Conflict and rent-seeking success functions: Ratio vs. difference models of relative success

  • Jack Hirshleifer


The rent-seeking competitions studied by economists fall within a much broader category of conflict interactions that also includes military combats, election campaigns, industrial disputes, lawsuits, and sibling rivalries. In the rent-seeking literature, each party’s success pi (which can be interpreted either as the probability of victory or as the proportion of the prize won) has usually been taken to be a function of the ratio of the respective resource commitments. Alternatively, however, pi may instead be a function of the difference between the parties’ commitments to the contest. The Contest Success Function (CSF) for the difference form is a logistic curve in which, as is consistent with military experience, increasing returns apply up to an inflection point at equal resource commitments. A crucial flaw of the traditional ratio model is that neither one-sided submission nor two-sided peace between the parties can ever occur as a Cournot equilibrium. In contrast, both of these outcomes are entirely consistent with a model in which success is a function of the difference between the parties’ resource commitments.


Difference Form Relative Success Election Campaign Reaction Curve Ratio Form 
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  1. 1.
    See, e.g., Hillman and Katz (1984), Corcoran and Karels (1985), Higgins, Shughart and Tollison (1985), Appelbaum and Katz (1986), Allard (1988), Hillman and Samet (1987).Google Scholar
  2. 2.
    A recent paper of Hillman and Riley (1988) makes use of still another family of contest payoff functions, in which — in contrast with the sharing rules analyzed here — the entire prize, as in an auction, goes to the high bidder. Their paper also allows for differing prize valuations.Google Scholar
  3. 3.
    In the standard Lanchester equations of military combat (Lanchester, 1916 (1956); Brackney, 1959), the outcome is also assumed to depend upon the ratio of the forces committed. But for Lanchester the battle result is always fully deterministic, in the sense that the side with larger forces (adjusted for fighting effectiveness) is 100% certain to win. This makes the CSF a step function, which jumps from p1 = 0 to p1 = 1 when C1 = C2. So Lanchester’s formula can be regarded as the limiting case of equation (2) as the mass effect parameter m goes to infinity. The same holds also for the auction-style payoffs in Hillman and Riley (1988).Google Scholar
  4. 4.
    As seen in the previous footnote, the Lanchester equations of combat take this to the extreme. The larger force is 100% certain of victory; the smaller force has no chance at all.Google Scholar
  5. 5.
    Compare T.N. Dupuy’s study of diminishing returns in combat interactions between Allied and German forces in World War II (Dupuy, 1987: Ch. 11). Dupuy’s curves generally show the inflection point displaced slightly from the “equal forces, equal success” point, owing (on his interpretation) to the superior unit effectiveness of the German army.Google Scholar
  6. 6.
    I thank David Levine and Michele Boldrin who independently discovered this generalization of the logistic Contest Success Function.Google Scholar
  7. 8.
    For the analogous result in a general-equilibrium context, see Hirshleifer (1988, Part B).Google Scholar
  8. 9.
    Dixit (1987) appears to assume, incorrectly, that all logit functions do lead to an interior Nash-Cournot asymmetrical equilibrium.Google Scholar
  9. 11.
    As suggested by the preceding discussion, this optimum is not at a smooth maximum (zero first derivative). Instead, player #2’s profit function has a negative first derivative throughout, leading him to cut back effort until the limit of zero is reached.Google Scholar
  10. 12.
    The key feature guaranteeing existence of a Nash-Cournot equilibrium is that the payoff functions are continuous, even though the Reaction Curves have discontinuities. See Debreu (1952) and Glicksberg ( 1952 ). I thank Eric S. Maskin for this point.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Jack Hirshleifer
    • 1
  1. 1.Department of EconomicsUniversity of CaliforniaLos AngelesUSA

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