Partnership Dissolution and the Willingness-to-Pay - Willingness-to-Accept Disparity

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1 Partnershi Dissolution and the Willingness-to-Pay - Willingness-to-Accet Disarity Alexander Heczko RWTH Aachen University This Version: June, 218 Working Paer Abstract A willingness-to-ay WTP) willingness-to-accet WTA) disarity strongly affects roerties of artnershi dissolution mechanisms. We identify a necessary and sufficient condition for the existence of an individually rational, ex ost efficient, budget balanced and incentive comatible dissolution mechanism for the rominent equal-share artnershi. In contrast to the standard case where WTP and WTA coincide, we find that the existence of such a mechanism cannot be guaranteed. JEL classifications: D44, C72, D82 Keywords: Mechanism design, artnershi dissolution, difference between willingnessto-ay and willingness-to-accet. 1 Introduction Differences between willingness-to-ay WTP) and willingness-to-accet WTA) have been widely studied and established in the literature. 1 A WTP WTA disarity is articularly interesting if the articiants in a mechanism do not know whether they will be a otential buyer or seller of an object because then the articiants are forced to assess both their WTP and their WTA. As we will see, this henomenon occurs in mechanisms to dissolve artnershis. In an indeendent rivate values environment Myerson and Satterthwaite [1983] rove that a mechanism ensuring efficient bilateral trade cannot exist unless some outsider is willing to ay subsidies to the articiants. Cramton, Gibbons, and Klemerer [1987] CGK) show that this imossibility result strongly deends on the initial ownershi structure. If the indivisible object to be traded is commonly owned by more than one agent then referred to as a artnershi and the ownershi rights are distributed sufficiently equally, efficient trade is ossible. A mechanism to dissolve a artnershi works as follows: Based on messages submitted by the agents, a dissolution mechanism assigns the artnershi to alexander.heczko@rwth-aachen.de 1 A review of WTP and WTA studies for a variety of goods can be found in Horowitz and McConnell [22]. The literature also refers to the WTP-WTA disarity as endowment effect. It describes that agents value an object they own more than they would if they did not own it. The endowment effect can be exlained by loss aversion. For a discussion on the existence of a WTP-WTA ga with a focus on the elicitation methods used in exeriments see Plott and Zeiler [25, 27], and Fehr, Hakimov, and Kübler [215]. A recent overview of exerimental evidence on the existence of an endowment effect can be found in Marzilli Ericson and Fuster [214]. 1

2 a subset of) the agents and determines transfers. We extend the artnershi dissolution model of CGK by allowing the agents to distinguish between their WTP for shares they do not own and their WTA for the share they own. Our main assumtion is that agents WTA exceeds their resective WTP. This can be exlained by the agents being loss averse according to the famous rosect theory of Kahneman and Tversky [1979] that exlains the difference between WTP and WTA by reference-deendent references. 2 However, it is also ossible to find a rational reason for this difference. The idea that agents erceive a first share as more valuable than an additional share is related to the roerty of diminishing marginal utility. The standard aroach suggests that the agents take a ste back in thought giving u their initial share and assign a value to the whole artnershi. 3 We think of the artnershi as an indivisible object that still can hyothetically be searated into disjoint arts, like a firm consisting of different deartments or a house with two seerate aartments. Hence, in a dissolution rocedure we assume that the agents think about acquiring an additional share in order to become the sole owner, or selling the share they initially own. This different normalization motivates a difference between WTP and WTA in a ossible artnershi dissolution rocess and rovides a rational exlanation for its existence. The considered setting includes a relevant examle: Imagine two agents commonly own a firm that consists of two deartments. Every agent is the head of one deartment and has greater knowledge about the workers, the machines, the oerations, and the rocesses in her resective deartment. Hence the agents assign a higher value to their deartment than to the deartment of the artner which can be exlained by ambiguity aversion. Following CGK we analyze a symmetric indeendent rivate values setting. We assume that the WTP and WTA are erfectly correlated, i.e., our model is a screening model with one-dimensional tyes. As a benchmark we consider the equal-share artnershi with two agents. 4 We identify a general necessary and sufficient condition for the existence of an ex ost efficient EF), Bayesian incentive comatible IC) dissolution mechanism that ex ante balances the budget EABB) and is interim individually rational IIR). In line with Williams [1999] we refer to such a mechanism as desirable. In contrast to CGK, where dissolution is always ex ost efficient, this is not true if WTP and WTA differ. For a certain subset of tyes it is ossible that neither agents WTP exceeds the other agent s WTA and ex ost efficiency rescribes maintaining the status quo. The existence of this non-trade region lowers the ossible gains from trade necessary to finance articiation and incentive comatibility. In articular, different to the results in CGK, the equal-share artnershi cannot necessarily be dissolved efficiently. 5 We show that it can be dissolved 2 A theory of reference-deendent references with endogenous reference oints is given by Kőszegi and Rabin [26]. 3 This normalization is used, amongst others, in Cramton et al. [1987], Fieseler, Kittsteiner, and Moldovanu [23] and Loertscher and Wasser [216]. 4 The two-agent equal-share artnershi is very common in reality. The equal-shares assumtion is made for convenience. However, the model includes any initial ownershi distribution with the roerty that the WTA of both agents exceeds their resective WTP. This is most likely if shares are distributed close to) equally. 5 In line with CGK we say that a artnershi can be dissolved efficiently, if a desirable dissolution mechanism exists. As for examle noted by Fieseler et al. [23] the desirable roerty efficiency should better be referred to as value-maximizing, because efficiency actually is a hyernym for all desirable 2

3 efficiently if the difference between WTP and WTA is not too large or trivially so large that no WTP exceeds any WTA. We follow the aroach described, for examle, in Krishna and Perry [1998], Makowski and Mezzetti [1994] and Williams [1999] to deduce a condition for the existence of a desirable mechanism and demonstrate that it can be alied to the considered framework. We first show that a ayoff-equivalence result holds Myerson [1979, 1981], Riley and Samuelson [1981]). 6 It imlies that any IC and EF mechanism is ayoff-equivalent to a Vickrey-Clarke-Groves VCG) mechanism. In articular, the exected ayoff and transfer of an agent in any IC and EF mechanism are inned down by the well-known VCG formula. The revelation rincile imlies that to any Bayesian Nash equilibrium of an arbitrary mechanism there exists a ayoff-equivalent direct IC mechanism. 7 Consequently, the exected ayoffs and the exected transfers of the agents in any Bayesian Nash equilibrium of an ex ost efficient mechanism are given by the VCG formula. This is a very owerful tool: It is sufficient to analyze the conditions under which a VCG mechanism manages to be ex ante budget balanced and interim individually rational to determine the conditons for the existence of a desirable mechanism in general. 8 Starting with CGK the literature on artnershi dissolution has exhibited that not only the ownershi structure affects the existence of a desirable dissolution mechanism, but also the outside otion, asymmetric) information, asymmetric) tye distributions, and a ossible interdeendence of valuations. Schweizer [26] derives a universal existence condition that does not necessarily deend on the distribution of tyes. In addition he discusses different outside otions. In an indeendent rivate values environment, Ornelas and Turner [27] show that the existence of a desirable dissolution mechanism with asymmetric ownershi structure deends on the distribution of control. They assume that the outside otion of a artnershi deends on a common) gross value. Particularly they assume that valuations of the agents are indeendent if the artnershi is dissolved and interdeendent if not. Turner [213] introduces the effectiveness of the cooeration between artners in a artnershi as another variable influencing the existence of a desirable dissolution mechanism. Figueroa and Skreta [212] analyze an indeendent rivate values setting with asymmetric tye distributions. They show that the ownershi structure that can be dissolved efficiently can then be extremely unequal. Jehiel and Pauzner [26] study a setting with interdeendent valuations and two agents, where only one agent learns the tye determining the value of both agents. The other agent stays uninformed. If there is no mechanism to dissolve the artnershi efficiently they solve for the second best mechanism and show that extreme ownershi can be second-best) welfare maximizing. Fieseler roerties. 6 Sometimes this result is referred to as the closely related Revenue-Equivalence Theorem. 7 Green and Laffont [1977] and Holmstrom [1979] show that any EF and dominant strategy incentive comatible mechanism is a VCG mechanism, i.e., the ayoff-equivalence equivalence result holds for dominant strategy incentive comatibility. Myerson [1979] and later in a more general version Williams [1999] show that this result also holds for Bayesian incentive comatibility, which is a weaker requirement. 8 A detailed derivation of this method can be found for examle in Williams [1999]. 3

4 et al. [23] generalize the findings of CGK by deriving a condition for the existence of a desirable dissolution mechanism for interdeendent valuations. They show that, deendent on the sign of the derivative of the valuation in the other agents tye, it can be both easier or more difficult to dissolve the artnershi efficiently as comared to the rivate values case. Esecially, if the valuations of the agents are increasing in the other agents tye, the equal-share artnershi cannot necessarily be dissolved efficiently. We show that this statement is also true if values are rivate and the WTP and WTA of the agents differ. Following the setu of Fieseler et al. [23], but focussing on individual rationality, Galavotti, Muto, and Oyama [211] derive a sufficient condition for the existence of an EF, IC, EABB and the stronger ostulation of ex ost individually rational dissolution mechanism. In addition they discuss imlications of ex ost quitting rights. Yenmez [212] allows the agents to jointly own more than one indivisible object. He assumes that initially every agent owns shares of the different objects that add u to one, and that ex ost every agent is assigned exactly one object. He shows that the set of initial ownershi structures for which a desirable dissolution mechanism exists is non-emty, convex and contains the structure with every agent initially owning equal shares of every object. Our assumtion that agents erceive the artnershi as searate objects, one they own and one they do not own is related to the exlicit assumtion of multi-artnershis. Yet, our setting is not catured by his model. In a general setting, Segal and Whinston [211] show that, if certain convexity conditions hold, and if the status quo ownershi shares are equal to the exected equilibrium assignment, IIR is ensured. The very comrehensive model of Loertscher and Wasser [216] imlies asymmetric tye distributions, arbitrary initial ownershi structures and interdeendent valuations. In addition to deriving the otimal second-best) dissolution mechanism with resect to social surlus and revenue, they solve for the otimal ownershi structure. The literature has focused on these causes influencing whether or not a artnershi can be dissolved efficiently or not. This aer introduces the WTP - WTA disarity as an imortant factor determining the existence of a desirable dissolution mechanism. 2 Model Consider two risk-neutral agents owning equal shares of a artnershi. We assume two different, erfectly correlated valuation functions. A function v a : [, 1] C a := [c a, c a ] R +, describing the willingness-to-accet WTA), and a function v : [, 1] C := [c, c ] R +, describing the willingness-to-ay WTP). Both are assumed to be strictly increasing and twice continuously differentiable. We analyze a symmetric setu, i.e., we assume that both agents have the same WTP and WTA function. The true tyes τ i, i = 1, 2 of the agents are distributed indeendently according to the same commonly known distribution function. We restrict the analysis to the case in which the true) tye τ i of agent i = 1, 2 is uniformly distributed on [, 1]. Furthermore, τ 1 and τ 2 are strictly increasing. This setu is without loss of generality, because we allow for arbitrary twice continuously 4

5 differentiable and strictly increasing valuation functions. 9 We assume v a τ i ) v τ i ) for all τ i [, 1] throughout this aer and leave the other cases to future research. This imlies that the agents erceive the share they own as more valuable than the share they do not own and is most likely if the agents own equal or close to equal) shares of the artnershi. A mechanism consists of a set of outcomes O, a message sace M, and a maing s, t) : M O, i.e., a mechanism determines an outcome based on messages submitted by the articiants that are not necessarily their reorted or true) tyes. 1 In a direct mechanism the message sace and the tye sace coincide, since the messages then are reorted) tyes. Throughout this aer we only consider direct mechanisms, which we refer to as a air s, t) for convenience. Consequently, s, t) mas the reorted tyes ˆτ i [, 1] of the agents to the set of outcomes, i.e., assignments and transfers. We denote the agents by i and j with i, j = 1, 2, i j. Definition 2.1. The assignment rule is a function s i : [, 1] [, 1] [, 1] [, ], ˆτ i, ˆτ j ) with s i ˆτ i, ˆτ j ) + s a j ˆτ j, ˆτ i ) =. s i ˆτ i, ˆτ j ) s a i ˆτ i, ˆτ j ) ). The function s i : [, 1]2 [, 1] mas the reorted tyes to the share of the artnershi that is allocated from agent j to agent i, whereas s a i : [, 1]2 [, ] mas the reorted tyes to the share that is allocated from agent i to agent j. The condition s i ˆτ i, ˆτ j )+s a j ˆτ j, ˆτ i ) = ensures that the artnershi is still owned by a subset of) the agents after the dissolution. For examle if s i ˆτ i, ˆτ j ) = 1 it follows that s a j ˆτ j, ˆτ i ) =. Consequently, the share of agent j is awarded to agent i, imlying that agent j loses her share: Ex ost agent i is the sole owner of the artnershi. Let t i : [, 1] 2 R be the transfer function of agent i. It ins down the ayments that agent i receives. We assume that the utility function of agent i is quasi-linear, i.e., the artnershis value and the transfer are additively searable. Given realizations of tyes the utility generated by the artnershi itself is v i τ i, s i ˆτ i, ˆτ j )) := v τ i )s i ˆτ i, ˆτ j ) + v a τ i )s a i ˆτ i, ˆτ j ) and the transfer is a function t i ˆτ i, ˆτ j ). Thus, the utility function of agent i is given by u i τ i, s i ˆτ i, ˆτ j ), t i ˆτ i, ˆτ j )) := v i τ i, s i ˆτ i, ˆτ j )) + t i ˆτ i, ˆτ j ). We define the desirable roerties of a mechanism. These are: Bayesian) incentive comatibility, ex ost efficiency, interim individual rationality and ex ante budget balancedness. Definition 2.2 Bayesian) Incentive Comatibility IC)). A mechanism s, t) is IC if truth-telling is a Bayesian Nash equilibrium. We give a characterization of IC mechanisms in Proosition See Aendix A.2. 1 Auctions, for examle, are mechanisms. The messages are bids and the ossible outcomes consist of assignments and ayments. Let the tye sace be given by [, 1], as in the considered setu. The message bid) sace, however, is usually given by R +, i.e., every weakly) ositive real number is a ossible bid. 5

6 Definition 2.3 Ex Post Efficiency EF)). The assignment rule s eff i τ i, τ j ), i = 1, 2 is ex ost efficient if it is given by 1, if v τ i ) > v a τ j ), s eff i τ i, τ j ) =, if v τ j ) > v a τ i ),, otherwise. In contrast to CGK not dissolving the artnershi can be efficient. However, swaing shares cannot be efficient because v a τ i ) v τ i ) for all τ i [, 1]. 11 We define the following functions: Let the exected assignment be a function ) S k ˆτ i ) := Eˆτj s k i ˆτ i, ˆτ j ), k = a,, let the interim) exected transfer to agent i be a function T i : [, 1] R, ˆτ i Eˆτj t i ˆτ i, ˆτ j )), and let the exected utility of agent i with tye τ i reorting her tye to be ˆτ i, assuming agent j reorts her tye truthfully, be given by U i : [, 1] 2 R, τ i, ˆτ i ) E τj u i τ i, s i, t i )) with U i τ i, ˆτ i ) = v τ i )S ˆτ i ) + v a τ i )S a ˆτ i ) + T i ˆτ i ). If both agents reort their tyes truthfully, for simlicity, we use the notation U i τ i ) := U i τ i, τ i ). Definition 2.4 Interim Individual Rationality IIR)). Let τ i arg min τ i [,1] U i τ i ) be a tye of agent i that is worst-off in an IC mechanism s, t). An IC mechanism is IIR if U i τ i ), i = 1, 2. Intuitively, in an IC mechanism that is IIR even the worst-off tyes of the agents look forward to articiating in the mechanism because they exect weakly) ositive utility. We define a transfer as an amount of money for examle) the agents receive. If the mechanism assigns the whole artnershi to agent i and she has to ay a comensation to 11 Because of the assumtion that v aτ i) v τ i) τ i we cannot simultaneously have v τ i) > v aτ j) and v τ j) > v aτ i). 6

7 agent j, a negative amount is transferred to her. This is in line with CGK and Fieseler et al. [23] and decisive for the formulation of ex ante budget balancedness. Definition 2.5 Ex Ante Budget Balancedness EABB)). An IC mechanism s, t) is ex ante budget balanced if 2 E τi [T i τ i )]. i=1 Ex ante budget balancedness ensures that the mechanism designer does not exect to run a deficit. The sum of ex ante exected transfers describes the outside) subsidy that is necessary to finance articiation and incentive comatibility. Definition 2.6 Desirable Mechanism). An IC mechanism s, t) is called desirable, if it meets all desirable roerties, i.e, if it is IIR, EF and EABB. 3 Existence Condition In this section we formulate the main result: A necessary and sufficient condition for the existence of a desirable mechanism. Since tyes are indeendent and identically distributed and agents are risk neutral we can show that the following ayoff-equivalence result holds for IC mechanisms. Proosition 3.1 Payoff-Equivalence). Let τ i, ˆτ i [, 1] and lim τ i ˆτ i s k i τ i, τ j ) = s k i ˆτ i, τ j ), k = a,, almost everywhere. i) Assume the mechanism s, t) is incentive comatible. Then the utility function for agent i with tye τ i [, 1] in the truth-telling equilibrium is given by τi U i τ i ) = U i τ) + v t)s t) + v at)s a t) ) dt, i = 1, 2, 1) τ where τ [, 1] is a constant. ii) If the utility functions in a mechanism s, t) are given by 1), and S a τ i ) and S τ i ) are non-decreasing in τ i, the mechanism s, t) is incentive comatible. 12 Proof. The roof is delegated to Aendix A.1. We define the family of VCG mechanisms by the efficient assignment rule s eff τ i, τ j ) and the transfer function v a τ j ) q i, if v τ i ) > v a τ j ), t VCG i τ i, τ j ) := v τ j ) q i, if v τ j ) > v a τ i ), q i, otherwise, 12 Note that if v aτ i) = v τ i) τ i[, 1], it suffices that S a τ i) + S τ i) is non-decreasing. 7

8 where q i R is an arbitrary constant. 13 Note that with this transfer function if we consider the constant searately as not art of t VCG i the IIR and EABB conditions can be written as U i τ i ) q i and 2 i=1 E τ i [T i τ i )] q i, resectively. The next roosition gives a general existence condition. Proosition 3.2. Let T i τ i ) be the interim exected transfers and U i τ i ) the utility of the worst-off tye of agent i = 1, 2 in a VCG mechanism. A desirable mechanism exists if and only if 2 U i τ i ) i=1 2 E τi [T i τ i )]. 2) i=1 Proof. See for instance Fieseler et al. [23], Theorem 2 or Williams [1999], Theorem 3. Both roofs are constructive. They show that the minimum subsidy of an IC, EF and IIR mechanism that is required to balance the budget is given by max{, 2 i=1 E τ i [T i τ i )] 2 i=1 U i τ i )}. The transfers in a VCG mechanism are inned down u to a constant q i, which for this roof sketch is assumed not to be art of the transfer function t VCG i. A VCG mechanism is EABB if 2 i=1 E τ i [T i τ i )] = 2 i=1 E [ [ ]] τ i Eτj t VCG i τ i, τ j ) q i. IIR then imlies U i τ i ) q i, i = 1, 2. Hence, a VCG mechanism fulfills IIR and EABB if and only if 2 i=1 U i τ i ) 2 i=1 q i 2 i=1 E τ i [T i τ i )]. Note that Proosition 3.2 can be formulated for arbitrary IC and EF mechanisms, but as aforementioned it suffices to identify a condition under which a VCG mechanism manages to be IIR and EABB to derive a general condition for the existence of a desirable mechanism. If inequality 2) holds, the VCG mechanism with q i = U i τ i ), i = 1, 2 is desirable. That imlies that we can achieve dominant strategy incentive comatibility. However, if inequality 2) does not hold there is no EF mechanism that is IIR and EABB among the larger set of Bayesian incentive comatible mechanisms. Proosition 3.2 imlies that the arbitrary constant q i has no influence on the existence condition. If not stated otherwise, let the VCG mechanism be the VCG mechanism with q i =. The functions v a : [, 1] C a and v : [, 1] C are continuous, strictly increasing and hence bijective, such that v x) for all x C and va y) for all y C a are well defined. The exected assignments are given by S va v τ i )) if τ i > v c a ), τ i ) = otherwise, and We assume S a 1 v v a τ i )) ) if τ i < va c ), τ i ) = otherwise. U iτ i ) = v τ i )va v τ i )) v aτ i ) 1 v v a τ i )) ) 13 The imortant roerty of the constant q i is q i τ i =. Hence in general q i can be a function of τ j. 8

9 to have a unique root in τ i v ) ca, v a c )), i.e., U i τ i) for all τ τ i with ca ), v a c )). 14 It imlies that the second order condition for a utility τ i, τ i v minimum holds for tyes of this range. 15 Proosition 3.2 imlies that the worst-off tyes in a VCG mechanism lay a crucial role for the existence of desirable mechanisms because we need to ensure that their utility exceed ex ante exected transfers. Lemma 3.1. Assume c a < c. If v given by ) ca < v a c ) a worst-off tye in the VCG mechanism exists and is imlicitly with τ i v ) ca, v a c ) ). If v worst-off. ca ) v a v τ i )va v τ i )) v a τ) 1 v v a τ i )) ) = c ) all elements of the closed interval [ v a The utility of the worst-off tye of agent i is given by Proof. v a v τ i )) v τ i ) v a t)) dt U i τ i ) = + v v a τ i )) v t) v a τ i )) dt if v if v ) ca < v a c ) ca ) v a c ). c ), v )] ca are Case 1: v c a ) τ i < va c ) v c a ) < τ i va c ). We consider tyes τ i with v a τ i ) C and v τ i ) C a. Any VCG mechanism is IC. Alying Proosition 3.1 gives U i τ i) = v τ i )S τ i ) + v aτ i )S a τ i ). With the efficient assignment rule s eff τ i, τ j ) the worst-off tye is imlicitly given by the equation v τ i )S τ i ) + v a τ i )S a τ i ) =, or, after lugging in S τ i ) = va v τ i )) and S a τ i ) = 1 v v a τ i )) ), v τ i )va v τ i )) v a τ) 1 v v a τ i )) ) =. 3) Equation 3) defines a critical tye, which always exists: Aly the intermediate value theorem to the continuous and non-decreasing, and therefore surjective) function U i τ i) = v τ i )va v τ i )) v aτ i )1 v v a τ i ))) on the closed interval [v ) ca, v a c )]. We 14 This is for examle true if v τ i)v a v τ i)) + v a v τ i) 2 v a v ) > τ i)) v a τ i) 1 v v ) aτ i)) v v aτ i) 2 v v ), aτ i)) i.e., if U i is convex. The inequality esecially holds if v a τ i) and v τ i). In contrast to the CGK we need to make this additional and natural) assumtion because we cannot directly infer that U i is convex if s, t) is IC. 15 We assume that v a VCG mechanism. ) ca < v a c ). Lemma 3.1 gives a full characterization of the worst-off tyes in 9

10 have U i )) v ca < and U i v a c ) ) >. Hence, there is a τ i v ) ca, v a c )) with U i τ i) =. Since U τ i ) for all τ i τ i, equation 3) defines a minimum with τ i v ca ), v a c ) ). Case 2: τ i > va c ) τ i > v c a ). We consider tyes with v a τ i ) > c and v τ i ) C a. It is U i τ i ) =v τ i ) va v τ i )) ) v a v τ i )) v a t)dt, U iτ i ) =v τ i ) va v τ i )) ) >, i.e., the utility is strictly increasing for tyes of this case. Case 3: τ i < va c ) τ i < v c a ). For tyes with v τ i ) < c a and v a τ i ) C we have U i τ i ) = v v aτ i )) v t)dt v a τ i ) 1 v v a τ i )) ), U iτ i ) = v aτ i ) 1 v v a τ i )) ) <, i.e., the utility is strictly decreasing for tyes of this case. Case 4: va c ) v c a ). Tyes with v a τ i ) c and v τ i ) c a do not to trade in a VCG mechanism. All tyes of this case have zero utility. The functions v a : [, 1] C a and v : [, 1] C are continuous, strictly increasing, and hence bijective, i.e., the inverse functions are well defined. Case 1 and 4 are mutually exclusive. Assume c a < c. 1. v c a ) < va c ). Cases 1, 2 and 3 imly that the worst-off tye is imlicitly given by 3). The utility of the global worst-off tye is given by U i τ i ) = v a v τ i )) 2. v c a ) va c ). Cases 2, 3 and 4 imly that U i τ i ) =. v τ i ) v a t)) dt + v t) v a τ i )) dt. v v a τ i )) In artnershi dissolution mechanisms those tyes tyically are worst-off who in exectation know the least about being a buyer or a seller. Agents with a low tye exect to 1

11 be sellers and have an incentive to overstate their true tye, whereas agents with a high tye exect to be buyers and have an incentive to understate their true tye. Agents that do neither exect to be the seller nor the buyer have neither an incentive to over- nor to understate their true tye. Hence the mechanism does not have to incentivize them to tell the truth. 16 This revalent intuition still holds if the WTP and WTA of the agents differ. Lemma 3.1 imlies that the VCG mechanism that is IC and EF also is IIR. It is well known that VCG mechanisms are not guaranteed to be both IIR and EABB, i.e., in the considered setting the VCG mechanism does not necessarily satisfy EABB. The following theorem is the main result of this section. It gives a necessary and sufficient condition for the existence of an IC, EF, IIR and EABB mechanism. Following CGK we use the notation of the Riemann-Stieltjes integral β α v k t) tdv l v k t)) = β α v k t) t v ) l vk t)) dt with Lebesgue density v ) l vk t)), k, l = a,, k l. 17 Theorem Assume c a < c. i) If v ) ca < v a c ) there exists a desirable mechanism if and only if v τ i )va v τ i )) v a τ i ) 1 v v a τ i )) ) + v v a τ i )) v a c ) v a c ) v t)dt v a v τ i )) v a t) tdv v a t)) + v a t) dt. v v a t)dt v t) tdva v t)) c a) 4) ii) If v ) ca v a c ) there is no desirable mechanism. 2. If c a c there always is a desirable mechanism. Proof. If c a c, the ex ante) exected transfers in the VCG mechanism are E τi [T i τ i )] =, because then trade is not efficient and hence does not occur. The utility of all tyes τ i [, 1] is given by U i τ i ) =. In this case the VCG mechanism imitates a nil mechanism and is therefore doing the job. 18 If c a < c, using integration by arts, the ex ante) exected transfers in the VCG 16 For examle CGK, Fieseler et al. [23], Fieseler et al. [23], and Loertscher and Wasser [216] allow for n 2 agents and an arbitrary initial ownershi structure α 1,..., α n) with n i=1 αi = 1. If the agent s tyes are distributed according ) to the same strictly increasing distribution function F, they show that the tye τ i = F n α 1 i that does neither exect to buy nor to sell is worst-off. 17 The integrals are well defined, because the functions v l v k t)), k, l = a,, k l are continuously differentiable and esecially absolutely continuous). It is β v α k t) t dv l v k t)) = β v α k t) t ds k t). 18 The nil mechanism maintains the status quo without any ayments. 11

12 mechanism can be written as v E τi [T i τ i )] = a c ) v c a) v a c ) = + v v v aτ i )) v a v τ i )) v τ j ) dτ j dτ i v a τ j ) dτ j dτ i v a τ i ) τ i dv v a τ i )) v τ i ) τ i dva c a) v a c ) v τ i )) v a τ j ) dτ j. 5) Lemma 3.1 imlies that the worst-off tye utility is given by U i τ i ) = + v a v τ i )) v v a τ i )) v τ i ) v a t)) dt v t) v a τ i )) dt if v ) ca < v a c ), if v Alying Proosition 3.2 finishes the roof. ca ) v a c ). In the VCG mechanism, the tye τ i = v c a ) can be interreted as the highest tye that buys with robability zero and τ i = va c ) as the lowest tye that sells with robability zero. In case 1.i) the interiors of the ranges of v a and v intersect, i.e., c a < c and v ) ca < v a c ). The utility of the worst-off tye, U i τ i ), and the exected transfers, E τi [T τ i )], are both ositive. Hence, we cannot directly assess whether there is a desirable mechanism. If condition 4) holds, a mechanism designer can extract the subsidy required to balance the budget from the worst-off tye utility without violating IIR. The VCG mechanism with q i = U i τ i ) then is desirable. Case 1.ii) assumes c a < c and v c a ) va c ). The utility of the worst-off tyes is zero, because tyes τ i [va c ), v c a )] do not trade. But still trade is efficient for tyes τ i [, va c )) v c a ), 1] and therefore occurs with ositive robability. That imlies E τi [T i τ i )] >. The amount that is needed in order to revent the mechanism from running a deficit is ositive and cannot be obtained from the worst-off tyes without violating IIR. Hence a desirable mechanism cannot exist. Case 2 is trivial, because the WTA of any tye of an agent exceeds the WTP of any tye of the other agent u to the null set c a = c ) and therefore trade is never efficient. In contrast to the famous result of CGK it can no longer be guaranteed that the equal-share artnershi can be dissolved efficiently if there is ga between WTP and WTA. The following examle shows that the result from CGK for equal ownershi shares is a secial case of the considered setu. 12

13 Examle 3.1. In the model of CGK we have vτ i ) := v a τ i ) = v τ i ) for all τ i, i = 1, 2, and hence C a = C. This imlies c a < c and v ) ca = < 1 = v a c ), i.e., it is art of Theorem 3.1 case 1.i). The worst-off tye is τ i = 1 2. Condition 4) reduces to 2 t 2 dvt) t) 2 dvt). The left hand side is strictly) ositive, i.e., there always is a desirable mechanism to dissolve the equal-share artnershi. The following examle analyzes the existence of a desirable mechanism deending on the difference between WTP and WTA in case of linear valuations that only differ by a constant c. Examle 3.2. Assume v τ i ) = τ i and v a τ i ) = τ i + c, c. There is a desirable mechanism if and only if 19 i.e., if c.21731, or if c c 1)2 1 [, 1 2 ]c) c)3, The range of v a and v is determined by the constant c. Figure 1 shows the inequality of examle 3.2 grahically. The utility of the worst-off tyes as well as the exected transfers are strictly decreasing in c; the larger c becomes the less trade occurs. 2 Alying Theorem 3.1 we consider the following cases: 1.i) For c < 1 2 we have that c a < c with v c a ) = c < 1 c = v c ). The unique) a worst-off tye τ i = 1 2 realizes strictly ositive utility, i.e., an agent with tye τ i = 1 2 trades with ositive robability. There is a desirable mechanism if the difference between WTP and WTA fulfills c ii) For 1 2 c < 1 we still have that c a < c, but va c ) = 1 c c = v c a ). In this case there is a symmetric set of worst-off tyes τ i [1 c, c] around 1 2 with zero utility. These tyes trade with robability zero. The ex ante exected transfers are still ositive, because tyes τ i [, 1 c) c, 1] trade. There is no desirable mechanism. 2. For c 1 we have that c a c and any tye has zero utility and the ex ante exected transfers are zero. Therefore there trivially is a desirable mechanism. 4 Conclusion CGK show that the imossibility of efficient bilateral trade exhibited by Myerson and Satterthwaite [1983] deends on the initial ownershi structure. Particularly, they rove 19 The function 1 Ax) is the characteristic function, i.e., 1 Ax) = 1 if x A and 1 Ax) = if x / A. 2 The arameter c can be interreted as a measure of an endowment effect or the degree of diminishing marginal valuation as it describes the difference between WTP and WTA. 13

14 2 i=1 U i τ i, c), 2 i=1 E τ i [T i τ i, c)].5.4 Worst-off Tye Utility Exected Transfers c Figure 1: Worst-off tye utility and exected transfers for v a τ i ) = τ i + c and v τ i ) = τ i deendent on c. The green area reresents the values of c for which a desirable mechanism exists, i.e., c < c 1. that the equal-share artnershi can always be dissolved efficiently. There is strong evidence in the literature that agents assess their WTP and WTA differently suggesting that this alies to real-life situations. In artnershi dissolution mechanisms agents do not know whether they will take the role of the seller or the buyer. A WTP WTA disarity then has a major imact on the existence of a desirable mechanism. We include in our artnershi dissolution model a ossible difference between the agents WTP and WTA and therefore extend the model of CGK. More recisely, we show that the existence of a desirable dissolution mechanism can no longer be guaranteed if the WTA of the agents exceeds their resective WTP. If the realized WTP of an agent does not exceed the realized WTA of the resective other agent, efficiency imlies that the artnershi is not dissolved. In these situations the agents do not realize the gains from trade that would be necessary to finance articiation and incentive comatibility. The roblem of constructing a second best mechanism in this environment is still unsolved and remains for future research. 14

15 A Aendix A.1 Proof of Proosition 3.1 Proosition Payoff-Equivalence). Let τ i, ˆτ i [, 1] and lim s k i τ i, τ j) = s k i ˆτ i, τ j), k = a,, almost everywhere. τ i ˆτ i i) Assume the mechanism s, t) is incentive comatible. Then the utility function for agent i with tye τ i [, 1] in the truth-telling equilibrium is given by τi U iτ i) = U iτ) + v t)s t) + v at)s a t) ) dt, i = 1, 2, 6) τ where τ [, 1] is a constant. ii) If the utility functions in a mechanism s, t) are given by 6), and Si a τ i) and S i τi) are non-decreasing in τ i, the mechanism s, t) is incentive comatible. Proof. Assume the mechanism s, t) is incentive comatible. The utility from lying must not be higher than the utility from truth-telling for every agent i = 1, 2 if the other agent reveals her tye truthfully, i.e., Inequality 7) can be written as U iτ i) := U iτ i, τ i) U iτ i, ˆτ i) ˆτ i [, 1]. 7) U iτ i) = E τj [v τ i)s i τi, τj) + vaτi)sa i τ i, τ j)] + T iτ i) = v τ i)s τ i) + v aτ i)s a τ i) + T iτ i) v τ i)s ˆτ i) + v aτ i)s a ˆτ i) + T i ˆτ i) = U i ˆτ i) U i ˆτ i) + v τ i)s ˆτ i) + v aτ i)s a ˆτ i) + T i ˆτ i) = U i ˆτ i) + v τ i) v ˆτ i))s ˆτ i) + v aτ i) v a ˆτ i))s a ˆτ i). Hence, U iτ i) U i ˆτ i) [v τ i) v ˆτ i)]s ˆτ i) + [v aτ i) v a ˆτ i)]s a ˆτ i). 8) Conversely, the following inequality must hold: U iˆτ i) = U iˆτ i, ˆτ i) U iˆτ i, τ i) τ i [, 1]. That imlies U iˆτ i) =v ˆτ i)s ˆτ i) + v a ˆτ i)s a ˆτ i) + T i ˆτ i) v ˆτ i)s τ i) + v a ˆτ i)s a τ i) + T iτ i) =U iτ i) + v ˆτ i) v τ i))s τ i) + v aˆτ i) v aτ i))s a τ i), or equivalently U iτ i) U i ˆτ i) [v τ i) v ˆτ i)]s τ i) + [v aτ i) v a ˆτ i)]s a τ i). 9) Consequently, after dividing by τ i ˆτ i for w.l.o.g. τ i > ˆτ i and combining inequalities 8) and 9) we obtain v τ i) v ˆτ i) S τ i) + τ i ˆτ i Uiτi) Uiˆτi) τ i ˆτ i v τ i) v ˆτ i) S ˆτ i) + τ i ˆτ i vaτi) va ˆτi) τ i ˆτ i S a τ i) vaτi) va ˆτi) S a ˆτ i). τ i ˆτ i 15

16 The dominated convergence theorem imlies that lim s k i τ i, τ j) = s k i ˆτ i, τ j) a.e. is equivalent to lim S k τ i) = τ i ˆτ i τ i ˆτ i S k ˆτ i), k = a,. Hence, we obtain U iτ i) U iˆτ i) lim = U i ˆτ i) = v ˆτ i)s ˆτ i) + v a ˆτ i)s a ˆτ i), 1) τ i ˆτ i τ i ˆτ i Integrating gives us the utility function where τ [, 1] is an arbitrary constant. ˆτi U iˆτ i) = U iτ) + v t)s t) + v at)s a t) ) dt, 11) τ We next show that if the utility functions are given by 6), and if S τ i) and S a τ i) are non-decreasing in τ i, the mechanism s, t) is incentive comatible. Assume w.l.o.g. τ i > ˆτ i. It is U iτ i) U iˆτ i) = = τi ˆτ i τi v t)s t) + v at)s a t)dt ˆτ i S t)dv t) + τi ˆτ i S a t)dv at) S ˆτ i) v τ i) v ˆτ i)) + S a ˆτ i) v aτ i) v aˆτ i)), or equivalently, U iτ i) S ˆτ i) v τ i) v ˆτ i)) + S a ˆτ i) v aτ i) v aˆτ i)) + U iˆτ i) = S ˆτ i)v τ i) + S a ˆτ i)v aτ i) + T ˆτ i) = U iτ i, ˆτ i). Analogously we can derive that U iˆτ i, ˆτ i) U iˆτ i, τ i). A.2 Transformation The following transformation shows that assuming uniform distribution of tyes and arbitrary valuation functions is without loss of generality. Moreover, it hels to comrehend that the results of this aer are in line with CGK. Transformation A.1. Given a distribution function F : Θ := [θ, θ] [, 1] with continuous density fθ) > for all θ Θ, we can always comose the WTP and WTA function with a distribution function F, such that ˆv j := v j F : Θ C j, with τ := F θ) [, 1], θ Θ. We exloit integration by substitution which follows from the fundamental theorem of calculus. This is ossible, because F and v j are strictly monotonically) increasing and therefore bijective. Hence, v j F : Θ C j is also bijective with inverse F v j : C j Θ. For g : [, 1] R continuous and a, b Θ integration by substitution gives A.3 Existence Condition F b) F a) gt)dt = = b a b a g F ϑ)) fϑ)dϑ g F ϑ)) df ϑ). The existence condition from Theorem 3.1 equals the existence condition given in CGK if we assume v aτ) = v τ) =: vτ) and v F = id Θ. Then, the worst-off tye is given by θ i = F 1 2 ). To show the 16

17 equivalence we use the substitution rule: Define g 1 : [, 1] R, x 1 x)vx) and g 2 : [, 1] R, x xvx) which are continuous functions. The existence condition 4) reduces to With F θ) θ 2 2 vt)t dt vt)1 t)dt. 1 t)vt)dt = v F ) ϑ) 1 F ϑ)) df ϑ) = FF 1 2 )) F 1 2 ) FF 1 2 )) F 2 1 ) tvt)dt = ϑf ϑ) df ϑ), F θ) θ θ ϑ) 1 F ϑ)) df ϑ), F 1 2 ) it gives us the existence condition derived by CGK: 2 i=1 θ θi ϑ 1 F ϑ)) df ϑ) ) θ i ϑf ϑ) df ϑ). θ References Cramton, P., R. Gibbons, P. Klemerer Econometrica 553) Dissolving a Partnershi Efficiently. Fehr, D., R. Hakimov, D. Kübler The willingness to ay willingness to accet ga: A failed relication of lott and zeiler. Euroean Economic Review Fieseler, K., T. Kittsteiner, B. Moldovanu. 23. Partnershis, lemons, and efficient trade. Journal of Economic Theory 1132) Figueroa, N., V. Skreta Asymmetric artnershis. Economics Letters 1152) Galavotti, S., N. Muto, D. Oyama On efficient artnershi dissolution under ex ost individual rationality. Economic Theory 481) Green, J., J.-J. Laffont Characterization of satisfactory mechanisms for the revelation of references for ublic goods. Econometrica: Journal of the Econometric Society Holmstrom, B Groves scheme on restricted domains. Econometrica 475) Horowitz, J. K., K. E. McConnell. 22. A review of wta/wt studies. Journal of Environmental Economics and Management 443) Jehiel, P., A. Pauzner. 26. Partnershi dissolution with interdeendent values. The RAND Journal of Economics 371) Kahneman, D., A. Tversky Prosect theory: An analysis of decision under risk. Econometrica 472) Kőszegi, B., M. Rabin. 26. A model of reference-deendent references. The Quarterly Journal of Economics 1214) Krishna, V., M. Perry Efficient mechanism design. Discussion aer, Penn State University. 17

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