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Splitting Games: Nash Equilibrium and the Optimisation Problem

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  • Ana Paula Martins

Abstract

This research states the stylised n (more than two) players’ splitting problem as a mathematical programme, relying on definitions of the values of the game and problem stationarity to generate tractable reduced forms, and derives the known solutions according to the properties of pertaining first-order conditions. On the one hand, boundary constraints are taken into consideration, required by the most general formulation possible with respect to the controls. On the other, distinction between FOC’s of optimizing behavior and equilibrium fitness is provided. Finally, the formal proof of the internal insufficiency of the usual approach to determine the equilibrium is advanced, and the imposing additional conditions – affecting cross multipliers - required for model solving forwarded and interpreted. Two different types of protocols (sets of rules of the game) were staged: alternate offers and synchronized ones. Perfect information (and foresight) of the players, infinite horizon, and offers exchange restricted to infinite-term settlements are always assumed. Each player makes a proposition of the division among the n participants. Periodic “outside” alternatives may differ according to whose offer is being analysed, and from those accruing to the players when none is forwarded. The alternate offers protocol is a generalization of the Rubinstein’s structure. At each round of negotiations, one and only one player, exogenously determined, can make an – the – offer. An agent must conciliate – and solve consistently – as many optimization problems as eventual proponents there are in the game.

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Paper provided by Economics and Econometrics Research Institute (EERI), Brussels in its series EERI Research Paper Series with number EERI_RP_2010_36.

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Date of creation: 10 Aug 2010
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Handle: RePEc:eei:rpaper:eeri_rp_2010_36

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Keywords: Non-Cooperative N-Person Games; Infinite Horizon; Mixed Strategy Games; Mixed Strategies under Perfect Information Games; Simultaneous Sequential Bargaining; Matching Equilibrium under Sequential Bargaining; Synchronous (Decisions) Equilibrium under Sequential Bargaining; Mechanism Design; Bargaining Protocols; Dynamic Programming; Stationary Problems (without State Variables).;

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References

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  1. D’Artis Kancs, 2005. "Can we use NEG models to predict migration flows? An example of CEE accession countries," Migration Letters, Transnational Press London, UK, vol. 2(1), pages 32-63, April.
  2. Julda Kielyte, 2008. "Estimating Panel Data Models in the Presence of Endogeneity and Selection," Journal of Economics and Econometrics, Economics and Econometrics Research Institute (EERI), Brussels, vol. 51(2), pages 1-19.
  3. Ana Paula Martins, 2010. "Splitting Games: Nash Equilibrium and the Optimisation Problem," Journal of Economics and Econometrics, Economics and Econometrics Research Institute (EERI), Brussels, vol. 53(1), pages 1-28.
  4. Ana Paula Martins, 2006. "On Simple Conditions for Mixed Equilibria in Dualistic Models. Part I: Degree of Mobility," EERI Research Paper Series EERI_RP_2006_03, Economics and Econometrics Research Institute (EERI), Brussels.
  5. Ana Paula Martins, 2002. "Reconciling the Nash and Kalai-Smorodinsky Cooperative Solutions: Generalized Maximands of CES Form," EERI Research Paper Series EERI_RP_2002_05, Economics and Econometrics Research Institute (EERI), Brussels.
  6. Nash, John, 1953. "Two-Person Cooperative Games," Econometrica, Econometric Society, vol. 21(1), pages 128-140, April.
  7. Busch, L-A. & Wen, Q., 1991. "Perfect Equilibria in a Negotiation Model," University of Western Ontario, The Centre for the Study of International Economic Relations Working Papers 9108, University of Western Ontario, The Centre for the Study of International Economic Relations.
  8. Sakovics Jozsef, 1993. "Delay in Bargaining Games with Complete Information," Journal of Economic Theory, Elsevier, vol. 59(1), pages 78-95, February.
  9. Ana Paula Martins, 2012. "On Depth and Retrospect: “I Forget, and Forgive – but I Discount”," Journal of Economics and Econometrics, Economics and Econometrics Research Institute (EERI), Brussels, vol. 55(3), pages 1-45.
  10. Ariel Rubinstein, 2010. "Perfect Equilibrium in a Bargaining Model," Levine's Working Paper Archive 252, David K. Levine.
  11. Ana Paula Martins, 2010. "Frontier Techniques: Contrasting the Performance of (Single-)Truncated Order Regression Methods and Replicated Moments," Journal of Economics and Econometrics, Economics and Econometrics Research Institute (EERI), Brussels, vol. 53(2), pages 75-93.
  12. Kancs, D’Artis & Kielyte, Julda, 2010. "Education in the East, Emigrating to the West?," European Review, Cambridge University Press, vol. 18(02), pages 133-154, May.
  13. Perry Motty & Reny Philip J., 1993. "A Non-cooperative Bargaining Model with Strategically Timed Offers," Journal of Economic Theory, Elsevier, vol. 59(1), pages 50-77, February.
  14. JÕzsef SÂkovics & Clara PonsatÎ, 1998. "Rubinstein bargaining with two-sided outside options," Economic Theory, Springer, vol. 11(3), pages 667-672.
  15. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
  16. d’Artis Kancs, 2006. "The economic geography of labour migration: Competition, competitiveness and development," EERI Research Paper Series EERI_RP_2006_01, Economics and Econometrics Research Institute (EERI), Brussels.
  17. Kancs, d'Artis & Kielyte, Julda, 2010. "European Integration and Labour Migration," European Integration online Papers (EIoP), European Community Studies Association Austria (ECSA-A), vol. 14, November.
  18. Ana Paula Martins, 2006. "On Simple Conditions for Mixed Equilibria in Dualistic Models. Part II: Degree of Coverage," EERI Research Paper Series EERI_RP_2006_04, Economics and Econometrics Research Institute (EERI), Brussels.
  19. Martin J. Osborne & Ariel Rubinstein, 2005. "Bargaining and Markets," Levine's Bibliography 666156000000000515, UCLA Department of Economics.
  20. d’Artis Kancs & Julda Kielyte, 2002. "Migration in the Enlarged European Union: Empirical Evidence for Labour Mobility in the Baltic States," EERI Research Paper Series EERI_RP_2002_04, Economics and Econometrics Research Institute (EERI), Brussels.
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Cited by:
  1. Ana Paula Martins, 2010. "Frontier Techniques: Contrasting the Performance of (Single-) Truncated Order Regression Methods and Replicated Moments," EERI Research Paper Series EERI_RP_2010_37, Economics and Econometrics Research Institute (EERI), Brussels.
  2. Ana Paula Martins, 2011. "Compliance with the Institutional Wage in Dualistic Models," EERI Research Paper Series EERI_RP_2011_15, Economics and Econometrics Research Institute (EERI), Brussels.
  3. d'Artis Kancs & Julia Kielyte, 2010. "European Integration and Labour Migration," EERI Research Paper Series EERI_RP_2010_27, Economics and Econometrics Research Institute (EERI), Brussels.
  4. James Anderson, 2001. "Migration, FDI, and the Margins of Trade," EERI Research Paper Series EERI_RP_2001_05, Economics and Econometrics Research Institute (EERI), Brussels.
  5. Ana Paula Martins, 2010. "Splitting Games: Nash Equilibrium and the Optimisation Problem," Journal of Economics and Econometrics, Economics and Econometrics Research Institute (EERI), Brussels, vol. 53(1), pages 1-28.

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