Advertisement

Top

, Volume 12, Issue 2, pp 331–350 | Cite as

The core and related solution concepts for infinite assignment games

  • Natividad Llorca
  • Joaquín Sánchez-Soriano
  • Stef Tijs
  • Judith Timmer
Article

Abstract

Assignment problems where both sets of agents that have to be matched are countably infinite, the so-called infinite assignment problems, are studied as well as the related cooperative assignment games. Further, several solution concepts for these assignment games are studied. The first one is the utopia payoff for games with an infinite value. In this solution each player receives the maximal amount he can think of with respect to the underlying assignment problem. This solution is contained in the core of the game.

Second, we study two solutions for assignment games with a finite value. Our main result is the existence of core-elements of these games, although they are hard to calculate. Therefore another solution, the f-strong ε-core is studied. This particular solution takes into account that due to organisational limitations it seems reasonable that only finite groups of agents will eventually protest against unfair proposals of profit distributions. The f-strong ε-core is shown to be nonempty.

Key Words

Cooperative games assignment problems infinite programs solutions 

AMS subject classification

91A12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aliprantis C.D. and Border K.C. (1999).Infinite Dimensional Analysis. Springer.Google Scholar
  2. Anderson E.J. and Nash P. (1987).Linear Programming in Infinite Dimensional Spaces. Wiley.Google Scholar
  3. Birkhoff G. (1946). Tres Observaciones sobre el Algebra Lineal.Revista de la Universidad Nacional de Tucumán A 5, 147–151.Google Scholar
  4. Borm P., Hamers H. and Hendrickx R. (2001). Operations Research Games: A Survey.TOP 9, 139–198.Google Scholar
  5. Cross W.P., Romeijn H.E. and Smith R.L. (1998). Approximating Extreme Points of Infinite Dimensional Convex Sets.Mathematics of Operations Research 23, 433–442.CrossRefGoogle Scholar
  6. Fragnelli V., Patrone F., Sideri E. and Tijs S. (1999). Balanced Games Arising from Infinite Linear Models.Mathematical Methods of Operations Research 50, 385–397.CrossRefGoogle Scholar
  7. Kaneko M. and Wooders M. (1986). The Core of a Game with a Continuum of Players and Finite Coalitions: The Model and Some Results.Mathematical Social Sciences 12, 105–137.CrossRefGoogle Scholar
  8. Köthe G. (1983).Topological Vector Spaces I. Springer.Google Scholar
  9. Llorca N., Tijs S. and Timmer J. (2003). Semi-Infinite Assignment Problems and Related Games.Mathematical Methods of Operations Research 57, 67–78.CrossRefGoogle Scholar
  10. Sánchez-Soriano J., Llorca N., Tijs S. and Timmer J. (2001). Semi-Infinite Assignment and Transportation Games. In: Goberna M.A. and López M.A. (eds.),Semi-Infinite Programming: Recent Advances. Kluwer Academic Publishers, 349–363.Google Scholar
  11. Shapley L.S. and Shubik M. (1966). Quasi-Cores in a Monetary Economy with Nonconvex Preferences.Econometrica 34, 805–827.CrossRefGoogle Scholar
  12. Shapley L.S. and Shubik M. (1972). The Assignment Game I: The Core.International Journal of Game Theory 1, 111–130.CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2004

Authors and Affiliations

  • Natividad Llorca
    • 1
  • Joaquín Sánchez-Soriano
    • 1
  • Stef Tijs
    • 2
  • Judith Timmer
    • 3
  1. 1.CIO and Department of Statistics and Applied MathematicsMiguel Hernández University Elche CampusElcheSpain
  2. 2.CentER and Department of Econometrics and Operations ResearchTilburg UniversityTilburgThe Netherlands
  3. 3.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

Personalised recommendations