Relationship‐Specific Investment and Multiple Interested Parties

This paper proposes a solution to the hold&#8208;up problem for situations where a seller interacts with several parties who have a common interest in inducing the seller to make a quality&#8208;enhancing investment. Since parties can observe purchase decisions, they are able to add a clause to the contract specifying that an extra payment should be made to the seller, conditional on verifiable previous purchases. Since previous purchases depend on observable but not verifiable quality, this &#8216;conditional multilateral contract&#8217; provides correct incentives to restore efficiency, and is feasible in the context of both, complete and incomplete information.


IntroductIon
In its classic version, a hold-up problem arises when a relationship-specific investment increases the value of the transaction among parties, but parties are unable to specify a complete contract defining the division of the returns from the investment. Anticipating that future negotiations may confer parts of the benefit from the customized investment to the non-investing party, the investing party will under-invest in comparison to the social optimum (Williamson, 1985;Holmström and Roberts, 1998;Hart and Moore, 1999). This paper proposes a solution to the hold-up problem for situations where a seller interacts-and potentially trades-with several parties who have a common interest in inducing the seller to make a quality-enhancing investment. The quality of the product is observable for the buyer(s) before purchase, but not verifiable. Thus, contracts contingent on quality are infeasible. Also, the level of the quality-enhancing investment is not verifiable, implying that contracts contingent on the investment level are not feasible, either. However, previous trades are observable and verifiable. This allows parties to agree on contracts that take advantage of the presence of multiple buyers and the sequence of purchase decisions. Specifically, parties are able to solve the hold-up problem, by making transfers paid by some party contingent on the verifiable purchase decision of other buyer(s). 1 More specific examples include SAP, developing software for firms operating in different industries, such as E.ON SE, one of the largest private energy companies in Europe, or B. Braun, one of the world's leading health-care suppliers (SAP, 2013). Air force authorities from different countries wishing to buy the Lockheed Martin F-35 Lightning II may sign a conditional multilateral contract as proposed here (GlobalSecurity.Org, 2011). Another example is Gore-Tex ® supplying materials to distributors targeting different audiences, such as Black Diamond Equipment or Burton (Gore-Tex ® , 2013a, 2013b). 2 The question might arise whether such a contract is in line with competition laws or not.
Since the contract (i) leads to increased overall welfare due to efficient investment and (ii) evidently does not harm consumers, it is in line with regulations both in the EU and in the US (EU: paragraph 101/102 of the EU Competition Law; US: the Sherman Act, the Federal Trade Commission Act and the Clayton Act). since pledges of the form 'I pay w to x if y number of people buy z' are feasible). A second example of successful crowdfunding through a contract similar to the proposed multilateral one is the development and production process of the fitness e-watch Pebble. While no party promised to pay a premium upon others purchasing the product, the purchase decision of each buyer (and the production decision of the firm) was conditional upon a sufficient number of supporters placing a pre-order. As a result, the campaign raised $10 million in (kickstarter.com, 2012Forbes.com, 2012;Money, 2013). As in the previous example, the form of the contract was similar to the one proposed here, although the purpose of the latter is different. Finally, certain forms of multi-sided markets Tirole, 2003, 2006) are also similar in structure to the one proposed here, such as video game platforms or any kind of media supported by advertising. No multilateral contract is involved-not even a contract conditional on the behaviour of other parties-and the incentives for one side to pay a premium are related to a consumption externality, rather than to a signal of the quality of the good. Yet, the two settings share an important characteristic. Both, in the case of multi-sided markets as in the proposed contract, only after sufficient (end-)customers from one side buy a unit, customers from the other side (game producers or advertisers) will be willing to pay (more) for a unit of the good. 4 Even among multi-sided markets, there are cases in which parties sign contracts which specify a premium that the company pays to the platform, conditional on the behaviour of other parties. Facebook, e.g. offers advertisement contracts where the advertiser pays, conditional on the number of views and clicks generated by the users. Similarly, one may find evidence in editorial contracts in which the publisher starts paying royalties to the author of a book, but only when sales cross a certain threshold.
In the model presented here, relation-specific investment generates direct benefits only to the seller's trading partners and not to the seller himself. Che and Hausch (1999) call such investments 'cooperative', to distinguish them from investments that benefit only the investor, which are called selfish. Che and Hausch find that contracting creates additional value to the parties when investment is cooperative, and committing not to renegotiate is possible. If renegotiation cannot be ruled out, Maskin and Moore (1999) and Segal and Whinston (2002) have shown that the first-best outcome cannot be attained. Hence, the question arises whether it is possible or not to commit not to renegotiate. One possibility to achieve such a commitment is to include a third party who is able to observe quality and may act as an intermediary or arbitrator, threatening to punish renegotiators (Dixit, 2004). In the contract presented here, renegotiation is not an issue since the contract is incentive compatible. The purchase decision of buyer one translates the unverifiable state of quality into a verifiable event, and the transfer paid by the second buyer (the 'third party') is contingent on this event. Renegotiating the price of the good cannot improve the positions of the respective parties.
Hence, the question whether the contract is feasible or not transforms into a question of whether collusion can be ruled out or not (Hart and Moore, 1999;Maskin and Tirole, 1999). Collusion is a potential problem here and needs to be ruled out, so that the solution holds. This latter property might be considered a major drawback. However, if the number of buyers is large, and / or the buyers are unable or unwilling to communicate (for instance, because they are firms in the same industry that respect the cartel law), then the assumption of 'no collusive behaviour' seems less problematic.
The present paper does not rely on other means that typically have been employed to remedy the hold-up problem. Demski and Sappington (1991) consider a moral hazard problem, where efforts of a principal and an agent are not verifiable, yet the principal observes the agent's efforts. Efficient effort levels can be induced by means of an option contract that specifies a price for which the principal can sell his firm to the agent. This paper proposes a solution that does not involve possible changes in asset ownership. Overall, no (vertical) integration or restructuring of firm boundaries is required (see also Klein et al., 1978;Grossman and Hart, 1986;Baker et al., 2002). In addition, the contract does not rely on repeated interactions with the same agent or within a group (Radner, 1981;Kandori, 1992;Dixit, 2003). The one-time interaction among the agents involved in the transaction suffices to induce efficient incentives.
Other papers have considered dynamic hold-up problems, as well. In Nöldeke and Schmidt (1998), a buyer and a seller sequentially invest cooperatively, both facing a hold-up problem. Giving the buyer the option to acquire the investing firm at a predetermined price at some later date induces efficient investment and explains the use of contingent ownership structures in joint ventures. Also, in the work of Pitchford and Snyder (2004), sequential investments by the seller and sequential payments by the buyer lead to efficiency. However, the present setting requires more than one interested party, which, however, implies that the downstream parties do not collude, that is they do not act as one player.
A paper related to the transformation of a non-verifiable event into a verifiable one is Nöldeke and Schmidt (1995). Focusing on investment that only directly affects the investor, they show that when the supply of the seller is observable, an option contract remediates the hold-up problem. In my model, while trade has to be verifiable, if no trade takes place, there is no need to distinguish between whether the seller refuses to supply or not and whether the buyer refuses to accept delivery or not.
The remaining part of the paper is organized as follows. Section 2 introduces the basic model under complete information. Section 3 extends the model to incomplete information. Section 4 concludes the paper.

Introduction Model
There are three players, a seller S and two buyers, B 1 and B 2 . 5 S produces two units of a good. The cost of producing a unit is normalized to zero. The two units are of the same quality, and quality depends on level e of a quality-enhancing investment chosen by S. Investment is costly for S and the investment cost, denoted as c(e), is a strictly increasing function. For the sake of simplicity, I assume a binary relationship between investment and quality, in that there exists an investment level e* such that for any e < e* the product is of low quality and for e ≥ e* the product is of high quality. 6 Each unit of the good can be sold either to one of the potential buyers (each is interested in at most one unit), or potentially to the market. The price in the market is exogenously given, and might depend on quality (but does not necessarily have to). Specifically, a unit of low-quality yields m l ≥ 0 and a unit of high quality yields m h ≥ m l in the market. The two (potential) buyers B 1 and B 2 both have valuations of zero for the low quality product 7 and a strictly positive valuation for the high quality product. Let the valuation for a unit of the high quality product be v i , with v i > m h for i = 1,2. Furthermore, the first part of this condition states that the cost of the investment is not higher than the additional value created by the high quality over the low quality for the two buyers-otherwise, investing would never be efficient. The second part states that the cost of investment is higher than the additional value created by high quality over low quality in the market-otherwise, no hold-up problem would exist.
All players are risk neutral and their utility payoffs are just the material payoffs. There are no transaction costs.
The timing of the game is shown in Fig. 1. At time t = 0, partners decide upon the terms of the contract. At t = 1, S chooses the level of investment, and production occurs. At t = 2, the two buyers B 1 and B 2 sequentially 5 For the sake of simplicity, in the rest of the paper, I will concentrate on two buyers. The extension to more than two buyers and the extension to one buyer and one investor is straightforward. 6 Appendix B presents a cleaner, continuous version of the relationship, where an increase in e increases the probability that the good is of high quality. 7 While this is not necessary for the result to go through, it simplifies the analysis greatly.
Furthermore, it seems very natural in many transactions involving relationship-specific investment or customized products (Lülfesmann, 2012).
decide whether to buy or not to buy at a price p, after having observed the quality of the good. 8 If investment was verifiable, but quality was not, an efficient outcome could be reached that guarantees each party at least the reservation payoff, e.g. by a contract that specifies an investment e = e* and a price p ∈ [ 1 2 c(e * ),v i ]. Also, if investment was not verifiable, but quality was, there exist incentive compatible contracts which induce efficient outcomes. Any contract which specifies a pair of prices ,2} and lump sum transfers ∈ R + to distribute profits induces an efficient outcome.
If neither quality nor investment is verifiable, a contract that specifies a price independent of the realized quality of the good does not induce an efficient level of investment, since S has no incentive to produce high quality. An option contract, which specifies a pair of prices with p h > m h (and p l ≥ 0) is not self-enforcing, and therefore, parties cannot provide the seller with enough incentives to invest e*.

Multilateral Contract
A multilateral contract leads to an efficient level of investment if neither quality nor investment is verifiable. The contract is the following: B 1 and B 2 each has the option to buy one unit of the good once they have observed its quality. B 1 has the option to buy a unit at price p = m h ≥ 0; if B 1 buys, then B 2 is required to pay ρ to S. Once B 1 has taken his decision, B 2 has the option to buy a unit at price p = m h . The timing is summarized in Fig. 2.
In case the good is of high quality, B 1 and B 2 execute the option and S obtains a payoff of 2p + ρ. The payment ρ can be specified such that S has an incentive to participate in the contract and to invest e*. B 1 and B 2 pay no more for the good than they would pay if S sold it to the market. B 2 is required to pay ρ contingent on B 1 buying the good. Depending on B 2 's valuation v 2 , the fact that he has to pay ρ may require B 1 to compensate B 2 so 8 All parties involved bargain simultaneously over the contract, deciding which of the buyers purchases first. Without loss of generality, assume B 1 buys first.
Timeline Two Independent Contracts that his participation constraints are fulfilled. x denotes the payment that B 1 makes to B 2 upon signing the contract. The timing of the game is crucial. Incentive compatibility is assured only by making payments conditional on prior transactions. Note that the sequential arrangement does not provide additional information-quality is observable at all times. Instead, it generates a verifiable variable (i.e. whether B 1 bought the good or not), which is perfectly correlated with quality.
self-enforcing and induces the optimum level of investment.
Proof First, consider incentive compatibility. Suppose the contract has been signed and quality of the good is high. Both buyers have the possibility to refuse to buy from S. In this case, S can sell the two units to the market at a price m h . Since there are zero transaction costs, if there is a market, B 1 and B 2 can then buy each at a price m h from the market. Once quality is observable, B i will buy from S as long as his payoff is not lower than what he gets while buying from the market, Also, B i will not buy a unit of the low-quality good if the price is not less than the value created. Since the value that a low quality good creates is zero, this implies that B i will not buy a low-quality good as long as With p = m h , inequalities (1) and (2) are satisfied for i ∈ {1,2}.
Knowing that she receives 2p and ρ if the good is of high quality, S will invest e* as long as her payoff from selling two units of the high quality good is not lower than her payoff from not investing, 9 (2) p ≥ 0. 9 I assume that parties do not collude (see below). As long as the first buyer does not receive a share of ρ for claiming that the good is of high quality when it is of low quality, he does not have an incentive to do so. Therefore, the seller's payoff from not investing is only the payment she receives for two low quality goods.
Second, consider individual rationality. S will only participate in the contract as long as inequality (3) is fulfilled. If the contract is not signed, S will not invest e*, and no high quality good will be produced. A unit of low quality has a value of 0 for B 1 and B 2 , hence the expected value of not signing the contract is 0 for each buyer. Having to pay x to B 2 upon signing the contract, B 1 participates in the contract if his expected value is not lower than not partaking: Similarly, receiving x upon signing the contract, B 2 will participate as long as (4) and (5) are fulfilled.
The contract is individual rational, but not immune to collusion between B 1 and B 2 . Obviously, the two buyers together would prefer to pay 2p instead of 2p + ρ for two high quality goods. However, as already mentioned in the introduction, if one regards B 1 as representing a large number of buyers, or assumes that B 1 and B 2 are unable or unwilling to collude (because of collusion entailing breaking competition laws, for example) then this property seems less problematic.
A related problem is collusion between S and B 1 at B 2 's expense: when S produces low quality and B 1 certifies the good as having high quality, S receives ρ from B 2 . Hence, S may transfer an amount τ ≤ ρ to B 1 for certifying high quality. Note, however, that for τ ≤ v 1 −p, B 1 is better off with a high quality good than with low quality plus a transfer of τ. With B 1 is better off with high quality even if S transfers the whole amount ρ. Hence, under this condition, B 1 has an incentive to commit ex ante not to collude with S, if a commitment technology is available.
The exact specification of ρ and x depends on the cost of investment, the additional profit created and the distribution of profits. As long as the relationship creates no additional surplus (i.e. v 1 + v 2 −c(e*) = 2m l ) the only possible contract specifies ρ = v 1 + v 2 −2m h and x = v 1 −m h . In this case, the overall profit for S equals her outside option (i.e. the profit when the contract does not materialize), 2m l . Also, for B 1 and B 2 , profits equal their outside options, which is a payoff of zero.
The contract will generate a positive surplus (to be distributed among the participating parties) only if v 1 + v 2 −2m l > c(e*). How the surplus is distributed depends on the bargaining shares of the parties, which are implicitly defined by the transfers ρ and x, and assumed to be exogenously given (Edlin and Reichelstein, 1996;Che and Hausch, 1999). With ρ = c(e*)−2m h + 2m l , all surplus goes to the buyers (they may divide the surplus by specifying x accordingly). 10 For illustrative reasons, consider the case when all additionally generated surplus goes to the seller: ρ = v 1 + v 2 −2m h , x = v 1 , and hence her payoff is v 1 + v 2 −c(e*). Obviously, other distributions of extra surplus are feasible.

Premium Contract
Under some circumstances-with v 2 large enough-efficiency can be achieved by means of bilateral contracts that do not entail any contractual relationship or payment between B 1 and B 2 . In this situation, the contract obliges B 2 to pay a premium, conditional on other buyer(s) purchasing a unit of the good. The contract is similar to the one proposed. Yet, it resembles better examples in which firms contract less directly as a group, e.g. when the number of buyers B 1 is large.
Again, B 1 and B 2 have the option to each buy one unit of the good once they have observed its quality. B 1 has the option to buy a unit at price p = m h . If B 1 buys, then B 2 is required to pay ρ to S. Once B 1 has taken his decision, B 2 has the option to buy a unit at price p = m h . In case the good is of high quality, B 1 and B 2 execute the option and S obtains a payoff of 2m h + ρ. Without loss of generality, assume B 2 has a 'high' valuation: he values one unit of the high quality good not less than its market price plus the payment S needs to receive to invest efficiently: Hence, B 2 does not need to be compensated to participate in the contract, and firms can contract bilaterally. Incentive compatibility constraints are given by the inequalities (1)-(3), and participation constraints are given by the inequalities (3)-(5), where in the latter two inequalities , all constraints are satisfied, and S and B 2 condition the exchange of a payment ρ on the external event that B 1 buys the good. In this case, the payment ρ is a kind of premium, which may honour the effort of the seller to attract other buyer(s). v 2 ≥ m h + .

IncoMplete InForMatIon
Consider the case that the own valuation of a unit of the high quality good is private information to each buyer; an assumption which is straightforward to include, to take the model closer to reality. In such a case, ρ and x may be made conditional on the declared types. As will be shown in the following, allowing for this type of private information does not change the results and the suggested contract induces truthful revelation of private information.
Assume that there are two buyers who, at the time of writing the contract, privately observe their type v k ,k ∈ {H,L} with v H ≥ v L > m h (and, as before, m h ≥ m l ≥ 0). The types v H and v L are identically and independently distributed, with Pr {v k i = v H } = q ∈ [0,1], the distribution being common knowledge. Let The first part of this condition states that the cost of the investment is not higher than the additional value created by the high quality over the low quality for the two buyers, when both buyers have low valuations of the high-quality unit. The second part states that the cost of investment is higher than the additional value created by high quality over low quality at the market. 11 Proposition 2 There exists a contract that induces truthful revelation of types and the optimal level of investment.
Proof See Appendix A.1.
If the additional benefit generated is minimal (i.e. c(e*) = 2v L −2m l ) then the solution is unique: ρ = 2v L −2m h and x = v L −m h . In this case-as in the case of full information-the profits of the seller and the buyers are equal to their reservation utility. Each buyer's profit is independent of what the other buyer reports. When being of a high type, the buyer in question gets an informational rent while reporting truthfully, equal to the difference in valuations.
If the additional benefit generated is not minimal (i.e. c(e*) < 2v L −2m l ), there exist a number of solutions, depending on the size of the benefit generated. In either case, both, the seller and the buyers receive at least their reservation utilities. Furthermore, again, each buyer's profit is independent of what the other buyer reports, and when being of a high type, the buyer in question gets an informational rent while reporting truthfully.

11
If c(e*) = v 1 + v 2 −2m l and v 1 and / or v 2 equals v H , and when buyers are of high types, there is not sufficient surplus to guarantee the informational rents required to truthfully report the types, see the proof in A.1. For a probabilistic version of this model with incomplete information, see Appendix C.

conclusIon
The presented solution to the hold-up problem applies to settings in which several parties want to incentivize a seller to make a quality-enhancing investment. More importantly, it also holds good when the outcome of investment is probabilistic, i.e. when the seller cannot be sure of producing high quality. The extension makes the contract applicable to settings of research and development, and this is illustrated in Appendix B.
The extension of the simple contract to more than two parties is straightforward. Together with the version of the multilateral contract entitled 'premium contract', such a setting best resembles the introductory example of a multi-sided market. In this case, the two (groups of) buyers may share different characteristics, which justifies the assumption of different valuations for the good-and justifies why one buyer/one group of buyers may be willing to pay a premium.
As highlighted in the introduction, contracts similar to the one proposed here have been used in many situations. Yet, the contract may not be immune to collusion among parties, even though this might be less of a problem with a large number of buyers-or in the version of the premium contract, when buyers have different valuations. If firms can exclude collusion, if they interact on just a one-time basis, or if they do not want to rely on reputational concerns, then, such a contract provides optimal means to restore efficiency. I show that inefficiencies associated with the hold-up problem can be eliminated easily with the proposed contract, with simple market transactions-there is no need to build sophisticated organizations (such as firms) to handle the problem.

A.1 Proof of Proposition 2
Let v i be the reported types, i ∈ {1,2}. Knowing that she receives 2p and ρ if the good is of high quality, S will invest e* as long as her payoff from selling two units of the high quality good is not lower than her payoff from not investing, Suppose the quality of the good is high. Once quality is observable, B i will buy from S as long as his payoff contracting is not lower than his payoff from buying from the market, He will not buy a unit of low quality as long as what he pays for it is more than the value created, p ≥ 0.

The participation and incentive compatibility constraint for S is
The participation constraints for B 1 and B 2 are, respectively Incentive compatibility regarding the reported type in (weakly) dominant strategies requires that a strategy must exist That is, for B 1 and B 2 respectively, Again, there are many different solutions to the specifications of the contract, which all include ∈ [c(e * ) − 2m h + 2m l ,v 1 + v 2 − 2m h ] and the xs specified such that inequalities (10)-(13) are satisfied. With the values specified in Section 3, constraints are fulfilled.

APPENDIX B. PROBABILISTIC REALIZATION OF QUALITY
Assume that the level of effort influences the probability of the good being of high quality π(e) ∈ [0,1], with π(e) being an increasing function. In order to derive a closed-form solution for the model, I employ specific functional forms for the probability and cost functions. 12 Namely, assume that π(e) = min{ηe,1}, with η > 0, and c(e) = α 2 e 2 , with α > 0. Moreover, assume that α is sufficiently high and η is sufficiently low to prevent S from choosing such a large investment level as to induce π(e) = 1; furthermore, assume m l = 0.
The welfare-maximising level of investment e solves 12 The result also goes through with more general functions. However, this and particularly the section on asymmetric information benefits a lot from the functional forms for a simplified presentation of the results.
and from the first order condition the optimal level of investment follows Assume that neither quality nor investment is verifiable. Anticipating a payment of p = m h for each unit of the good, S invests Hence, the induced investment level is A multilateral contract leads to an efficient level of investment in case neither quality nor investment is verifiable. Differently from the deterministic case, it may not suffice that B 1 pays a transfer x to B 2 to fulfil B 2 's participation constraints, but also S has to pay a transfer. x S and x B denote the payments that S and B 1 make in favour of B 2 upon signing the contract for that reason.
Proposition 3 A multilateral contract that is self-enforcing and induces the optimal level of investment exists.
Proof Suppose the contract has been signed. Now S maximizes From the first order condition, it follows that is the optimal level of investment for S, given that the contract has been signed. For e IC ≡ α 2m h .
Second, consider individual rationality. If the contract is not signed, S will invest e IC , and the buyers each pay m h at the market if the good is of high quality. Hence, the expected value of not contracting computes to π(e IC )(v i −m h ) for the buyers. Having to pay x B upon signing the contract, B 1 is willing to participate in the contract if his expected value from contracting is not lower than not partaking, Similarly, B 2 will partake as long as S partakes as long as her expected value from contracting is not lower than her reservation payoff, There are many solutions to the contract. They all include p = m h , ρ = v 1 + v 2 −2m h and

REALIZATION OF QUALITY
The contract for the probabilistic realization of quality can also be extended to situations in which the valuation of one unit of the good is private information for each buyer. Before the contract is signed, each buyer privately observes his type v k ,k ∈ {H,L}, with v H ≥ v L ≥ m h . The types v H and v L are identically and independently distributed, with Pr{v i = v H } = q ∈ [0,1], the distribution being common knowledge. Let v i be the reported types, i ∈ {1,2}.
Proposition 4 There exists a contract that induces truthful revelation and the optimal level of investment.

C.3 Truthful Reporting
Incentive compatibility regarding the reported type in (weakly) dominant strategies requires that there exists a strategy v i = v k i , ∀ i ∈ {1,2} such that That is, for B 1 and B 2 , respectively, Replacing functional forms, p = m h , (v k 1 ,v