The Land Acquisition Problem: a Mechanism Design Approach

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1 The Land Acquisition Problem: a Mechanism Design Approach Soumendu Sarkar February 2012 (Preliminary and Incomplete. Please do not quote.) Abstract The land acquisition problem is an exchange problem with one buyer and multiple sellers with the buyer haing demand for more than one units and each seller holding one unit of a homogeneous good. We show that under incomplete information, no mechanism designed for the problem can be fully satisfactory. Then we characterize different second best mechanisms to address this problem. My superisor Professor Arunaa Sen encouraged me to to look into the problem of land acquisition. I owe both him and Professor Debasis Mishra for their inaluable adice, inspiration and patience. Thanks also to my colleagues Anup Pramanik, Dushyant and Mridu Prabal Goswami for their aluable suggestions. Errors and omissions are only mine. Indian Statistical Institute and TERI Uniersity, New Delhi; sarkarsoumendu@gmail.com 1

2 1 Introduction Land acquisition generally refers to the act of acquiring priately owned land by the state either for its own purposes or in the interest of some other priate party. There are elaborate laws controlling land acquisition in different parts of the world, mostly in the spirit of Eminent Domain as included in the British common law. The State can acquire land for public purposes using eminent domain, like construction of railroads, highways etc.; it can also transfer the rights to priate parties which can utilize the property for public or ciic use or for economic deelopment. Land acquisition using eminent domain is quite common in British history as well as in countries like the United States or India who inherited the eminent domain along with the British common law. In recent times, eminent domain has been in focus particularly after the 2005 Kelo s. City of New London case where the right of the municipality to acquire priate property and then transfer it to a priate deeloper for a nominal amount was questioned. At Singur, India, the local goernment acquired about 1000 acres of mostly farmland to set up a car manufacturing unit of the Tatas in Although the goernment was claiming that the compensation package offered to the eicted was more than fair, there were protests from the opposition parties and a section of intelligentsia. The unit was then relocated to another part of the country. The Land Acquisition problem may be described as follows there is typically a number of indiiduals holding plots of land which is of some alue to them; there is another indiidual who is interested in haing a specific number of plots. The latter does not gain anything by haing more than the specific number of plots, but anything strictly less would yield him zero alue. Neither the buyer nor any seller has information regarding the aluations of agents other than himself. The problem is to find a satisfactory means to facilitate exchange between the buyer and the sellers. 1.1 Mechanisms and Eminent Domain Informally speaking, mechanisms are rules of allocation applied to economic situations which inole asymmetric information among agents. Typically, a mechanism asks the agents for some information, and gien this information, it allocates the objects inoled according to a rule announced in adance, and in some situations, asks for payments from the agents according to another pre-announced rule. The eminent domain system can be iewed as a mechanism because it contains the following: an allocation rule that assigns all the goods to the buyer irrespectie of the aluations of the buyer and sellers; a payment rule that makes the buyer pay and the sellers receie an amount which is at least equal to an exogenously gien market rate. 2

3 A casual examination of this mechanism reeals both the inadequacy of the eminent domain system and the requirements of satisfactory mechanism design. In particular, (a) the eminent domain law is in general inefficient because it is insensitie to the aluations of the buyers and sellers it will allocate all plots to the buyer een when the aggregate alue of the land to the sellers is higher than that of the buyer. (b) There may be parties who would gain by not participating in the exchange at all if the market rate is relatiely low, a seller may be better off by just cultiating it as farmland but he is forced to participate and accept a low payoff. At the same time, it is adantageous from a control point of iew, because (c) neither party can affect the outcome by misreporting their aluations the same reason that makes eminent domain inefficient, makes it non-manipulable as well. (d) The controller of an eminent domain transaction does not hae to use outside funds for the payments what the buyer pays is what the sellers receie, regardless of the fact whether it is sufficient to compensate the loss of land or not. 1.2 Structure of the Paper The goals of satisfactory mechanism design, as corresponding to items (a)-(d) in the last subsection are, efficiency or maximize total gain, indiidual rationality or oluntary participation, incentie compatibility or non-manipulation, and budget balance or no-deficit condition. The goal of this paper is to find the most satisfactory mechanism for the land acquisition problem. We show the non-existence of a mechanism which satisfies all the four criteria. Then we alternately relax one of the criteria and identify the best mechanism in the corresponding class. Throughout, the Bayesian approach to mechanism design (in the spirit of Myerson and Satterthwaite (1981); Myerson (1981)) The next section briefly reiews the literature in the area. The third section introduces the notation and preliminary concepts. Section 4 presents and discusses the impossibility result. Sections 5-7 present the second best mechanisms. The concluding section summarizes the work and lays down some future directions of research. 2 Literature Apart from a ery recent paper by Kominers and Weyl (2010), the literature has not specifically focused on problems relating to land acquisition. Howeer, the problem of bilateral trade under incomplete information, which is a special case of the land acquisition problem, has been studied in considerable detail. The literature on bilateral trade under incomplete information began with Chatterjee and Samuelson (1983). Assuming that aluations are drawn from mutually independent distributions, they looked at a specific mechanism called the k-double auction, which implies that trade takes place if buyer s reported aluation exceeds that of the seller s, and at a price equal to the aerage of these two reports. They looked at the equilibria with differentiable 3

4 strategies and showed the possibility of inefficiency for a range of aluations. This result was formalized by?. They also showed that with uniformly and identically distributed aluations, the double auction mechanism in fact maximized ex-ante gains from trade when incentie compatibility and indiidual rationality are imposed. A large literature spawned later making attempts to address either of the two issues one, characterizing the secondbest among mechanisms by relaxing one or more assumptions of the Myerson-Satterthwaite setup and two, characterizing double auction mechanisms. In the first strain of works, couple of works concentrate on dominant strategy incentie compatibility. McAfee (1992) constructed a dominant strategy truthful mechanism that results in occasional surplus, and calculated the rate at which the mechanism conerges to ex-post efficiency as the number of agents become large. Hagerty and Rogerson (1987) showed that in the Myerson-Satterthwaite setup, only the posted price mechanisms satisfy dominant strategy incentie compatibility, ex post indiidual rationality and budget balance. For other contributions in this area, we refer to the works by T. A. Gresik listed in the reference. Major contributions in the second strain include Wilson (1981),Satterthwaite and Williams (1989), Williams (1991) and Satterthwaite and Williams (2002). Kominers and Weyl (2010) are interested in the case where the number of units demanded by the buyer is exactly equal to the number of sellers, each of whom hold a unit of the good in question. They propose a family of mechanisms which they call concordance mechanisms which can be iewed as a modified eminent domain scheme. Proceeds of the sale, if it occurs are to be distributed among the sellers in a pre-specified manner. Each seller can report either his share of the sale proceeds or his aluation. But if he reports the latter, he has to pay a VCG-like payment. Trade takes place if the buyer s bid exceeds the sum of the sellers reports and truthfulness is ensured by the payment scheme. Other ariants of the model simply specify different payment schemes that ensure truthfulness in dominant strategies or in Bayesian sense. It is easy to see that these mechanisms are not efficient, budget balanced or indiidually rational. The authors show that these mechanisms conerge to efficiency as the number of sellers increase, they are collectiely rational and approximately indiidually rational. The present work allows the number of potential sellers to be larger than the number of units demanded by the buyer. Further, we are only interested in incentie compatible mechanisms which satisfy exactly two of the properties among indiidual rationality, budget balance and efficiency. We characterize such classes of mechanisms and then look for a mechanism in each such class which comes closest to satisfying the property which is relaxed. Thus, the mechanisms we come up with, satisfy more number of desirable properties than the concordance class. 4

5 3 Preliminaries There is one buyer, indexed by 0, with the following aluation pattern: his total aluation is 0 if he gets less than k units of the good and it is 0 if he gets k units of the good (k n) or more. We assume that 0 [ 0, 0 ] and 0 G( 0 ). There are n sellers, indexed by i, each holding one unit of an indiisible good for which his aluation is some i [, ] and iid i F ( i ), i = 1,, n. The aluations of the buyer and the sellers are independent. All aluations are strictly positie. Own aluations are priate information, the distributions of others aluations are common knowledge. The buyer and the sellers report their indiidual aluations or any signal thereof to an auctioneer who decides on a mechanism to allocate the goods in question. Such a mechanism consists of an allocation rule and a transfer rule. We will consider only direct mechanisms where all agents directly report their aluations. To make things interesting, we do not want alues such that trade is always beneficial or it is always bad. Therefore, we may assume that k > 0 and k < 0. Definition 1 (Allocation Rule) An allocation is any realization of a random ector X = (X 0, X 1,, X n ) where X i, i 0 takes alue -1 if trade takes place between the i-th seller and the buyer, and 0 otherwise, X 0 is 1 if i 0 X i k and 0 otherwise. We call the set of 2 n possible allocations X. An allocation rule P : [ 0, 0 ] [, ] n X specifies a probability distribution oer X as a function of the reported aluations of the agents. Definition 2 (Transfer Rule) Any rule t : [ 0, 0 ] [, ] n R n+1, relating the aluations of the agents to transfers t j gies a transfer rule. Definition 3 (Direct Mechanism) Any pair (P, t) is a direct mechanism. When defining utility of an agent, we will distinguish among the different stages of priate information agents hae. The stage where all information is public is called ex post, the stage where only own aluations are priate information to the respectie agents is interim, the one where the entire state is unknown to all the agents is ex ante. This terminology is due to Holmstrom and Myerson (1983). Definition 4 (Utility) The (ex post) utility of agent j with aluation j reporting ˆ j under mechanism (P, t) is U j (ˆ j, j j ) = j P j (ˆ j, j )) t j (ˆ j, j ) where j denotes the n dimensional random ector representing the aluations of agents other than j. It follows therefore, that the utility of agent j under honest reporting is U j ( j, j ) = j P j ( j, j ) t j ( j, j ) (1) An important equilibrium concept in mechanism design, iz., dominant strategy incentie compatibility has to do with ex post payoffs. 5

6 Definition 5 (Dominant Strategy Incentie Compatible) A direct mechanism is Dominant Strategy Incentie Compatible or DSIC if honest reporting forms a equilibrium in dominant strategies, i.e., U j ( j, j ) U j (ˆ j, j j ) j, j, j, ˆ j The participation condition corresponding to ex-post payoffs is outlined below. Definition 6 (Indiidual Rationality) A gien mechanism is (ex post) indiidually rational if for each agent, U j ( j, j ) 0 j, j, j When only own aluations are priate information, indiiduals calculate payoffs by taking expectation oer the unknown alues. Definition 7 (Expected Utility) The expected utility of an agent is the expected net gain receied under a direct mechanism (P, t) for a particular report gien his aluation, the expectation being taken oer all the agents other than the specific agent in concern, i.e., for each agent j with aluation j reporting ˆ j, EU j ( j, ˆ j ) = j E j (P j (ˆ j, j )) E j (t j (ˆ j, j )) where j denotes the n dimensional random ector representing the aluations of agents other than j and E j ( ) denotes expectation taken oer aluations other than that of the i-th agent. It follows therefore, that the expected utility of agent j under honest reporting is EU j ( j ) = j E j (P j ( j, j )) E j (t j ( j, j )) (2) The equilibrium concept corresponding to interim payoffs is Bayesian Incentie Compatibility. Definition 8 (Bayesian Incentie Compatibility) A direct mechanism is Bayesian Incentie Compatible or BIC if honest reporting forms a Bayes-Nash equilibrium (Harsanyi, ), i.e., EU j ( j ) EU j ( j, ˆ j ) j, ˆ j. The participation condition is also modified accordingly. Definition 9 (Interim Indiidual Rationality) A gien mechanism is interim indiidually rational if for each agent j and for eery j, EU j ( j ) 0 6

7 When agents do not hae any information about the realized state of the world whatsoeer, the appropriate concept of utility is the ex ante one. Definition 10 (Ex Ante Utility) The ex ante utility of an agent is the expected net gain receied under a direct mechanism (P, t), the expectation being taken oer all possible aluation profiles. In symbols, for each agent j, AU j = E (P j ( j, j )) E (t j ( j, j )) The releant participation condition becomes as follows. Definition 11 (Ex Ante Indiidual Rationality) A gien mechanism is ex ante indiidually rational if for each agent j, AU j 0 Efficiency is often a criterion to ealuate a mechanism. We can also distinguish between ex-post, interim and ex-ante concepts of efficiency. Definition 12 (Efficiency) An allocation rule P is ex post efficient if for any gien ector of aluations, j P j () j P j() for any allocation rule P j j An allocation rule P is interim efficient if for any gien ector of aluations, j E j P j () j E j P j() for any allocation rule P j j An allocation rule P is ex ante efficient if E ( j P j ()) E ( j P j()) for any allocation rule P j j The concepts of incentie compatibility at different stages of information is related to each other in an asymmetrical way, and so are the concepts of indiidual rationality and efficiency. The following statements (Holmstrom and Myerson, 1983) can be easily erified: A dominant strategy incentie compatible mechanism is also Bayesian incentie compatible, but the conerse is not true in general. Ex-post indiidual rationality implies interim indiidual rationality, which in turn implies ex-ante indiidual rationality, but none of the conerses are true in general. Ex-post efficiency implies interim efficiency, which in turn implies ex ante efficiency, but the conerses are not true in general. 7

8 We also impose a restriction on the transfers, which is known as budget balance. Definition 13 (Budget Balance) A mechanism (P, t) satisfies (strong) budget balance if for all profiles, t j ( j, j ) = 0 j There is also a weak ersion of this condition where the equality aboe is replaced by a weak inequality. Here, gien a ector of aluations 0,, n, we hae: { 0 + n i=1 j P j () = ix i if n i=1 X i k n i=1 ix i otherwise j So, in order to find the ex-post efficient mechanism we need to specify the probabilities Pr(X i = 1) in a way that j jp j () is maximum. Let us now re-arrange the seller aluations in descending order and let i be the i-th highest among these n numbers. Then, for the case under consideration, the ex post efficient rule is the following: Pr(X i = 1) = For the buyer, the releant condition is: { { Pr(X 0 = 1) = 1 if i k + 1,, n and 0 > n j=k+1 j 0 otherwise 1 if 0 > n j=k+1 j 0 otherwise i.e., under the efficient allocation rule, if the sum of aluations of the k lowest-aluation sellers is less than the buyer s aluation, exactly these k sellers trade with the buyer and no more; otherwise, nobody trades. We will denote the efficient rule as P. (3) (4) 4 Proing the Impossibility In this section, we proe that there exist no mechanism for our problem which is simultaneously ex post efficient, Bayesian incentie compatible, interim indiidually rational and budget balancing. We use a result due to Krishna (2002): the surplus of expected payments under the VCG mechanisms is the highest among all incentie compatible, indiidually rational and efficient mechanisms. We will show that the VCG surplus is negatie in our setup. The Vickrey-Clarke-Groes mechanism is a generalization of the well-known second price auction scheme. It is efficient, indiidually rational, and dominant strategy incentie compatible, and hence lies in the wider class of Groes mechanisms. 8

9 Let SW () = n j Pj () j=0 i. e., SW () is the sum of aluations that results from the efficient allocation rule P (). Let α j be the aluation of j for which his truthtelling payoff is minimal. As we will see, the incentie compatibility condition will impose monotonicity for the utility functions, so that for the buyer, α j is simply his lowest aluation, and for the sellers, it is their highest aluation. Definition 14 (VCG Mechanism) The VCG mechanism is gien by the pair (P, t V ) where P is the efficient allocation rule, and t V is a payment function defined as: t V j () = SW (α j, j ) SW j () (5) The VCG payment can be interpreted as the externality imposed by the corresponding player. For the single-good auction problem, it is exactly equal to the second highest bid. Under the VCG scheme, the utility of agent j with aluation j who reports ˆ j is: j P j (ˆ j, j )) + SW j (ˆ j, j ) SW (α i, j ) The third term is independent of the action chosen by i, and by efficiency, the sum of the other two terms is maximized by choosing ˆ i = i. Hence the VCG scheme is incentie compatible in dominant strategy. Also,the VCG scheme is indiidually rational: in a truthtelling equilibrium, the type α j gets a payoff of 0 and by definition this is the minimum payoff for all types of j. The following lemma is an important result in mechanism design theory. Lemma 1 Indiidual expected payoffs under any incentie compatible mechanism is determined by the allocation rule upto an additie constant. Proof : If a mechanism is incentie compatible, then for any two aluations j and ˆ j of j, we hae, and Therefore, EU j ( j ) = j E j (P j ( j, j )) E j (t j ( j, j )) j E j (P j (ˆ j, j )) E j (t j (ˆ j, j )) EU j (ˆ j ) = ˆ j E j (P j (ˆ j, j )) E j (t j (ˆ j, j )) ˆ j E j (P j ( j, j )) E j (t j ( j, j )) EU j (ˆ j ) EU j ( j ) ˆ j E j (P j ( j, j )) E j (t j ( j, j )) j E j (P j ( j, j )) + E j (t j ( j, j )) = (ˆ j j )E j (P j ( j, j )) 9

10 and, EU j (ˆ j ) EU j ( j ) ˆ j E j (P j (ˆ j, j )) E j (t j (ˆ j, j )) j E j (P j (ˆ j, j )) + E j (t j (ˆ j, j )) = (ˆ j j j)e j (P j (ˆ j, j )) which gies us the following inequality: (ˆ j j )E j (P j ( j, j )) EU j (ˆ j ) EU j ( j ) (ˆ j j j)e j (P j (ˆ j, j )) (6) From (6), (ˆ j j )(E j P j (ˆ j, j )) E j (P j ( j, j ))) 0, implying that is E j (P j ( j, j )) is increasing in j, therefore being Riemann integrable. Further, EU j( j ) = E j (P j ( j, j )) at almost eery j and EU j ( j ) = EU j (α j ) + j α j E j (P j (z, j ))dz (7). This result implies that if the same allocation rule is being used for two different mechanisms which are incentie compatible, the payoffs and payments can differ only by an additie constant. This is commonly known as Reenue Equialence. We are now going to present a theorem which appeared in Krishna (2002). We will use it to proe the impossibility result of this section. Theorem 1 [Krishna (2002)] There exists an efficient, incentie compatible and indiidually rational mechanism that balances the budget if and only if the the VCG mechanism results in an expected surplus, i.e., if and only if E [ n t V j ()] 0 j=0 Proof : Let us consider any efficient, incentie compatible and indiidually rational mechanism, (P, t ). Since type α j of player j gets the payoff of 0 under VCG, by indiidual rationality of (P, t ), we hae, for all j: EU j (α j ) 0 = EU V j (α j ) (8) Since the efficient allocation rule is being used in both cases, by the main equation of the equialence principle (7), we get, for all j and j : which implies that for all j and j : EU j ( j ) EU V j ( j ) (9) E j t j() E j t V j () (10) 10

11 Taking expectation oer j s and summing oer j, we get: E ( j t j()) E ( j t V j ()) i.e. the expected sum of payments of the VCG mechanism is the highest among all efficient, incentie compatible, and indiidually rational mechanisms. Therefore, if there exists a mechanism which is efficient, incentie compatible, indiidually rational and budget balancing, then we must hae: n E [ t V j ()] 0 j=0 Hence the necessary part. To proe sufficiency, we will construct a mechanism with the four properties gien E [ n j=0 tv j ()] 0. The Arrow-d Aspremont-Gérard-Varet mechanism (d Aspremont and Gérard-Varet, 1979) has the property that it is efficient, Bayesian incentie compatible, and budget-balancing but not necessarily indiidually rational. It is gien by the efficient allocation rule and the following payment rule: t A j = 1 E s l [SW l ( l, s l )] E s j [SW j ( j, s j )] (11) n l j so that for all, n j=0 ta j () = 0. To check that the AGV mechanism is incentie compatible, we obsere that the expected payoff to j from reporting j when his type is j and other agents are reporting truthfully is: E j [ j P j ( j, j ) + l j l P l ( j, j )] 1 n E j [ l j E l [SW l ( l, l )]] Since the third term is independent of j, this is maximized by setting j = j. By the payoff equialence lemma (1), there exist constants c A j and c V j such that: EU A j ( j ) = E j [SW ()] c A j EU V j ( j ) = E j [SW ()] c V j If the VCG mechanism runs an expected surplus, i.e., which implies that n E [ t V j ()] 0 j=0 11

12 n n E [ t V j ()] E [ t A j ()] j=0 j=0 then it follows that: n j=0 c V j For all j > 0, let c j = c A j c V j and let c 0 = n j=1 c j, and consider the mechanism (P, µ) defined by: n j=0 c A j µ j (w) = t A j (w) + c j Clearly, (P, µ) balances the budget. It is incentie compatible because the payments differs from the AGV payment by only constants. To check that it is indiidually rational, for all j > 0, the expected payoff to agent j with aluation j, say EŪj( j ), is gien by: EŪj( j ) = EUj A ( j ) + c j = EUj A ( j ) + c A j c V j = EUj V ( j ) 0 Obsering that,, we get: n c 0 = c j = j=1 n (c V j c A j ) c A 0 c V 0 j=1 EŪ0( 0 ) = EU A 0 ( 0 ) + c 0 EU A 0 (w 0 ) + c A 0 c V 0 = EU V 0 ( 0 ) 0 showing that (P, µ) is indiidually rational. Thus we hae constructed an efficient, indiidually rational, incentie compatible and budget balanced mechanism showing the sufficient part. We will now state and proe the main result of this section. 12

13 Theorem 2 There exists no mechanism for the land acquisition problem which is ex post efficient, Bayesian incentie compatible, interim indiidually rational and budget balancing. Proof : We will show that the expected sum of payments under the VCG mechanism is negatie. Then the result follows by Theorem 1. We recall that alue of the buyer is 0 if he gets at least k units and 0 otherwise., and that for the seller i is i if he sells a unit, and 0 otherwise. If we denote the set of sellers who sell in a particular allocation by T, then total alue is 0 j T j if T k and j T j if T < k. Gien that all i s are non-negatie, the sum of alues or gains from trade is maximized by the following efficient rule: the sellers with the k smallest aluations trade with the buyer when the sum of their alues is greater than the alue of the buyer, and no trade occurs otherwise. So, if the buyer s alue is 0 and the sellers alues are ordered as 1 n, then the efficient surplus or gains from trade is gien by: SW () = { 0 n i=n k+1 i if 0 > n i=n k+1 i 0 otherwise If the bid of the buyer is ˆ 0 and those of the sellers are ordered as ˆ 1 ˆ n, the VCG payment rule for the problem is: Therefore, we get the following: and, t V i = t V 0 (ˆ) = SW ( 0, ˆ 0 ) SW 0 (ˆ) t V i (ˆ) = SW (, ˆ i ) SW i (ˆ) 0 if ˆ 0 0 n t V i=n k+1 ˆ i; 0 = n i=n k+1 ˆ i if ˆ 0 n i=n k+1 ˆ i 0 ; 0 otherwise 0 if ˆ 0 n i=n k+1 ˆ i and i / {n k + 1,, n}; if ˆ 0 n i=n k+1 ˆ i and i {n k + 1,, n}; (ˆ 0 n j=n k+1 ˆ j ) if ˆ 0 n j=n k+1 ˆ j, i {n k + 1,, n} j i but ˆ 0 n j=n k+1 j i 0 if ˆ 0 n i=n k+1 ˆ i ˆ j + ; From the statement of the payment scheme aboe, it follows that in any profile of bids where trade occurs, the buyer pays in total the maximum of 0 and n i=n k+1 ˆ i while the sellers who sell receie either or ˆ 0 n j=n k+1 ˆ j which always sum to more than what the j i buyer pays in total. No payments are made when trade does not occur, and no non-trading 13

14 seller receies anything. Therefore, the sum of payments for any profile is either zero or strictly negatie. Therefore, the expected sum of payments is also negatie. E [ n t V j ()] < 0 j=0 In the absence of a completely satisfactory mechanism for land acquisition, the problem of finding most satisfactory mechanisms is a important agenda. In the next section and the following two, we look for the best possible or second best solutions to the mechanism design problem. 5 Relaxing Budget Balance In this section, we try to find out the mechanisms which satisfy efficiency, indiidual rationality and incentie compatibility and achiee the highest possible budget balance. From the theorems 1 and 2, it is clear that there does not exist any ex-post efficient, Bayesian incentie compatible, interim indiidually rational mechanism that results in an expected sum of payments greater than that of the VCG mechanism. In other words, the VCG mechanism characterizes the best outcome in terms of expected budget surplus. It may be noted that in our setup, the expected VCG surplus is negatie. When we replace indiidual rationality and incentie compatibility in the interim sense with their corresponding ex-post counterparts, an appropriate way to compare budget balance performance of mechanisms is worst-case optimality. Consider the class of mechanisms which are Dominant Strategy Incentie Compatible (DSIC), efficient(eff), Indiidually Rational (IR) and defined oer a conex domain (CD)of type profiles, V. The VCG mechanism falls in this class. We ask the following question: Does there exist a mechanism which is in this class and gies strictly higher budget balance in the worst case? In symbols, we are looking for a mechanism (P, t M ) which is DSIC, EFF and IR and min V i t M i () > min V t V i () (12) i Theorem 3 There does not exist a DSIC, EFF and IR mechanism which satisfies (12). Proof : It is known (e.g. Green and Laffont (1977)) that any DSIC and EFF mechanism defined oer a conex domain has the form: t M i () = gi M ( i ) SW i ( i, i ) (13) 14

15 where, gi M ( i ) is any arbitrary function of i. The VCG mechanism defined aboe in (5) being in this class has the same form. Let arg min t M i () (14) V We hae, By IR of M, This implies, arg min V i t V i () (15) i tm i ( ) i tm i ( ) by (14) = i gm i ( i) i SW i( i, i) = i gm i ( i) i SW (α i, i) +SW (α i, i) i SW i( i, i) = i gm i ( i) i SW (α i, i) + i tv i ( ) i (16) (17) Ui M () = i Pi ( i, i ) gi M ( i ) + SW i ( i, i ) = SW ( i, i ) gi M ( i ) 0 i, i SW (α i, i) g M i ( i) 0 g M i ( i) SW (α i, i) 0 i gm i ( i) i SW (α i, i) 0 i gm i ( i) i SW (α i, i) + i tv i ( ) i tv i ( ) (18) From (16) and (18),we hae: which directly contradicts (12). i t M i ( ) i t V i ( ) (19) This shows that no DSIC, EFF and IR mechanism has a higher worst case budget surplus than the VCG mechanism. It may be noted that this result is true under any single dimensional conex domain problem with priate alues. 6 Sacrificing Indiidual Rationality The class of ex-post efficient, Bayesian incentie compatible, and budget balancing mechanisms is non-empty. We will offer some examples of mechanisms in this class, and then discuss some properties which are apparent from these examples. We will then construct a mechanism which comes closest to satisfy indiidual rationality. Unless otherwise mentioned, the results are alid for any n-agent mechanism design problem with independent priate alues. The releant results can be immediately applied to the Land Acquisition problem. 15

16 The d Aspremont-Gerard-Varet(AGV) mechanism (P, t A ) where t A is defined in (11), is one important mechanism in the class. Like the VCG mechanism, it is well defined for the land acquisition problem as well, but while the VCG does not satisfy budget balance, the AGV mechanism does not satisfy indiidual rationality. The following example shows this for the setup with one buyer, one seller, k = 1 and both aluations being independently and identically distributed in U[0, 1]. Example 1 With one buyer indexed 0 and one seller indexed 1, the alues of whom are distributed iid U[0, 1], we hae, EU A 0 ( 0 ) = EU A 1 ( 1 ) = Clearly, either of the payoffs can be negatie; for instance, 0 = 1 gies a negatie payoff to 0, while 1 = 1 gies a negatie payoff to 1. We now construct a mechanism in this class which satisfies the indiidual rationality condition for n 1 players. Example 2 Let us consider a mechanism (p, t {S,l} ) where l N, defined in the following way: { t {S,l} t A i () c i, i l i () = t A l () + i l c (20) i otherwise where c i = E i t A i () E i t V i () (We recall from the Reenue Equialence result that the term on the right hand side is a constant since both the mechanisms apply the efficient allocation rule. We obsere that, i t{s,l} i () = i ta i () = 0, hence it is budget balancing. Further, EU {S,l} i ( i ) = EU A i ( i ) + c i = EU V i ( i ) 0, i n, i EU {S,l} l ( i ) = EUl A ( i ) i l c i = EU V l ( i ) i c i Therefore, (P, t {S,l} ) satisfies indiidual rationality for all i l. This mechanism adds only constants to a Bayesian incentie compatible payment schedule, and is, therefore, Bayesian incentie compatible itself. It is efficient as it uses the rule p. In the aboe example, we obtained a BIC, BB and EFF mechanism, by adding to the d Aspremont-Gerard-Varet mechanism a set of constants which add up to 0. But we could choose any other set of constants which add up to 0, and thus obtain another BIC, BB and EFF mechanism. Since there are infinitely many sets of such constants, there are infinitely many such mechanisms. An important question to ask, therefore, whether the class of mechanisms obtained in this manner is equialent to the entire BIC, BB and EFF class. 16

17 Theorem 4 A direct mechanism satisfies BIC, BB and EFF in an independent priate alues model if and only if it is of the form (P, t A + c) where P is the efficient allocation rule, t A is the payment schedule of the d Aspremont Gerard-Varet mechanism, and c is a ector with the same dimension as the number of agents and i c i = 0. Proof : The sufficient part is easy and apparent from the discussion in the preceding example and the discussion which follows. To check the necessary part, consider a mechanism (P, t M ) which satisfies BIC, BB and EFF. We can write: t M i () = t A i () + g M i () where i gm i () = 0 by budget balance. We will show that each gi M () is a constant. Since (P, t M ) satisfies BIC and EFF, by the reenue equialence of the two mechanisms, we must hae: E i t M i () = E i t A i () + c i, where c i is a constant E i t A i () + E i gi M () = E i t A i () + c i E i gi M () = c i This implies that g M i is a function of i alone. But, i gm i ( i ) = 0 g M k ( k) = j k g j( j ) But while the function on left hand side does not hae k as an argument, the one on the right hand side includes k as an argument in j. Therefore, j k g j( j ) must not hae k as an argument. Repeating this logic for different k s we get that each gi M is a constant. This structure of the class of mechanisms satisfying BIC, BB and EFF has an important implication regarding the mechanism (P, t {S,l} ). Theorem 5 Among all mechanisms which are BIC, BB and EFF and satisfy IR for all agents but l, (P, t {S,l} ) gies the highest expected utility to agent l. Proof : Let M be a mechanism which is BIC, BB and EFF and satisfies IR for all agents but l. Then, ( i ) = EUi A ( i ) + c M i EU M i Since EU V i ( i ) attains a minimum of 0 at i = α i, in order to hae M satisfy indiidual 17

18 rationality for all agents but l, we must hae c M i 0 for i l. It follows that, EU M l ( l ) = EU A l ( l ) i l c M i = EU {S,l} l ( l ) + i l c i i l c M i = EU {S,l} l ( l ) + i l EU {S,l} i ( l ) t M i () i l t V i () The first equality is by the preceding theorem; the second one follows from the relationship between EU {S,l} l ( l ) and EUl A ( l ) as gien in the example; the third one follows from the definition of c i s and c M i s; the last inequality follows from (10). Usually, we like to compare mechanisms in terms of the payoffs they generate for agents. It is of interest to ask whether there exists a mechanism which dominates other mechanisms in terms of expected payoffs. Definition 15 We say that mechanism M dominates mechanism M in expected payoffs, if EUi M ( i ) EU M i ( i ) i, i and j : EUj M ( j ) > EU M ( j ) j Theorem 6 No mechanism in the class which satisfies BIC, BB and EFF dominates another mechanism in expected payoffs. Proof : Suppose not, i.e., there exist two mechanisms M and M in the class satisfying BIC, BB and EFF such that: EUi M ( i ) EU M i ( i ) i, i j : EUj M ( j ) EU M j ( j ) j Then by using theorem 20, we must hae sets of constants c M and c M such that i cm i = = 0 and i cm i j EUi A ( i ) c M i EUi A ( i ) c M i EUj A ( j ) c M j > EUj A ( j ) c M j i c M i > i c M i i, i j But we hae zero on both sides of a strict inequality which is not possible. Hence contradiction. 18

19 The aboe result points out to the fact that we cannot use expected payoffs to rank mechanisms in this class. Therefore, we use a different approach. We consider the class of mechanisms satisfying BIC, EFF and IR, and ask if there exists a mechanism which constitutes the lower boundary of this class in terms of expected payoffs. If so, then by reenue equialence, we can select a member in the class of mechanisms satisfying BIC, EFF and BB which has minimum distance from this boundary. Theorem 7 The expected payoff of any agent under VCG is the lowest among all mechanisms satisfying BIC, EFF and IR at any gien profile. Proof : By (9) Gien the fact that the VCG payoffs constitute a lower boundary of the expected payoffs for mechanisms in the class satisfying BIC, EFF and IR, we want now to find a mechanism in the class satisfying BIC, EFF and BB which results in expected payoffs which are at the minimum distance from the VCG payoffs. If M is a mechanism in the latter class, by theorem 20, it can be written as (p, t A + c M ), so that EUi M ( i ) = EUi A ( i ) c M i, for all i. Consequently, we can characterize any mechanism in this class by the corresponding ector of constants. We will now choose the ector of constants which minimizes the (Euclidean) distance between the payoff ector corresponding to the VCG mechanism and that corresponding to the mechanism related to the ector of constants. The only constraint on the ector of constants is that the sum of its elements must be zero. The problem is represented as the following: Minimize i (EU i V ( i ) EUi M ( i )) 2 where EUi M ( i ) = EUi A ( i ) c M i and i cm i = 0 Since we want the set of constants which minimizes the distance, this can be re-written as: Minimize {c M i } n i=1 i (EU i V ( i ) EUi A ( i ) + c M i ) 2 (22) where i cm i = 0 The expected equilibrium payoffs are monotonic functions of the aluations and therefore bounded by the payoffs at the limits of the aluations. The objectie function is therefore conex and continuous in c M i s, a unique minimum exists by Weierstrass theorem and it is characterized by the first order conditions of minimizing the corresponding Lagrangian. This Lagrangian is: (21) = i (EU V i ( i ) EU A i ( i ) + c M i ) 2 λ i c M i (23) where λ is the Lagrangian multiplier. 19

20 The first order conditions for minimization are: c M i : 2(EU V i ( i ) EU A i ( i ) + c M i ) λ = 0 λ : i cm i = 0 (24) Summing oer the i first order conditions corresponding to c M i, we obtain: 2 i (EU i V ( i ) EUi A ( i ) + c M i ) nλ = 0 λ = 2 n i (EU i V ( i ) EUi A ( i )) (25) Substituting for λ in the first order condition corresponding to c M i get the optimal c M i s, denoted (c M i ) :, and re-arranging, we (c M i ) = 1 (EUi V ( i ) EUi A ( i )) EUi V ( i ) + EUi A ( i )) (26) n i Using the expressions for EU V i ( i ) and EU A i ( i ), this may be re-written as: (c M i ) = 1 n i ( 1 n 1 1 n 1 l i E s l [SW l ( l, s l )] SW (α i, i )) l i E s l [SW l ( l, s l )] + SW (α i, i ) The aboe discussion can be summarized in the following theorem. Theorem 8 The mechanism satisfying BIC, BB and EFF which lies at the minimum distance from the minimally indiidually rational mechanism in the class of mechanisms satisfying BIC, EFF and IR is unique, and is characterized by the mechanism (p, t A +(c M ) where p is the efficient allocation rule, t A is the d Aspremont Gerard-Varet payment rule and (c M ) is the n-ector defined by (27). (27) 7 Ex Ante Efficiency In settings of incomplete information, ex-ante efficient mechanisms maximize weighted sums of expected utilities, i.e., these are concerned about haing the highest welfare aeraged across states and agents. The optimal mechanism by?, iz., the one which maximize sum of expected utilities, is definitely an ex-ante efficient mechanism which puts equal weights on all agents. But this is not the only ex-ante efficient mechanism. Following Williams (1999), who characterized the set of efficient mechanisms for the bilateral trade problem, we attempt to characterize efficient mechanisms for the land acquisition problem. A buyer wants to buy k units of land or more, but each unit is owned by a separate seller. We want to find the set of ex ante efficient mechanisms for this problem when all the agents hae priate aluations. 20

21 7.1 Indiidual Rationality and Incentie Compatibility: Characterization With respect to lemma 1, we note that for the buyer j = 0, α j = 0 and for each of the seller, α j =. Therefore,. Further, we obtain, EU 0 ( 0 ) = EU 0 ( 0 ) + EU i ( i ) = EU i ( ) j 0 E 0 (P 0 (z, 0 ))dz (28) j E i (P i (z, i ))dz (29) 0 0(P 0 ())f( 1 ) f( n )g( 0 )d 0 d n n 0 ip i ()f( 1 ) f( n )g( 0 )d 0 d n 0 = 0 0 n i=1 0 0 i= n i=1 n i=1 0 = 0 0 [ n i=1 = 0 0 0(P 0 ())f( 1 ) f( n )g( 0 )d 0 d n t i()f( 1 ) f( n )g( 0 )d 0 d n t i()f( 1 ) f( n )g( 0 )d 0 d n ip i ()f( 1 ) f( n )g( 0 )d 0 d n { 0(P 0 ()) n [ 0 0 }{{} n 1 times U 0 ( 0 )g( 0 )d 0 + n = U 0 ( 0 ) n i=1 [U i( ) + i=1 t i()}f( 1 ) f( n )d 1 d n ]g( 0 )d 0 {t i () i P i ()}g( 0 ) f( 1)...f( n) f( i ) i=1 U i( i )f( i )d i 0 E 0 (P 0 (z 0, 0 ))dz 0 g( 0 )d 0 d 0...d n d i ]f( i )d i i E i (P i (z i, i ))dz i f( i )d i ] = U 0 ( 0 ) + n i=1 U i( ) + n i=1 [ E i(p i ( i, i ))F ( i )d i ] + 0 E 0 (P 0 ( 0, 0 ))(1 G( 0 ))d 0 0 = U 0 ( 0 ) + n i=1 U i( ) + n 0 i= (P 0 0()) 1 G( 0) g( 0 f( ) 1 ) f( n )g( 0 )d 0 d n U 0 ( 0 ) + n i=1 U i( ) = 0 (P 0 0())[ 0 1 G( 0) g( 0 ]f( ) 1 ) f( n )g( 0 )d 0 d n n 0 i=1 P i()[ i + F ( i) ]f( f( i ) 1) f( n )g( 0 )d 0 d n 0 P i() F ( i) f( i ) f( 1) f( n )g( 0 )d 0 d n (30) The first equality is obtained by first subtracting and then adding the second term on the right hand side; the second equality uses the facts that real alued functions of independent random ariables are independent and that the expectation of a sum of independent random ariables is the sum of expectations of those random ariables. The third equality uses the definitions of the appropriate expectations; the fourth one uses equations (29) and (28); 21

22 the next one uses integration by parts to remoe the second integral; the sixth equality is obtained by using the expression for the expectations. The final expression is obtained by transposing and re- arranging terms on both sides. From (30), we get the following Results: Lemma 2 For any incentie compatible mechanism, U 0 ( 0 ) + n i=1 U i( ) = min 0 [ 0, 0 ] U 0 ( 0 ) + n i=1 min i [, ] U i ( i ) = 0 [P 0 0()( 0 1 G( 0) g( 0 )} n ) i=1 {P i() (31) ( i + F ( i) )}] f( f( i ) 1) f( n )g( 0 )d 0 d n Lemma 3 A mechanism is incentie compatible and indiidually rational only if, 0 0 [{P 0 0()( 0 1 G( 0) g( 0 )} n ) i=1 {P i()( i + F ( i) )}] f( i ) (32) f( 1 ) f( n )g( 0 )d 0 d n The conerse is also true, i.e., suppose (32) holds, E 0 (P 0 ()) is increasing in 0 and E i (P i ()) s are decreasing in respectie i s; then we can find a payment function t i () such that (P, t) is indiidually rational and incentie compatible. We may consider the following payment function among many other possibilities: t i () = 1 0 n z 0 z = 0 d(e 0 (P 0 (z 0, 0 ))) i z 0 z i = id( E i (P i (z i, i ))) + E i (P i (, i )) 1 E n i( 0 z 0 z = 0 d(e 0 (P 0 (z 0, 0 )))) 0 + z z i = id( E i (P i (z i, i ))) Since E 0 (P 0 ()) is increasing in 0 and E i (P i ()) s are decreasing in respectie i s, the first two terms on the right hand side of (33) are positie; the first term is a function of 0 alone, the second term is a function of i alone, and the rest are constant terms chosen to make U i ( ) = 0 for all i = 1,, n. To check that this payment function leads to Bayesian incentie compatibility, U 0 ( 0 ) U 0 ( 0, ˆ 0 ) = 0 E 0 (P 0 ( 0, 0 )) 0 E 0 (P 0 (ˆ 0, 0 )) E 0 ( n i=1 t i( 0, 0 ) n i=1 t i(ˆ 0, 0 )) = 0 E 0 (P 0 ( 0, 0 )) 0 E 0 (P 0 (ˆ 0, 0 )) E 0 [ 0 z 0 z = 0 d(e 0 (P 0 (z 0, 0 ))) n 0 i=1 ˆ 0 z 0 = 0 z 0 d(e 0 (P 0 (z 0, 0 ))) + n i=1 0 = 0 z 0 =ˆ 0 d(e 0 (P 0 (z 0, 0 ))) 0 = 0 z 0 =ˆ 0 ( 0 z 0 )d(e 0 (P 0 (z 0, 0 ))) 0 i z i = z id( E i (P i (z i, i ))) i z i = z id( E i (P i (z i, i )))] z 0 =ˆ 0 z 0 d(e 0 (P 0 (z 0, 0 ))) The first equality uses definition 7, the second substitutes (33) for t i s; since the second and the fourth term under the square braces cancel out, the first term less the third one is 22 (33)

23 only a function of 0, and hence taking expectation oer aluations other than 0 does not affect it consequently, we are left with the right hand side of the third equality. The inequality follows from the fact that if 0 > ˆ 0 then 0 z 0 is positie throughout the range of the integral, and the sign of the differential is positie by assumption; if on the other hand, 0 < ˆ 0, 0 z 0 is negatie, so we can interchange the limits of the integral, replace 0 z 0 with z 0 0 to obtain a positie sign. For the sellers, we hae: U i ( i ) U i ( i, ˆ i ) = E i (t i ()) i E i (P i ()) E i (t i (ˆ i, i )) + i E i (P i (ˆ i, i )) = E i ( ˆ i z z i = id( E i (P i (z i, i ))) i z z i = id( E i (P i (z i, i ))) i E i (P i ()) + i E i (P i (ˆ i, i )) = E i ( ˆ i z i = i z i d( E i (P i (z i, i ))) i ( ˆ i z i = i d( E i (P i (z i, i ))) = ˆ i z i = i (z i i )d( E i (P i (z i, i ))) 0 The first equality obtains by the definition 7, the second one is obtained by substituting for the payment scheme in concern and then obsering that certain terms cancel out. The rest is by arguments similar to the case of the buyer. Since (P, t) is incentie compatible, (31) is satisfied. Since U i ( ) = 0 for all i = 1,, n and (32) is pre-assumed, we must hae U 0 ( 0 ) 0. Since U 0 and U i s for all i = 1,, n are increasing and decreasing respectiely, indiidual rationality obtains for all agents. This obseration leads to the following result: Lemma 4 If E 0 (P 0 ()) is increasing in 0 and E i (P i ()) s are decreasing in respectie i s and (32) holds, then there exists a payment function t i () such that the mechanism (P, t) is indiidually rational and Bayesian incentie compatible. These results are summarized in the following theorem. Theorem 9 (Myerson-Satterthwaite Extended) For any incentie compatible mechanism, U 0 ( 0 ) + n i=1 U i( ) = min 0 [ 0, 0 ] U 0 ( 0 ) + n i=1 min i [, ] U i ( i ) = 0 [{P 0 0()( 0 1 G( 0) g( 0 )} n ) i=1 {P i() ( i + F ( i) )}] f( f( i ) 1) f( n )g( 0 )d 0 d n Further, there exists payment functions t i () such that (P, t) is Bayesian incentie compatible and indiidually rational if and only if E 0 (P 0 ()) is increasing in 0 and E i (P i ()) s are decreasing, and g( 0 ) f( 1 ) f( n )g( 0 )d 0 d n [{P 0()( 0 1 G( 0) )} n i=1 {P i()( i + F ( i) f( i ) )}] 23

24 The next theorem is going to be useful when we discuss efficiency. The proof is immediate from the aboe discussion. Theorem 10 Let us call the expression in (31), Γ(P ). If P is such that E 0 (P 0 ()) is increasing in 0, E i (P i ()) s are decreasing and Γ(P ) 0, then there exists a t such that (P, t) is incentie compatible, indiidually rational and for each q [0, Γ(P )] such that nq < Γ(P ), t can be chosen in a manner such that U i ( ) = q i = 1,, n. This implies U i ( i ) = q + U 0 ( 0 ) = Γ(P ) nq + i E i P i ()f( i )d i (34) 0 0 E i P 0 ()g( 0 )d 0 (35) Therefore, mechanisms which are indiidually rational and incentie compatible can be written as pairs (P, q). 7.2 Deriing the Optimal Mechanism The expected total gains from trade is 0 0 = 0 0 U 0 ( 0 )g( 0 )d 0 + n } {{ } n times i=1 U i( i )f( i )d i n j=0 jp j ()f( 1 ) f( n )g( 0 )d 0 d n which we want to maximize with respect to incentie compatibility and indiidual rationality constraints. For any α 0, let c i ( i, α) = i + α F i( i ) f i ( i ) c 0 ( 0, α) = 0 α 1 G( 0) g( 0 ) Then, using (32), our problem can be written as (36) (37) max 0 0 } {{ } such that 0 0 n times n j=0 jp j ()f( 1 ) f( n )g( 0 )d 0 d n } {{ } n times n j=0 c j( j, 1)P j ()f( 1 ) f( n )g( 0 )d 0 d n 0 We construct the Lagrangian objectie function as follows. 24

25 Λ = 0 0 } {{ } n times = (1 + λ) 0 0 n j=0 [ j + λc j ( j, 1)]P j ()f( 1 ) f( n )g( 0 )d 0 d n } {{ } n times n j=0 c j( j, λ 1+λ )P j()f( 1 ) f( n )g( 0 )d 0 d n (38) Any P that satisfies constraint (32) with equality and maximizes the Lagrangian objectie function (38) for some λ 0 must be a solution to our problem. Now consider the following allocation rule P α : P α 0 () = P α i () = { { 1 if c 0 ( 0, α) n j=n k+1 c j( j, α) 0 otherwise 1 if c 0 ( 0, α) n j=n k+1 c j( j, α) and i {n k + 1,, n} 0 otherwise (39) We obsere that P α maximizes Λ for α = λ. Now we need to show that P α meets the 1+λ conditions of Lemma 4, and there exists an α for which the constraint (32) is satisfied with equality. If c 0 (, 1) and c i (, 1) s are increasing on [ 0, 0 ] and [, ], then for any α [0, 1], c 0 (, α) and c i (, α) are increasing. Therefore both P0 α and Pi α are increasing in 0 and decreasing in i and consequently E 0 P0 α is increasing in 0 and E i Pi α is decreasing in i. Let G(α) = 0 0 } {{ } n times n c j ( j, 1)Pj α ()f( 1 ) f( n )g( 0 )d 0 d n (40) j=0 By the argument in the preious paragraph, G(1) 0. Further, P0 α is increasing in α and Pi α s are decreasing in α. Therefore, if α < β, G(α) and G(β) will be different only in profiles such that P β () 0 but P0 α () = 1, Pi α () = 1 i {n k + 1,, n}. This implies that c 0 ( 0, β) < j {n k+1,...,n} c j( j, β) and so c 0 ( 0, 1) < j {n k+1,...,n} c j( j, 1). This implies that G(α) is non-decreasing. If each c j ( j, 1) is increasing so is eery c j ( j, α), α < 1. Therefore the equation c 0 ( 0, α) = j {n k+1,...,n} c j( j, α) has at most one solutions in j s which aries continuously in j s and α. Therefore, G(α) is continuous, increasing and G(1) 0. But G(0) < 0 by?. Hence there exists an α (0, 1] such that G(α) = 0. The following theorem summarizes this subsection. Theorem 11 The optimal incentie compatible and indiidually rational mechanism for the 25

26 land acquisition problem is gien by the following form of allocation rule: { P0 α 1 if c 0 ( 0, α) n j=n k+1 () = c j( j, α) 0 otherwise { Pi α 1 if c 0 ( 0, α) n j=n k+1 () = c j( j, α) and i {n k + 1,, n} 0 otherwise where c i ( i, α) = i + α F i( i ) f i ( i ), c 0( 0, α) = 0 α 1 G( 0) g( 0 ) and α [0, 1] 7.3 Efficiency The Constrained Problem Expected Gains for buyer and sellers in (P, q) are: Ū 0 (P, q) = Ū i (P, q) = 0 0 U 0 ( 0 )g( 0 )d 0 U i ( i )f( i )d i If P ( ) is allowed to be any ector alued function and q R, each Ūj( ) is linear in P ( ) and q. This (P, q) determine an incentie compatible and and indiidually rational (ICIR) allocation rule if and only if E j (P j ( )) s hae the appropriate sign of deriaties and 0 q Γ(p). This implies that ICIR allocation rules constitute a closed and conex subset of the set of all reals and real alued functions. Such a rule is efficient if and only if it maximizes γ 0 Ū 0 ( ) + + γ n Ū n ( ) for some n-ector γ such that n j=0 γ j = 1. Therefore, we can write the set of ICIR efficient rules as the set of (P, q) which is the solution set of the following problem : max n j=0 γ jūj( ), n j=0 γ j = 1 such that (i) P is an an admissible allocation rule, (ii) 0 q Γ(P ), nq Γ(P ), (iii) E j P j () s hae appropriate sign of deriaties. 26

27 7.3.2 The Less Constrained Problem Consider the preious problem without constraint (iii). We call it the Less Constrained(LC) problem. max n j=0 γ jūj( ), n j=0 γ j = 1 such that (i) P is an an admissible allocation rule, i.e. P 0 {0, 1}, P i {0, 1} i 0, P 0 = 1 j 0 P j = k P is IC and IR (ii) 0 q Γ(P ), nq < Γ(P ) Consider allocation rules φ t,s defined as follows: φ t,s 0 = φ t,s i = where S is some k-member subset of N. The following lemma will be useful. { { 1 if c 2 ( 2, s) j S c j( j, t j ) 0 otherwise. 1 if c 2 ( 2, s) j S c j( j, t j ), i S 0 otherwise. Lemma 5 Suppose (P, q) is a solution to LC for some γ. If 0 γ 0 1, then q = 0 and if 2 1 < γ 2 0 1, then q = Γ(P )/n. Proof : We note that, where M = γ n j=0 γ jūj(p, q) 0 = γ 0 [U 0 ( 0 0 ) + 0 P0 (z 0 )dz 0 ]d j 0 γ j [U j( ) + Pj j (z j )dz j ]d j = γ 0 (Γ(P ) nq) + j 0 (γ 0 jq) + γ 0 + j 0 γ j Pj j (z j )dz j d j = γ 0 (Γ(P ) nq) + j 0 (γ jq) + M 0 0 P0 (z 0 )dz 0 + j 0 γ j P0 (z 0 )dz 0 j Pj (z j )dz j ]d j is a constant. This is maximized by choosing q = Γ(P ) n if 1 γ 0 > γ 0 and q = 0 if 1 γ 0 γ 0. Equipped with this lemma, we will now try to find solution of the LC problem. We only proide the argument for case 1 2 < γ 0 1. The other case is similar. 27