Skip to main content
SpringerLink
Log in

Fundamentals

  • Chapter
  • First Online:
Vertical Cooperative Advertising in Supply Chain Management

Part of the book series: Contributions to Management Science ((MANAGEMENT SC.))

  • 1163 Accesses

Abstract

This chapter shall give some theoretical fundamentals which are necessary for the following work. This includes introducing general principles of game theory as well as a general cooperative advertising model that is successively adapted to the changing requirements during this work.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See von Neumann and Morgenstern (1944): Theory of games.

  2. 2.

    Cf. Berninghaus et al. (2010): Strategische Spiele, p. 1 and Rieck (2010): Spieltheorie, p. 21.

  3. 3.

    Cf. von Neumann and Morgenstern (1953): Theory of games, pp. 1–6.

  4. 4.

    For a summary of the history of game theory, see Berninghaus et al. (2010): Strategische Spiele, pp. 3–9, and Peters (2008): Game theory, pp. 1 et seq.

  5. 5.

    Myerson (1997): Game theory, p. 1.

  6. 6.

    Cf. Riechmann (2008): Spieltheorie, p. 18.

  7. 7.

    Cf. Holler and Illing (2006): Spieltheorie, p. 1.

  8. 8.

    Cf. Borgwardt (2001): Optimierung, p. 509.

  9. 9.

    This explanation is based on Holler and Illing (2006): Spieltheorie, pp. 31–42, and Peters (2008): Game theory, pp. 73 et seq.

  10. 10.

    Utility functions were introduced by von Neumann and Morgenstern (1944), who firstly proposed an axiomatic system that allows to quantify utility numerically (see von Neumann and Morgenstern (1953): Theory of games, pp. 15–31). Thereby, a utility function is used to assign an individual utility value of a person to a certain event. This also allows to compare different utility values mathematically (cf. Laux (2005): Entscheidungstheorie, p. 26).

  11. 11.

    Cf. Rieck (2010): Spieltheorie, pp. 102–104.

  12. 12.

    Cf. Borgwardt (2001): Optimierung, p. 514.

  13. 13.

    Cf. Holler and Illing (2006): Spieltheorie, pp. 43–50, and Peters (2008): Game theory, p. 59.

  14. 14.

    Cf. Jost (2001): Spieltheorie in der Betriebswirtschaftslehre, pp. 21–25.

  15. 15.

    Cf. Rieck (2010): Spieltheorie, p. 162. Please note that matrices are mostly used for 2-person games, but are also defined for n-player games theoretically.

  16. 16.

    Cf. Holler and Illing (2006): Spieltheorie, p. 270, and Borgwardt (2001): Optimierung, p. 513.

  17. 17.

    Cf. Berninghaus et al. (2010): Strategische Spiele, pp. 28–34, and Rieck (2010): Spieltheorie, p. 80.

  18. 18.

    Cf. Cachon and Netessine (2004): Game theory, p. 36, and Sieg (2010): Spieltheorie, p. 91.

  19. 19.

    Please note that only cooperative (or axiomatic) bargaining theory is considered in this work, which can be used to determine a fair division of pay-offs as explained above. In addition, there is also a research field called non-cooperative, strategic or behavioristic bargaining theory, which refers to the actual bargaining process. Examples of this area are the Zeuthen-Harsanyi game (see Harsanyi (1977): Rational behavior, pp. 162–164, cited in Holler and Illing (2006): Spieltheorie, p. 252), or the Rubinstein game (see Rubinstein (1982): Perfect equilibrium). For a more detailed elaboration, we refer the reader to Berninghaus et al. (2010): Strategische Spiele, pp. 198–229, and Holler and Illing (2006): Spieltheorie, pp. 240–266.

  20. 20.

    Cf. Holler and Illing (2006): Spieltheorie, pp. 189 & 267.

  21. 21.

    See Holler and Illing (2006): Spieltheorie, pp. 78–87.

  22. 22.

    Cf. Leng and Parlar (2005): Game theoretic applications, p. 189.

  23. 23.

    The limitation on bargaining games instead of coalition games results from the fact that only the 2-player game in Chap. 4 additionally requires the application of cooperative game theory. Hence, the analysis of coalitions is dispensable in this context, wherefore we refer interested readers to, e.g., Peleg and Sudhölter (2007): Theory of cooperative games for further information.

  24. 24.

    See Nash (1950b): Equilibrium points in n-person games, Nash (1951): Non-cooperative games, and Cournot (1838): Recherches. Please note that the term Nash equilibrium characterizes the solution of a so-called Nash game.

  25. 25.

    A strategy fulfilling this condition is called a dominant strategy. For more information on the solution technique of strategic dominance see, e.g., Riechmann (2008): Spieltheorie, pp. 25–32.

  26. 26.

    Cf. Sieg (2010): Spieltheorie, p. 15.

  27. 27.

    In the following chapters, we use the superscript N instead of an asterisk to denote a Nash equilibrium in order to distinguish the equilibria introduced in this section.

  28. 28.

    Cf. Holler and Illing (2006): Spieltheorie, pp. 57–67, Berninghaus et al. (2010): Strategische Spiele, pp. 23 et seq., and Riechmann (2008): Spieltheorie, pp. 34 et seq.

  29. 29.

    More examples related to firms competing in prices or quantities can be found in Pfähler and Wiese (2008): Unternehmensstrategien im Wettbewerb, pp. 53–102.

  30. 30.

    See von Stackelberg (1934): Marktform und Gleichgewicht.

  31. 31.

    Cf. Berninghaus et al. (2010): Strategische Spiele, pp. 144–146, Cachon and Netessine (2004): Game theory, pp. 27 et seq., Leng and Parlar (2005): Game theoretic applications, pp. 191 et seq., and Riechmann (2008): Spieltheorie, pp. 132 et seq.

  32. 32.

    Another example regarding a Stackelberg game with firms competing in quantity can be found in Pfähler and Wiese (2008): Unternehmensstrategien im Wettbewerb, pp. 150–157.

  33. 33.

    In the following chapters, we use the superscript S to denote a Stackelberg equilibrium.

  34. 34.

    Please note that this concept is sometimes also named collusion or cartelization in economics. To avoid any misunderstandings, it shall again be pointed out that the notion non-cooperative game only refers to the prohibition of preliminary agreements between the players, which does not forbid to act in cooperation.

  35. 35.

    Cf. here and in the following Riechmann (2008): Spieltheorie, pp. 123–125.

  36. 36.

    In the following chapters, we use the superscript C to denote a Cooperation.

  37. 37.

    Cf. Berninghaus et al. (2010): Strategische Spiele, pp. 158 et seq. Please note that bargaining theory directly refers to utility values v p , while non-cooperative games are characterized by the players utility functions u p (s) as explained in the previous section.

  38. 38.

    Please note that this is a modification of the notation \(\mathcal{B} = (\mathcal{V},c)\), which can be found in, e.g., Holler and Illing (2006): Spieltheorie, p. 191, and Sieg (2010): Spieltheorie, p. 92. Here, c denotes the disagreement point of the bargaining game, i.e., the pay-off value each player receives when no agreement can be settled. However, since this disagreement point is dispensable in this work, it is not considered further. In order to ensure comparability to the notation of non-cooperative games, \(\mathcal{G} = (\mathcal{N},\mathcal{S},\mathcal{U})\), we also include the set of participating players \(\mathcal{N}\).

  39. 39.

    See Nash (1950a): Bargaining problem.

  40. 40.

    Cf. Berninghaus et al. (2010): Strategische Spiele, pp. 162–178, Riechmann (2008): Spieltheorie, pp. 171–174, and Holler and Illing (2006): Spieltheorie, pp. 195–205.

  41. 41.

    This form of utility function is applied in Xie and Neyret (2009): Co-op advertising, p. 1383, and Xie and Wei (2009): Co-op advertising, p. 789.

  42. 42.

    In general, the risk behavior of an utility function can be determined by means of the Arrow-Pratt measure of absolute risk aversion, which is defined by \(\mathbb{R}(y) = -(\partial ^{2}u(y)/\partial y^{2}) \cdot (\partial u(y)/\partial y)\) (cf. Pratt (1964): Risk aversion, p. 122). \(\mathbb{R}(y) > 0\) represents risk aversion, \(\mathbb{R}(y) = 0\) risk neutrality, and \(\mathbb{R}(y) < 0\) a risk-seeking behavior. For more information on these terms, see Bamberg et al. (2008): Betriebswirtschaftliche Entscheidungslehre, pp. 81–84.

  43. 43.

    Cf. Holler and Illing (2006): Spieltheorie, p. 215. For more information on the term bargaining power, we refer the reader to Kunter (2009): Absatzkanalkoordination, pp. 25–27.

  44. 44.

    See Harsanyi and Selten (1972): Generalized Nash and Kalai (1977): Nonsymmetric Nash.

  45. 45.

    Cf. Holler and Illing (2006): Spieltheorie, pp. 215–217.

  46. 46.

    An alternative way to integrate bargaining power into a bargaining model can be found in Eliashberg (1986): Arbitrating a dispute. Here, the total utility is calculated via \(v_{\mathcal{N}} =\sum _{p}\lambda _{p}v_{p}\). This model is used in, e.g., Yue et al. (2006): Coordination of cooperative advertising, p. 79, together with an exponential utility function of the shape \(u_{p}(y_{p}) = 1 - e^{-\mu _{p}y_{p}}\). However, as the resulting mathematical expressions a far more complex than those deriving from the Asymmetric Nash bargaining solution, a further discussion of this concept is omitted in this work.

  47. 47.

    See the literature review in Chap. 3, especially Table 3.2, for a detailed analysis of existing research.

  48. 48.

    This approach can be found in, e.g., Huang and Li (2001): Co-op advertising models, where margins are, however, exogenously given, or later on in Chap. 4, where margins are decision variables.

  49. 49.

    This approach can be found in, e.g., SeyedEsfahani et al. (2011): Vertical co-op advertising and Xie and Wei (2009): Coordinating advertising.

  50. 50.

    Please note that retailer j’s margin can be calculated via \(m_{\mathit{ijk}} = p_{\mathit{jk}} - w_{\mathit{ijk}} - c_{\mathit{jk}}\) if necessary.

  51. 51.

    Cf., e.g., Huang and Li (2001): Co-op advertising models, pp. 529 et seq.

  52. 52.

    See, e.g., SeyedEsfahani et al. (2011): Vertical co-op advertising, p. 265, and Xie and Neyret (2009): Co-op advertising, p. 1376.

  53. 53.

    For a more detailed classification, see the literature review in Chap. 3.

Bibliography

  • Bamberg, G., Coenenberg, A. G., & Krapp, M. (2008). Betriebswirtschaftliche Entscheidungslehre (14th ed.). München: Vahlen.

    Google Scholar 

  • Berninghaus, S. K., Ehrhart, K.-M., & Güth, W. (2010). Strategische Spiele: Eine Einführung in die Spieltheorie (3rd ed.). Berlin: Springer.

    Book  Google Scholar 

  • Borgwardt, K. H. (2001). Optimierung, Operations Research, Spieltheorie: Mathematische Grundlagen. Basel: Birkhäuser.

    Book  Google Scholar 

  • Cachon, G. P., & Netessine, S. (2004). Game theory in supply chain analysis. In D. Simchi-Levi, S. D. Wu, & Z. J. Shen (Eds.), Handbook of quantitative supply chain analysis: Modeling in the e-business era (pp. 13–66). Boston: Kluwer Academic Publishers.

    Chapter  Google Scholar 

  • Cournot, A. (1838). Recherches sur le principes mathématiques de la théorie des richesses. Paris: Hachette.

    Google Scholar 

  • Eliashberg, J. (1986). Arbitrating a dispute: A decision analytic approach. Management Science, 32(8), 963–974.

    Article  Google Scholar 

  • Harsanyi, J. C. (1977). Rational behavior and bargaining equilibrium in games and social situations. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Harsanyi, J. C., & Selten, R. (1972). A generalized Nash solution for two-person bargaining games with incomplete information. Management Science, 18(5), 80–106.

    Article  Google Scholar 

  • Holler, M. J., & Illing, G. (2006). Einführung in die Spieltheorie (6th ed.). Berlin: Springer.

    Google Scholar 

  • Huang, Z., & Li, S. X. (2001). Co-op advertising models in manufacturer-retailer supply chains: A game theory approach. European Journal of Operational Research, 135(3), 527–544.

    Article  Google Scholar 

  • Jost, P.-J. (2001). Die Spieltheorie in der Betriebswirtschaftslehre. Stuttgart: Schäffer-Poeschl.

    Google Scholar 

  • Kalai, E. (1977). Nonsymmetric Nash solutions and replications of 2-person bargaining. International Journal of Game Theory, 6(3), 129–133.

    Article  Google Scholar 

  • Kunter, M. (2009). Absatzkanalkoordination durch Hersteller-Handels-Konditionen. Wiesbaden: Gabler.

    Book  Google Scholar 

  • Laux, H. (2005). Entscheidungstheorie (6th ed.). Berlin: Springer.

    Google Scholar 

  • Leng, M., & Parlar, M. (2005). Game theoretic applications in supply chain management: A review. Information Systems and Operational Research, 43(3), 187–220.

    Google Scholar 

  • Myerson, R. B. (1997). Game theory: Analysis of conflict. Harvard: Harvard University Press.

    Google Scholar 

  • Nagarajan, M., & SoÅ¡ić, G. (2008). Game-theoretic analysis of cooperation among supply chain agents: Review and extensions. European Journal of Operational Research, 187(3), 719–745.

    Article  Google Scholar 

  • Nash, J. (1951). Non-cooperative games. The Annals of Mathematics, 54(2), 286–295.

    Article  Google Scholar 

  • Nash, J. F. (1950a). The bargaining problem. Econometrica, 18(2), 155–162.

    Article  Google Scholar 

  • Nash, J. F. (1950b). Equilibrium points in n-person games. In Proceedings of the National Academy of Sciences of the United States of America, 36(1), 48–49.

    Google Scholar 

  • Peleg, B., Sudhölter, P., Leinfellner, W., & Eberlein, G., (Eds.). (2007). Introduction to the theory of cooperative games. Theory and Decision Library (Vol. 34, 2nd ed.). Berlin: Springer.

    Google Scholar 

  • Peters, H. (2008). Game theory: A multi-leveled approach. Berlin: Springer.

    Book  Google Scholar 

  • Pfähler, W., & Wiese, H. (2008). Unternehmensstrategien im Wettbewerb: Eine spieltheoretische Analyse (3rd ed.). Berlin: Springer.

    Google Scholar 

  • Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica, 32(1/2), 122–136.

    Article  Google Scholar 

  • Riechmann, T. (2008). Spieltheorie (2nd ed.). München: Vahlen.

    Google Scholar 

  • Rieck, C. (2010). Spieltheorie: Eine Einführung (10th ed.). Eschborn: Christian Rieck Verlag.

    Google Scholar 

  • Rubinstein, A. (1982). Perfect equilibrium in a bargaining model. Econometrica, 50(1), 97–109.

    Article  Google Scholar 

  • SeyedEsfahani, M. M., Biazaran, M., & Gharakhani, M. (2011). A game theoretic approach to coordinate pricing and vertical co-op advertising in manufacturer-retailer supply chains. European Journal of Operational Research, 211(2), 263–273.

    Article  Google Scholar 

  • Sieg, G. (2010). Spieltheorie (3rd ed.). München: Oldenbourg.

    Google Scholar 

  • von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior (1st ed.). Princeton: Princeton University Press.

    Google Scholar 

  • von Neumann, J., & Morgenstern, O. (1953). Theory of games and economic behavior (3rd ed.). Princeton: Princeton University Press.

    Google Scholar 

  • von Stackelberg, H. (1934). Marktform und Gleichgewicht. Wien: Springer.

    Google Scholar 

  • Xie, J., & Neyret, A. (2009). Co-op advertising and pricing models in manufacturer-retailer supply chains. Computers & Industrial Engineering, 56(4), 1375–1385,

    Article  Google Scholar 

  • Xie, J., & Wei, J. C. (2009). Coordinating advertising and pricing in a manufacturer-retailer channel. European Journal of Operational Research, 197(2), 785–791.

    Article  Google Scholar 

  • Yue, J., Austin, J., Wang M.-C., & Huang, Z. (2006). Coordination of cooperative advertising in a two-level supply chain when manufacturer offers discount. European Journal of Operational Research, 168(1), 65–85.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Industrial Management, TU Dresden, Dresden, Germany

    Gerhard Aust (Chair)

Authors
  1. Gerhard Aust
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Aust, G. (2015). Fundamentals. In: Vertical Cooperative Advertising in Supply Chain Management. Contributions to Management Science. Springer, Cham. https://doi.org/10.1007/978-3-319-11626-6_2

Download citation

Publish with us

Policies and ethics

Search

Navigation