ; ) d for [a; ] : Similarly, h ( ) x (; ) d for [a; ] : To derive the symmetric optimal mechanism, we derive an upper bound for the symmetric M F. Then show that such upper bound can be achieved by a unique feasible symmetric mechanism. The upper bound is derived by dividing the objective functions into two parts and deriving the upper bound of each parts respectively. S-I: We derive the upper bound for the rst part of the objective function below: F ( ) f ( ) h ( ) df ( ) F ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) h ( ) df ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) f ( ) x ( ; ) df ( ) df ( ) df ( ) df ( ) f ( ) x ( ; ) df ( ) 6

17 = F ( ) f ( ) x ( ; ) F ( ) f ( ) x ( ; ) df ( ) df ( ) df ( ) The inequality is due to above disscussion. obtained when x ( ; ) = x ( ; ) = over [; ] [; ]. Due to Lemma, the maximum value is Notice that the allocation rule we determined depends on the value of x i over [; a] ; and ; [; a]. It is possible that to maximize the upper bound of the remaining part of the objective function, such values may need to be changed which makes the above argument invalid. However, in the following, we rule out such possibility. Hence, the validity of the above argument guaranteed. S-II: We determine the upper bound for the remaining part of the objective function: F ( ) f ( ) F ( ) f ( ) F ( ) f ( ) h ( ) df ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) By symmetry, h i should also satisfy: h ( ) df ( ) : h () = h () = x (; ) d t (; ) x (; ) d t (; ) : Let g ( ) = g ( ) = x ( ; ) d x (; ) d: n o n o Hence, h ( ) min g ( ) ; and h ( ) min g ( ) ; for i [ ; ]. This 7

18 yields an upper bound for the above remaining term, F ( ) f ( ) F ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) F ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) f ( ) min g ( ) ; df ( ) min g ( ) ; df ( ). Before we choose x i to maximize the upper bound. We rst compare the upper bounds using either g i () or alone. If g ( ) ; g ( ) >, the upper bound becomes (T) (T) (T3) F ( ) f ( ) F ( ) f ( ) F ( ) f ( ) df ( ). x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) Given Lemma, term (T3) is maximized at x i =. x i = too. To see it, over area [ ; ] [; ], >, F ( ) f( ) < F ( ) f( ) And term (T) is maximized by F ( ) f( ) is decreasing, then. To maximize this term, we should let x be as large as possible. But from previous discussion we also know that within the area > the largest value can be assigned to x is. Similar argument can also be applied to term (T). To sum up, if g ( ) ; g ( ) >, the upper bound is maximized by having x ( ; ) = x ( ; ) =. If g ( ) ; g ( ) <, the upper bound becomes 8

19 F ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) F ( ) x ( ; ) F ( ) f ( ) f ( ) x ( ; ) df ( ) df ( ) F ( ) f ( ) x ( ; ) F ( ) x ( ; ) df ( ) df ( ) f ( ) F ( ) x ( ; ) F ( ) f ( ) f ( ) x ( ; ) df ( ) df ( ) df ( ). It is maximized by having x = if > and x = if > : Compare the above two upper bounds using either g i () or alone; x, when >, using g ( ) and as an upper bound requires to assign x i with di erent values. Meanwhile, the the choice of x i ( ; ) also a ects the choice of g ( ) and as minimum value used. If x ( ; ) = for <, g ( ) = R o x ( ; ) d is less than. Then, ng can not be used as min ( ) ;. If x ( ; ) = for <, g ( ) is greater than. Then g ( ) cannot be used as g ( ). o WLOG, we start with the upper bound using ng as min ( ) ;. Fix, that means we need to increase the value of x ( ; ) = to for some when > to prevent g ( ) <. To do that, one can only increase x ( ; ) to from the smallest due to monotonicity and up to the point ( ; ) ; otherwise g ( ) becomes greater than which means increasing x ( ; ) = to will decrease the the value of the upper bound. As a result the upper bound is maximized by having x ( ; ) = x ( ; ) = over [ ; ][ ; ], x ( ; ) = over [; ][ ; ] and x ( ; ) = over [ ; ][; ]. To put the two parts of the maximized value of the upper bound together, we have F ( ) F ( ) d ( F ( )) where = E (). Now, we optimize the upper bound over. Take derivative of we have f ( ) ( F ( )) ( F ( )) f ( ) = 9

20 It is easy to see that the upper bound is optimized when =. Moreover, such upper bound is obtainable by using allocation rule: x ( ; ) = x ( ; ) = for ( ; ) (; ) (; ) [ (; ) (; ), x ( ; ) = for ( ; ) [ ; ] [; ] and x ( ; ) = for ( ; ) [; ] [ ; ]. STEP 3: The corresponding deterministic allocation rule is x ( ; ) = for ( ; ) [ ; ] [; ] and x ( ; ) = for ( ; ) [; ] [ ; ] and either x or x equals to over (; ) (; ) [ (; ) (; ). Depends on the value of x i chosen over (; ) (; ) [ (; ) (; ), the mechanisms are corresponding to the two x price mechanisms and two option mechanisms. Q.E.D.. Optimal Mechanisms when Stochastic Mechanisms Allowed Next, we further enlarge admissible mechanisms by allowing stochastic mechanisms as well as withholding by the designer. The stochastic mechanisms allows the designer to allocate the good with certain probability at some pro les ( ; ). Intuitively, withholding the good by not giving out to any agent at some realized pro les ( ; ) may hurt the total utility. However, by doing so, the bene t from improved incentives may be able to compensate the loss. As shown in Theorem, we fully characterize the structure of all optimal mechanisms with fx i ( ; ) ; i = ; jx () x () and x i () [; ]g. As a trade-o, we cannot do that under a class of distributions as in Theorem. Hence, we only restrict to uniform distribution. The fundamental di culty remains the same as in Theorem. But the speci c property of uniform distribution allows us to say more in this situation. Theorem If agents types are distributed uniformly, every optimal mechanism is a convex combination of the two xed-price mechanisms and the two option mechanisms with p = =. Proof We will divide the proof into two parts. In Part I we show max MF T U(M) = 5 8. In Part II we demonstrate that any mechanism that satis es T U(M) = 5 8 must be a convex combination of the four mechanisms. Part I First us estimate the upper-bound of T U (M) = ( x ( ; ) x ( ; ) t ( ; ) t ( ; ))d d :

21 That is T U(M) = = = = ( x ( ; ) x ( ; ) t ( ; ) t ( ; ))d d (( )x ( ; ) ( )x ( ; )) d d h ( )d h ( )d (x ( ; ) x ( ; )) d d ( x ( ; ) x ( ; )) d d (x ( ; ) x ( ; )) d d h ( )d h ( )d h ( )d h ( )d ( x ( ; ) x ( ; ) t ( ; ) t ( ; )) d d (x ( ; ) x ( ; )) d d h ( )d T U(M): h ( )d Hence, (E) T U(M) (x ( ; ) x ( ; )) d h ( )d h ( )d A : The equation holds in (E) if and only if R R (t ( ; ) t ( ; )) d d =. The next Lemma gives us an estimate of the second term on the right hand side of (E): Lemma 3 Let A be the area in that is below the minor diagonal = : h ( ) d h ( ) d 3 4 A (x ( ; ) x ( ; ))d d : Proof of Lemma 3: Let us consider the feasibility inequality on the minor diagonal

22 = ; and = : h ()h ( ) x (; ) x (; ) d( ) x (; ) x (; ) d: We integrate the above inequality over [; ], = h () d x (; )d h ( ) d ( )x (; )d x (; (x (; ) ( )x (; )) d x (; ) dd ) dd x (; ) dd x (; ) dd: The rst term is easy to estimate: Since we have max fx (; ) ( )x (; )g = x x ( ; ; ; (x (; ) ( )x (; )) d ( )d d = 3 4 : Through the change of variables, we can express the double integrals in the second and the third term as integrals over the area A, = = A x (; ) dd ~ x ; ~ dd ~ (x ( ; ) x ( ; ))d d : x (; ) dd x (; ) dd

23 Putting these two inequalities together, we prove the lemma. Finally, we can apply the lemma to (E) to obtain the desired estimate T U(M) = [;][;]na 3 8 = 5 8 : (x ( ; ) x ( ; ))d d (x ( ; ) x ( ; ))d d! (x ( ; ) x ( ; ))d d h ( )d A h ( )d (x ( ; ) x ( ; ))d d Since T U(M) = 5 8 for both the xed price mechanisms and the option mechanisms, these mechanisms are all optimal. Part II We now show that any mechanism that satis es T U(M) = 5 8 must be a convex combination of the xed price mechanisms and the option mechanisms. We can see from the proof above that any mechanism M satis es T U(M) = 5 8 must also satisfy (B) (B) h ( ) h ( ) = x ( ; ) R x (; ) d x ( ; ) x ( ; ) x ( ; ) = on [; ] [; ]na: R x ( ; ) d; We now divide [; ] [; ] into four small squares of equal size and study M on each of them separately. Part II- Consider rst the upper-left square (; ] ( ; ]. On this area, agent s type is always higher than agent s type. It is intuitive that the good should and could be given to agent, i.e., x ( ; ) = on ; ;. Let us present a formal proof. We begin with the upper half of ; ; and prove x ( ; ) = for ( ; ) with < and > : Suppose, on the contrary, x ( ; ) with > for some ( ; ) with < and >. Since x ( ; ) is non-decreasing in and x ( ; ) is non-increasing in 3

24 Figure 4 (since x ( ; ) = x ( ; ) on this region), we must have x ( ; ) for all ( ; ) with and (see Figure 4): In particular, this inequality holds on the minor diagonal = ; and =. Repeating a part of the proof of Lemma 3, we have when = h () d x (; )d h ( ) d x (; ) dd x (; ) dd (x (; ) ( )x (; )) d (x (; ) ( )x (; )) d A ( )x (; )d (x (; ) ( )x (; )) d (x ( ; ) x ( ; ))d d 4

25 < 3 4 A ( )d ( ( )( )) d (x ( ; ) x ( ; ))d d (x ( ; ) x ( ; ))d d : A When we plug this into the estimation of T U(M), we have d T U(M) < 5 8 : This contradicts the assumption that M is optimal. Thus x ( ; ) = for ( ; ) with < and > : Now we show that x ( ; ) = also holds for the other half of [; ] ( ; ] in which > and. First, for any > >, we can nd some with < and >. Then h ( ) h ( ) = x ( ; ) d and h h ( ) = Taking the di erence of the two, we have x ( ; ) d: h h ( ) = x ( ; ) d = : Hence, h ( ) is a constant h on ;. A similar argument also shows that h ( ) is a constant h on ;. Consider any ( ; ) with > and. Since x is zero to the right of the diagonal, x is also zero on this area for x is non-increasing in. Hence h ( ) h ( ) = x ( ; ) x ( ; ) d: 5

26 Choosing any >, we have h h ( ) = x ( ; ) d: Subtracting one from the other yields x ( ; ) = x ( ; )d x ( ; )d = x ( ; )d : This shows x ( ; ) =. Hence, (II-) x ( ; ) = on ; ; : Part II- When we work with the lower-right square ; ;, we can show that h ( ) is a constant h on ; and that h ( ) is a constant h on ;, and (II-) x ( ; ) = on ; ; : Part II-3 Now consider the upper-right square ; ;. (B) implies h h = x ( ; ) x (; ) d x ( ; ) x ( ; ) d: Since x (; ) = for <, and x ( ; ) = for <, h h = x ( ; ) x (; ) d x ( ; ) x ( ; ) d: If we let both! and!, we obtain h h =. Now plug it back into the equation above, = x ( ; ) x (; ) d x (; ) d x ( ; ) x ( ; ) d = x ( ; ) x ( ; ) d; or x ( ; ): 6

27 Since x ( ; ) is non-decreasing in and x ( ; ) is non-decreasing in, the equation above implies x (; ) = x ( ; ); for all ; ; and x ( ; ) = x ( ; ); for all ; : It is easy to verify that this holds for all ( ; ) ; ; if and only if there are two constants c and c with c c = such that (II-3) x ( ; ) = c and x ( ; ) = c on ; ; : Part II-4 Lastly, we consider the lower-left square ; ;. We know h h = x ( ; ) x (; ) d x ( ; ) x ( ; ) d: When let both! and!, we obtain h h =. Hence, x ( ; ) x ( ; ) = x (; ) d x ( ; ) d: So we can conclude that there are two constants d and d such that (II-4) x ( ; ) = d and x ( ; ) = d on ; ; : To nd out more about d and d, we put together we have shown about h s and x s. First, on the upper-left square, we have h h = x ( ; ) Similarly, on the lower-right square x ( ; ) d = d d = ( d ) : h h = ( d ) : 7

28 Figure 5 On the other hand, we already know h h = and h h = : These four equations together lead to d d =. Hence, any optimal mechanism can be characterized by two parameters c [; ] and d [; ] with the allocation probabilities given by the table in Figure 5: There are four di erent combinations of extreme values of c and d: (i) c = and d = : this corresponds to the xed price mechanism M in which agent two is the seller; (ii) c = and d = : this corresponds to the xed price mechanism M in which agent one is the seller; (iii) c = and d = : this corresponds to the option mechanism M in which agent one is given the good and agent two is given the option; and (iv) c = and d = : this corresponds to the option mechanism M in which agent two is given the good and agent one is given the option. 8

29 Any other optimal mechanism M is just a convex combination of these four mechanisms: M = c M (d c) M ( c d) M ; for c d; or M = d M (c d) M ( c d) M ; for c > d: This provides the characterization optimal mechanisms among all feasible strategy-proof mechanisms. Q.E.D. 3 Discussions Before closing, we discuss more about our result in comparison with other known results in the literature and explore some possible extensions of our result for future research. Every feasible strategy-proof mechanism consists of two parts: the rst part is the rule of assigning the good to the agents, and the second part transfers that are necessary to force the agents to report their types truthfully. The loss of e ciency may come from both sources: either the good is not assigned to the agent with the higher valuation, or the total transfer leads to a money out ow. A very natural and important question is: What is the optimal trade-o between these two types of ine ciency? However, this issue has never been addressed formally by others in the previous literature. Most papers in the literature either focus on V-C-G mechanisms or on trading mechanisms in which money changes hands between two agents. These two classes of mechanisms are mutually exclusive by de nition: all V-C-G mechanisms are immune from the rst type of ine ciency, and most trading mechanisms including the xed-price mechanisms are immune from the second type of ine ciency. Hence, it is impossible to discuss the potential trade-o within between the two types of ine ciency in either model. In order to address this issue, we must adopt a more general setting that includes both V-C-G mechanisms and trading mechanisms as subclasses of admissible mechanisms. We have done it successfully in this paper. The nding is somewhat surprising: although it might be expected that an optimal mechanism entails a compromise of both types of ine ciency, our result indicates the second type of ine ciency seems more damaging. Furthermore, it must be completely absent at any optimal mechanism for a class of well-behaved environments. Our model also di ers from the model of Hagerty and Rogerson, and other models in bilateral bargaining, in another dimension for we do not impose individual rationality on mechanisms under consideration in our model. Since this enlarges the class of admissible 9

30 mechanisms, this makes our result even stronger. In addition, we are able to discover the two optimal option mechanisms, which have never been studied before largely because they do not satisfy individual rationality when one agent the designated seller and the other agent the designated buyer 5. In many allocation problems where neither agent originally owns the good, such as the o ce assignment problem, the option mechanisms are better alternatives than the xed-price mechanisms as they are more equitable ex ante. advantage is not huge as it disappears once lotteries are admissible. This Once we break away from the bilateral bargaining model, it is then natural to consider a model in which an indivisible good (or even multiple units of the good) are allocated among more than two agents. We still do not have any formal results in such a model yet. Admittedly, it will be quite di cult to obtain a complete characterization of all optimal mechanisms. However, it is reasonable to believe that we can still derive some partial results. While we are unsure how to extend the xed-price mechanism, we have found a generalization of the option mechanism. We assign the good to an agent, say agent one, conditionally and direct him to run a second-price auction of the good with the other n agents with a reservation price. This mechanism always balances the budget since money just changes hands from one agent to another. It should also be reasonably e cient, depending on the value of. The best value of can be found by maximizing the sum of the expected utilities of all agents (the transfers are absent since the budget is always balanced): = arg n yd(y n ) A : Hence, =. Note that this optimal reservation value is the same as that in Myerson s mechanism in which the expected revenue of the seller is maximized. It would be remarkable that the maximum e ciency in allocating an indivisible good among n agents can be achieved when we give the good to one of the agents and direct him to conduct a revenue maximizing auction with the other n agents as buyers. Of course, this is just a conjecture at this point, and further research is needed to yield a formal answer. Finally, we return to an issue we already mentioned when we set up the basic model. Although we have derived the optimal mechanisms under a class of distributions, one may still ask what will happen to our main nding if di erent distributions are used. Although 5 Nevertheless, that the option mechanisms do satisfy the weak individual rationality condition that no agent has negative utility at any pro le. 3

31 it is clear that optimal mechanisms will change, we have not been able to derive a full characterization of all optimal mechanisms for a general distribution function of agents types. The distributions used in Theorem are only su cient to guarantee the budget balance mechanisms are optimal. In a separate paper (Shao and hou, 8), we undertake a more modest task. Instead of including all feasible strategy-proof mechanisms, we consider only the V-C-G mechanisms and the xed-price mechanisms (or the option mechanisms), two classes of mechanisms that are most prominent in the literature. Assuming that the distributions of agents types are independent and symmetric, we manage to nd separately the best mechanism among all V-C-G mechanisms and the best mechanism among all xed-priced mechanisms. Then we compare these two mechanisms to see which one is better. For some distributions, the best V-C-G mechanism actually outperforms the best xed-price mechanism. However, in two important cases when the distribution function of types is either concave or convex, we show that the best xed-price mechanism beats the best V-C-G mechanism. We should point out that the model of quasi-linear preferences with a general distribution of types is mathematically equivalent to the model of more general preferences with a uniform distribution of types. Copic and Ponsati (8) have reported some results for the latter model in the bilateral trade framework. While their work has made some progresses in dealing with non-linearity of preferences, it still shares the similar weakness that Hagerty and Rogerson s work exhibits. For instance, it assumes budget-balanced-ness so it excludes the V-C-G mechanisms as well as many other potentially mechanisms from consideration. Hence, it cannot even compare the e ciency of the xed-priced mechanisms and the V-C-G mechanisms. This being said, their work is already a rather complicated mathematical exercise. We certainly cannot underestimate the di culty we shall face when we try to nd optimal feasible strategy-proof mechanisms when we assume a much more general distribution of agents types. References [] S. Athey and D. Miller. E ciency in repeated trade with hidden valuations. Theoretical Economics (3), (7). [] D. Bergemann and S. Morris. Robust Mechanism Design. Econometrica 73(6), (5). 3

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