On surplus-sharing in partnerships

Manuscript version: Author’s Accepted Manuscript The version presented in WRAP is the author’s accepted manuscript and may differ from the published version or Version of Record. Persistent WRAP URL: http://wrap.warwick.ac.uk/115148 How to cite: Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it. Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available. Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher’s statement: Please refer to the repository item page, publisher’s statement section, for further information. For more information, please contact the WRAP Team at: [email protected]. warwick.ac.uk/lib-publications Manuscript Click here to download Manuscript: PartnershipR2-PAPER.pdf On Surplus-Sharing in Partnerships∗ Özgür Kıbrıs† Arzu Kıbrıs Sabancı University September 1, 2015 Abstract For investment or professional service partnerships (in general, for partnerships where measures of the partners’ contributions are available), we consider a family of partnership agreements commonly used in real life. They allocate a fixed fraction of the surplus equally and the remains, proportional to contributions; and they allow this fraction to depend on whether the surplus is positive or negative. We analyze the implications of such partnership agreements on (i) whether the partnership forms in the first place, and if it does, (ii) the partners’ contributions as well as (iii) their welfare. We then inquire which partnership agreements are productively e¢cient (i.e. maximizes the partners’ total contributions) and which are socially e¢cient, (i.e. maximizes the partners’ social welfare as formulated by the two seminal measures of egalitarianism and utilitarianism). Keywords: partnership agreements, proportional surplus shares, equal surplus shares, productive e¢ciency, egalitarian social welfare, utilitarian social welfare. JEL classification codes: C72, D33, D86, J33, L29 ∗ We would like to thank Jordi Brandts, Benjamin Brooks, İsa Hafalır, Kevin Hasker, Colin Raymond and seminar participants at Bilkent University, İstanbul Technical University, TED University, The 2014 Conference of the Society for Social Choice and Welfare, and The 2014 Murat Sertel Workshop for comments and suggestions. We would also like to thank the associate editor and two anonymous reviewers of this journal for their comments and suggestions. This research was funded in part by the Scientific and Technological Research Council of Turkey (TUBITAK) under grant 114K060. All errors are ours. † Corresponding author: Faculty of Arts and Social Sciences, Sabanci University, 34956, Istanbul, Turkey. E-mail: [email protected] Tel: +90-216-483-9267 Fax: +90-216-483-9250 1 1 Introduction Imagine a group of lawyers forming a partnership or a group of investors partnering up to undertake a financial endeavor. As a first step, the partners need to agree on (i) how to allocate positive surplus in case of profits and (ii) how to allocate negative surplus in case of losses. This is a very important choice for the partnership since it in turn a§ects the partners’ contributions as well as their welfare from the partnership. In this paper, we focus on the implications of this choice. More specifically, we analyze the (dis)advantages of some partnership agreements that are commonly used in real life professional service partnerships (such as in law, accounting, medicine, or real estate) as well as in investment partnerships. Farrell and Scotchmer (1988) and Lang and Gordon (1995) describe three basic systems law partnerships use to allocate surplus. In the first one, called the lock-step system, all partners of the same seniority receive the same surplus share. The lock-step system is used by most law firms with 2 or 3 partners, which approximately constitute 2/3 of all law firms in the US, though less than half the lawyers (also see Curran, 1985; Flood, 1985). In the second system, called the objective performance-related system, an explicit formula (using variables such as the number of hours billed, cases won, or business brought in) is used to determine each partner’s contribution. The partners’ surplus shares are then determined in proportion to these contributions. The third basic system, called the subjective performance-related system, is only di§erent from the second in the sense that it is now the firm’s founders who evaluate each partner’s contribution. The above case of law partnerships demonstrates the two most common surplus sharing methods in real life: equal (or in general, fixed) shares versus shares proportional to contributions (also called the piece-rate). Gaynor and Pauly (1990) mention that it is also common in professional service partnerships to combine these two methods by allocating a fixed fraction of the surplus equally and allocating the rest proportional to contributions as a bonus. A partnership agreement can additionally fix di§erent fractions in cases of positive and negative surplus. 1 The following is an example of such a partnership agreement: Partners Johnson and Smith agree that (i) if their partnership makes a positive surplus, 60% of this positive surplus will be allocated equally while the remaining 40% will be allocated in proportion to each partner’s contribution and (ii) if their partnership 1 Legal regulations on partnerships recognize the usage of di§erent fixing di§erent fractions) in cases of positive and negative surplus. tice managers’ medical guide practices in to co-ownership Australia, agreements, prepared by partnerships, “McMasters’ and Training surplus sharing associateships”, Pty Ltd”, a available http://www.medicalpracticemanagement.com.au/practice_manager_s_guides/guide5/guide_5. 2 rules (as in For example see “The pracguide for online at makes a negative surplus, all of this negative surplus will be allocated equally. The framework of our study is as follows. First, we take a partnership agreement as a pair of surplus allocation rules (used for positive and negative surplus respectively), focusing on the class of rules discussed in the previous paragraph. Second, we analyze environments where measures of the partners’ contributions are available. As already exemplified for law partnerships, such measures are commonly used in professional service partnerships. Similarly, partners’ monetary contributions are routinely used to allocate surplus in investment partnerships. Third and last, we assume that there is a stochastic component to the success of the partnership. Whether the partnership makes positive or negative surplus depends on some factors external to it, such as the state of the economy or the performance of the competitors. Several previous studies on partnerships make a similar assumption (e.g. Morrison and Wilhelm (2004), Comino, Nicolo and Tedeschi (2010), Li and Wolfstetter (2010)). In the confines of this framework, we model a simple “partnership game” and using it, we analyze the implications of a partnership agreement on (i) whether the partnership forms in the first place, and if it does, (ii) the partners’ contribution choices as well as (iii) their resulting welfare. Armed with these answers, we then inquire which partnership agreements are productively e¢cient (Gaynor and Pauly, 1990), that is, maximize the partners’ total contributions. We also inquire which partnership agreements are socially e¢cient, that is, maximize the partners’ social welfare as formulated by the two seminal measures of egalitarianism and utilitarianism. Overall, we observe a trade-o§ between maximizing total contributions and social welfare. To maximize total contributions, two “opposite” surplus allocation rules need to be used in cases of positive and negative surplus. However, (i) maximizing egalitarian social welfare requires choosing the same surplus-allocation rule in cases of both positive and negative surplus, and furthermore (ii) a numerical analysis detailed in Section 3.4 obtains a similar result for utilitarian social welfare on a significant part of our parameter space. In the Conclusion, we present a short discussion of the mechanism behind this trade-o§. In many countries, legal regulations include a partnership act, that is, a statutory agreement that applies to any partnership that does not have a written agreement. This default agreement typically allocates both positive and negative surplus equally. Also, if the partners have only specified the surplus-sharing rule to be used in case of positive or negative surplus, the legal default is that the same surplus-sharing rule is used in the other case as well. Our analysis thus shows that the state partnership acts have picked the welfare side of the trade-o§ mentioned in the previous paragraph. The paper is organized as follows. In Subsection 1.1, we discuss the related literature. In Section 3 2, we present the model. Section 3 contains our findings. In Subsection 3.1, we analyze acceptable partnership agreements (which we argue to be intimately linked to partnership formation). In Subsection 3.2, we characterize the equilibrium contributions in a formed partnership. In Subsection 3.3, we compare partnership agreements in terms of the total contributions and in Section 3.4, we compare them in terms of individual and social welfare. We summarize our findings and conclude in Section 4. The proofs are relegated to Section 5. 1.1 Literature There are two strands of theoretical literature related to our paper. The first follows the seminal papers by Alchian and Demsetz (1972) and Holmström (1982) to discuss the design of incentives in partnerships where the partners’ contributions are not observable (and thus, contribution-sensitive sharing schemes like proportionality are not available). In contrast to Alchian and Demsetz (1972) who argue that e¢ciency can only be restored by bringing in a principle who monitors the agents, Holmström (1982) shows that group incentives can remove the free-rider problem.2 The following literature focuses on the same question under alternative assumptions. Kandel and Lazear (1992) analyze the e§ect of peer pressure, Legros and Mathews (1993) analyze the e§ect of limited liability, Miller (1997) and Strausz (1999) analyze cases where a partner can observe the e§ort exerted by a subset of other partners, and Morrison and Wilhelm (2004) discuss moral hazard problems associated with intergenerational transfer of human capital. Hart and Holmström (2010) and Hart (2011) adopt the “contracts as reference points” approach to discuss shading and e¢cient partnership contracts. Farrell and Scotchmer (1988) analyze the e¢ciency costs of equal-sharing in a theoretical model of partnership formation. The above literature focuses on a stylized asymmetric information environment where contributionsensitive surplus allocation rules such as the proportional are not available; the common intuition being that if informational constraints permitted it, proportional surplus sharing would solve incentive problems. We contribute to this literature by providing a formal discussion of this intuition in an environment where there is possibility of negative as well as positive surplus. Our results show that (i) a move towards more proportional surplus-shares does not necessarily increase a partner’s contributions (e.g. see figures 1 and 2) and (ii) by using di§erent surplus sharing rules 2 While we work under di§erent informational assumptions, Holmström’s question is similar to this paper. Quoting (pg 326): “The question is whether there is a way of fully allocating the joint outcome so that the resulting noncooperative game among the agents has a Pareto optimal Nash equilibrium.” Holmström shows that the free rider problem can be solved as follows. One sets an output objective (by utilizing the observable information about the agents’ costs of e§ort). If it is not met, all partners receive zero as punishment. Otherwise, they share the produced value. 4 in cases of positive and negative surplus, a partnership can improve total contributions over simple proportionality. The second strand of theoretical literature related to our paper is on axiomatic resource allocation. The partnership agreements that we consider are based on two principles (proportional versus equal sharing) central in the surplus sharing literature. See O’Neill (1982), Aumann and Maschler (1985) and the following literature (reviewed in Thomson 2003 and 2008) for axiomatic studies on allocating negative surplus (referred to as claims or bankruptcy problems by this literature). On the other hand, Moulin (1987) and the following literature provides an axiomatic study for positive surplus. There also is a smaller literature that covers both cases simultaneously. For example, Chun (1988) proposes characterizations of classes of rules that mix the proportionality and equal awards principles in both cases of positive and negative surplus. Herrero, Maschler and Villar (1999) propose and analyze a “rights-egalitarian solution” which uses the equal awards principle in case of positive surplus and the equal losses principle in case of negative surplus. The axiomatic literature analyzes a much larger class of rules in comparison to the one following Holmström (1982). However, studies in this literature focus on normative questions and typically remain silent on strategic issues, particularly the role of incentives in the formation of surplus. By focusing on this latter question and by analyzing the structure of productively and socially e¢cient partnership agreements, our paper contributes to this literature. Some of our modeling choices are related to earlier studies as follows. First, there are many earlier papers that, like us, model output as stochastic. For example, see Huddart and Liang (2003), Comino, Nicolo and Tedeschi (2010). Again similar to us, several earlier studies argue that the partners’ expectations on their shares in case the partnership fails will have an e§ect on the partners’ e§ort choices. For example, see Comino, Nicolo and Tedeschi (2010) or Li and Wolfstetter (2010). Finally, almost all the theoretical literature following Holmström (1982) uses additively separable utility functions (quasilinear preferences). Similar to those studies, we measure contributions in monetary units. But we alternatively assume that the agents have constant absolute risk aversion (CARA) utilities. Since we consider a stochastic production function, the CARA family provides us a good way to measure the e§ect of the agents’ risk attitudes on the outcome. Finally it is useful to mention Kıbrıs and Kıbrıs (2013), where a similar modeling approach is used to analyze the investment implications of bankruptcy laws. While the two studies consider two separate economic institutions and contribute to two distinct strands of literature, they both analyze the incentive implications of resource allocation mechanisms in an environment with uncertainty and, in that sense, are technically related to each other. In terms of this relation, it is useful to note that this paper analyzes a more complicated problem than Kıbrıs and Kıbrıs (2013). In that study, 5 the allocation problem is restricted only to the “bad outcome” (in that case, bankruptcy) whereas here, it concerns both outcomes. Thus, while for the special case where the positive surplussharing rule is purely proportional the findings of Kıbrıs and Kıbrıs (2013) can be adapted to calculate individual and total contributions, they remain silent on partnership agreements that use an infinite number of other surplus-sharing rules (involving mixtures of proportionality and equal surplus-shares), all of which are analyzed here. As a result, central issues in this paper such as the e§ect of changes in the positive/negative surplus-sharing rule on individual/total contributions and on the acceptability of the partnership agreement, or how the size of these e§ects depends on the other surplus-sharing rule in use and on the number of partners, are outside the confines of the analysis carried out in Kıbrıs and Kıbrıs (2013). In terms of individual and social welfare, there is even less relationship between the two papers. Welfare comparisons in Kıbrıs and Kıbrıs (2013) are restricted to the two-agent case and can only be adapted to compare social welfare under two extreme partnership agreements. This paper however allows an arbitrary number of agents and involves both individual and social welfare comparisons for all of the continuum of rules that we consider. For additional discussion, please see Remark 1 at the end of Section 2. 2 Model The set of partners is N = {1, ..., n}. Each partner i 2 N has a Constant Absolute Risk Aversion (CARA) utility function ui : R ! R on money: ui (x) = −e−ai x . Assume the partners are risk averse and are ordered according to risk aversion: 0 < a1 ≤ ... ≤ an . Each partner i chooses his contribution to the partnership, si 2 R+ . We measure contributions in monetary units (or equivalently assume a constant marginal cost normalized to 1). The total P contribution of the partners is then N sj . With success probability p 2 (0, 1), this value brings P P a return r 2 (0, 1] and becomes (1 + r) N sj , creating a positive surplus of r N sj for the P partners. With the remaining (1 − p) probability, the partnership’s value becomes β N si where β 2 (0, 1) is the fraction that survives failure. In this case, the partnership makes a negative P surplus of (1 − β) N si . A partnership agreement is a pair of rules F, G to be used in case of positive and negative surP plus, respectively. The positive-surplus rule F allocates the gross returns (1 + r) sj according P to the vector of contributions s, partner i’s share being Fi (s, (1 + r) sj ). The negative-surplus P rule G, on the other hand, allocates the amount that survives failure β sj according to the vector P of contributions s, partner i’s share being denoted as Gi (s, β sj ). The following partnership agreements are based on two central surplus-sharing rules commonly 6 used in real life. Suppose the partnership creates value V. (From previous discussion, we know V P P is either (1 + r) sj or β sj . But the next two definitions will be independent of what V is.) The proportional surplus-sharing rule, P, allocates the surplus proportional to the partners’ P contributions. The share of a typical agent is then Pi (s, V ) = Psisj V = si + Psisj (V − sj ) (where P V − sj is the surplus). The equal surplus-sharing rule, E, allocates the surplus equally. The share of an agent is then Ei (s, V ) = si + P V − sj . n Gaynor and Pauly (1990) mention that the following “mixtures” of P and E are also commonly used, especially in professional service partnerships. For each ρ 2 [0, 1], the PE [ρ] rule first reimburses each partner for his contributions. Then, it allocates (1 − ρ) part of the surplus equally among the partners and uses the remaining fraction ρ to give bonuses in proportion to contributions: P E[ρ]i (s, V ) = ρPi (s, V ) + (1 − ρ) Ei (s, V ) & " X $ % si 1 . ρ P + (1 − ρ) = si + V − sj sj n Geometrically, these rules span all convex combinations of the proportional and equal surplus-share allocations. As noted in the introduction, a partnership agreement can specify di§erent rules to be used in cases of positive and negative surplus. The class of partnership agreements that we analyze, therefore combine a positive-surplus rule P E [γ] and a negative surplus rule P E [α] where α, γ 2 [0, 1] and α 6= γ is allowed.3 We will refer to such a partnership agreement as P E [γ, α]. Given the partnership agreement P E [γ, α] , the partners simultaneously choose their contributions. Agent i’s (expected) payo§ from a contribution profile s 2 Rn+ is P E[γ,α] Ui " " $ " " $ X $ X $ (s) = pui Fi s, (1 + r) sj − si + (1 − p)ui Gi s, β sj − si where Fi (s, (1 + r) P sj ) − si and Gi (s, β negative surplus, respectively. Let P U P E[γ,α] induced by P E [γ, α] is then defined as sj ) − si are his surplus shares in cases of positive and " $ P E[γ,α] P E[γ,α] = U1 , ..., Un . The partnership game P E[γ,α] G P E[γ,α] = hRN i. +,U Let ϵ(G P E[γ,α] ) denote the set of Nash equilibria of G P E[γ,α] . To measure the partners’ social welfare from a partnership agreement, we will resort to two leading measures in the literature. The egalitarian social welfare induced by P E [γ, α] is the 3 The parameter γ (respectively, α) determines which fraction of positive (respectively, negative) surplus is allocated proportionally. 7 minimum utility an agent obtains at the Nash equilibrium of the partnership game induced by P E [γ, α]: " $ EG P E[γ,α] (p, r, β, a1 , ..., an ) = min Ui (ϵ GP E[γ,α] ). i2N The utilitarian social welfare induced by P E [γ, α] is the total utility the agents obtain at the Nash equilibrium of the partnership game induced by P E [γ, α]: UT P E[γ,α] (p, r, β, a1 , ..., an ) = X i2N " $ Ui (ϵ GP E[γ,α] ). Remark 1 As noted at the end of the previous section, the findings of Kıbrıs and Kıbrıs (2013) can be adapted to calculate individual and total contributions for γ = 1 (though they remain silent on partnership agreements where γ 2 [0, 1)). This is a special case that contains no interaction among the agents via positive returns. (This can be verified in Equation 2 in the Appendix where taking γ = 1 makes the first part of the utility function of agent i independent of the other agents’ contributions.) In this paper on the other hand, with the exception of boundary cases where α = 1 or γ = 1, strategic interaction takes place via both positive and negative returns. As will be detailed in the next section, this enriches the analysis and leads to a number of interesting conclusions. It is also useful to reiterate that welfare comparisons in Kıbrıs and Kıbrıs (2013) are restricted to the two-agent case and can only be adapted to compare social welfare under the two extreme partnership agreements P E [1, 1] versus P E [1, 0]. This paper however allows an arbitrary number of agents and involves both individual and social welfare comparisons for all of the continuum of rules that we consider. 3 Results As defined in the previous section, each partnership agreement P E [γ, α] induces a partnership game among the agents. We next analyze the Nash equilibria of these games to discuss equilibrium contributions and productive as well as social e¢ciency. 3.1 Acceptable Agreements In this section, we characterize partnership agreements that induce all partners to contribute to the partnership. To this end, we say that a partnership agreement P E [γ, α] is acceptable for N if at the Nash equilibrium of the partnership game, all partners choose a positive contribution level. Acceptable partnership agreements are of special importance for two reasons. The first is technical: acceptable partnership agreements induce interior Nash equilibria at which the partners’ 8 equilibrium strategies and payo§s are di§erentiable with respect to the game’s parameters. Therefore, they facilitate comparative statics analyses. The second reason is empirical: real life data o§ers strong evidence that positive contributions by all partners is rather the norm in partnerships. Given that it is precisely the acceptable partnership agreements that induce positive contributions by all partners, this empirical regularity constitutes supportive evidence for the hypothesis that partnerships only form under acceptable partnership agreements. The empirical evidence we o§er comes from a rich dataset on legal partnerships, which are noted by Lang and Gordon (1995) to be the most common form of partnership in the US. This comprehensive dataset which has been extensively employed by the literature is based on two national surveys of lawyers in the US carried out by the American Bar Association in 1984 and 1990.4 The data show how many billable hours each lawyer in the survey has reported in one month. The average report is 187.88 (with a standard deviation of 46.87) and the minimum report among all lawyers in the survey is 32 hours per month.5 That is, no lawyer in the dataset has chosen to make zero contributions to the partnership (s)he works for. To determine whether a partnership agreement is acceptable, two intuitive conditions turn out to be important. The first condition, profitability, requires: & % pr (nγ − γ + 1) > 0. ln (1 − p) (1 − β) (nα − α + 1) (Profitability) $ " ' ( This condition, which can be rewritten as p γr + (1 − γ) nr > (1 − p) α (1 − β) + (1 − α) (1−β) , n ( ' r simply compares the return on unit contribution in case of positive surplus, γr + (1 − γ) n , weighted by the probability of success, p, with the loss incurred on unit contribution in case of " $ negative surplus, α (1 − β) + (1 − α) (1−β) , weighted by the probability of failure, (1 − p). Posn itive contributions are optimal if the returns in case of success outweigh the losses incurred in case of failure.6 Note that the Profitability condition does not make any reference to the partners’ risk attitudes. That will be the concern of our next condition. 4 The full survey data is available from the University of Michigan based Inter-university Consortium for Political and Social Research (ICPSR) at their webpage: http://www.icpsr.umich.edu/icpsrweb/ICPSR/studies/8975. 5 Billable hours do not include administrative or clerical work or working on client development (Lang and Gordon, 1995; page 621). Therefore, the billable hours data is an understatement of a lawyer’s contribution to the partnership. " ! 6 The left hand side expression γr + (1 − γ) nr has two parts. The γ weighted part r is the partner’s return on unit contribution under proportional surplus-sharing and the (1 − γ) weighted part nr is his return under equal $ # (1−β) again has two parts. The α weighted surplus-sharing. The right hand side expression α (1 − β) + (1 − α) n part of this expression, (1 − β) is the loss incurred for unit contribution in case of proportional surplus-sharing and the (1 − α) weighted part 1−β n is the loss incurred in case of equal surplus-sharing. 9 The second condition, homogeneity, requires that the agents are not too di§erent in terms of their risk attitudes : 1 an 1 n "P 1 N aj $ >1− γr + α (1 − β) . r+1−β (Homogeneity) The left hand side of this inequality has played an important role in previous studies such as Wilson (1968) and Huddart and Liang (2003). It is interpreted as agent n’s risk tolerance relative to the average risk tolerance of the partnership (e.g. see Wilson’s interpretation for the case of syndicates). Since agent n is the most risk averse partner (i.e. a1 ≤ ... ≤ an ), the left hand side is less than or equal to 1 (and it is equal to 1 precisely when a1 = ... = an ). For the same reason, if agent n were to be replaced with any other agent, the left hand side would increase in value, making the inequality less binding. This is why the Homogeneity condition is stated for agent n, even though it applies to all partners. The right hand side of the inequality depends on how distant P E [γ, α] is from pure proportionality, P E [1, 1]. The denominator of the fraction shows how P E [1, 1] allocates positive surplus (r) and negative surplus (1 − β). The numerator, on the other hand, shows that under P E [γ, α], only γ fraction of positive surplus and α fraction of negative surplus is allocated proportionally (γr and α (1 − β)). When both α and γ are 1, that is for P E [1, 1], the right hand side is zero and thus, not binding. As either of the two surplus sharing rules move towards equal shares however, that is, as α or γ goes down, the right hand side increases, becoming more binding. When α = γ = 0 (i.e. when the partnership agreement allocates both positive and negative surplus equally), the right hand side reaches its maximum value of 1. The reader will note an interesting distinction between α and γ. An increase in γ increases the partnership’s profitability and homogeneity simultaneously. Yet, an increase in α decreases the partnership’s profitability while increases its homogeneity. Proposition 1 (Acceptable Agreements) A partnership agreement P E[γ, α] with max{α, γ} > 0 is acceptable for N if and only if both Profitability and Homogeneity conditions are satisfied. The partnership agreement P E[0, 0] is acceptable for N if and only if Profitability is satisfied and the Homogeneity condition holds with a weak inequality. Note that when α = γ = 0, the right hand side of the Homogeneity condition is 1. The maximum value for the left hand side, achieved when a1 = ... = an , is also 1. Thus, when α = γ = 0, the Homogeneity condition holds with a weak inequality if and only if all agents have identical risk attitudes. This is precisely the case when the partnership game has multiple Nash equilibria and for that reason, it will require special attention, as can be seen below. 10 3.2 Equilibrium Contributions In this section, we analyze the equilibrium contributions of partners in a formed partnership. As can be seen in the following proposition, equilibrium contributions are unique under all partnership agreements but P E[0, 0]. Proposition 2 (Equilibrium contributions under P E [γ, α]) If the agreement P E[γ, α] with max{α, γ} > 0 is acceptable for N, the resulting partnership game has a unique Nash equilibrium s∗ where s∗i = " n (r + 1 − β) a1i − ((1 − γ) r + (1 − α) (1 − β)) "P 1 N aj n (r + 1 − β) (γr + α (1 − β)) $$ ln " pr(nγ−γ+1) (1−p)(1−β)(nα−α+1) $ (1) for each i 2 N. On the other hand, if P E [0, 0] is acceptable for N, the partnership game has a continuum of Nash equilibria: any contribution profile s∗ ≥ 0 such that $ " pr n ln X (1−p)(1−β) s∗i = an (1 − β + r) N is a Nash equilibrium. Note that the ln term in Equation (1) is the one used in the Profitability condition. Also, as can easily be checked, the denominator of the first term in Equation (1) is always positive. The Homogeneity condition guarantees that the numerator is of positive sign as well. As stated in Proposition 1, under P E [0, 0] a partnership forms if and only if a1 = ... = an . Proposition 2 then tells us that this symmetric game has a continuum of Nash equilibria. # " Nevertheless, the symmetric equilibrium among them (where for each i 2 N, s∗i = ln pr (1−p)(1−β) ai (1−β+r) ) is more robust than the rest in the following sense. Imagine a sequence of partnership agreements, each satisfying max{α, γ} > 0, but converging to P E [0, 0] . As can be seen from Proposition 2, the corresponding sequence of unique equilibrium contributions will also be converging, and it will converge precisely to this symmetric equilibrium under P E [0, 0] . No other equilibrium under P E [0, 0] satisfies this property. Therefore, in welfare comparisons, we will focus on this symmetric equilibrium when analyzing P E [0, 0] and a1 = .. = an . Since a1 ≤ ... ≤ an , Equation 1 implies s∗1 ≥ ... ≥ s∗n . That is, agent i is a “bigger partner” than agent j whenever i ≤ j. A corollary of Proposition 2 identifies conditions under which the partnership game has a dominant strategy equilibrium.7 Partnership agreements that induce dominant strategy equilibria 7 It follows from Equation (3) in the Appendix that the partnership games induced by P E [γ, α] agreements admit dominant strategy equilibria if and only if (1 − γ) r + (1 − α) (1 − β) = 0 (in which case, partner i’s best response is 11 Figure 1: Partner 1’s (left) and Partner 2’s (right) equilibrium contributions, as a function of α. Parameter values are r = 0.3, p = 0.8, β = 0.7, a1 = 1, a2 = 1.5. Also α = 0.3 and γ = 0.5 when not a variable. are advantageous to those that do not since it is possible to make a stronger prediction about how the partners will behave. Corollary 3 (Dominant strategy equilibrium under P E [1, 1]) The partnership game induced by the agreement P E [1, 1] has a dominant strategy equilibrium (in strictly dominant strategies). No other partnership agreement induces dominant strategy equilibria. To provide the reader with a better understanding of the above propositions, we conclude this section with a two-partner numerical example that demonstrates how individual contributions depend on the partnership agreement P E [γ, α] . In the example, the parameter values are r = 0.3, p = 0.8, β = 0.7, a1 = 1, a2 = 1.5, γ = 0.5. Figure 1 plots how individual contributions change as a function of α, the percentage of negative surplus allocated proportionally. As can be seen, an increase in α decreases Partner 1’s contribution. This might seem surprising at first glance, since it is commonly argued in the literature that a shift from equal to proportional surplus shares will increase individual contributions. However, the reader will note after a closer inspection that an increase in α decreases the marginal return on contributions in case of negative surplus (by making losses more sensitive to contributions, as can be seen in Footnote 5). It thereby induces both partners to contribute less. independent of the others’ strategies). This equality holds if and only if α = γ = 1. In Equation (1), this equality ensures that partner i’s equilibrium strategy is independent of the others’ risk attitudes. 12 Maybe more surprisingly, Figure 1 shows that α has a non-monotonic e§ect on the contribution of the smaller partner, Partner 2, who first increases and then decreases his contribution. This nonmonotonicity is caused by two competing e§ects. The first, direct e§ect is already mentioned in the previous paragraph. The second, indirect e§ect is due to the fact that the two partners’ contributions are strategic substitutes. Thus, as Partner 1 decreases his contribution in response to an increase in α, partner 2 is inclined to increase his own contribution in response. The figure shows that the latter a§ect is dominant for small values of α. But for high α values, the first direct e§ect overtakes the second.8 The nonmonotonicity of s∗2 in α is not a knife-edge case. In this example, unilateral changes in γ or r do not disturb this nonmonotonicity at all; a unilateral change in p disturbs it only when p > 0.87 (making s∗2 an increasing function) and a unilateral change in β disturbs it only when β > 0.95 (making s∗2 a decreasing function). It is also useful to note that, for the above parameter values, the value of α that maximizes s2 is decreasing in γ (the percentage of positive surplus allocated proportionally). This shows that the incentives Partner 2 faces are not straightforward, but are determined through an interplay of the positive-surplus and negative-surplus rules. In the same example, we next fix α = 0.3 and let γ vary. Figure 2 demonstrates that, as claimed by the previous literature, an increase in γ (the percentage of positive surplus allocated proportionally) in turn increases Partner 2’s contributions.9 However, it also shows that the e§ect of γ on Partner 1 is non-monotonic. (The discussion, similar to the case of α, is omitted.) Thus, contrary to what the previous literature suggests, even when allocating positive surplus, moving from fixed surplus-shares towards proportionality (the piece-rate) does not necessarily increase individual contributions of all partners. 3.3 Productive E¢ciency In this section, we compare partnership agreements in terms of the total contribution that they induce in equilibrium, that is, in terms of their productive e¢ciency. As demonstrated in the previous section, a look at individual contributions suggests no clear prediction as to how total contributions would be a§ected from changes in the underlying partnership agreement. On the other hand, Figure 3 suggests a clear ordering in our numerical example. First, an increase in γ in 8 As can be more formally seen in Equation (3) in the Appendix, both partners have linear best response functions (with a positive intercept and a negative slope). An increase in α a§ects both best response functions in the same way: it decreases the intercept and decreases the slope in absolute value, making best responses less sensitive to the other partner’s choices. It is because of this that the strategic substitutes property matters less at high values of α. 9 Figure 2 also demonstrates that, for γ ≤ 0.1, the partnership agreement P E [γ, α] is not acceptable and, as discussed in the previous section, the partnership does not form. 13 Figure 2: Partner 1’s (left) and Partner 2’s (right) equilibrium contributions, as a function of γ. Parameter values are the same as in Figure 1. turn increases the partners’ total contributions. This confirms the common belief that a move from equal surplus-sharing towards proportionality increases total contributions. However, the figure also shows that a similar move in the allocation of negative surplus has exactly the opposite e§ect. The following theorem generalizes what we observe in this numerical example to the whole parameter space. Theorem 1 Equilibrium total contributions under P E [γ, α] is (i) increasing in γ (the fraction of positive surplus allocated proportionally) and (ii) decreasing in α (the fraction of negative surplus allocated proportionally). Furthermore, both e§ects are increasing in the number of partners in the partnership. In terms of what it says regarding the positive-surplus rule, the theorem supports the general view that moving from fixed surplus shares to proportionality increases total contributions. For the negative-surplus rule, however, the theorem identifies that a move towards proportionality now decreases total contributions. The theorem, thus, shows us that a way to improve over the commonly-used piece rate agreement is to change the surplus-sharing rule used in case of negativesurplus; a move towards equal surplus-shares helps productive e¢ciency. While such a change does not incentivize every partner to contribute more (e.g. see Partner 2 in Figure 1), its aggregate e§ect is certain. It is interesting to note that, even in symmetric partnerships (i.e. when all partners have identical risk attitudes), the ordering of partnership agreements in terms of total contributions is 14 Figure 3: The e§ect of α (left) and γ (right) on total contributions. Parameter values are the same as in Figure 1. still as above. Particularly, P E [1, 0] still remains as the unique productively e¢cient agreement. It is also important to reiterate that the e§ect of the agreement on total contributions is emphasized in partnerships with a greater number of partners. Thus, bigger partnerships would be more likely to pick greater γ and smaller α parameters. Theorem 1 implies that the partnership agreement P E [1, 0] is the unique productively e¢cient agreement in the P E [γ, α] family. However, as our findings in Subsection 3.1 demonstrate, there are partnerships where this agreement will not be acceptable. In such partnerships, P E [1, 0] violates either the Profitability or the Homogeneity condition. First, it is straightforward to see that if P E [1, 0] violates Profitability, every other partnership agreement also does so. Thus, in such cases the partnership will not form under any P E [γ, α] agreement. The more interesting case is when P E [1, 0] violates Homogeneity. Then, an increase in α helps to satisfy the inequality while a decrease in γ does not. Thus, keeping γ = 1, there is a critical value 1 α∗ = 1 − r + 1 − β an P 1 1−β N a j where the set of acceptable agreements are P E [1, α] such that α > α∗ .10 As α decreases, productive e¢ciency increases and simultaneously the most risk averse Partner n’s contribution decreases. At the limit α = α∗ , Partner n picks a zero contribution making P E [1, α∗ ] unacceptable. 10 To see this, note that 1 a P n1 N aj >1− r+α(1−β) r+1−β i§ α > (r+1−β) (1−β) 15 % 1− 1 a P n1 N aj & − r (1−β) i§ α > 1 − 1 r+1−β a P n1 1−β N aj . Figure 4: Utility of Partner 1 (solid line) and Partner 2 (dashed line) as a function of α (left) and γ (right). Parameter values are the same as in Figure 1. 3.4 Individual and Social Welfare In this section, we look at the individual and social welfare levels induced by alternative partnership agreements. We make an analytical comparison in terms of egalitarian social welfare. Additionally, we carry out a numerical analysis in terms of utilitarian social welfare. Figure 4 demonstrates how equilibrium welfare of the two partners in our example changes in response to P E [γ, α]. The following observations are in order. First, in both pictures Partner 1 (the bigger partner) receives a greater utility than Partner 2 if and only if α < γ, that is, when a higher proportion of positive than negative surplus is allocated proportionally. Thus, egalitarian social welfare is equal to the utility of Partner 1 when α > γ and to the utility of Partner 2 when α < γ. As both pictures demonstrate, when α = γ, the two partners receive equal payo§. Second, this egalitarian social welfare increases as α and γ gets closer to each other, and is maximized at α = γ. Surprisingly, both of the above points are generalizable to an arbitrary number of agents and to all parameter values we consider. The following proposition orders the agents according to their equilibrium welfare. Proposition 4 Under P E [γ, α], the partners are ordered according to their equilibrium utilities as 1, 2, ..., n. If α > γ, the least risk-averse Partner 1 always receives the smallest utility and the most risk-averse Partner n always receives the highest utility. If α < γ, the ordering is reversed, Partner 1 now receiving the highest utility and Partner n, the smallest. If α = γ, all partners receive the same utility level. 16 The above proposition implies that the egalitarian social welfare is equal to the equilibrium payo§ of either the most or the least risk averse partner, depending on the α-γ relationship in their partnership agreement. The following theorem shows that this egalitarian social welfare is maximized at α = γ. Theorem 2 Under P E [γ, α], egalitarian social welfare is decreasing in |α − γ|, being maximized when α = γ = x. In this case, all partners’ payo§s are equal and this common payo§, which is also the egalitarian social welfare under the P E [x, x] partnership agreement, is independent of x. While all P E [x, x] partnership agreements induce the same egalitarian social welfare level, they might be di§erent in other aspects. The first that comes to mind is the agents’ contribution choices. It turns out that all P E [x, x] partnership agreements induce the same total contribution in equilibrium. (Thus, maximizing total contributions among P E [x, x] agreements does not restrict x at all.) These agreements, however, di§er in terms of the individual contributions that they induce in equilibrium. Partners who are less (more) risk averse than the average decrease (increase) their contributions in response to an increase in the common x, keeping total contributions constant. (For a proof, please see Claim 1 in Section 5.) Due to di§erences in individual contribution choices, it might be that some P E [x, x] agreements are acceptable while the others are not (as discussed in Subsection 3.1). It is straightforward to check that the Profitability condition does not distinguish among the P E [x, x] agreements; either they all satisfy or violate it. The Homogeneity condition, on the other hand, partitions the set of P E [x, x] agreements. There is a critical value 1 x∗ = 1 − Pan 1 N aj where an agreement P E [x, x] is acceptable if and only if x > x∗ . As the common x decreases in an acceptable agreement, the most risk averse Partner n’s contribution will also decrease, reaching zero at x = x∗ . We conclude this section with a discussion of utilitarian social welfare. For this case, the ordering of partnership agreements in terms of utilitarian social welfare depends on the underlying parameter values. This makes a general analytical result as in the case of egalitarian social welfare not possible. We first carry out a numerical analysis for the case of two partners. We allow the following parameter values: β, p, r, α, γ 2 {0.01, 0.11, 0.21, 0.31, 0.41, 0.51, 0.61, 0.71, 0.81, 0.91} , a1 , a2 2 {0.01, 0.51, 1.01, 1.51, 2.01, 2.51, 3.01, 3.51, 4.01, 4.51} and a2 ≥ a1 . 17 This grid produces (i) 23 469 combinations of β, p, r, a1 , a2 at which both the Profitability and Homogeneity conditions are satisfied under some α-γ combinations, that is, the partnership has acceptable agreements available11 and (ii) 53 721 combinations of α, γ, β, p, r, a1 , a2 where the α-γ combination maximizes utilitarian social welfare under β, p, r, a1 , a2 . Surprisingly, at 36 280 (that is, 67.5%) of these parameter combinations, utilitarian social welfare is maximized when γ = α. And, at 40 794 (that is, 75.9%) of these combinations, utilitarian social welfare is maximized when α and γ di§er by at most one grid point. It is also interesting to note that, among the remaining parameter combinations, α > γ is observed more than twice as much as γ > α (precisely, at 8 981 versus 3 946 combinations). We also carried out a numerical analysis for the case of three partners. Since the computer could not handle the above grid, we switched to a slightly coarser grid of β, p, r 2 {0.01, 0.16, 0.31, 0.46, 0.51, 0.66, 0.71, 0.86, 0.91} , α, γ 2 {0.01, 0.11, 0.21, 0.31, 0.41, 0.51, 0.61, 0.71, 0.81, 0.91} , a1 2 {0.1, 0.7, 1.3, 1.9, 2.5, 3.1, 3.7, 4.3, 4.9} and a3 ≥ a2 ≥ a1 . This grid produces (i) 24 286 combinations of β, p, r, a1 , a2 , a3 at which both the Profitability and Homogeneity conditions are satisfied under some α-γ combinations, that is, the partnership has acceptable agreements available12 and (ii) 44 442 combinations of α, γ, β, p, r, a1 , a2 , a3 where the α-γ combination maximizes utilitarian social welfare under β, p, r, a1 , a2 , a3 . At more than half (specifically 25 680, that is, 57.8%) of these parameter combinations, utilitarian social welfare is maximized when γ = α. And, at more than two thirds (specifically 31 244, that is, 70.3%) of these combinations, utilitarian social welfare is maximized when α and γ di§er by at most one grid point. Finally, among the remaining parameter combinations, α > γ is observed twice as much as γ > α (precisely, at 8 785 versus 4 413 combinations). These numerical findings should be interpreted with caution. The γ = α finding, in a significant number of the cases, is due to the grid that we impose on the parameter space. Thus, we can only deduce from this analysis that quite frequently, utilitarian social welfare is maximized at α, γ values that are close to each other and that, maximizing utilitarian social welfare does not create agreements that systematically di§er from those that maximize egalitarian social welfare. 11 12 This corresponds to 1 324 692 combinations of α, γ, β, p, r, a1 , a2 . This corresponds to 1 029 914 combinations of α, γ, β, p, r, a1 , a2 , a3 . 18 4 Conclusion Our analysis compares a family of partnership agreements (i.e. surplus allocation rules) in terms of total contributions and social welfare that they induce in equilibrium of a noncooperative partnership game. Our findings are as follows: (i) Equilibrium total contributions induced by a partnership agreement increases as the positivesurplus rule gets closer to proportionality and the negative-surplus rule gets closer to equal surplusshares. Using proportionality in case of positive surplus and equal-surplus shares in case of negative surplus (i.e. γ = 1, α = 0) maximizes total contributions whenever this agreement is acceptable. Otherwise, the partners pick γ = 1 and α as small as acceptability permits. (ii) Egalitarian social welfare increases as the percentages of positive and negative surplus allocated proportionality (i.e. γ and α) get closer to each other. Partnership agreements where γ = α all maximize egalitarian social welfare. Such agreements give all agents the same welfare and produce the same amount of total contributions. They, however, di§er in terms of individual contribution choices that they induce. (iii) The ordering of partnership agreements in terms of utilitarian social welfare depends on the parameter values. Thus a general statement as in egalitarian social welfare or total contributions can not be made. However, a numerical analysis shows that the utilitarian optimal partnership agreements are not systematically di§erent from egalitarian optimal ones. Simulations for two and three agent partnerships show that 60 to 70 % of the utilitarian optimal partnership agreements exhibit γ = α. (iv) In symmetric games (where a1 = ... = an ), the egalitarian optimal agreements described in (ii) additionally Pareto dominate all other agreements. (v) There always is a unique dominant strategy equilibrium under the purely proportional agreement P E [1, 1]. No other partnership agreement induces dominant strategies. Overall, we observe a trade-o§ between maximizing total contributions and social welfare. To maximize total contributions, two opposite surplus allocation rules needs to be used in cases of positive and negative surplus (i.e. γ = 1 and α = 0). However, (i) maximizing egalitarian social welfare requires choosing the same surplus-allocation rule in cases of both positive and negative surplus (i.e. γ = α), and furthermore (ii) a numerical analysis detailed in Section 3.4 obtains a similar result for utilitarian social welfare on a significant part of our parameter space. The mechanism behind this trade-o§ can be explained as follows. An increase in γ increases the marginal return of contributing by making returns in case of positive surplus more sensitive to contributions. A decrease in α has a similar e§ect by decreasing the sensitivity of losses (made in case of negative surplus) to contributions. Due to this reason, an increase in γ or a decrease in α provides all agents 19 with a direct incentive to contribute more, thus leading to an increase in total contributions, as demonstrated in Figure 3.13 However, an increase in γ or a decrease in α also transfers wealth from the smaller (than average) partners to the bigger ones and the high γ low α combinations which induce high total contribution levels thus make the bigger partners much better o§ than the smaller ones as can be seen in Figure 4. As the figure also shows, bringing the two parameters closer to each other (by decreasing γ or increasing α) makes the bigger partners worse o§ and the smaller partners better o§, thereby bringing the partners’ welfare levels closer to each other and increasing egalitarian social welfare. References [1] Alchian, A. and Demsetz, H. (1972), “Production, Information Costs, and Economic Organization, American Economic Review, 62(5), 777-795. [2] Aumann, R. J. and Maschler, M. (1985), “Game Theoretic Analysis of a Bankruptcy Problem from the Talmud”, Journal of Economic Theory, 36, 195-213. [3] Chun, Y. (1988) “The proportional solution for rights problems”, Mathematical Social Sciences, 15, 231—246. [4] Comino, S., Nicolo, A. and Tedeschi, P. (2010), “Termination clauses in partnerships”, European Economic Review, 54, 718-732. [5] Curran, B. (1985), The Lawyer Statistical Report: A Statistical Profile of the U.S. Legal Profession in the 1980s, American Bar Foundation, dist. Hein and Co. Inc. [6] Farrell, J. and Scotchmer, S. (1988), “Partnerships”, Quarterly Journal of Economics, 103(2), 279-297. [7] Flood, J. (1985), The Legal Profession in the United States, The American Bar Foundation, Chicago, IL. [8] Gaynor, M. and Pauly, M. V. (1990), “Compensation and Productive E¢ciency in Partnerships: Evidence from Medical Group Practice”, Journal of Political Economy, 98(3), 544-573. 13 There is a second indirect e§ect on individual contributions stemming from the fact that the partners’ contribu- tions are strategic substitutes. A partner increasing his contribution incentivizes the other partners to decrease their contributions in return. A combination of these two e§ects can thus create nonmonotonic individual contribution responses to changes in α and γ as seen in figures 1 and 2. Yet, as our theorem shows, when aggregated over agents, this first e§ect overrides the second. 20 [9] Hart, O. (2011), “Thinking about the Firm: A Review of Daniel Spulber’s ’The Theory of the Firm’.”, Journal of Economic Literature, 49(1), 101-113. [10] Hart, O. and Holmström, B. (2010), “A Theory of Firm Scope”, Quarterly Journal of Economics, 125(2), 483-513. [11] Herrero, C., Maschler, M. and Villar, A. (1999), “Individual rights and collective responsibility: the rights—egalitarian solution”, Mathematical Social Sciences, 37(1), 59-77. [12] Holmström, B. (1982), “Moral Hazard in Teams”, Bell Journal of Economics, 13, 324-340. [13] Huddart, S. and Liang, P. J. (2003), “Accounting in Partnerships”, American Economic Review, 93:2, 410-414. [14] Kandel, E. and Lazear, E. (1992), “Peer Pressure in Partnerships” Journal of Political Economy, 100(4), 801-817. [15] Kıbrıs, Ö. and Kıbrıs, A. (2013). “On the investment implications of bankruptcy laws”, Games and Economic Behavior, 80, 85-99. [16] Lang, K. and Gordon, P.J. (1995), “Partnership as insurance devices: theory and evidence”, RAND Journal of Economics, 26(4), 614-629. [17] Legros, P. and Matthews, S.A. (1993), “E¢cient and Nearly-E¢cient Partnerships”, The Review of Economic Studies, 60(3), 599-611. [18] Li, J. and E. Wolfstetter (2010), “Partnership failure, complementarity, and investment incentives”, Oxford Economic Papers, 62, 529—552. [19] Miller, N.H. (1997), “E¢ciency in Partnerships with Joint Monitoring”, Journal of Economic Theory, 77, 285 299. [20] Morrison, A. and Wilhelm, W.J. (2004), “Partnership Firms, Reputation, and Human Capital”, American Economic Review, 94(5), 1682-1692. [21] Moulin, H., (1987), “Equal or proportional division of a surplus, and other methods”, International Journal of Game Theory, 16:3, 161—186. [22] O’Neill, B. (1982) “A Problem of Rights Arbitration from the Talmud”, Mathematical Social Sciences, 2, 345-371. 21 [23] Strausz, R. (1999) “E¢ciency in sequential partnerships”, Journal of Economic Theory, 85(1), 140-156. [24] Thomson, W. (2003) “Axiomatic and Game-Theoretic Analysis of Bankruptcy and Taxation Problems: a Survey”, Mathematical Social Sciences, 45, 249-297. [25] Thomson, W. (2008), How to Divide When There Isn’t Enough: From Talmud to Game Theory, unpublished manuscript. [26] Wilson , R. (1968), “The Theory of Syndicates”, Econometrica, 36(1), 119-132. 5 Appendix We will start this section by calculating the Nash equilibrium of the partnership game. Under the family P E [γ, α] , the utility function of partner i is 0 P E[γ,α] Ui (s) B −ai B @γrsi +(1−γ) = −pe rsi +r P N \i n sj 1 C C A " − (1 − p)e (1−β)(1+(n−1)α) n # P (1−α)(1−β) ai N \i sj ai si + n . (2) The unconstrained maximizer of this expression is si = σ i (s−i ) = " $ 1 0 pr(nγ−γ+1) n ln (1−p)(1−β)(nα−α+1) X r (1 − γ) + (1 − β) (1 − α) @ − sj A . (3) ai ((1 − α + nα) (1 − β) + (n − 1) rγ + r) (1 − α + nα) (1 − β) + (n − 1) rγ + r N \i r(1−γ)+(1−β)(1−α) (1−α+nα)(1−β)+(n−1)rγ+r 2 [0, 1] , the slope of this expression is negative.14 Also, the sign of its $ " pr(nγ−γ+1) . This ln term is nothing but the constant term is determined by the sign of ln (1−β)(1−p)(nα−α+1) Since left hand side of the Profitability condition. Partner i’s best response is bi (s−i ) = max{0, σ i (s−i )}. Solving the system in Expression 3 gives for each i 2 N (Expression 1 of Proposition 2) s∗i = " "P $$ 1 % & n (r − β + 1) a1i − (r − α − β − rγ + αβ + 1) N aj pr (nγ − γ + 1) ln (4) n (r − β + 1) (α + rγ − αβ) (1 − β) (1 − p) (nα − α + 1) which, under certain conditions, will give us the unique Nash equilibrium of the partnership game. Proof. (Proposition 1) Case 1: The partnership agreement is P E [γ, α] such that max {α, γ} > 0. 14 This expression is equal to 0 if and only if α = γ = 1 and equal to 1 if and only if α = γ = 0. The former is trivial. To see the latter, note that r(1−γ)+(1−β)(1−α) (1−α+nα)(1−β)+(n−1)rγ+r equality if and only if α = γ = 0. 22 ≤ 1 simplifies to 0 ≤ nα (1 − β) + nrγ, achieved with (=)) Assume P E [γ, α] is acceptable for N . To see that Profitability and Homogeneity are satisfied, first suppose Profitability is violated. Then, for each i 2 N and for all s−i , σ i (s−i ) < 0. Thus, the unique Nash equilibrium is s = (0, ..., 0), contradicting acceptability of P E [γ, α] . Next, suppose Profitability is satisfied but Homogeneity is violated. Then, (n (r − β + 1) a1n − (r − α − $ "P 1 ∗ β − rγ + αβ +1) N aj ) ≤ 0 and thus sn < 0, contradicting acceptability of P E [γ, α] . ((=) Assume Profitability and Homogeneity conditions are satisfied. By Profitability, we have $ " pr(nγ−γ+1) > 0 and by Homogeneity, we have (n (r − β + 1) a1n − (r − α − β − rγ + αβ ln (1−β)(1−p)(nα−α+1) $ "P 1 ∗ ∗ +1) N aj ) > 0. This guarantees s > 0 where s (Expression 4) is then the unique Nash equilibrium. Thus, P E [γ, α] is acceptable. Case 2: The partnership agreement is P E [0, 0]. (=)) Assume P E [0, 0] is acceptable for N . We want to show that Profitability holds and Homogeneity holds with a weak inequality. First suppose Profitability is violated. Then, as noted above, the unique Nash equilibrium is s = (0, ..., 0), contradicting acceptability of P E [0, 0] . Next, suppose Profitability is satisfied but Homogeneity is violated. Since α = γ = 0, we then have " $ 1 1 P 1 < N aj . This implies, a1 < an . Again due to α = γ = 0, Expression 3 simplifies to an n n ln si = σ i (s−i ) = " pr (1−β)(1−p) ai (1 − β + r) $ 0 −@ X N \i 1 sj A . (5) Since a1 < an and since each agent i’s best response is the maximum of zero and σ i (s−i ), agent n picks zero contributions in equilibrium, contradicting acceptability of P E [0, 0]. ((=) Assume Profitability and the weaker form of Homogeneity are satisfied. We want to show "P $ 1 that P E [0, 0] is acceptable for N . By the weaker form of Homogeneity, a1n ≥ n1 which in N aj n ln " pr (1−β)(1−p) # > 0. Thus, the best response # $ "P n ln expression of every agent i can be written as si = σ i (s−i ) = an (1−β+r) − N \i sj . Thus, all turn implies a1 = ... = an . By Profitability, we have an (1−β+r) " pr (1−β)(1−p) s∗ ≥ 0 such that P ∗ N si = n ln " pr (1−β)(1−p) an (1−β+r) # is a Nash equilibrium. Since a continuum among these equilibria satisfy s∗ > 0, we conclude that P E [0, 0] is acceptable. Proof. (Proposition 2) Case 1: The partnership agreement is P E [γ, α] such that max {α, γ} > 0. Assume that P E[γ, α] is acceptable for N. By Proposition 1, both Profitability and Homogeneity conditions hold. To see that the resulting partnership game has a unique Nash equilibrium s∗ which is given by Expression 1, note that s∗ solves the system in Expression 3. By Profitability, we have $ " pr(nγ−γ+1) > 0 and by Homogeneity, we have (n (r − β + 1) a1i − (r − α − β − rγ + αβ ln (1−β)(1−p)(nα−α+1) 23 +1) "P 1 N aj $ ) > 0. This guarantees that that s∗ > 0. It is thus the unique Nash equilibrium of the partnership game under P E [γ, α]. Case 2: The partnership agreement is P E [0, 0]. Assume P E [0, 0] is acceptable for N . By Proposition 1, Profitability and the weaker form of Homogeneity are satisfied. The ((=) #part in Case 2 of the previous proof then shows that all " pr n ln P is a Nash equilibrium of the partnership game. s∗ ≥ 0 such that N s∗i = an(1−β)(1−p) (1−β+r) r(1−γ)+(1−β)(1−α) Proof. (Corollary 3) In the Expression 3, the slope is: − (1−α+nα)(1−β)+(n−1)rγ+r . If this expres- sion is zero, the best response of partner i is independent of s−i , making it a strictly dominant strategy. Now note that the denominator of this expression is always positive. And its numerator r (1 − γ) + (1 − β) (1 − α) = 0 if and only if α = γ = 1. Therefore, P E [1, 1] is the only partnership agreement that always induces a dominant strategy equilibrium. Proof. (Theorem 1) Total contribution is %X & % & X pr (nγ − γ + 1) 1 1 ∗ ln .15 si = r−β+1 ai (1 − β) (1 − p) (nα − α + 1) The derivative of this expression with respect to α is "P $ 1 P ∗ − N aj (n − 1) @ ( si ) = < 0. @α (nα − α + 1) (r − β + 1) Thus, a decrease in α increases total contributions. Now let % at the e§ect $ us look $ of adding a% partner on this derivative. desired conclusion. (n−1) Since (nα−α+1) P @ ( s∗ ) That is, @α i − < n (nα+1) , we have P 1 N aj (n−1) (nα−α+1)(r−β+1) − > P 1 + N a1 an+1 j (n) (nα+1)(r−β+1) , the is increasing in absolute value as the number of agents increases. Now, let us look at the derivative of total contributions respect to γ : "P $ 1 P ∗ N aj (n − 1) @ ( si ) = > 0. @γ (nγ − γ + 1) (r − β + 1) So, an increase in γ increases total contributions. Now let us $ look %at the e§ect $ of adding a%partner on this desired P 1 N aj (n−1) (n) (n−1) derivative. As above, (nγ−γ+1) < (nγ+1) implies (nγ−γ+1)(r−β+1) < P @ ( s∗i ) conclusion. That is, @γ is increasing in the number of agents. (Proposition 4) Introducing s∗ into partner i’s utility function, we P 1 + N a1 an+1 j (n) (nγ+1)(r−β+1) , the P E[γ,α] Proof. obtain Ui (s∗ ) = 0 1 % $ $ −rγ " P 1 α+rγ−αβ pr(nγ−γ+1) % & rai (1−β)(γ−α) −p n(r−β+1)(α+rγ−αβ) a j pr (nγ − γ + 1) (1−β)(1−p)(nα−α+1) B C @ " $ α(1−β) A× (1 − β) (1 − p) (nα − α + 1) α+rγ−αβ pr(nγ−γ+1) −(1 − p) (1−β)(1−p)(nα−α+1) 15 Note that, this expression gives total contribution when α = γ = 0 as well. Even though there is multiplicity of equilibria in this case, they all have the same total contribution level given by this expression. 24 " determines the $ P 1 rai (1−β)(γ−α) aj n(r−β+1)(α+rγ−αβ) P E[γ,α] ∗ e§ect of ai on Ui (s ) . If All components of have determinate signs except γ − α, whose sign P E[γ,α] γ − α > 0, an increase in ai decreases Ui (s∗ ) . As a result, Partner 1 receives the highest utility and Partner n, the lowest. The welfare ordering of the partners is exactly the opposite when γ − α < 0. And if γ − α = 0, ai does not a§ect P E[γ,α] Ui (s∗ ) . Thus, all agents receive the same utility. Proof. (Theorem 2) Proposition 4 establishes that 8 P E[γ,α] ∗ > U1 (s ) if > < P E[γ,α] ∗ P E[γ,α] EG (p, r, β, a1 , ..., an ) = Un (s ) if > > : U P E[γ,α] (s∗ ) = ... = U P E[γ,α] (s∗ ) if n 1 γ<α " $ P E[γ,α] min U (ϵ G ). γ>α i i2N γ=α We will treat each case separately. First, assume γ = α. In this case, the individual utility functions simplify to P E[γ,α] Ui 0 (s∗ ) = @−p % pr (1 − β) (1 − p) & −r 1+r−β − (1 − p) % pr (1 − β) (1 − p) & (1−β) 1+r−β 1 A Two observations are in line. First, Partner i’s equilibrium utility is independent of ai . Therefore, all partners receive identical utility. Second, the expression is independent of the common value of γ = α. That is, all γ = α partnership agreements produce the same level of egalitarian social welfare. This establishes the second sentence of the theorem. P E[γ,α] ∗ To see the first sentence, first assume γ > α and i = n. Then @Un @α (s ) = " $ P pr(nγ−γ+1) γr (1 − β) (r − β + 1) (nα − α + 1) ((n − 1) pα + 1) (( j aanj ) − n) ln (1−β)(1−p)(nα−α+1) P − (n − 1) (α + rγ − αβ) (pr (α − γ) ( j aanj ) (r (nγ − γ + 1) + (1 − β) (1 − α + nα))−n (r − β + 1) pr (α − γ)). $ $ ""P an − n ≥ 0, the first term is nonnegative. And it is strictly positive unless a1 = Since j aj ... = an . The second term is also positive since 1 1 0 0 X an A (r (nγ − γ + 1) + (1 − β) (1 − α + nα)) − n (r − β + 1) pr (α − γ)A < 0.16 @pr (α − γ) @ aj j This establishes that egalitarian social welfare is increasing in α when γ > α. Next assume γ < α and i = 1. Then, P E[γ,α] @U1 (s∗ ) = " $ P a1 pr(nγ−γ+1) γr (1 − β) (r − β + 1) (nα − α + 1) ((n − 1) pα + 1) (( j aj ) − n) ln (1−β)(1−p)(nα−α+1) P − (n − 1) (α + rγ − αβ) (pr (α − γ) ( j aa1j ) (r (nγ − γ + 1) + (1 − β) (1 − α + nα))−n (r − β + 1) pr (α − γ)). 16 @α For brevity of presentation, calculations that prove this and similar secondary claims have been skipped. However, they all are available from the authors upon request. 25 The first term is nonpositive since ""P a1 j aj $ $ − n ≤ 0. And it is strictly negative unless a1 = ... = an . The second term is also negative since 1 1 0 0 X a1 A (r (nγ − γ + 1) + (1 − β) (1 − α + nα)) − n (r − β + 1) pr (α − γ)A > 0. @pr (α − γ) @ aj j This establishes that egalitarian social welfare is decreasing in α when γ < α. Similar calculations show that the egalitarian social welfare is decreasing in γ when γ > α and increasing in γ otherwise. Claim 1 All P E [x, x] agreements induce the same amount of total contributions. However, a partner more (less) risk averse than average responds to an increase in x by increasing (decreasing) his contributions. Proof. Under P E [x, x], the total contribution expression (used in the proof of Theorem 1) "P $ " $ P ∗ pr 1 1 ln simplifies to si = r−β+1 ai (1−β)(1−p) . Note that the expression is independent of x, proving$the first claim. $ %% n a1 −(1−x) i P 1 N aj ln nx(r+1−β) " respect to x, we obtain For the second claim, note that Expression 1 simplifies to s∗i = pr (1−p)(1−β) " "P 1 n # N under P E [x, x]. Taking the derivative of this expression with # pr $ $ ln" (1−p)(1−β) 1 1 . The second part of the expression is posaj − ai x2 (r−β+1) itive (by Profitability). Thus, the sign is determined by the first part. If agent i is more (less) risk averse than average, this term is positive (negative), the desired conclusion. 26