Consensus dynamics, network interaction, and Shapley indices in the Choquet framework

  • Silvia BortotEmail author
  • Ricardo Alberto Marques Pereira
  • Anastasia Stamatopoulou


We consider a set \(N = \{ 1,\ldots ,n \}\) of interacting agents whose individual opinions are denoted by \( x_{i}\), \(i \in N \) in some domain \({\mathbb {D}}\subseteq {\mathbb {R}}\). The interaction among the agents is expressed by a symmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The interacting network structure is thus that of a complete graph with edge values in (0, 1). In the Choquet framework, the interacting network structure is the basis for the construction of a consensus capacity \(\mu \), where the capacity value \(\mu (S)\) of a coalition of agents \(S \subseteq N\) is defined to be proportional to the sum of the edge interaction values contained in the subgraph associated with S. The capacity \(\mu \) is obtained in terms of its 2-additive Möbius transform \(m_{\mu }\), and the corresponding Shapley power and interaction indices are identified. We then discuss two types of consensus dynamics, both of which refer significantly to the notion of context opinion. The second type converges simply the plain mean, whereas the first type produces the Shapley mean as the asymptotic consensual opinion. In this way, it provides a dynamical realization of Shapley aggregation.


Consensus reaching Linear dynamical models Network interaction Choquet capacities Möbius transforms Shapley power and interaction indices 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by the authors.


  1. Abelson RP (1964) Mathematical models of the distribution of attitudes under controversy. In: Frederiksen N, Gulliksen H (eds) Contributions to mathematical psychology. Holt, Rinehart, and Winston, New York, pp 142–160Google Scholar
  2. Anderson NH (1981) Foundations of information integration theory. Academic Press, New YorkGoogle Scholar
  3. Anderson NH (1991) Contributions to information integration theory. Lawrence Erlbaum, HillsdaleGoogle Scholar
  4. Anderson NH, Graesser CC (1976) An information integration analysis of attitude change in group discussion. J Pers Soc Psychol 34:210–222CrossRefGoogle Scholar
  5. Beliakov G, Pradera A, Calvo T (2007) Aggregation functions: a guide for practitioners. Studies in Fuzziness and Soft Computing, vol 221. Springer, HeidelbergGoogle Scholar
  6. Beliakov G, Bustince Sola H, Calvo T (2016) A practical guide to averaging functions. Studies in Fuzziness and Soft Computing, vol 329. Springer, HeidelbergCrossRefGoogle Scholar
  7. Berger RL (1981) A necessary and sufficient condition for reaching a consensus using DeGroot’s method. J Am Stat Assoc 76:415–418MathSciNetzbMATHCrossRefGoogle Scholar
  8. Berrah L, Clivillé V (2007) Towards an aggregation performance measurement system model in a supply chain context. Comput Ind 58(7):709–719CrossRefGoogle Scholar
  9. Berrah L, Mauris G, Montmain J (2008) Monitoring the improvement of an overall industrial performance based on a Choquet integral aggregation. Omega 36(3):340–351CrossRefGoogle Scholar
  10. Bortot S, Marques Pereira RA (2013) Inconsistency and non-additive capacities: the analytic hierarchy process in the framework of Choquet integration. Fuzzy Sets Syst 213:6–26MathSciNetzbMATHCrossRefGoogle Scholar
  11. Bortot S, Marques Pereira RA (2014) The binomial Gini inequality indices and the binomial decomposition of welfare functions. Fuzzy Sets Syst 255:92–114MathSciNetzbMATHCrossRefGoogle Scholar
  12. Calvo T, Kolesárova A, Komorníková M, Mesiar R (2002a) Aggregation operators: properties, classes and construction methods. In: Calvo T, Mayor G, Mesiar R (eds) Aggregation operators: new trends and applications. Physica-Verlag, Heidelberg, pp 3–104zbMATHCrossRefGoogle Scholar
  13. Calvo T, Mayor G, Mesiar R (2002b) Aggregation operators: new trends and applications. Studies in Fuzziness and Soft Computing, vol 97. Springer, HeidelbergGoogle Scholar
  14. Chateauneuf A, Jaffray JY (1989) Some characterizations of lower probabilities and other monotone capacities throught the use of Möbius inversion. Math Soc Sci 17(3):263–283zbMATHCrossRefGoogle Scholar
  15. Chatterjee S (1975) Reaching a consensus: some limit theorems. In: Proceedings of International Statistics Institute, pp 159–164Google Scholar
  16. Chatterjee S, Seneta E (1977) Towards consensus: some convergence theorems on repeated averaging. J Appl Probab 14:89–97MathSciNetzbMATHCrossRefGoogle Scholar
  17. Choquet G (1953) Theory of capacities. Annales de l’Institut Fourier 5:131–295MathSciNetzbMATHCrossRefGoogle Scholar
  18. Clivillé V, Berrah L, Mauris G (2007) Quantitative expression and aggregation of performance measurements based on the MACBETH multi-criteria method. Int J Prod Econ 105(1):171–189CrossRefGoogle Scholar
  19. Davis JH (1973) Group decision and social interaction: a theory of social decision schemes. Psychol Rev 80:97–125CrossRefGoogle Scholar
  20. Davis JH (1996) Group decision making and quantitative judgments: a consensus model. In: Witte EH, Davis JH (eds) Understanding group behavior: consensual action by small groups. Lawrence Erlbaum, Mahwah, pp 35–59Google Scholar
  21. Deffuant G, Neau D, Amblard F, Weisbuch G (2000) Mixing beliefs among interacting agents. Adv Complex Syst 3(1):87–98CrossRefGoogle Scholar
  22. DeGroot MH (1974) Reaching a consensus. J Am Stat Assoc 69(345):118–121zbMATHCrossRefGoogle Scholar
  23. Denneberg D (1994) Non-additive measure and integral. Kluwer, DordrechtzbMATHCrossRefGoogle Scholar
  24. Dittmer JC (2001) Consensus formation under bounded confidence. Nonlinear Anal 47:4615–4621MathSciNetzbMATHCrossRefGoogle Scholar
  25. Dong Y, Zha Q, Zhang H, Kou G, Fujita H, Chiclana F, Herrera-Viedma E (2018a) Consensus reaching in social network group decision making: research paradigms and challenges. Knowl-Based Syst 162:3–13CrossRefGoogle Scholar
  26. Dong Y, Zhan M, Kou G, Ding Z, Liang H (2018b) A survey on the fusion process in opinion dynamics. Inf Fusion 43:57–65CrossRefGoogle Scholar
  27. Emrouznejad A, Marra M (2014) Ordered weighted averaging operators 1988–2014: a citation-based literature survey. Int J Intell Syst 29:994–1014CrossRefGoogle Scholar
  28. Fedrizzi M, Fedrizzi M, Marques Pereira RA (1999) Soft consensus and network dynamics in group decision making. Int J Intell Syst 14(1):63–77zbMATHCrossRefGoogle Scholar
  29. Fedrizzi M, Fedrizzi M, Marques Pereira RA (2007) Consensus modelling in group decision making: a dynamical approach based on fuzzy preferences. New Math Nat Comput 3(2):219–237MathSciNetzbMATHCrossRefGoogle Scholar
  30. Fedrizzi M, Fedrizzi M, Marques Pereira RA, Brunelli M (2008) Consensual dynamics in group decision making with triangular fuzzy numbers. In: Proceedings of the 41st Hawaii international conference on system sciences, pp 70–78Google Scholar
  31. Fedrizzi M, Fedrizzi M, Marques Pereira RA, Brunelli M (2010) The dynamics of consensus in group decision making: investigating the pairwise interactions between fuzzy preferences. In: Greco S et al (eds) Preferences and decisions, Studies in Fuzziness and Soft Computing, vol 257. Physica-Verlag, Heidelberg, pp 159–182zbMATHGoogle Scholar
  32. Fodor J, Roubens M (1994) Fuzzy preference modelling and multicriteria decision support. Kluwer Academic Publishers, DordrechtzbMATHCrossRefGoogle Scholar
  33. Fodor J, Marichal JL, Roubens M (1995) Characterization of the ordered weighted averaging operators. IEEE Trans Fuzzy Syst 3(2):236–240CrossRefGoogle Scholar
  34. French JRP (1956) A formal theory of social power. Psychol Rev 63:181–194CrossRefGoogle Scholar
  35. French S (1981) Consensus of opinion. Eur J Oper Res 7:332–340MathSciNetzbMATHCrossRefGoogle Scholar
  36. Friedkin NE (1990) Social networks in structural equation models. Soc Psychol Q 53:316–328CrossRefGoogle Scholar
  37. Friedkin NE (1991) Theoretical foundations for centrality measures. Am J Sociol 96:1478–1504CrossRefGoogle Scholar
  38. Friedkin NE (1993) Structural bases of interpersonal influence in groups: a longitudinal case study. Am Sociol Rev 58:861–872CrossRefGoogle Scholar
  39. Friedkin NE (1998) A structural theory of social influence. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  40. Friedkin NE (1999) Choice shift and group polarization. Am Sociol Rev 64:856–875CrossRefGoogle Scholar
  41. Friedkin NE (2001) Norm formation in social influence networks. Soc Netw 23(3):167–189CrossRefGoogle Scholar
  42. Friedkin NE, Johnsen EC (1990) Social influence and opinions. J Math Sociol 15(3–4):193–206zbMATHCrossRefGoogle Scholar
  43. Friedkin NE, Johnsen EC (1997) Social positions in influence networks. Soc Netw 19:209–222CrossRefGoogle Scholar
  44. Friedkin NE, Johnsen EC (1999) Social influence networks and opinion change. In: Thye SR, Macy MW, Walker HA, Lawler EJ (eds) Advances in group processes, vol 16. JAI Press. Greenwich, pp 1–29Google Scholar
  45. Friedkin NE, Johnsen EC (2011) Social influence network theory: a sociological examination of small group dynamics, vol 33. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  46. Grabisch M (1995) Fuzzy integral in multicriteria decision making. Fuzzy Sets Syst 69(3):279–298MathSciNetzbMATHCrossRefGoogle Scholar
  47. Grabisch M (1996) The application of fuzzy integrals in multicriteria decision making. Eur J Oper Res 89(3):445–456zbMATHCrossRefGoogle Scholar
  48. Grabisch M (1997a) k-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst 92(2):167–189MathSciNetzbMATHCrossRefGoogle Scholar
  49. Grabisch M (1997b) Alternative representations of discrete fuzzy measures for decision making. Int J Uncertain Fuzziness Knowl-Based Syst 5(5):587–607MathSciNetzbMATHCrossRefGoogle Scholar
  50. Grabisch M, Roubens M (1999) An axiomatic approach to the concept of interaction among players in cooperative games. Int J Game Theory 28(4):547–565MathSciNetzbMATHCrossRefGoogle Scholar
  51. Grabisch M, Labreuche C (2001) How to improve acts: an alternative representation of the importance of criteria in MCDM. Int J Uncertain Fuzziness Knowl-Based Syst 9(2):145–157MathSciNetzbMATHCrossRefGoogle Scholar
  52. Grabisch M, Labreuche C (2005) Fuzzy measures and integrals in MCDA. In: Figueira J, Greco S, Ehrgott M (eds) Multiple criteria decision analysis. Springer, Heidelberg, pp 563–604CrossRefGoogle Scholar
  53. Grabisch M, Labreuche C (2008) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR 6(1):1–44Google Scholar
  54. Grabisch M, Labreuche C (2010) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann Oper Res 175(1):247–286MathSciNetzbMATHCrossRefGoogle Scholar
  55. Grabisch M, Miranda P (2015) Exact bounds of the Möbius inverse of monotone set functions. Discrete Appl Math 186:7–12MathSciNetzbMATHCrossRefGoogle Scholar
  56. Grabisch M, Nguyen HT, Walker EA (1995) Fundamentals of uncertainty calculi with applications to fuzzy inference. Kluwer Academic Publishers, DordrechtzbMATHCrossRefGoogle Scholar
  57. Grabisch M, Murofushi T, Sugeno M (eds) (2000) Fuzzy measure and integrals: theory and applications. Physica-Verlag, HeidelbergGoogle Scholar
  58. Grabisch M, Kojadinovich I, Meyer P (2008) A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: applications of the Kappalab R package. Eur J Oper Res 186(2):766–785MathSciNetzbMATHCrossRefGoogle Scholar
  59. Grabisch M, Marichal JL, Mesiar R, Pap E (2009) Aggregation functions. Encyclopedia of mathematics and its applications, vol 127. Cambridge University Press, CambridgeGoogle Scholar
  60. Grabisch M, Marichal JL, Mesiar R, Pap E (2011) Aggregation functions: means. Inf Sci 181(1):1–22MathSciNetzbMATHCrossRefGoogle Scholar
  61. Graesser CC (1991) A social averaging theorem for group decision making. In: Anderson NH, Hillsdale NJ (eds) Contributions to information integration theory, vol 2. Lawrence Erlbaum, Mahwah, pp 1–40Google Scholar
  62. Harary F (1959) A criterion for unanimity in French’s theory of social power. In: Cartwright D (ed) Studies in social power. Institute for Social Research, Ann Arbor, pp 168–182Google Scholar
  63. Hegselmann R, Krause U (2002) Opinion dynamics and bounded confidence models, analysis and simulation. J Artif Soc Soc Simul 5(3):1–33Google Scholar
  64. Jia P, MirTabatabaei A, Friedkin NE, Bullo F (2015) Opinion dynamics and the evolution of social power in influence networks. SIAM Rev 57(3):367–397MathSciNetzbMATHCrossRefGoogle Scholar
  65. Kelly FP (1981) How a group reaches agreement: a stochastic model. Math Soc Sci 2:1–8MathSciNetzbMATHCrossRefGoogle Scholar
  66. Lehrer K (1975) Social consensus and rational agnoiology. Synthese 31:141–160CrossRefGoogle Scholar
  67. Lehrer K, Wagner K (1981) Rational consensus in science and society. Reidel, DordrechtzbMATHCrossRefGoogle Scholar
  68. Marichal JL (1998) Aggregation operators for multicriteria decision aid. Ph.D. Thesis. University of Liège, Liège, BelgiumGoogle Scholar
  69. Marques Pereira RA, Bortot S (2001) Consensual dynamics, stochastic matrices, Choquet measures, and Shapley aggregation. In: Proceedings of 22nd Linz seminar on fuzzy set theory: valued relations and capacities in decision theory, Linz, Austria, pp 78–80Google Scholar
  70. Marques Pereira RA, Ribeiro RA, Serra P (2008) Rule correlation and Choquet integration in fuzzy inference systems. Int J Uncertain Fuzziness Knowl-Based Syst 16(5):601–626MathSciNetzbMATHCrossRefGoogle Scholar
  71. Marsden PV, Friedkin NE (1993) Network studies of social influence. Sociol Methods Res 22:127–151CrossRefGoogle Scholar
  72. Marsden PV, Friedkin NE (1994) Network studies of social influence. In: Wasserman S, Galaskiewicz J (eds) Advances in social network analysis, vol 22. Sage, Thousand Oaks, pp 3–25Google Scholar
  73. Mayag B, Grabisch M, Labreuche C (2011) A representation of preferences by the Choquet integral with respect to a 2-additive capacity. Theor Decis 71(3):297–324MathSciNetzbMATHCrossRefGoogle Scholar
  74. Mayag B, Grabisch M, Labreuche C (2011) A characterization of the 2-additive Choquet integral through cardinal information. Fuzzy Sets Syst 184(1):84–105MathSciNetzbMATHCrossRefGoogle Scholar
  75. Mesiar R, Kolesárová A, Calvo T, Komorníková M (2008) A review of aggregation functions. In: Bustince H, Herrera F, Montero J (eds) Fuzzy sets and their extensions: representation, aggregation and models, Studies in Fuzziness and Soft Computing, vol 220. Springer, Heidelberg, pp 121–144zbMATHCrossRefGoogle Scholar
  76. Miranda P, Grabisch M (1999) Optimization issues for fuzzy measures. Int J Uncertain Fuzziness Knowl-Based Syst 7(6):545–560MathSciNetzbMATHCrossRefGoogle Scholar
  77. Miranda P, Grabisch M, Gil P (2005) Axiomatic structure of k-additive capacities. Math Soc Sci 49(2):153–178MathSciNetzbMATHCrossRefGoogle Scholar
  78. Murofushi T (1992) A technique for reading fuzzy measures (I): the Shapley value with respect to a fuzzy measure. In: 2nd Fuzzy Workshop, Nagaoka, Japan, pp 39–48 (in Japanese)Google Scholar
  79. Murofushi T, Sugeno M (1989) An interpretation of fuzzy measures and the Choquet integral with respect to a fuzzy measure. Fuzzy Sets Syst 29(2):201–227MathSciNetzbMATHCrossRefGoogle Scholar
  80. Murofushi T, Soneda S (1993) Techniques for reading fuzzy measures (III): interaction index. In: 9th Fuzzy system symposium, Sapporo, Japan, pp 693–696 (in Japanese)Google Scholar
  81. Murofushi T, Sugeno M (1993) Some quantities represented by the Choquet integral. Fuzzy Sets Syst 2(56):229–235MathSciNetzbMATHCrossRefGoogle Scholar
  82. Murofushi T, Sugeno M (2000) Fuzzy measures and fuzzy integrals. In: Grabisch M et al (eds) Fuzzy measures and integrals: theory and applications. Physica-Verlag, Heidelberg, pp 3–41zbMATHGoogle Scholar
  83. Nurmi H (1985) Some properties of the Lehrer–Wagner method for reaching rational consensus. Synthese 62:13–24CrossRefGoogle Scholar
  84. Rota GC (1964) On the foundations of combinatorial theory I. Theory of Möbius functions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebeite 2(4):340–368zbMATHCrossRefGoogle Scholar
  85. Sen A (1982) Choice, welfare, and measurement. Basil Blackwell, OxfordzbMATHGoogle Scholar
  86. Shapley LS (1953) A value for \(n\)-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, vol II. Annals of Mathematics Studies, Princeton University Press NJ, pp 307–317Google Scholar
  87. Sugeno M (1974) Theory of fuzzy integrals and its applications. Ph.D. Thesis. Tokyo Institut of Technology, JapanGoogle Scholar
  88. Taylor M (1968) Towards a mathematical theory of influence and attitude change. Hum Relats XXI:121–139CrossRefGoogle Scholar
  89. Torra V, Narukawa Y (2007) Modeling decisions: information fusion and aggregation operators. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  90. Ureña R, Kou G, Dong Y, Chiclana F, Herrera-Viedma E (2019) A review on trust propagation and opinion dynamics in social networks and group decision making frameworks. Inf Sci 478:461–475CrossRefGoogle Scholar
  91. Wagner C (1978) Consensus through respect: a model of rational group decision-making. Philos Stud 34:335–349MathSciNetCrossRefGoogle Scholar
  92. Wagner C (1982) Allocation, Lehrer models, and the consensus of probabilities. Theor Decis 14:207–220MathSciNetzbMATHCrossRefGoogle Scholar
  93. Wang Z, Klir GJ (1992) Fuzzy measure theory. Springer, New YorkzbMATHCrossRefGoogle Scholar
  94. Yager RR, Kacprzyk J (eds) (1997) The ordered weighted averaging operators. Theory and applications. Kluwer Academic Publisher, DordrechtzbMATHGoogle Scholar
  95. Yager RR, Kacprzyk J, Beliakov J (ed) (2011) Recent Developments in the ordered weighted averaging operators: theory and practice. Studies in Fuzziness and Soft Computing, vol 265. Springer, HeidelbergGoogle Scholar
  96. Yager RR (1988) On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybern 18(1):183–190zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Economics and ManagementUniversity of TrentoTrentoItaly
  2. 2.Department of Industrial EngineeringUniversity of TrentoPovoItaly

Personalised recommendations