Efficient Rent-Seeking pp 97-107 | Cite as

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

## Abstract

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 p_{i} (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, p_{i} 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.

## Keywords

Difference Form Relative Success Election Campaign Reaction Curve Ratio Form## Preview

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## Notes

- 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.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.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 p_{1}= 0 to p_{1}= 1 when C_{1}= C_{2}. 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.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.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.I thank David Levine and Michele Boldrin who independently discovered this generalization of the logistic Contest Success Function.Google Scholar
- 8.For the analogous result in a general-equilibrium context, see Hirshleifer (1988, Part B).Google Scholar
- 9.Dixit (1987) appears to assume, incorrectly, that all logit functions do lead to an interior Nash-Cournot asymmetrical equilibrium.Google Scholar
- 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
- 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