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Bertrand Equilibria and Sharing Rules

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  • Hoernig, Steffen

Abstract

We analyse how sharing rules affect Nash equilibria in Bertrand games, where the sharing of profits at ties is a decisive assumption. Necessary conditions for either positive or zero equilibrium profits are derived. Zero profit equilibria are shown to exist under weak conditions if the sharing rule is ‘sign-preserving’. For Bertrand markets we define the class of ‘expectation sharing rules’, where profits at ties are derived from some distribution of quantities. In this class the winner-takes-all sharing rule is the only one that is always sign-preserving, while for each pair of demand and cost functions there may be many others.

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Paper provided by C.E.P.R. Discussion Papers in its series CEPR Discussion Papers with number 4972.

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Date of creation: Mar 2005
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Handle: RePEc:cpr:ceprdp:4972

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Keywords: Bertrand games; expectation sharing rules; sharing rule; sign-preserving sharing rules; tie-breaking rule;

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  1. Xavier Vives, 2001. "Oligopoly Pricing: Old Ideas and New Tools," MIT Press Books, The MIT Press, edition 1, volume 1, number 026272040x, December.
  2. Leo K. Simon and William R. Zame., 1987. "Discontinuous Games and Endogenous Sharing Rules," Economics Working Papers, University of California at Berkeley 8756, University of California at Berkeley.
  3. Osborne, Martin J. & Pitchik, Carolyn, 1986. "Price competition in a capacity-constrained duopoly," Journal of Economic Theory, Elsevier, vol. 38(2), pages 238-260, April.
  4. Todd R. Kaplan & David Wettstein, 2000. "The possibility of mixed-strategy equilibria with constant-returns-to-scale technology under Bertrand competition," Spanish Economic Review, Springer, Springer, vol. 2(1), pages 65-71.
  5. Baye, Michael R. & Morgan, John, 1999. "A folk theorem for one-shot Bertrand games," Economics Letters, Elsevier, vol. 65(1), pages 59-65, October.
  6. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, Econometric Society, vol. 67(5), pages 1029-1056, September.
  7. Harrington, Joseph Jr., 1989. "A re-evaluation of perfect competition as the solution to the Bertrand price game," Mathematical Social Sciences, Elsevier, Elsevier, vol. 17(3), pages 315-328, June.
  8. Hoernig, Steffen H., 2002. "Mixed Bertrand equilibria under decreasing returns to scale: an embarrassment of riches," Economics Letters, Elsevier, vol. 74(3), pages 359-362, February.
  9. John Morgan & Michael R. Baye, 2002. "Winner-take-all price competition," Economic Theory, Springer, Springer, vol. 19(2), pages 271-282.
  10. Sharkey, William W. & Sibley, David S., 1993. "A Bertrand model of pricing and entry," Economics Letters, Elsevier, vol. 41(2), pages 199-206.
  11. Dasgupta, Partha & Maskin, Eric, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 53(1), pages 1-26, January.
  12. Dastidar, Krishnendu Ghosh, 1995. "On the Existence of Pure Strategy Bertrand Equilibrium," Economic Theory, Springer, Springer, vol. 5(1), pages 19-32, January.
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Cited by:
  1. Hoernig, Steffen H., 2002. "Mixed Bertrand equilibria under decreasing returns to scale: an embarrassment of riches," Economics Letters, Elsevier, vol. 74(3), pages 359-362, February.

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