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

Abstract

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.

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Notes

  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.

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Correspondence to Daniel Beimborn.

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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

Appendix

Appendix

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)

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Beimborn, D. The stability of cooperative sourcing coalitions ‐ game theoretical analysis and experiment. Electron Markets 24, 19–36 (2014). https://doi.org/10.1007/s12525-013-0128-4

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Keywords

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

JEL classification

  • D74
  • D8
  • G34
  • L1
  • L24