Skip to main content

The stability of cooperative sourcing coalitions ‐ game theoretical analysis and experiment


Cooperative sourcing is when multiple firms form a coalition to merge similar processes using shared-service processing in order to leverage economies of scale and skill. This paper discusses how coalition partners should allocate coalition costs and benefits to ensure a stable coalition which minimizes the incentives for current or future coalition partners to withdraw from the coalition. The paper develops a formal cost allocation and benefit model which is then used in game theoretical analyses and bargaining experiments. The results show that proportional cost allocation is key to avoiding unstable coalitions. However, if coalition partners do not disclose their cost structures to facilitate a cooperative decision in favor of proportional cost allocation, but rather negotiate shared costs based on “closed books”, proportional cost allocation is rarely achieved. In such cases, the Shapley value has high predictive power and negotiators are in danger of making poor decisions that can lead to instable coalitions. These findings help sensitize managers to the structural threats inherent to multilateral negotiations of cooperative sourcing coalitions.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. 1.

    Cooperative sourcing arrangements are also common, e.g., in the insurance industry (Focke et al. 2004; Lemaire 1984), in travel and retail logistics (Flåm and Jourani 2003), in the public sector (Borman 2010; Ulbrich 2010), and in manufacturing (Gebauer 1996; Hart and Moore 1990).

  2. 2.

    Such as requests for proposals and other instruments for matching market partners and determining prices.

  3. 3.

    While the first assumption is unproblematic as any cross-firm cost allocation requires contracts and can thus be taken for granted in all outsourcing arrangements, the second assumption is more problematic. But since a critique of the concept of an exchange economy and the underlying neo-classical paradigm is not part of this work and since many other game-theoretical works have built on this assumption, this premise is accepted as given in this work. For a discussion, see e.g. (Kelly 1978).

  4. 4.

    Game theory provides further concepts for determining such undominated coalition allocations, e.g. the concepts of stable sets (von Neumann and Morgenstern 1953), nucleolus (Schmeidler 1969), or kernel (Owen 1995; Spinetto 1974), however, the core has shown to be the most popular concept, by far, and will be used in the following.

  5. 5.

    In addition to the schemes used in this paper, Lemaire (1984) discusses two further concepts: equal distribution of non-marginal benefits and proportional allocation of non-marginal costs. Due to the particular structure of the cooperative sourcing model introduced in the next section, the first concept is not applicable, while the second in this special case is identical to the proportional cost allocation because the model assumes linear cost functions.

  6. 6.

    In reality, there are many different approaches for setting up a cooperative sourcing coalition. One might be a traditional outsourcing-insourcing relationship, as described above; others might be the creation of joint subsidiaries (shared service centers) which are jointly owned by the coalition partners. However, in the latter case the model assumption still fits because the partner with the most efficient processing environment will most likely contribute to the set-up of the joint venture’s processing infrastructure.

  7. 7.

    According to (Rapoport and Kahan 1976, p. 255), the cooperative sourcing model can be categorized as non-constant, super-additive, positive n-person bargaining game.

  8. 8.

    Formally, this allocation slightly differs from the one introduced in the previous section because here the cost are allocated proportionally based on individual process volumes rather than on individual cost.

  9. 9.

    For example \( K_i^F=\left\{ {100,200,1000} \right\},c_i^P=\left\{ {1,2,3} \right\},{x_i}=100\forall i \) would lead to an unstable coalition, because the group rationality constraint would be violated for Firm 2. That is, firms 1 and 3 would be better off if they formed their own sub-coalition without Firm 2.

  10. 10.

    A more detailed embodiment of different forms of transaction cost into this model can be found in (Beimborn 2006).

  11. 11.

    DeAngelo (1981) and Ewert and Wagenhofer (2008) have shown that firms with access to the capital market are able to act on a risk-neutral rationale because they can hedge any existent risk aversion by capital market transactions if the conditions of spanning and competitivity hold. (Spanning refers to the set of existing financial instruments available at the capital market containing all possible cash flow structures which can be created by the firm’s own activities. Competitivity refers to a static valuation system for market value – in a certain environment comparable to a stable interest rate for maximizing the net present value.)

  12. 12.

    This game design matches Rapaport and Kahan’s (1976) design in terms of round mechanism and explicit set of possible actions for negotiation. However, it differs from this and many other games in terms of non-disclosure of the actors’ cost information to the other players.

  13. 13.

    Behaving cooperatively during negotiations means “moving” towards a compromise with offers and counter-offers, instead of too rigidly insisting on an offer made.


  1. Ang, S., & Straub, D. W. (1998). Production and transaction economies and IS outsourcing: a study of the U.S. banking industry. MIS Quarterly, 22:4, 535–552.

    Article  Google Scholar 

  2. Aumann, R. J. (2008). “Game Theory,” In: The Palgrave Dictionary of Economics, S. N. Durlauf & L. E. Blume (Eds.). pp. 529–555.

  3. Bachrach, Y., Kohli, P., & Graepel, T. (2011). “Rip-off: Playing the Cooperative Negotiation Game,” Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems.

  4. Beimborn, D. (2006). A model for simulation analyses of cooperative business sourcing in the banking industry. 39th Hawaii International Conference on System Sciences, Kauai.

  5. Bergeron, B. P. (2003). Essentials of shared services. Hoboken: Wiley.

    Google Scholar 

  6. Bolton, G. E., Chatterjee, K., & McGinn, K. L. (2003). How communication links influence coalition bargaining: a laboratory investigation. Management Science, 49:5, 583–598.

    Article  Google Scholar 

  7. Bongartz, U. (2004). Transaktionsbanking quo vadis? In H.-J. Lamberti, A. Marlière, & A. Pöhler (Eds.), Management von Transaktionsbanken (pp. 39–57). Heidelberg: Springer.

    Chapter  Google Scholar 

  8. Borman, M. (2007). “Recognising the need for a context sensitive decision making framework for cosourcing - a case study in the financial service sector,” European Conference on Information Systems (ECIS). Switzerland: St. Gallen.

    Google Scholar 

  9. Borman, M. (2010). Characteristics of a successful shared services centre in the Australian public sector. Transforming Government: People, Process and Policy, 4:3, 220–231.

    Article  Google Scholar 

  10. Borman, M., & Janssen, M. (2012). “The design and success of shared services centres,” 45th Hawaii International Conference on System Sciences (HICSS), Hawaii.

  11. Daberkow, M., & Radtke, I. (2008). Der Zahlungsverkehr der Postbank als Beispiel für die Industrialisierung im Finanzdienstleistungssektor. In B. Kaib (Ed.), Outsourcing in Banken. Gabler: Wiesbaden.

    Google Scholar 

  12. DeAngelo, H. (1981). Competition and unanimity. American Economic Review, 71:1, 18–27.

    Google Scholar 

  13. Dowling, M. J., Roering, W. D., Carlin, B. A., & Wisnieski, J. (1996). Multifaceted relationships under coopetition: description and theory. Journal of Management Inquiry, 5:2, 155–167.

    Article  Google Scholar 

  14. Dubey, P., & Shapley, L. S. (1984). Totally balanced games arising from controlled programming problems. Mathematical Programming, 29, 245–267.

    Article  Google Scholar 

  15. Earl, M. J. (1996). The risks of outsourcing IT. Sloan Management Review, 37:3, 26–32.

    Google Scholar 

  16. Ewert, R., & Wagenhofer, A. (2008). Interne Unternehmensrechnung (7th ed.). Heidelberg: Springer.

    Google Scholar 

  17. Flåm, S. D., & Jourani, A. (2003). Strategic behavior and partial cost sharing. Games and Economic Behavior, 43:1, 44–56.

    Article  Google Scholar 

  18. Focke, H., Kremlicka, R., Freudenstein, G., Gröflin, J., Pratz, A., Röckemann, C., & West, A. (2004). Tendenz steigend: Transaction Banking auf dem Weg zu Service und Innovation, A.T. Kearney Transaction-Banking-Studie 2004. Frankfurt: A.T. Kearney.

    Google Scholar 

  19. Friedman, D., & Sunder, S. (1994). Experimental methods. A primer for economists. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  20. Gebauer, J. (1996). “Virtual organizations from an economic perspective,” 4th European Conference on Information Systems (ECIS), J .D. Coelho, T. Jelassi, W. König, H. Krcmar, R. O’Callagham & M. Säälsjarvi (Eds.), Lisbon, Portugal.

  21. Gellings, C. (2006). “Outsourcing Relationships: Designing ‘Optimal’ Contracts – A Principal-Agent-Theoretic Approach,” 12th Americas Conference on Information Systems (AMCIS), Acapulco.

  22. Gillies, D. B. (1959). Solutions to general non-zero-sum games. In A. W. Tucker & R. D. Luce (Eds.), Contributions to the theory of Games IV (pp. 47–85). Princeton: Princeton University Press.

    Google Scholar 

  23. Granot, D. (1986). A generalized linear production model: a unifying model. Mathematical Programming, 34, 212–222.

    Article  Google Scholar 

  24. Grant, G., McKnight, S., Uruthirapathy, A., & Brown, A. (2007). Designing Governance for Shared Services Organizations in the Public Service. Government Information Quarterly, 24:3, 522–538.

    Article  Google Scholar 

  25. Hamoir, O., McCamish, C., Nierderkorn, M., & Thiersch, C. (2002). “Europe’s banks: verging on merging,” In: McKinsey Quarterly. pp. 116–125.

  26. Harsanyi, J. C. (1966). A general theory of rational behavior in game situations. Econometrica, 34, 613–634.

    Article  Google Scholar 

  27. Hart, O., & Moore, J. (1990). Property rights and the nature of the firm. Journal of Political Economy, 98:6, 1119–1158.

    Article  Google Scholar 

  28. Jamal, K., & Sunder, S. (1991). Money vs. gaming: effects of salient monetary payments in double oral auctions. Organizational Behavior and Human Decision Processes, 49:1, 66–151.

    Google Scholar 

  29. Janssen, M., & Joha, A. (2004). “Issues in relationship management for obtaining the benefits of a shared service center,” 6th International Conference on Electronic Commerce, Delft.

  30. Janssen, M., & Joha, A. (2006). Motives for establishing shared service centers in public administrations. International Journal of Information Management, 26:2, 102–115.

    Article  Google Scholar 

  31. Kelly, J. S. (1978). Arrow impossibility theorems. New York: Academic.

    Google Scholar 

  32. Knol, A.J., & Sol, H.G. (2011). “Sourcing with Shared Service Centres: Challenges in the Dutch Government,” European Conference on Information Systems (ECIS), Helsinki.

  33. Lammers, M. (2004). Make, buy or share - Combining resource based view, transaction cost economics and production economies to a sourcing framework. Wirtschaftsinformatik, 46:3, 204–212.

    Article  Google Scholar 

  34. Lehrer, E. (2003). Allocation processes in cooperative games. International Journal of Game Theory, 31, 341–351.

    Article  Google Scholar 

  35. Lemaire, J. (1984). An application of game theory: cost allocation. ASTIN Bulletin, 14:1, 61–81.

    Google Scholar 

  36. Liebowitz, S. J., & Margolis, S. E. (1995). Path dependence, lock-in, and history. Journal of Law, Economics and Organization, 11:4, 205–225.

    Google Scholar 

  37. Loh, L., & Venkatraman, N. (1992). Determinants of information technology outsourcing: a cross-sectional analysis. Journal of Management Information Systems, 9:1, 7–24.

    Google Scholar 

  38. Longwood, J., & Harris, R. G. (2007). Leverage BPO Lessens to Build a Successful Shared Business Service Organization. Stamford: Gartner Research.

    Google Scholar 

  39. Medlin, S. M. (1976). Effects of grand coalition payoffs on coalition formation in three-person games. Behavioral Science, 21:1, 48–61.

    Article  Google Scholar 

  40. Montero, M., Sefton, M., & Zhang, P. (2008). Enlargement and the Balance of Power: an Experimental Study. Social Choice & Welfare, 30, 69–87.

    Article  Google Scholar 

  41. Münstermann, B., & Weitzel, T. (2008). “What is process standardization?,” International Conference on Information Resource Management (Conf-IRM). Ontario: Niagara Falls.

    Google Scholar 

  42. Murnighan, J. K., & Roth, A. E. (1977). “The effects of communication and information availability in an experimental study of a three-person game,” Management Science (23:12).

    Google Scholar 

  43. Myerson, R. B. (1984). “Cooperative games with incomplete information”. International Journal of Game Theory, 13:2, 69–96.

    Article  Google Scholar 

  44. Nash, J. F. (1950). The bargaining problem. Econometrica, 18, 155–162.

    Article  Google Scholar 

  45. Owen, G. (1975). On the core of linear production games. Mathematical Programming, 9, 358–370.

    Article  Google Scholar 

  46. Owen, G. (1995). Game theory (3rd ed.). New York: Academic.

    Google Scholar 

  47. Pieske, R. (2005). Hypotheken-Portfolios effizient verwalten. Versicherungswirtschaft, 21:17, 1300–1302.

    Google Scholar 

  48. Quan, J., Quan, Q. H., & Hart, P. J. (2003). Information technology investments and firms’ performance - A duopoly perspective. Journal of Management Information Systems, 20:3, 121–158.

    Google Scholar 

  49. Quélin, B., & Duhamel, F. (2003). Bringing together strategic outsourcing and corporate strategy: outsourcing motives and risks. European Management Journal, 21:5, 647–661.

    Article  Google Scholar 

  50. Quinn, B., Cooke, R., & Kris, A. (2000). Shared services: Mining for corporate gold. London: Prentice-Hall.

    Google Scholar 

  51. Rapoport, A. (2001). N-person game theory: Concepts and applications. Mineola: Dover Publications.

    Google Scholar 

  52. Rapoport, A., & Kahan, J. P. (1976). When three is not always two against one: coalitions in experimental three-person cooperative games. Journal of Experimental Social Psychology, 12, 253–273.

    Article  Google Scholar 

  53. Riera, A., Davies, R., Wurzel, G., & Schwarz, J. E. (2003). “Banking à la Nike and Dell: achieving scale without acquisition premiums,” Boston Consulting Group.

  54. Riker, W. H. (1967). Bargaining in a Three-Person Game. American Political Science Review, 61, 642–656.

    Article  Google Scholar 

  55. Roth, A. E., & Verrecchia, R. E. (1979). The shapley value as applied to cost allocation: a reinterpretation. Journal of Accounting Research, 17:1, 295–303.

    Article  Google Scholar 

  56. Samet, D., & Zemel, E. (1984). On the core and dual set of linear programming games. Mathematics of Operations Research, 9:2, 309–316.

    Article  Google Scholar 

  57. Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on Applied Mathematics, 17:6, 1163.

    Article  Google Scholar 

  58. Schulz, V., & Brenner, W. (2010). Characteristics of Shared Service Centers. Transforming Government: People, Process and Policy, 4:3, 210–219.

    Article  Google Scholar 

  59. Selten, R. (1987). Equity and coalition bargaining in experimental three-person games. In A. E. Roth (Ed.), Laboratory experiments in economics: Six points of view. Cambridge: Cambridge University Press.

    Google Scholar 

  60. Shapley, L. (1953). A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games (pp. 307–317). Princeton: Princeton University Press.

    Google Scholar 

  61. Spinetto, R. (1974). The geometry of solution concepts for N-person cooperative games. Management Science, 20:9, 1292–1299.

    Article  Google Scholar 

  62. Staw, B. M. (1984). The escalation of commitment to a course of action. Academy of Management Review, 6, 577–587.

    Google Scholar 

  63. Suijs, J., Borm, P., De Waegenaere, A., & Tijs, S. (1999). “Cooperative games with stochastic payoffs”. European Journal of Operational Research, 113:1, 193–205.

    Article  Google Scholar 

  64. Thatcher, M. E., & Oliver, J. R. (2001). The impact of technology investments on a firm’s production efficiency, product quality, and productivity. Journal of Management Information Systems, 18:2, 17–45.

    Google Scholar 

  65. Timmer, J., Borm, P., & Tijs, S. (2005). Convexity in stochastic cooperative situations. International Game Theory Review, 7:1, 25–42.

    Article  Google Scholar 

  66. Ulbrich, F. (2006). Improving shared service implementation: adopting lessons from the BPR movement. Business Process Management Journal, 12:2, 191–205.

    Article  Google Scholar 

  67. Ulbrich, F. (2010). Adopting shared services in a public-sector organization. Transforming Government: People, Process and Policy, 4:3, 249–265.

    Article  Google Scholar 

  68. Vinacke, W. E., & Arkoff, A. (1957). An experimental study of coalition in the triad. American Sociological Review, 22, 406–414.

    Article  Google Scholar 

  69. von Neumann, J., & Morgenstern, O. (1953). Theory of games and economic behavior. Princeton: Princeton University Press.

    Google Scholar 

  70. Waegenaere, A., Suijs, J., & Tijs, S. (2005). “Stable profit sharing in cooperative investments”. OR Spectrum, 27:1, 85–93.

    Article  Google Scholar 

  71. Wang, E. T. G. (2002). Transaction attributes and software outsourcing success: an empirical investigation of transaction cost theory. Information Systems Journal, 12:2, 153–181.

    Article  Google Scholar 

  72. Weitzel, T., Son, S., & König, W. (2001). Infrastrukturentscheidungen in vernetzten Unternehmen: Eine Wirtschaftlichkeitsanalyse am Beispiel von X.500 Directory Services. Wirtschaftsinformatik, 43:4, 371–381.

    Article  Google Scholar 

  73. Westen, T. E., & Buckley, J .J. (1974). “Toward an Explanation of Experimentally Obtained Outcomes to a Simple, Majority Rule Game,” Journal of Conflict Resolution (18:198).

    Google Scholar 

  74. Williamson, O. E. (1975). Markets and hierarchies: Analysis and antitrust implications. A study in the economics of internal organization. London: Free Press.

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Daniel Beimborn.

Additional information

An earlier version of this work was presented as Beimborn, D., Lamberti, H.-J., Weitzel, T. “Game theoretical analysis of cooperative sourcing scenarios,” 39th Hawaii International Conference on System Sciences, Kauai, 2006.

Responsible Editors: Frank Ulbrich and Mark Borman



A – equal distribution of gain

In a three-firm coalition scenario with Firm 1 as current insourcer and Firm 2 as optimal insourcer after Firm 1 would have left, the group rationality condition to be satisfied is:

$$ \begin{array}{*{20}c} {\left( {K_1^F+c_1^P\cdot {x^M}} \right)-\left( {K_2^F+c_2^P\cdot \left( {{x_2}+{x_3}} \right)} \right)\,\,\leq C_1^P-\frac{{K_1^F+c_1^P\cdot {x_1}+K_2^F+c_2^P\cdot {x_2}+K_3^F+c_3^P\cdot {x_3}-K_1^F-c_1^P\cdot {x^M}}}{3}} \hfill \\ {\Leftrightarrow 3\left( {c_1^P-c_2^P} \right)\,\left( {{x_2}+{x_3}} \right)-2K_2^F\leq -K_3^F-c_2^P\cdot {x_2}-c_3^P\cdot {x_3}+c_1^P\cdot \left( {{x_2}+{x_3}} \right)} \hfill \\ {\Leftrightarrow 2K_2^F+2c_2^P{x_2}+3c_2^P{x_3}\geq K_3^F+c_3^P\cdot {x_3}+2c_1^P\cdot \left( {{x_2}+{x_3}} \right)} \hfill \\ {\Leftrightarrow K_2^F+c_2^P\left( {{x_2}+{x_3}} \right)\geq \frac{{K_3^F+\left( {c_3^P-c_2^P} \right)\cdot {x_3}}}{2}+c_1^P\cdot \left( {{x_2}+{x_3}} \right)} \hfill \\ \end{array} $$

Since Firm 2 is the best insourcer for the remaining coalition, the following condition holds true:

$$ K_2^F+c_2^P\left( {{x_2}+{x_3}} \right)\leq K_3^F+c_3^P\left( {{x_2}+{x_3}} \right) $$

If we assume a situation where all c p are equal, it becomes clear that the constraint for ensuring group rationality will not be generally fulfilled for \( K_2^F<<K_3^F \).

B – proportional allocation of cost – three-firm scenario

As outlined above, the initial proof needs to be repeated under the assumption that if Firm 1 left the trilateral coalition, Firm 3 would become the insourcer, whereas Firm 2 is the insourcer otherwise:

$$ K_2^F+{c_2}{x^M}-K_3^F-{c_3}\left( {{x^M}-{x_1}} \right)\leq {x_1}\cdot \left( {\frac{{K_2^F}}{{{x^M}}}+c_2^P} \right)\leq K_i^F+c_i^P{x_i} $$

The right condition has already been proven to be true in the main text above. The left condition can be reformulated to:

$$ \begin{array}{*{20}c} \frac{{{x_2}+{x_3}}}{{{x^M}}}K_2^F+\left( {{c_2}-{c_3}} \right)\left( {{x_2}+{x_3}} \right)-K_3^F\leq 0\Leftrightarrow \hfill \\ \frac{{{x_2}+{x_3}}}{{{x^M}}}K_2^F+{c_2}\left( {{x_2}+{x_3}} \right)\leq K_3^F+{c_3}\left( {{x_2}+{x_3}} \right)\Leftrightarrow \hfill \\ \frac{{K_2^F}}{{{x^M}}}+{c_2}\leq \frac{{K_3^F}}{{{x_2}+{x_3}}}+{c_3} \hfill \\\end{array} $$

In order to fulfill the assumption that Firm 2 dominates Firm 3 in the trilateral coalition, the following condition also has to be met:

$$ K_2^F+{c_2}{x^M}\leq K_3^F+{c_3}{x^M}\Leftrightarrow \frac{{K_2^F}}{{{x^M}}}+{c_2}\leq \frac{{K_3^F}}{{{x^M}}}+{c_3} $$

Since x 2 + x 3 <x M= x 1 + x 2 + x 3, this inequation is more restrictive than the condition above. Thus, the lower border is fulfilled under any given parameter configuration.

As a third possible constellation which has to be tested, we have to assume that i itself is the insourcer. For determining the group rationality constraint (lower border), another insourcer has to be determined again. The insourcer must fulfill the following conditions (cf. Eq. 13), substituting \( \sum\limits_{{j\in M}} {{x_j}} \) by x M and \( \sum\limits_{\begin{subarray}{l} j\in M \\ \,j\ne i \end{subarray}} {{x_j}} \) by \( {x^{M-i }}( {={x^M}-{x_i}} ): \)

$$ {x^{M-i }}\left( {={x^M}-{x_i}} \right):\left( {{K^{FM }}+{c^{PM }}\cdot {x^M}} \right)-\left( {{K^{FM-i }}+{c^{PM-i }}\cdot {x^{M-i }}} \right)\leq \left( {{K^{FM }}+{c^{PM }}\cdot {x^M}} \right)\cdot \frac{{{x_i}}}{{{x^M}}}\leq \left( {K_i^F+c_i^P{x_i}} \right) $$

Since i is the insourcer, the following additional substitutions can be made:

$$ {K^{FM }}=K_i^F\,and\,\,{c^{PM }}=c_i^P $$

The right constraint is obviously always fulfilled:

$$ K_i^F\frac{{{x_i}}}{{{x^M}}}\leq K_i^F\,\Leftrightarrow {x_i}\leq {x^M} $$

Because of \( {x^{M-i }}={x^M}-{x_i} \) the left condition can be reformulated to

$$ \begin{array}{*{20}c} {{K^{FM }}\cdot \frac{{{x^{M-i }}}}{{{x^M}}}+{c^{PM }}\cdot {x^{M-i }}-{K^{FM-i }}-{c^{PM-i }}\cdot {x^{M-i }}\leq 0\;\Leftrightarrow {K^{FM }}\cdot \frac{{{x^{M-i }}}}{{{x^M}}}-{K^{FM-i }}\leq \left( {{c^{PM-i }}-{c^{PM }}} \right)\cdot {x^{M-i }}} \hfill \\ {\Leftrightarrow \frac{{{K^{FM }}}}{{{x^M}}}-\frac{{{K^{FM-i }}}}{{{x^{M-i }}}}\leq {c^{PM-i }}-{c^{PM }}\,\,\,\Leftrightarrow\,\,\,\frac{{{K^{FM }}}}{{{x^M}}}+{c^{PM }}\leq \frac{{{K^{FM-i }}}}{{{x^{M-i }}}}+{c^{PM-i }}} \hfill \\ \end{array} $$

Since the insourcer provides lowest cost for x M, the following property holds true:

$$ {K^{FM }}+{c^{PM }}\cdot {x^M}\leq {K^{FM-i }}+{c^{PM-i }}\cdot {x^M}\Leftrightarrow\,\frac{{{K^{FM }}}}{{{x^M}}}+{c^{PM }}\leq \frac{{{K^{FM-i }}}}{{{x^M}}}+{c^{PM-i }} $$

This condition is more restrictive than the lower border of the core. Consequently, it can be stated that a trilateral coalition is always stable (under the theoretical assumptions adopted) if a proportional cost allocation scheme is adopted.

C – proportional allocation of cost – n-firms-scenario

Compared to the trilateral coalition, the lower border (i.e., group rationality constraint) in the n-lateral coalition must additionally hold true for every possible sub-coalition S of coalition M. We distinguish between sub-coalition S and the remaining set of actors M/S. It has to be proven that every possible sub-coalition S within the grand coalition M has to bear at least those cost that resemble the marginal value (i.e., cost savings) it contributes to the coalition (= lower border of the core) in order to ensure that the remaining coalition members M/S have no incentive to leave.

$$ \begin{array}{*{20}c} {\left( {{K^{FM }}+{c^{PM }}\cdot {x^M}} \right)-\left( {{K^{{{FM \left/ {S} \right.}}}}+{c^{{{PM \left/ {S} \right.}}}}\cdot {x^{{{M \left/ {S} \right.}}}}} \right)\leq \sum\limits_{{i\in S}} {\left( {\left( {{K^{FM }}+{c^{PM }}\cdot {x^M}} \right)\cdot \frac{{{x_i}}}{{{x^M}}}} \right)\,}\,\,\,\,\,\,\,\forall S\subset M} \hfill \\ {\Leftrightarrow\,\,{K^{FM }}+{c^{PM }}\cdot {x^M}-{K^{{{FM \left/ {S} \right.}}}}-{c^{{{PM \left/ {S} \right.}}}}\cdot {x^{{{M \left/ {S} \right.}}}}\leq \left( {{K^{FM }}+{c^{PM }}\cdot {x^M}} \right)\cdot \frac{{{x^S}}}{{{x^M}}}\,\,\,\,\,\,\,\,\forall S\subset M} \hfill \\ {\Leftrightarrow\,\,{K^{FM }}+{c^{PM }}\cdot {x^M}-{K^{{{FM \left/ {S} \right.}}}}-{c^{{{PM \left/ {S} \right.}}}}\cdot {x^{{{M \left/ {S} \right.}}}}\leq {K^{FM }}\cdot \frac{{{x^S}}}{{{x^M}}}+{c^{PM }}\cdot {x^S}\,\,\,\,\,\,\,\,\forall S\subset M} \hfill \\ {\Leftrightarrow\,\,{K^{FM }}+{c^{PM }}\cdot {x^{{{M \left/ {S} \right.}}}}-{K^{{{FM \left/ {S} \right.}}}}-{c^{{{PM \left/ {S} \right.}}}}\cdot {x^{{{M \left/ {S} \right.}}}}\leq {K^{FM }}\cdot \frac{{{x^S}}}{{{x^M}}}\,\,\,\,\forall S\subset M} \hfill \\ {\Leftrightarrow\,\,{K^{FM }}-{K^{{{FM \left/ {S} \right.}}}}+\left( {{c^{PM }}-{c^{{{PM \left/ {S} \right.}}}}} \right)\cdot {x^{{{M \left/ {S} \right.}}}}\leq {K^{FM }}\cdot \frac{{{x^S}}}{{{x^M}}}\,\,\,\,\,\,\,\,\forall S\subset M} \hfill \\ {\Leftrightarrow\,\,{K^{FM }}\cdot \frac{{{x^{{{M \left/ {S} \right.}}}}}}{{{x^M}}} + \left( {{c^{PM }}-{c^{{{PM \left/ {S} \right.}}}}} \right)\cdot {x^{{{M \left/ {S} \right.}}}}\leq {K^{{{FM \left/ {S} \right.}}}}\,\,\,\,\,\,\,\forall S\subset M} \hfill \\ {\Leftrightarrow\,\,\frac{{{K^{FM }}}}{{{x^M}}} + {c^{PM }}\leq \frac{{{K^{{{FM \left/ {S} \right.}}}}}}{{{x^{{{M \left/ {S} \right.}}}}}} + {c^{{{PM \left/ {S} \right.}}}}\,\forall S\subset M} \hfill \\ \end{array} $$

This is always fulfilled due to the linear form of the cost function and the larger process volume on the left side (monotonically decreasing average costs).

D – Shapley allocation

For testing the Shapley allocation (Shapley 1953), Eq. 13 is reformulated to a three-firm scenario assuming Firm 1 is the insourcer and examining the Shapley value for firm i = 2. In the following, we focus on the lower border:

$$ \begin{array}{*{20}c} {c_1}{x_2}\leq \frac{{0!\cdot 2!}}{3!}\left( {K_2^F+{c_2}\cdot {x_2}} \right)+\frac{{1!\cdot 1!}}{3!}\left( {Min\left( {C_1^P\left( {{x_1}+{x_2}} \right);C_2^P\left( {{x_1}+{x_2}} \right)} \right)-C_1^P\left( {{x_1}} \right)} \right) \hfill \\ + \frac{{1!\cdot 1!}}{3!}\left( {Min\left( {C_2^P\left( {{x_2}+{x_3}} \right);C_3^P\left( {{x_2}+{x_3}} \right)} \right)-C_3^P\left( {{x_3}} \right)} \right)+\frac{{2!\cdot 0!}}{3!}\left( {K_1^F+{c_1}\left( {{x_1}+{x_2}+{x_3}} \right)-Min\left( {C_1^P\left( {{x_1}+{x_3}} \right);C_3^P\left( {{x_1}+{x_3}} \right)} \right)} \right) \hfill \\\end{array} $$

Assuming \( C_1^P(x)<C_2^P(x)<C_3^P(x)\,\,\,\,\forall x \) leads to

$$ \begin{array}{*{20}c} {{c_1}{x_2}\leq \frac{1}{3}\left( {C_2^P\left( {{x_2}} \right)} \right)+\frac{1}{6}C_1^P\left( {{x_1}+{x_2}} \right)-\frac{1}{6}C_1^P\left( {{x_1}} \right)+\frac{1}{6}C_2^P\left( {{x_2}+{x_3}} \right)-\frac{1}{6}C_3^P\left( {{x_3}} \right)+\frac{1}{3}C_1^P\left( {{x_1}+{x_2}+{x_3}} \right)-\frac{1}{3}C_1^P\left( {{x_1}+{x_3}} \right)} \\ {\Leftrightarrow\,\,0\leq \frac{1}{2}C_2^P\left( {{x_2}} \right)-\frac{1}{6}C_3^P\left( {{x_3}} \right)-\frac{1}{2}c_1^P\cdot {x_2}+\frac{1}{6}c_2^P\cdot {x_3}\;\Leftrightarrow 0\leq K_2^F-\frac{1}{3}K_3^F+c_2^P\cdot {x_2}-\frac{1}{3}c_3^P{x_3}-c_1^P{x_2}+\frac{1}{3}c_2^P{x_3}} \\ \end{array} $$

No one can easily see that if \( K_3^F>>K_2^F \), the right side of the in will become negative, i.e. the group rationality constraint will be violated.

Further, the insourcer must fulfill the following conditions:

$$ \begin{array}{*{20}c} \left( {{K^{FM }}+{c^{PM }}\cdot {x^M}} \right)-\left( {{K^{FM* }}+{c^{PM-i }}\cdot {x^{M-i }}} \right)\leq \hfill \\ \sum\limits_{{S\subset M}} {\frac{{\left( {\left| S \right|-1} \right)!\left( {\left| M \right|-\left| S \right|} \right)!}}{{\left| M \right|!}}} \left[ {\left( {{K^{FS }}+{c^{PS }}\cdot {x^S}} \right)-\left( {{K^{FS-i }}+{c^{PS-i }}\cdot \left( {{x^S}-{x_i}} \right)} \right)} \right]\leq \left( {K_i^F+c_i^P{x_i}} \right) \hfill \\\end{array} $$

In the following three-firm scenario, Firm 1 is assumed to be the insourcer and Firm 2 to be the second-best insourcer (after i = 1 would have left the coalition). The analysis focuses on the group rationality constraint.

$$ \begin{array}{*{20}c} \left( {K_1^F+c_1^P\cdot {x^M}} \right)-\left( {K_2^F+c_2^P\cdot \left( {{x_2}+{x_3}} \right)} \right)\leq \frac{{0!\cdot 2!}}{3!}\left( {C_1^P\left( {{x_1}} \right)} \right)+\frac{{1!\cdot 1!}}{3!}\left( {Min\left( {C_1^P\left( {{x_1}+{x_2}} \right);C_2^P\left( {{x_1}+{x_2}} \right)} \right)-C_2^P\left( {{x_2}} \right)} \right) \hfill \\ +\frac{{1!\cdot 1!}}{3!}\left( {Min\left( {C_1^P\left( {{x_1}+{x_3}} \right);C_3^P\left( {{x_1}+{x_3}} \right)} \right)-C_3^P\left( {{x_3}} \right)} \right)+\frac{{2!\cdot 0!}}{3!}\left( {C_1^P\left( {{x_1}+{x_2}+{x_3}} \right)-Min\left( {C_2^P\left( {{x_2}+{x_3}} \right);C_3^P\left( {{x_2}+{x_3}} \right)} \right)} \right) \hfill \end{array} $$

Assuming again that \( C_1^P(x)<C_2^P(x)<C_3^P(x)\,\,\,\,\forall x \) leads to

$$ \begin{array}{*{20}{c}} {\frac{2}{3}C_1^P\left( {{x^M}} \right)\leq \frac{1}{3}C_1^P\left( {{x_1}} \right)-\frac{1}{6}C_2^P\left( {{x_2}} \right)-\frac{1}{6}C_3^P\left( {{x_3}} \right)+\frac{1}{6}C_1^P\left( {{x_1}+{x_2}} \right)+\frac{1}{6}C_1^P\left( {{x_1}+{x_3}} \right)+\frac{2}{3}C_2^P\left( {{x_2}+{x_3}} \right)\Leftrightarrow } \hfill \\ {0\leq \frac{1}{2}C_2^P\left( {{x_2}} \right)-\frac{1}{6}C_3^P\left( {{x_3}} \right)-\frac{1}{2}c_1^P\cdot \left( {{x_1}+{x_2}} \right)+\frac{2}{3}c_2^P\cdot {x_3}\,\Leftrightarrow\,\,0\leq K_2^F-\frac{1}{3}K_3^F+{x_2}\left( {{c_2}-{c_1}} \right)+\frac{{{x_3}}}{3}\left( {4c_2^P-c_3^P} \right)-c_1^P\cdot {x_1}} \hfill \\ \end{array} $$

If \( K_3^F>>K_2^F \), the right side can become negative, again, and thus violates the condition. We can conclude that the Shapley allocation to linear cost functions (with strictly positive fixed costs) does not generally lead to stable coalitions in situations with three (and more) members.

E – results from game experiment (rounds 2 and 3)

Table 12 Cost allocation results from theoretical allocations and from the experiment (second game variant)
Table 13 Cost allocation results from theoretical allocation and from experiment (incl. transaction costs) (third game variant, only trilateral coalitions)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Beimborn, D. The stability of cooperative sourcing coalitions ‐ game theoretical analysis and experiment. Electron Markets 24, 19–36 (2014).

Download citation


  • Cooperative sourcing
  • Shared services
  • Benefit allocation
  • Stability of equilibria
  • Game theory
  • Negotiation experiment

JEL classification

  • D74
  • D8
  • G34
  • L1
  • L24