Dissoling (In)effectie Partnerships John L. Turner March 2012 Abstract This paper studies the problem of partnership dissolution in the context of asymmetric information. Past work shows that the initial share allocation, interdependence of partners aluations, and asymmetric control all affect the possibility of efficient dissolution. In this paper, I show, in a noel class of cooperatie partnerships characterized by ex ante interdependence of aluations, that effectieness is significantly more important than the initial share allocation. Intuitiely, as the effectieness of cooperation between partners (and thus partnership alue) increases, the gains from dissoling decrease but the informational rents remain constant, so efficient dissolution is more difficult to achiee. For sufficiently high effectieness, efficient dissolution is impossible for any initial share allocation. For sufficiently low effectieness, howeer, efficient dissolution is possible for all initial share allocations. The possibility of efficient bargaining depends on the initial share allocation only for moderately effectie partnerships. JEL Codes: C72, D82, L14 Keywords: Mechanism design, trading, dissolution, partnerships. Department of Economics, Uniersity of Georgia, Brooks Hall 5th Floor, Athens, GA 30602-6254. tel: 706-542-3682, e-mail:jlturner@terry.uga.edu. Thanks to Emanuel Ornelas for extremely helpful comments.
1. Introduction The theoretical economics literature on dissoling partnerships in the presence of asymmetric information has helped to explain ownership patterns of closely-held corporations and joint entures. In a symmetric, independent-priate-alue (IPV) model, Cramton, Gibbons and Klemperer (1987) show that an equal-shares partnership can always be dissoled efficiently with an auction, which suggests that there are organizational economies to equal ownership. Segal and Whinston (2011) proe similar results in a more general setting. If the decision space is conex and utility is conex in the number of shares allocated, then partnerships with initial shares that are the expected efficient allocation which is equal-shares under symmetric priate information always permit efficient bargaining. 1 This efficiency property is consistent with the work of Hauswald and Hege (2003), who find that around two thirds of two-parent US joint entures formed during 1985-2000 had 50/50 ownership. Howeer, some important puzzles remain unsoled. While the results of Cramton, Gibbons and Klemperer (1987) suggest that equal-shares partnerships can and should be dissoled using an efficient mechanism such as an auction, partnerships often persist for long periods of time. In addition, lawyers frequently recommend that partnership agreements specify simple mechanisms for dissolution (such as the Texas Shootout ) that are inefficient in the IPV setting. 2 Clearly, the IPV setting does not capture all key features of partnerships. In this paper, I add realistic characteristics of partnerships, to the IPV model, that are partly motiated by the theoretical literature focusing on the formation and functioning of partnerships (Holm- 1 For these results to hold, efficient decision rules must also satisfy cross congruence. This condition is always satisfied in the IPV setting. Intuitiely, Segal and Whinston (2011) show that an efficient allocation yields higher expected utility than a random allocation that has the same distribution as the equilibrium (efficient) allocation. When partners are symmetric and utility is conex in the number of shares awarded, this random allocation yields (by Jensen s inequality) higher utility than a deterministic initial allocation with aerage numbers of shares for all partners (the expected efficient allocation) under the status quo. This guarantees the possibility of efficient dissolution. See Proposition 1 (p. 111-12) and Proposition 2 (p. 115). 2 For example, it is considered malpractice among some legal scholars and practitioners for a lawyer superising a partnership formation not to recommend the simplest ersion of the Texas Shootout (Brooks, Landeo and Spier 2010). Under the Texas Shootout, one partner proposes a price for the firm. The other partner then can buy or sell at that price, but must choose one of those options. It is inefficient wheneer the proposer s price prompts the chooser to buy when his aluation is not highest, or sell when his aluation is not lowest (McAfee 1992). De Frutos and Kittsteiner (2008) show that a more elaborate ersion of this mechanism can be efficient. 1
ström 1982; Farrell and Scotchmer 1988; Legros and Matthews 1993; Lein and Tadelis 2005). 3 Specifically, the performance of a partnership depends on the characteristics of multiple owners, while the performance of a sole proprietorship does not. As a result, the priate per-share alue of ownership should typically change when there is a change from joint to single ownership. This is not captured in the IPV setting, where indiidual per-share aluations are the same under joint and single ownership. To formalize these ideas, I introduce and analyze a class of cooperatie partnerships. Let each of two partners hae priate information, a signal, about the quality of their best ideas for running the firm. Let the firm s alue, when each partner owns some shares, be a conex combination of the two signals. When alue is more heaily weighted toward the highest signal, I say the partnership is more effectie at generating alue. Let the partnership be sufficiently mature that the partners know the partnership s effectieness, i.e., how well they would work together should the partnership remain. Gien the possibility that they may dissole the partnership, the partners do not share each other s plans for the future and do not know each other s signals. Let the firm s alue be a function of only the owner s signal under single ownership. Efficient dissolution is Pareto optimal. Consistent with the IPV model, I show that equal-shares cooperatie partnerships are the easiest to dissole, in that if there is a non-empty set of initial share allocations such that the partnership can be dissoled efficiently, then the equal-shares partnership is in the set. For sufficiently high effectieness, howeer, there is no initial share allocation such that the partnership can be dissoled efficiently. Intuitiely, when partners routinely choose and execute the best ideas, there is little to gain by dissoling the partnership and it is difficult to dissole efficiently. On the other hand, when effectieness is sufficiently low, efficient dissolution is possible for any initial share allocation. The basic results from the IPV model, that the possibility of efficient dissolution depends on the initial share allocation, obtain only for moderately effectie partnerships. Hence, the equal-shares partnership retains some efficiency properties in the cooperatie setting, but the initial share allocation is relatiely unimportant for efficient bargaining. 3 This literature grew out of the analysis of team production problems by Holmström (1982). It focuses on incenties of partners to contribute to production and the resulting inefficiencies (see also Legros and Matthews 1993), the characteristics of partners who choose to join forces (Farrell and Scotchmer 1988) and the determinants of when people prefer the partnership to the corporate form (Lein and Tadelis 2005). 2
The key technical feature of my model is that the firm s alue changes discretely as ownership is allocated entirely to one partner. Because of this, total surplus is a non-conex function of the share allocation. As a result, the possibility results of Segal and Whinston (2011) do not apply. More intuitiely, status quo payoffs in the cooperatie partnership setting are functions of the partnership s alue. In a highly-effectie partnership, expected status quo payoffs are worth more, collectiely, than a partner s aerage signal. In contrast, status quo payoffs in the IPV setting are like payoffs in a lottery that picks one partner as a sole proprietor. In expectation, such status quo payoffs are collectiely worth the aerage signal. To see this, consider the mid-1980s proposal made by the US Federal Communications Commission (FCC) to allocate licences for cellular telephone franchises by lottery. Motiated, in part, by the question of how efficient this mechanism would be, Cramton, Gibbons and Klemperer (1987) instead argued that it would be efficient for all possible winners of the FCC lottery to pool their chances, win for sure, then auction off the licenses. In this setting, partnering amounts to the formation of the pool and the commitment to the rules of the dissolution mechanism. Partners outside options reflect payoffs that would occur under the lottery, not under a partnership. Hence, the requirement of indiidually rational participation in the cooperatie partnership case may set a higher bar than in the IPV case. As a result, cooperatie partnerships cannot always be dissoled efficiently and may tend to persist. Perhaps more importantly, people forming partnerships must be aware that it may be impossible to contract for efficient dissolution. Future work should consider this possibility in explaining the popularity of real-world dissolution mechanisms. 4 Moldoanu (2002), Fieseler, Kittsteiner and Moldoanu (2003) and Jehiel and Pauzner (2006) model aluations as interdependent one partner s priate information may affect another partner s aluation. Interdependence strongly affects which partnerships may be dissoled efficiently. Fieseler, Kittsteiner and Moldoanu (2003), in particular, show that efficient dissolution may be impossible for any initial share allocation. 5 Howeer, changes 4 I discuss this further in the conclusion. 5 Jehiel and Pauzner focus on cases where only one partner is informed about the alue of the co-owned asset. In such a setting, they identify a wide class of situations where efficient dissolution is not achieable. Fieseler, Kittsteiner and Moldoanu (2003) and Moldoanu (2002) find that, when information is ex ante 3
in the functions that determine the interdependence in these models affect aluations under sole ownership and aluations under the partnership in the same way. Hence, the equal-share status quo allocation in these settings remains a lottery payoff for all partners. In assuming that each partner s independent priate signal represents the firm s profit (and his own payoff) under his sole ownership and that both partners signals determine the profit under a jointly-owned partnership (so the partners payoffs are interdependent), cooperatie partnerships are a hybrid of the IPV case and the interdependent-aluations case. This hybrid case has already yielded results quite distinct from the related literature. 6 For instance, Ornelas and Turner (2007) specify one partner s sole-proprietor aluation to be the same as the status quo alue of the partnership and the other partner s aluation to be different (we term this arrangement a silent partnership ), and show that efficient dissolution is impossible if there are only two partners. 2. The Model 2.1. Preliminaries Consider a partnership of two risk-neutral people, facing no wealth constraints, that owns a firm. 7 Partner 1 is endowed with share r 1 (0, 1) and partner 2 owns share r 2 = 1 r 1. symmetric, a partnership is more difficult to dissole if a gien partner s aluation is increasing in the types of the other partners, while the opposite is true if the partner s aluation is decreasing in the types of the other partners. In the former case, the equal-shares partnership may not be dissolable efficiently, while in the latter case een efficient bilateral exchange may be possible. 6 Related work studies the properties of specific mechanisms. Seeral authors study the reenue properties of particular auction mechanisms in the IPV setting (Engelbrecht-Wiggans 1994; Singh 1998; Goeree, Maasland, Onderstal and Turner 2005; Lengwiler and Wolfstetter 2005; Engers and McManus 2007), as reenue equialence does not generally hold. Auctions are generally efficient when the initial share allocation is symmetric, howeer. In an incomplete contracts model of bilateral exchange, Matouschek (2004) shows that the integration decision of two firms influences continuation or disagreement payoffs, and that the optimal organizational structure balances a greater likelihood of trade ersus worse disagreement payoffs when trade breaks down. 7 These assumptions yield the familiar specification of utility that is linear in earnings from firm ownership and wealth, and where unlimited side payments are permitted. In the absence of asymmetric information, there would be no transaction costs, so the Coase Theorem would imply the possibility of efficient bargaining. Hence, this specification isolates cases where efficient bargaining may be derailed by (transaction) costs that emerge from the demands of running an incentie compatible mechanism. With risk-aerse partners or wealth constraints, efficient dissolution would be more difficult to achiee. 4
Let r (r 1, r 2 ). Let π 0 be the net return resulting from actions that (from experience) both partners agree should be taken. Each partner i has priate information i [, ], drawn independently from distribution F, which is common knowledge and has positie continuous density f. Each i gies the net returns to partner i s ideas about the best new plans to implement going forward. Denoting ( 1, 2 ), let the firm s random profit be gien by π(, r 1, r 2 ). Under joint ownership, profit depends on both partners. Define ṽ max{ 1, 2 } and ṽ min{ 1, 2 }, and we hae the following. Assumption 1. In a Cooperatie Partnership r, α, F, the firm earns random profit π(, r 1, r 2 ) = π 0 + αṽ + (1 α)ṽ. I term α the effectieness of the partnership. 8 It is common knowledge. Intuitiely, it gies the probability that the best ideas would be implemented under joint ownership. Note that α can also be seen as measuring the negatie effects of team coordination problems and moral hazard (i.e., low-α partnerships suffer more). When α = 1, the partnership achiees the highest possible profit of π(, r 1, r 2 ) = π 0 + ṽ. Since the common knowledge part of profit is realized regardless of whether the partnership remains intact, it is without loss of generality to henceforth set π 0 = 0. 9 Note that while the partners know exactly how well they would work together, they do not know the alue of the other partner s ideas for the future. This is natural in a setting where the partnership may ultimately be dissoled. Should partner 1 acquire full ownership (r 1 = 1), profit would be π(, 1, 0) = 1 + π 0. Under partner 2 s full ownership, profit would be π(, 0, 1) = 2 +π 0. Importantly, for α > 0, ṽ < π(, r 1, r 2 ) ṽ, so profit is not conex in the allocation. Appealing to the reelation principle, I study a direct reelation game where partners 8 We restrict attention to two partners to fix ideas around our main points about the importance of effectieness. Fully extending the analysis to the n-partner case introduces a wide array of possible modeling structures that could capture effectieness, but there is not a single natural analog to α as a measure of effectieness. For example, one could justify making expected profit a conex combination of the highest and lowest aluations or a combination of the highest and second-highest aluations. 9 Implicitly, we assume that the partnership was, at some point in the past, necessary to realize π 0. Thus, for the initial formation of the partnership to be efficient, one would need π 0 > 0. For simplicity, our model treats the partnership s initial formation as a fait accompli and sets π 0 = 0 to economize on notation. 5
report types simultaneously, then a mechanism allocates shares s() = {s 1, s 2 } and determines transfer payments t() = {t 1, t 2 } to the partners. I refer to s, t as a trading mechanism. If truthful reelation is a Bayesian-Nash equilibrium under mechanism s, t, then it is incentie compatible. A mechanism is (ex post) efficient if the firm is allocated to the partner with the highest type with probability 1, namely, 1 if i = ṽ s i () = 0 if i < ṽ. A mechanism is (ex ante) budget balanced if the mechanism designer does not expect to pay a positie subsidy to the partners, i.e., E{ t i ()} 0. Under an efficient mechanism s, t, partner i obtains utility i s i + t i and, conditional on i, expects to receie shares and transfers S i ( i ) E i {s i ()} and T i ( i ) E i {t i ()}, respectiely, where E i { } denotes the expectation operator with respect to i. His interim expected utility from such a mechanism is therefore M i ( i ) = i S i ( i ) + T i ( i ). By contrast, partner i s expected utility under joint ownership is his share r i times Π i ( i ) = E i {αmax{ 1, 2 } + (1 α)min{ 1, 2 }} (1) = α [ i F ( i ) + i udf (u) ] + (1 α) [ i (1 F ( i )) + i udf (u)]. I define the interim expected net utility from dissoling instead of maintaining the partnership as U i ( i ) M i ( i ) r i Π i ( i ). The worst-off type of partner i in s, t earns the least expected net utility, i.e., U i ( i ) U i ( i ) for all i. Formally, I call a mechanism interim indiidually rational if expected net utility is nonnegatie, i.e., U i ( i ) 0, for all types of both partners. 10 2.2. Efficient Dissolution The technical features of this problem fall within the class analyzed by Makowski and Mezzetti (1994) and Williams (1999). 11 Hence, without loss of generality, I restrict attention 10 Where there is no confusion, we drop the modifier interim. 11 Krishna and Perry (2000) proe results similar to Williams for this class. 6
to Vickrey-Clarke-Groes (VCG) mechanisms. This class specifies transfers k i if s i () = 1 t i () = ṽ k i if s i () = 0, (2) where k i is a real number. 12 VCG mechanisms are incentie compatible, and any two of them yield, up to a constant, the same expected transfers. Gien incentie compatibility, I can use the enelope theorem to derie U i ( i ) = U i ( i ) + i i [ S i (u) r i ( dπi (x) dx ) x=u ] du for all i. (3) I now identify the worst-off types of partners. The first-order condition for (3) is ( ) S i (i dπi (x) ) = r i. (4) dx x=i The efficient allocation rule implies S i ( i ) = F ( i ). Hence, condition (4) reduces to F ( i ) = r i(1 α) 1 + r i (1 2α). (5) Since expected net utility is conex in i, this fully characterizes i. A partnership can be dissoled efficiently if there exists an ex post efficient mechanism s, t that is incentie compatible, indiidually rational and ex ante budget balanced. The following proposition gies the conditions goerning whether a cooperatie partnership can be dissoled efficiently. 13 See the appendix for a proof. 14 Proposition 1. A partnership r, α, F can be dissoled efficiently if and only if {[ 2 ] } u[2f (u)]df (u) i F (i ) + udf (u) r i Π i ( i = 1 i i ), (6) 12 This need not be a constant term; its expectation conditional on agent i s reported type is constant. 13 Makowski and Mezzetti (1994) also allow for a status quo alue for the partnership that depends on all types, so this proposition can be iewed as an application of their Theorem 3.1. Note that this is the same condition that would emerge if ex post budget balance were required. Using the technique of d Aspremont and Gérard-Varet (1979), it is straightforward to specify the constant terms in (2) so that ex post budget balance is satisfied. 14 All proofs are in the appendix. 7
where i satisfies condition (5) for i {1, 2}. To see the intuition, suppose that a risk-neutral broker charges entry fees to both partners, who surrender their shares and submit to a VCG mechanism s rules. To be incentie compatible, a VCG mechanism sets its transfers so that it promises the full ex post alue of the partnership, ṽ, to each of the two partners. Since the mechanism produces that alue only once, it runs a budget deficit, of expected size E[ṽ] = u[2f (u)]df (u). Efficient dissolution is possible if and only if the designer can coer this cost through entry fees, while ensuring full participation. Each partner is willing to pay an entry fee because he expects a payoff to participating. The term in brackets is what a partner with alue i expects ṽ to be. The term r i Π i ( i ) is the expected alue of the shares that this partner surrenders in participating. The difference between these terms is his expected net payoff to participating, i.e., the highest entry fee he will pay. The sum on the right-hand side of (6) is therefore the most the mechanism designer can charge in entry fees while ensuring the participation of the worst-off types. 3. Characterization To simplify the exposition, I first note that it is without loss of generality to call partner 1 s initial share r, partner 2 s share 1 r, and define the net surplus from dissoling, {[ 2 ] V (r, α) = udf (u) + i F (i ) r i Π i (i ) i = 1 i } F (u)udf (u), (7) where 1, 2, Π 1 and Π 2 depend on r and α. The condition for the possibility of efficient dissolution in Proposition 1 is equialent to V (r, α) 0. To compare the results here with those from the IPV case, I define its corresponding net surplus for two partners similarly: { 2 V IP V (r) = udf (u) i = 1 where 1 = F 1 (r) and 2 = F 1 (1 r). The following definition also proes useful. 8 i } F (u)udf (u), (8)
Definition 1. Partnership r, α, F is easier to dissole efficiently than partnership r, α, F if and only if V (r, α) > V (r, α ). Intuitiely, a partnership is easier to dissole if the amount a broker is willing to pay to design the mechanism is higher. If V (r, α) > V (r, α ), then it is neer the case that r, α, F can be dissoled efficiently while r, α, F cannot. If neither can be dissoled efficiently, then a lower outside subsidy is needed to dissole r, α, F. I now state the key characterization results. First, holding the partnership s effectieness constant, net surplus changes with the initial share allocation. The following result shows that net surplus is maximized for the equal-shares partnership. Proposition 2. For any α, net surplus is maximized at the equal-shares partnership, r = 1. 2 The set of initial share allocations for which a partnership can be dissoled efficiently is either empty or is a symmetric subset of the unit interal, centered around the equal-shares partnership. Intuitiely, an increase in r lowers the net utility from participating for partner 1 (who gies up more ownership to participate) at rate Π 1 (1) and raises the net utility from participation for partner 2 (who gies up less to participate) at rate Π 2 (2). When r < 1, we hae 2 2 > 1, so that Π 2 (2) > Π 1 (1). Hence, an increase in r helps net surplus, as partner 2 s net utility increases by more than partner 1 s net utility decreases. The opposite is true when r > 1. 2 In contrast to the IPV case, howeer, the equal-shares partnership is not always dissolable and it is possible that all partnerships may be dissolable. The key determining factor is α, the effectieness of the partnership. Proposition 3. If α = 1, the partnership cannot be dissoled efficiently for any r. Partnership r, α, F is easier to dissole than partnership r, α, F if and only if α < α. If α = 0, the partnership can be dissoled efficiently for all r. Taken together, Propositions 2 and 3 characterize three different categories of partnerships based on α. Define α H as the highest leel of effectieness such that the equal-shares cooperatie partnership can be dissoled efficiently and α L as the highest leel of effectieness 9
such that any cooperatie partnership can be dissoled efficiently. Propositions 2 and 3 proe that 0 < α L < α H < 1, which yields an intuitie partition. For high effectieness α (α H, 1], there is no initial share allocation for which efficient dissolution is possible. 15 For moderate effectieness α (α L, α H ], the equal-shares partnership can be dissoled, but others cannot be, as in the IPV case. For low effectieness α [0, α L ], efficient bargaining is possible for any initial share allocation. Propositions 2 and 3 also imply that, for any r, there exists some α(r) [α L, α H ] such that V (r, α) < 0 if and only if α α(r), and that α(r) is smaller for more extreme r. When α = 1, dissoling introduces strictly positie informational rents but yields no gains from trade. Hence, efficient bargaining is impossible for all initial share allocations. To get efficient bargaining, the minimum positie outside subsidy is V (r, 1) = F (u)[f (u) 1]du, the same as the subsidy needed to implement efficient bilateral exchange for symmetric distributions of types (Myerson and Satterthwaite 1983, p. 272). The worst-off types in the α = 1 case equal regardless of r, so these types do not expect any surplus from the mechanism. Since the VCG mechanism is the same here as under bilateral exchange (except for constant terms), expected informational rents are the same. Now consider the second part of Proposition 3. As α falls, the status quo alue of the partnership falls, so the worst-off types expected gains from trade from dissoling increase. A broker can charge the partners more to participate in the mechanism, while still obtaining full participation. The informational rents from the mechanism are unchanged, howeer, because α does not affect incentie compatibility. Hence, V increases. Now let α = 0. The cooperatie partnership is more difficult to dissole as r becomes more extreme. 16 In the limit as r approaches 1, net surplus approaches the same leel as the equal-shares partnership in the IPV setting. This partnership can always be dissoled 15 The first sentence of Proposition 3 extends to the case where the highest of n partners aluations determines the profit of the firm. See Turner (2011, p. 14) for details. 16 This comes out of the proof of Proposition 2. V (r, 0) decreases as r approaches 0 or 1. 10
1 0.9 0.8 0.7 0.6 alpha 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r Figure 1: Efficient Dissolution with Uniform Distributions. efficiently. 17 That is { lim V (r, 0) = 2 udf (u) r 1 F 1 ( 1 2 ) } ( ) 1 F (u)udf (u) = V IP V > 0. 2 4. An Example Let the partners types be distributed uniformly on [0, 1]. Then the worst-off types are i = r i(1 α) 1 + r i (1 2α), and, letting r 1 = r, the condition for efficient dissolution is 2 3(( 1) 2 + ( 2) 2 ) 6 α 2 + (2α 1)(r ( 1) 2 2 + (1 r) ( 2) 2 2 ) +(1 α)(r( 1) + (1 r)( 2)) ( 1) 2 ( 2) 2. (9) 17 Note that if the initial share allocation is r = 0 or r = 1, so that there is no partnership under the status quo, the model is identical to Myerson and Satterthwaite (1983), so that efficient bargaining is impossible. 11
0.1 Cooperatie Case IPV Case 0 0.1 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r Figure 2: Comparing V (r, 1 2 ) to V IP V (r). The set of efficiently dissolable partnerships are those inside the solid lines in Figure 1. For high effectieness α (.567, 1] (approximately), no partnerships are dissolable. The moderate leels of effectieness are α ( 1,.567], while the low range is α [0, 1 ]. Hence, 3 3 the range of α where the initial share allocation matters is relatiely small. When α = 1, the firm s expected status quo alue, just the aerage of the maximum and 2 minimum types, equals 1, the same as the expected status quo alue of a two-person IPV 2 partnership with uniform types. 18 Both V (r, 1 2 ) and V IP V (r) are plotted in Figure 2. The set of dissolable partnerships is the same in both cases. 19 The moderately effectie partnership α = 1 2 is, essentially, equialent to an IPV partnership from a dissolability standpoint. The set of dissolable partnerships is not generally the same in these two cases. distributions of the form F () = β, β > 0, I find that the range of the initial share allocation, for which efficient dissolution is possible, is larger in the cooperatie (α = 1 ) case than in the 2 IPV case when β > 1, and smaller otherwise. The equal-shares partnership is not dissolable in the cooperatie case when β <.62 (approximately). 18 Note this expected status quo alue also obtains in the case of interdependent aluations where aluations are additiely separable functions of independent signals. Fieseler, Kittsteiner and Moldoanu (2003, section 4) use this specification, i.e., i (θ 1,..., θ n ) = g(θ i ) + j i h(θ j) with θ i θ j. 19 This set, defined by r 2 + (1 r) 2 2 3, coers all r [.211,.789] (approximately). For 12
5. Conclusion My results suggest that settings where the first-best allocation is not achieable are more prealent than originally thought. This may help to explain why partnerships persist. Moreoer, it motiates future work studying why mechanisms that are inefficient in simple static settings are nonetheless popular in partnership agreements. And in practice, there are interesting dynamics. In a typical partnership agreement, partners know neither the date when they will dissole nor which partner will attempt to buy out the other. In an IPV setting, these dynamics are irreleant, because it is always adantageous to dissole immediately. With a cooperatie partnership structure, howeer, immediate dissolution is adantageous only with a relatiely ineffectie partnership. Hence, the cooperatie partnership model could be a useful laboratory for studying dynamic decisions to dissole. 20 My results also reinforce the iew of Moldoanu (2002) that the construction of (secondbest) incentie efficient mechanisms and the identification of well-performing mechanisms whose rules do not depend on aluation functions and distributions are crucially important tasks for the mechanism design literature. Kittsteiner (2003) and Chien (2007), for example, show that the efficiency of an auction differs from the (much more complicated) second-best mechanism in an enironment where the first best is not possible. 21 I look forward to further progress in the area. 20 For example, if a Texas Shootout were used to dissole a cooperatie partnership, then a partner with a ery low i would not want to propose a price. 21 Chien (2007) studies the case where interdependence of aluations follows additiely-separable functions of priate single-dimensional signals [preiously studied by Fieseler, Kittsteiner and Moldoanu (2003)], and shows that when there are two partners and symmetric distributions, second-best mechanisms allocate the partnership to the partner with the highest ex ante ownership more often. Jehiel and Pauzner (2006) also study second-best mechanisms. 13
Appendix Proof of Proposition 1. Williams (1999, Theorem 3, p. 166) shows that, for n = 2, an interim indiidually rational, ex ante budget balanced mechanism exists if and only if the expected total alue under efficient bargaining is no greater than the sum of expected net utilities of the worst-off types in the basic Groes mechanism, where k i = 0 for all i for the transfers defined in (2). This yields the condition 2 E(ṽ) {i F (i ) + E i [ṽ1(i < ṽ)] r i Π i (i )}, i = 1 where 1( ) is the indicator function. This expression can be rewritten as which is (6). QED {[ 2 ] } 2uF (u)df (u) i F (i ) + udf (u) r i Π i ( i = 1 i i ), Proof of Proposition 2 It is clear that V (r, α) = V (1 r, α), as partners 1 and 2 are ex ante identical except for their shares. It remains to show that V (r, α) is maximized at r = 1 2. Differentiating (7), and doing some algebra, we hae dv dr = Π 2( 2) Π 1 ( 1), where the algebra is greatly simplified by the enelope theorem (using the definition of i ). Clearly, dv dr is positie if 2 > 1, zero if 2 = 1, and negatie if 2 < 1. These three cases correspond, respectiely, to the cases of r < 1, r = 1, and r > 1. Hence, V is quasiconcae 2 2 2 in r for r (0, 1) and is maximized at r = 1. If V ( 1, α) < 0, then the set of dissolable 2 2 partnerships is empty. If V ( 1 2, α) > 0, this set is clearly a symmetric interal about r = 1 2. QED Proof of Proposition 3 When α = 1, the worst-off types are the lowest possible types: i =. Thus, we can write 2 V (r, 1) = i = 1 2 u[1 F (u)]df (u) r i udf (u). 14 i = 1
Simple algebra and integration by parts then yields V (r, 1) = F (u)[f (u) 1]du < 0. For the second sentence of the proposition, it suffices to show that V is strictly decreasing in α. Differentiating, we find dv dα = [ 2 ] i r i (i u)df (u) + (u i = 1 i i )df (u) < 0. For the third sentence of the proposition, let α = 0 and consider the limit as r approaches 1. Then 1 approaches F 1 ( 1 2 ) and 2 approaches. We hae lim V (r, 0) = r 1 { Simplifying, we hae F 1 ( 1 2 ) udf (u) F (u)udf (u) + { lim V (r, 0) = 2 udf (u) r 1 F 1 ( 1 2 ) u[1 F (u)]df (u) F (u)udf (u) } F 1 ( 1 2 ) udf (u) }. Noting that the right-hand side equals V IP V ( 1), the result lim 2 r 1 V (r, 0) > 0 follows from the proof of Proposition 1 by Cramton, Gibbons and Klemperer (1987), applied to the case of 2 partners. By Proposition 2, we hae V (r, 0) lim r 1 V (r, 0) > 0, completing the proof. QED References Brooks, Richard, Claudia Landeo and Kathryn Spier. Trigger Happy or Gun Shy? Dissoling Common-Value Partnerships with Texas Shootouts, RAND Journal of Economics 41, 2010, 649-73. Chien, Hung-Ken. Incentie Efficient Mechanisms for Partnership Dissolution, manuscript, 2007. 15
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