Core Allocations in Co-investment Problems

In a theoretical co-investment problem, a set of agents face a surplus-sharing situation with a single input and a single output exhibiting increasing average returns. All agents contribute their respective inputs and expect part of the collective output. Focusing on the core of the problem, we analyze whether a core allocation of the output is acceptable or compatible with a variation of input contributions, where larger payoffs are expected by those agents whose contribution has increased. We state a necessary and sufficient condition for a core allocation to be acceptable. We also introduce and study the acceptable core, that is, those core allocations acceptable for any possible increase of inputs. Finally, we axiomatically characterize when a set-solution that contains acceptable core allocations shrinks into the proportional allocation.


Introduction
In many economic and social situations agents cooperate exerting some level of effort in order to obtain a common profit. The implication of each participant in the joint project depends on the payoff agents expect to receive. The negotiation process determines the level of effort and the allocation of the total gain. In this paper, we focus on specific resource-pooling situations that we name co-investment problems.

Co-investment Problems
General co-investment problems are situations where a set of agents pool their resources to obtain a surplus. In these problems, the effort of an agent is represented by the corresponding resource contribution and the common profit is the surplus obtained. Examples of these situations are the joint investment in infrastructures by the operators in the telecommunication sector (see Bourreau et al. 2012 andBourreau et al. 2018), cooperation at the supply chain level between firms (see Kogan and Tapiero 2012), or the situation where agents pool their money to make a joint investment.
In corporate finance, it is remarkable the fast and quite recent development of private equities firms (PE) that collect capital from investors who act as limited partners (LPs). On behalf of their LPs, the fund manager of the private equity firm (the general partner or GP) selects portfolio companies in which they invest capital and retain them with the aim to increase their value during a limited period, after which they exit. The literature on this topic is reviewed in Tykvová (2018). Recently, we can find more sophisticated funding figures that require the co-investment between private equities and other external limited partners that enable qualified investors to invest alongside a private equity firm without paying the usual fees charged by a fund manager (see Braun et al. 2020 ). Another co-investment structure is the syndication among Venture Capital firms (see Khavul and Deeds 2016 ).
One of the most relevant aspects of private equity is how profits are distributed. Essentially, limited partners pay an annual fee (management fee) to the general partner. This payment is typically 2.0% of the fund's committed capital 1 . The compensation agreement structure also specifies a hurdle rate (most often 8%) or minimum rate of return required, before the fund manager (GP) is awarded a performance fee called carried interest (typically 20% of profits). There are other aspects and combinations of them that affect how and when this compensation is awarded, and thus that defines different possible distributions schemes of profits. In all of them, the aggregate payoff that goes to the LPs is distributed among them proportionally to their capital contribution. A detailed description of different compensation agreements and a study of the performance of different funds raised between 1993 and 2006 can be found in Metrick and Yasuda (2010).

A Model of Co-investment
In this paper we propose a theoretical model of co-investment that is mainly addressed, but not only, to financial applications. We analyze cooperative investment situations where a group of agents, each of them endowed with a certain amount of capital, has the opportunity to jointly invest them to obtain higher rewards. Furthermore, we assume the returns are given by a function that exhibits increasing average 1 3 Core Allocations in Co-investment Problems returns that facilitates the stability of cooperation. The model aims to analyze possible allocation of the joint profit obtained.
As we have pointed out in the previous subsection, the reward for the limited partners in a PE is proportional to their capital contribution and hurdle rates are usually fixed between the range from 6% to 10% (Metrick and Yasuda 2010). Moreover, the final rate of return to LPs strongly depends on the carry compensation to the GP, that it is tradition to fix at the level of 20% 2 . In our model, we do not suppose any rate of return, but we impose stability and incentive principles to validate the payoff to agents. We next develop these two principles.
One way of providing agents with incentives to cooperate is to guarantee that the payoff that each of them will receive, is larger than the individual gain each participant would obtain acting alone. However, this might not be enough to convince agents to fully cooperate since sub-coalitions of agents could also evaluate whether the aggregate payoff received by their members compensate the joint effort exerted by them. Thus, a necessary condition for a sub-coalition of agents to fully cooperate is to propose to its members an allocation that guarantees a joint payoff larger than the profit the sub-coalition might obtain by itself. In the economic literature (see Telser 1994), and specifically in the field of cooperative game theory, the above argument requires the allocation to be in the core of the problem. This is a stability principle.
Another reasonable principle is the following: any eventual increase in the effort exerted by each individual of a group of agents should be rewarded with a strictly higher payoff. This is an incentive principle based on the economic criterion that no agent will accept any redistribution of wealth that worsens his initial position. To characterize allocation rules, related incentive properties are considered in Friedman (2004), for surplus sharing problems, and in Casajus and Huettner (2014), applied to cooperative games. Friedman (2004) names the incentive property as demand monotonicity. There are three main differences between this property and the concept used in our paper. First, the definition of demand monotonicity applies to single-valued solutions (or rules), while in our model we use monotonicity to test and validate arbitrary core allocations. Second, Friedman's concept of monotonicity considers the effect of an input increase of only one agent (ceteris paribus). Finally, the payoff to this agent must weakly increase, and not strictly increase as in our model. Casajus and Huettner (2014) introduce the concept of weak-monotonicity to characterize the egalitarian Shapley value for cooperative games. According to weak-monotonicity, a player's payoff weakly increases whenever her marginal contribution and the grand coalition worth weakly increase. If we interpret our model as a cooperative game, weak-monotonicity refers to the case when only one agent increases her input contribution. Our paper aims to overlap the above two principles to test the fairness and robustness of core allocations in a specific model of cooperation.
Co-investment problems are introduced as a cooperative games in Rafels (1996, 2001). In these papers, it is proved the non-emptiness of the core and its coincidence with some different bargaining sets. Some economic extensions of this model are developed in Borm et al. (2001), De Waegenaere et al. (2005 and Gulick et al. (2010).
In Borm et al. (2001) agents invest their capital in a bank deposit whose rate of return depends on the term of the investment, on the capital invested or on both. Generically, they name these situations as deposit games and they analyze under which situations a deposit game is balanced (i.e. with a non-empty core).
The other two papers analyze profit allocations where investors can select different investment projects to meet capital requirements for long term investment problems. Furthermore, investors may reinvest intertemporal gains from existing projects into new projects. Stable profit schemes are analyzed by using linear programming techniques.

An Explanatory Example
Let us illustrate the two principles discussed in the previous subsection through an example. Consider three individuals having respectively $1000, $1000 and $2000 to invest for this year. Suppose one-year deposits earn 2% interest per year if the capital deposited is strictly smaller than $3000 and 4% interest per year in other case. Investing their money separately, any individual would obtain the 2% of interest; this way, both agent 1 and 2 would get $20 and agent 3 would get $40. Pooling all their savings, individuals can jointly obtain the 4% of interest resulting in a total reward of $160. From the individual point of view, any allocation of the $160 should assign the first and second agent a payoff larger than $20 and the third agent a payoff larger than $40. However, the stability principle extends this requirement to two-agent groups of individuals. For instance, if agent 2 and agent 3 jointly invest their savings, they can get 4% and a potential joint gain of $120. The same analysis applies to the group of agents 1 and 3 with a joint potential gain of $120. However, the joint capital of the group formed by agents 1 and 2 does not reach the threshold of $3000 and thus the potential coalitional gain is just the sum of individual gains, $20+$20=$40.
The core of the problem is formed by those allocations that provide each oneagent and each two-agent groups a joint payoff larger than the corresponding potential gains. Many allocations satisfy the stability or core principle. For instance, if we allocate $30 to agents 1 and 2 and $100 to agent 3, that is, if we take the allocation (30,30,100) it is easy to check that stability or core conditions are all satisfied. The question is whether this allocation (30,30,100) is compatible with other related deposit problems where the interest structure is the same, but capital contributions are different. Consider another co-investment problem, with the same interest rate structure, but now capitals invested are $2000 by agent 1, $1000 by agent 2 and $2100 by agent 3. In this new situation, only agents 1 and 3 have increased their investment. It is easy to check the total gain is $204 and the unique core allocation is (80,40,84) (see Proposition 1). Notice that in this case, despite having increased the capital contribution, the payoff to agent 3 must necessarily decrease if we want to preserve stability. The stability principle and the incentive principle come into conflict. In this case, we say that the core allocation (30, 30, 100) is not acceptable.

Contributions and Results
A key concept of the paper is that of acceptable core. The acceptable core contains those core allocations that are consistent with both the stability and incentive principles: if some agents increase their contribution, it is possible to reward these agents with an extra payoff without breaking the stability of the allocation. Consequently, the originality of our approach is to discriminate between core allocations using an incentive principle. So far, to the best of our knowledge, no such discrimination has been used before.
The main results of the paper are the following. We begin showing that the proportional allocation is always an acceptable core allocation (see Proposition 3). However, Example 2 reveals that the acceptable core might contain multiple allocations and also illustrates that the core can be strictly larger than the acceptable core. After this preliminary analysis, we deal with two interesting questions. First, given an arbitrary co-investment problem, which is the set of all acceptable core allocations? Second, under what conditions can we guarantee the coincidence of both the acceptable core and the core? The first question is addressed by Theorem 1 and Proposition 2 of Sect. 3. In Theorem 1 we characterize when a core allocation can sustain the incentive principle. Besides, Proposition 2 studies the interesting situation where exactly only one agent increases the input contribution. The answer to the second question about the coincidence of the core and the acceptable core is addressed in Proposition 4 of Sect. 4. It states that the convexity of the output function it is enough to guarantee that coincidence. The interpretation of this result is that, as long as a payoff allocation is in the core of a convex co-investment problem, any increase in the investment level can be always encouraged. In Sect. 5, Theorem 2, we analyze which are the properties that restrict the subset of admissible allocations to contain just the proportional one. If we add to stability and to the incentive principle, the property of core invariance (if two co-investment problems have the same core, then admissible allocations are the same), then the unique possibility is to pick out the proportional allocation.

Preliminaries
Given N = {1, 2, … , n} , we denote 3 by ℝ N the n-dimensional space whose components are indexed by N. Moreover, let ℝ N + and ℝ N ++ , stands for the space of non-negative and positive vectors in ℝ N , respectively. Given a non-empty coalition S ⊆ N and a vector x ∈ ℝ N , we denote x S = ∑ i∈S x i , being x ∅ = 0 . Furthermore, given two vectors x, y ∈ ℝ N we denote by x ≤ y the case where x i ≤ y i , for all i ∈ N.
A co-investment problem is represented by a triplet (N, f , ) where N = {1, 2, … , n} , n ≥ 2 , is the set of agents, = ( 1 , 2 , … , n ) ∈ ℝ N ++ is the vector of resource endowments, where i is the amount of resource owned by agent i ∈ N , and the function f ∶ ℝ + → ℝ + represents the technology that transforms x units of input into f(x) units of output, with the following assumptions: Condition (a) states that no output can be produced with no input contribution and that some output is produced if all agents contribute. Condition (b) formalizes the classical idea of increasing average returns. We emphasize that neither continuity nor convexity is assumed in our model; in some examples we will use step-wise functions with discontinuities. As a consequence of (a) and (b) we have that f is a monotonic increasing function, being strictly monotonic in the positive range.
The problem to solve is how to distribute the output f ( N ) among the members of N.
In this case we say vector x satisfies efficiency. The core of a co-investment problem (N, f , ) is defined as It coincides with the classical definition of the core of a cooperative game. It is easy to check from condition (1) that the proportional allocation with respect to the initial endowments of input is always a core element, that is, At the proportional allocation, the return per unit contributed is constant and so different amounts of input receive the same average return. However, the core of a coinvestment problem is in general wider than the proportional allocation. Next proposition reveals when this happens since it states a necessary and sufficient condition to reduce the core into a singleton. The condition says that the average return of all n − 1 person coalitions is the same as the average return for the grand coalition.
Proposition 1 For any co-investment problem (N, f , ) the following statements are equivalent. (2) Proof (1) → (2). For any agent i ∈ N we define the allocation x i = x i k k∈N ∈ ℝ N as follows: We claim that vector x i , for each i ∈ N , is a core allocation, x i ∈ C(N, f , ) . It is efficient Moreover, let S ⊆ N be an arbitrary coalition of N. If agent i ∉ S then, by (1), where all the inequalities come from (1). As a consequence of the above claim we obtain the implication we want to prove since, if (2) → (1). It is known that for any arbitrary core element x ∈ C(N, f , ) the payoff to any agent i ∈ N , x i , is bounded above by its corresponding marginal contribution 4 , i.e. (3) , and The marginal contribution of an agent i ∈ N measures the effect on the total output of adding agent . It is easy to check that for any core element x ∈ C(N, f , ) , the payoff x i of any agent i ∈ N cannot exceed its marginal contribution; otherwise , but this would contradict x to be a core element.
where the last equality follows from hypothesis (2) of the proposition. Hence, by efficiency of any core allocation, we reach x = P (N, f , ) . ◻ In the proof of this result we have shown that vector x i defined in (3) is in the core of the problem, for any arbitrary agent i ∈ N . This shows that the marginal contribution of any agent i ∈ N , f ( N ) − f ( N⧵{i} ) , is always attainable as a payoff to this agent in the core of any co-investment problem.

Acceptable Core Allocations
As we have commented in the introduction, we are interested in a subset of the core of the co-investment problem (N, f , ) . This subset includes all core allocations that meet the following incentive test: if some agents increase their initial investment, their expectations of receiving a higher payoff can be supported by the existence of a new core element that guarantees this larger payoff. Next proposition shows that any core allocation satisfies this requirement if just one agent increases the input contribution.
. Taking this fact into account, take x ′ as: Recall that we have already argued in the proof of Proposition 1 that x i * is a core allocation, that is And we end the proof. ◻ Unfortunately, Proposition 2 cannot be generalized to the case where two or more agents increase their initial contribution. Next example shows that, if two agents increase their contribution at the same time, not every initial core allocation can be adapted to the new problem, that is, both players cannot benefit simultaneously from increasing their initial contribution.
Example 1 Let = (1, 2, 3) be the vector of initial endowments for a three-agent coinvestment problem where f (x) = x , for 0 ≤ x < 5 , and f (x) = 1.5 ⋅ x , for 5 ≤ x . The allocation x = (1, 5, 3) is in the core of the problem, C (N, f , ) . Let us suppose that players 1 and 2 increase their initial contribution by 1 unit, that is � = (1 + 1, 2 + 1, 3) = (2, 3, 3) . Notice that for any core element The above example shows that the core vector x = (1, 5, 3) is not acceptable since agent 2 has received an excessive amount at the initial distribution: it is not possible to increase his payoff without breaking the stability of the allocation. The next definition formally states when an allocation is acceptable given ′ .
Definition 1 Given a co-investment problem (N, f , ) and a vector � ∈ ℝ N , with ′ ≥ , we say that a core allocation We denote the set of all acceptable core allocations with respect to ′ , ′ ≥ , by A � (N, f , ).
Notice that if � = the acceptable core coincides with the core. Furthermore, as we have already pointed out in the introduction, the next proposition shows that the proportional allocation satisfies the interesting property of being always acceptable. (N, f , ) , the proportional distribution P (N, f , ) is acceptable with respect to any � ∈ ℝ N , ′ ≥ .

Proof
Recall P(N, f , ) ∈ C(N, f , ) . Let us assume ′ ≥ , ′ ≠ , since the case And we are done. ◻ Next theorem gives an algebraic description of the acceptable core set with respect to ′ , ′ ≥ . This characterization is useful and allows to develop interesting properties and numerical examples. For instance, we will use it in Example 2 to check that the acceptable core might be a proper subset of the core. (N, f , ) be a co-investment problem and � ∈ ℝ N an input vector such that ′ ≥ . The acceptable core of (N, f , ) with respect to ′ , A � (N, f , ) , is described by the core elements x ∈ C(N, f , ) satisfying the strict inequalities

Theorem 1 Let
Proof If � = , then S * = ∅ and the result holds trivially. From now on, let us suppose ′ ≠ and thus S * ≠ ∅ . First of all, for any acceptable core element Hence, for all nonempty R ⊆ S * , R ≠ S * , we have where the last inequality follows since x � ∈ C(N, f , � ) . Therefore, x satisfies the inequalities given in the statement of the theorem.
To see the other inclusion, let x ∈ C(N, f , ) be a core element. Notice that Then, define vector z ∈ ℝ N as follows: Notice is well defined by (5), (6) and the hypothesis of the theorem. We claim vector z satisfies: (i) for all nonempty R ⊆ S * , . For proving (i) notice that for all R ⊆ S * , R ≠ ∅ , we have For proving (ii) notice that Next we define the set 5

3
Core Allocations in Co-investment Problems The next claim states that for any coalition R in D S * , there are increasing average adjusted returns outside R. The proof can be found in the Appendix.
Claim For any R ∈ D S * and for any pair of coalitions S 1 and S 2 such that ∅ ≠ S 1 ⊆ S 2 ⊆ N ⧵ R and any Q ⊆ R we have We use this claim to finish the proof of the theorem. We consider two cases: Case 1.-D S * = ∅ . In this case, just by definition of D S * , it follows that shows vector x is in the acceptable core with respect to ′ .
Case 2.-D S * ≠ ∅ . Let R * be a maximal coalition with respect to the inclusion in D S * . Denote R * = {i 1 , … , i r * } where * = (i 1 , … i r * ) is the permutation of the set R * that fulfills all the inequalities of D S * . We first check that R * ≠ N . To see it, if R * = N , then R * = {i 1 , i 2 , … , i n } and, by definition of D S * , we would have which implies z N ≥ f ( � N ) and contradicts (8). Thus, we conclude R * ⊆ S * and R * ≠ ∅, N. Now define the vector x � ∈ ℝ N as follows: To complete the proof we must show that x ′ i > x i for all i ∈ S * and that x � ∈ C(N, f , � ) . The strict inequalities follow since, by definition of vector x ′ , we have where the first strict inequality in the last expression comes from the maximality of coalition R * in the set (D S * , ⊆).
For proving where the second equality holds since, for any Q ⊆ R * , Q ≠ R * , we have which implies that To prove the core inequalities of the vector x ′ , we must check that To this aim recall R * = {i 1 , … , i r * } and * = (i 1 , … , i r * ) . Then, two subcases are considered: Case 2.a. Coalition T satisfies T ⊆ R * = {i 1 , … , i r * } . If T = {i 1 } , then since R * ∈ D S * , we have If T ⊆ R * , T ≠ {i 1 } , let i t be the latest agent in T with respect to the permutation * = (i 1 , i 2 , … , i r * ) of coalition R * = {i 1 , i 2 , … , i r * } . In other words i t is the unique agent of T satisfying T ⧵ {i t } ⊆ {i 1 , … , i t−1 } . Then, we have

3
Core Allocations in Co-investment Problems and thus we obtain the inequality x � T ≥ f ( � T ) . In the above argument, the first inequality comes from the definition of D S * , and the second one follows by (9) In this case, we have where the first inequality comes from (9) taking R = R * , S 1 = T ∩ (N ⧵ R * ) ≠ ∅ and S 2 = N ⧵ R * and the second inequality taking Q = T ∩ R * . And the proof of the theorem concludes ◻ Let us remark that Proposition 2 is a special case of Theorem 1, when |S * | = 1 ; for this case no strict inequality needs to be checked and hence, the result of the theorem is unconditional. Nevertheless, the proof of Proposition 2 contributes with a direct and explicit core allocation that improves the payoff to the single agent that increases his contribution.
The above characterization offers a new inside to understand the trade off between core stability and incentive monotonicity. It states a positive and useful tool to check whether an allocation is acceptable in the context of co-investment problems. For instance, coming back to Example 1, it is easy to check that the acceptable core with respect to � = (2, 3, 3) is described by the set In Fig. 1 we represent this set, illustrating how the core is reduced. The core is the convex region determined by the points A, B, C and D ( ). The acceptable core with respect to � = (2, 3, 3) is the convex hull determined by the allocations A, B, C and E ( ). Notice that the segment CE is not included in the acceptable core since the strict inequality x 2 < 4.5 applies; on the other way around, this example also shows how some of the constraints that determine the acceptable core do not apply (in the example, the inequality x 1 < 3).

3 4 The Acceptable Core
In the previous section we have characterized when a specific core allocation is acceptable for a given input vector ′ , ′ ≥ . Now, we want to analyze the core allocations that are acceptable no matter the increase in the input contribution.

Definition 2
The acceptable core of a co-investment problem (N, f , ) is the set This is, a core allocation is acceptable if it is acceptable for any ′ , ′ ≥ . Let us point out that the acceptable core of a co-investment problem is, by definition, a subset of its core. Moreover, we have already stated in Proposition 3 that the proportional allocation is always acceptable; thus, the acceptable core is nonempty since it contains at least this allocation. Summarizing these preliminaries The following example shows that the acceptable core might include several allocations, not only the proportional allocation, but it might be strictly smaller than the core.
In Fig. 2 you can check that the acceptable core is larger than the proportional allocation but strictly smaller than the core. Indeed, the acceptable core of this numerical example is the convex hull of the extreme points A = (3, 3, 12) , B = (1, 5, 12) , C = (1, 8, 9) and D = (3, 8, 7).
Since the acceptable core is included in the core, let us describe a sufficient condition for the coincidence of both sets. As the next proposition states, the condition is based on assuming the function f to be convex. It is interesting to remark that when the function f is convex, the output of the different coalitions (f ( S )) S⊆N can be interpreted as a convex cooperative game (Shapley 1971), although the converse is not true as Example 1 illustrates.

Proposition 4 Let (N, f , ) be a co-investment problem and let f be a convex function. Then,
Proof Since f is a convex function it holds that 6 , for all 0 < x < y and z > 0, Now, let (N, f , ) be a co-investment problem and take a coalition S * ⊆ N , We must first prove that inequalities (5) in Theorem 1 hold, for any x ∈ C (N, f , ) . To this aim consider an arbitrary coalition ∅ ≠ R ⊆ S * , R ≠ S * and let us suppose to the contrary A(N, f , ) = C (N, f , ).

3
Core Allocations in Co-investment Problems where the second inequality comes from (11) taking Moreover, the strict inequality holds since � N − ( � S * ⧵R − S * ⧵R ) > N and f ( N ) > 0 , and the last inequality, since x ∈ C (N, f , ) . Hence, we reach a contradiction. We conclude The previous result shows a particular situation where the acceptable core is as large as possible and coincides with the core. On the other side, the next proposition states a sufficient condition for the acceptable core to just contain the proportional distribution. The condition requires that the average return remains constant beyond N . (N, f , ) . For any agent i ∈ N we take i = ( i k ) k∈N ∈ ℝ N as i i = i + , where > 0 , and i k = k + N , for all k ∈ N , k ≠ i . By Theorem 1 we obtain where we have used that f is average stationary at N . Being > 0 arbitrary, we obtain , or equivalently, x = P(N, f , ) . ◻

Proposition 5 Let (N, f , ) be a co-investment problem where the function f is average stationary at
The above proposition shows that the acceptable core of Example 1 reduces to the proportional distribution P(N, f , ) = (1.5, 3, 4.5) ; see Fig. 1 for a graphical location.

The Proportional Allocation: A Characterization Result
The proportional allocation is always acceptable (see Proposition 3). However, the acceptable core might include many other allocations. In this section, we look at setsolutions that assigns to any co-investment problem a subset of the acceptable core and we characterize when this subset just contains the proportional allocation.
Let us denote by a set-solution that assigns to any co-investment problem (N, f , ) a nonempty subset of allocations of the problem (N, f , ) ; this is We propose the following three properties 7 for .
Next, select an arbitrary pair of different agents i and j and, given > 0 , define Furthermore, since satisfies core selection we have that x � ∈ C(N, f � , � ( )) and thus x ′ i is bounded above by the marginal contribution of agent i, this is 7 These properties are adapted from Di Luca et al. (2013). Since As agents i and j have been arbitrarily selected, we conclude that Finally, since x is an allocation for the problem (N, f , ) we conclude for all i ∈ N and thus x = P (N, f , ). ◻ Let us point out that the three properties are logically independent. The core, i.e. (N, f , ) = C(N, f , ) satisfies core selection, core invariance, but, as we have analyzed, it is not incentive monotonic. The set solution that contains only the equal division allocation, (N, f , ) = x i = f ( N ) |N| i∈N , satisfies core invariance, incentive montonicity, but it is not a core selection. Finally, the solution defined as } , if f is not convex, satisfies core selection, incentive monotonicity, but it is not core invariant.

Conclusions
In this paper, we have described and analyzed co-investment problems and the compatibility between a stability principle and an incentive principle. The two principles collaborate to reinforce cooperation: the first one avoids breaking cooperation; the second one guarantees the effort of each agent is rewarded accordingly. The analysis carried out in the paper focuses on testing core allocations. A core allocation is validated if, in the eventual situation where some arbitrary agents increase their contribution, there is another core allocation that gives those agents a larger payoff.
There is another interesting aspect that is not analyzed in this paper and refers to the study of the compatibility of the two principles applied to allocation rules. That is, given a rule that assigns to each co-investment problem a unique stable allocation, does this rule assign a larger payoff to some agents whenever they increase their contribution? Does it exist at least one such a rule? The answer to this last question is in the positive: the proportional rule with respect to the initial endowments of input satisfies these requirements and it is one of the most used rules. However, the proportional rule assigns the same payoff per unit contributed and do not discriminate between agents with different contributions. It might be interesting to describe and characterize other possible allocation rules.
Another aspect not covered by the paper is the impact of the increase in the resource contribution made by some group of agents on the rest of agents. In some cases, if we want to preserve stability, the increase in the payoff to some agents implies, with no option, the decrease in the payoff to other agents. This work is left for further research and its main interest relies on the fact that new investment rounds sometimes need a unanimity approval. Furthermore, if we interpret a co-investment situation as a cooperative game, the associated game is an average monotonic game (Izquierdo and Rafels 2001). Average monotonicity has been analyzed and generalized in different areas of game theory as in Corcho and Ferreira (2003) , Sagara and Vlach (2011), Liu et al. (2013), Liu and Tian (2014) or Hong (2016).
In Corcho and Ferreira (2003), generalized externality games are introduced. Two interesting economic situations fall inside this model: provision of public goods and a joint venture among firms. Despite they do not fit in our model of co-investment, generalized externality games are average monotonic games and a parallel analysis similar to the one carried out in this paper gets its interest. Sagara and Vlach (2011) extend the notion of average monotonicity to cooperative games with a continuum of agents, reaching the non-emptiness of its core and leaving open the coincidence between the core and some bargaining sets.
In Liu et al. (2013) it is proved that all the results for average monotonic cooperative games are suitable to be replicated in the field of cooperative fuzzy games including examples and models of financial applications. In Liu and Tian (2014) they exhibit, for the first time, the model of NTU fuzzy game associated with a financial cooperative problem. Finally, in Hong (2016), the author introduces and develops the Average monotonic fuzzy interval cooperative games. In many of these works a similar analysis like the one made in this paper could be applied.
Finally, we would like to point out two potential practical implications for our work. First, compensation schemes in private equity firms could be revised looking at the potential expected profits of sub-coalitions of partners. This will allow discriminating between large and small investors, and not only applying rates of returns by default or just the proportional allocation. Second, it seems interesting to introduce our analysis in the field of group buying problems or cooperative purchasing situations, see Anand and Aron (2003) or Schaarsberg et al. (2013). Focusing on bundling order quantities to obtain discounts, the problem faced by a group of agents analyzing purchasing problems with decreasing average costs is the counterpart of co-investment problems.
Proof The proof will be argued by induction on the cardinality of the size of coalition R. If |R| = 1 , i.e. R = {j 1 } ∈ D S * there are just two cases. If Q = ∅ then (14) is and it follows just by (1). If Q = {j 1 } then (14) is To check this last inequality, notice that where the first inequality follows by (1) and the second one since, by definition of the set D S * , we have z j , and thus Suppose now that for any R ∈ D S * such that 1 ≤ |R| ≤ k and all Q ⊆ � R we have for all ∅ ≠ � S 1 ⊆ � S 2 ⊆ N ⧵ � R . Let R ∈ D S * , with |R| = k + 1 , that is R = {j 1 , j 2 , … , j k , j k+1 } where = (j 1 , j 2 , … , j k , j k+1 ) is the ordering in which the inequalities required to be in the set D S * are satisfied. Let us remark that the coalition R ⧵ {j k+1 } is also in D S * . (14)