The solution to the Tullock rent-seeking game when R > 2: Mixed-strategy equilibria and mean dissipation rates

  • Michael R. Baye
  • Dan Kovenock
  • Casper G. De Vries


In Tullock’s rent-seeking model, the probability a player wins the game depends on expenditures raised to the power R. We show that a symmetric mixed-strategy Nash equilibrium exists when R > 2, and that overdissipation of rents does not arise in any Nash equilibrium. We derive a tight lower bound on the level of rent dissipation that arises in a symmetric equilibrium when the strategy space is discrete, and show that full rent dissipation occurs when the strategy space is continuous. Our results are shown to be consistent with recent experimental evidence on the dissipation of rents.


Nash Equilibrium Dissipation Rate Symmetric Equilibrium Strategy Space Pure Strategy Equilibrium 
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  1. 3.
    The null hypotheses should be interpreted with caution because the experimental setup of Millner and Pratt (1989) is not entirely congruent with the simultaneous move requirement (neither does it fit the alternating move version studied in Leininger, 1990; Leininger and Yang (1990).Google Scholar
  2. 4.
    Baye, Tian, and Zhou (1993) show that one cannot generally blame the non-existence of a pure-strategy equilibrium on the failure of payoff functions to be quasi-concave or upper semi-continuous.Google Scholar
  3. 5.
    Although Millner and Pratt claim to be testing the Tullock model, the experiment actually allows the rent-seekers to expend resources continuously over a small time interval. Hence, the experiment does not formally test the original one-shot simultaneous-move Tullock game. This problem is corrected in the experiments of Shogren and Baik (1991), who do not reject the theoretical prediction when R = 1.Google Scholar
  4. 6.
    The continuous strategy space (infinite game) is dealt with below.Google Scholar
  5. 7.
    The mixed strategies may be degenerate, i.e., in the case of a pure strategy equilibrium.Google Scholar
  6. 8.
    Shogren and Baik (1991) state that the behavioral inconsistency reported in Millner and Pratt.. is due to the nonexistence of a Nash equilibrium. In this case there is no predictable behavioral benchmark to measure the experimental evidence against. Our Theorem 2, however, provides such a benchmark. Shogren and Baik are referring to the non-existence of a symmet-ric pure strategy Nash equilibrium.Google Scholar
  7. 9.
    In future work it may be of interest to repeat the experiment for R = 3 and Q small such that all the properties of the symmetric equilibrium can be evaluated, i.e., the values of the py’s.Google Scholar
  8. 10.
    Calculations are based on (3.34 – 3.9)/s1 = −5.11 and (84 – 97.5)/s2 = −2.73, where s1 and s2 were calculated from Millner and Pratt (1989) using (3.34 – 6)/s1 = −4.28 and (84 – 150)/s2 = −13.37.Google Scholar
  9. 11.
    See also Shogren and Baik, who run a related experiment for R = 1 and find that the Nash equilibrium dissipation hypothesis cannot be rejected at the 90 percent level.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Michael R. Baye
    • 1
  • Dan Kovenock
    • 2
  • Casper G. De Vries
    • 3
  1. 1.Dept. of EconomicsThe Pennsylvania State UniversityUSA
  2. 2.Dept. of EconomicsPurdue UniversityWest LafayetteUSA
  3. 3.Tinbergen InstituutErasmus Universiteit RotterdamRotterdamThe Netherlands

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