Abstract
Apart from the identification of improved solutions within an acceptable number of iterations, a further requirement for practicable coordination schemes is their applicability by rational, self-interested parties, which implies that the schemes can be embedded into suitable coordination mechanisms.
In the following sections, we outline three contractual frameworks that form building blocks for the resulting mechanisms in combination with the schemes proposed in Chap. 4. All frameworks rely on compensation payments among parties as incentives for the implementation of coordinated solutions. First, these payments are necessary for ensuring individual rationality in the mechanisms. Often, the implementation of coordinated solutions involves cost increases for at least one party. Such increases necessarily occur if a party acts as the leader and unilaterally determines the allocation of the central resources in the default solution.1 Unless several optimal solutions exist for the leader’s problem, the implementation of a coordinated proposal will force the leader to deviate from his individually optimal solution, and, hence, to implement a solution with increased costs. Second, such payments are a straightforward way to align parties’ incentives with the actions required by the schemes.2
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Notes
- 1.
An example for this is upstream planning with forced compliance (see p. 31), where the buyer acts as the leader.
- 2.
A renunciation of such payments – as sporadically advocated in the literature, e.g., by Gjerdrum et al. (2002, p. 592) – might only be an option if no party incurs any losses from implementing a coordinated proposal. This may occur to a limited extent in voluntary compliance settings, as indicated by our computational tests of Sect. 6.4. Even then, however, it seems difficult to ensure that parties will actually follow the rules of the underlying scheme. The main problem is that – depending on the rule for determining the proposal implemented – parties will have incentives not to accept proposals with small own savings in order to increase the probability of the acceptance of proposals that are more lucrative for these parties (though less advantageous for the other parties). If all parties pursued this strategy, the overall coordination performance would suffer strongly since a great number of favorable proposals would be declined then.
- 3.
We omit the index i when we consider a single cost-reporting party.
- 4.
Note that this possibility has been mentioned by Dudek and Stadtler (2007, p. 478).
- 5.
See Kahneman et al. (1986, p. 728).
- 6.
See, e.g., Kagel and Roth (1995, p. 256) and Camerer (2003, p. 59).
- 7.
See Camerer (2003, p. 113).
- 8.
- 9.
- 10.
- 11.
As an illustration for the effects of distorted reporting of cost changes, see Example 5.2 in the next section.
- 12.
A discussion of potential learning effects with a repeated application of the mechanism is provided at the end of this section.
- 13.
See e.g., Aviv (2001, p. 1327).
- 14.
A Stackelberg game is a dynamic game in which one party is the (Stackelberg) leader and moves before the follower, see, e.g., Myerson (1991, p. 187).
- 15.
See Assumption 5.1. We discuss an alternative assumption about the information status of the RP below.
- 16.
This assumption is natural since most standard probability distributions, e.g., normal, uniform, gamma, and Weibull distributions, show this property, see Lariviere and Porteus (2001, p. 296).
- 17.
An IGFR for f(S) means that \(Lf(L)/{\int \nolimits \nolimits }_{L}^{\infty }f(S)dS\) is increasing in L. Since \({\int \nolimits \nolimits }_{L}^{\infty }f\left (S\right )dS\) is decreasing in L, \({\int \nolimits \nolimits }_{L}^{\infty }f\left (S\right )dS - Lf\left (L\right )\) also decreases then, provided that \({\int \nolimits \nolimits }_{L}^{\infty }f\left (S\right )dS - Lf\left (L\right ) > 0\).
- 18.
A potential increase of the RHS of (5.2) for ranges where the RHS takes negative values does not affect the optimality of L ∗. The best value for L within such a range is the lower bound of this range, which either corresponds to a or to the upper bound of the adjacent positive range.
- 19.
Values of L ∗ > b can be excluded since the gains of RP become zero there.
- 20.
- 21.
For a risk-averse RP, L ∗ is lower. Risk averseness can be modeled straightforwardly here, e.g., analogously to the modeling by Chatterjee and Samuelson (1983, p. 848) for their sealed bid double auction.
- 22.
- 23.
We provide an example for such a setting below (Example 5.5).
- 24.
- 25.
See also Sect. 3.3.1.
- 26.
- 27.
- 28.
See Sect. 3.3.1.
- 29.
See Sect. 4.1.2.
- 30.
Note that, like in the other mechanisms presented in this work, we assume the existence of a default solution, which is implemented without coordination.
- 31.
Note that these “savings” can also take negative values if the implementation of a proposal involves a cost increase for a party. For ease of exposition, however, we keep this terminology also in this case.
- 32.
Such markdowns are always chosen by rational parties under mild conditions. This result has been proven by Myerson and Satterthwaite (1983, p. 265) for the bilateral trade mechanism and is directly valid for the setting considered here if | Π | = 1.
- 33.
In contrast to the previous section, a more detailed prior knowledge (i.e., for each proposal) is natural here since parties submit bids about the savings for all proposals within the auction, and these bids will become globally known afterwards.
- 34.
E.g., Myerson and Satterthwaite (1983, p. 265).
- 35.
See Sect. 3.3.1 for more details on this mechanism.
- 36.
- 37.
See Sect. 4.1.2.
- 38.
See also Sect. 3.3.1.
- 39.
For the definition of the set packing problem and an analysis of its properties, see, e.g., Balas and Padberg (1976, p. 710).
- 40.
See, e.g., Andersson et al. (2000, p. 6).
- 41.
Analogously to Theorem 5.2, with constant markups chosen by parties, the lower bound is \(1 -\sum\limits_{i\in P}E[{L}^{i}]/(4{S}^{sys})\) of the systemwide surplus. Further note that the sum of parties’ leeways may exceed S sys or go below L sys, depending on the complementarities of the central resource use in coordinated solutions; for an analogous discussion, see Sect. 5.2.
- 42.
For literature on learning and signaling strategies, see the corresponding discussion of the preceding section.
- 43.
For experimental studies of learning in repeated (anonymous, i.e., without signaling effects) sealed bid double auctions under the standard assumption of common knowledge about the distributions of parties’ reservation values, see Rapoport et al. (1998, p. 221), Seale et al. (2001, p. 177), and Daniel et al. (1998, p. 133).
- 44.
See also our computational tests of Sect. 6.1, p. 168.
- 45.
See e.g., Elmaghraby (2004, p. 214) for examples of auctions in B2B marketplaces and Hohner et al. (2003, p. 23) for a case study about the use of procurement auctions at Mars.
- 46.
See, Agndal and Nilsson (2008, p. 154) for a compilation of approaches for open-book accounting in supply chains.
- 47.
See our computational results of Sect. 6.1.
- 48.
See also Sect. 2.1.2.
- 49.
Such leeway is often specified by a flexibility range, e.g., Tsay (1999, p. 1341).
- 50.
Of course, such conflicts only arise if the supplier holds too little inventory to fulfill the buyer’s orders without a significant increase in his costs. Hence, a further interpretation of the mechanisms proposed is that of instruments to reduce inventory at decentralized parties.
- 51.
For ease of exposition, the existence of a frozen horizon is omitted here.
- 52.
See Dudek (2004, p. 116).
- 53.
See Dudek (2004, p. 120).
- 54.
See Sect. 4.4.3.
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Albrecht, M. (2010). New Coordination Mechanisms. In: Supply Chain Coordination Mechanisms. Lecture Notes in Economics and Mathematical Systems, vol 628. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02833-5_5
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