Axiomatizing additive multi-effort contests

Abstract

A rent seeking model is axiomatized where players exert multiple additive efforts which are substitutable in the contest success function. The axioms assume the sufficiency of exerting one effort, and that adding an amount to one effort and subtracting the same amount from a second equivalent substitutable effort keeps the winning probabilities unchanged. In contrast, the multiplicative Cobb–Douglas production function in the earlier literature requires players to exert all their complementary efforts. The requirement follows from assuming a homogeneity axiom where an equiproportionate change in two players’ matched efforts does not affect the winning probabilities. This article abandons the homogeneity axiom and assumes an alternative axiom where the winning probabilities remain unchanged when a fixed positive amount is added to all players’ efforts. This article also assumes a so-called summation axiom where the winning probabilities remain unchanged when a player substitutes an amount of effort from one effort into another effort. The summation axiom excludes multiplicative production functions, and furnishes a foundation for additive production functions.

A Review of Axioms for Group Contest Success Functions

“…Interestingly, most of the relevant axioms for group contest success functions already exist in the literature. It is clarified which earlier axiomatizationsapplyforgroupcontestsuccessfunctionsandhowtheyapply.Themostrelevant earliercontributionsareMünster's(2009)axiomatizationofgroupcontestsuccessfunctions, Hausken's (2021)…”

“…Münster's(2009)Axiom8isnotgenerallyadopted. Hausken (2021)showsthatAxiom8issatisfiedforadditiveeffortsifmultipleeffortsbyaplayerare substitutablewithequivalentimpactinthecontestsuccessfunction,andthatAxiom8isnot satisfiedfortheCobb-Douglasproductionfunction.…”

“…Axiom17aisaspecialcaseofMünster's(2009)Axiom8andcorrespondstoHausken's(2021) Axiom9.Itstatesthatiftwoeffortsx ig andx jg byplayersi and j aresubstitutablewithequivalent impactinthecontestsuccessfunction,thensubtracting ∆ from x ig andadding ∆ to x jg doesnot affect the winning probabilities. Hausken (2021) shows that the multiplicative Cobb-Douglas productionfunctiondoesnotsatisfyAxiom17a.…”

“…. (Hausken (2021) makes an analogous assumptiontoenabledifferentplayerstoexertdifferentnumbersofefforts.Thesuperfluous efforts for the player exerting fewest efforts are set to zero.) This means that group g where m K g < hasatleastonedummymemberwhichismatchedagainstthelastmember inthelargestgroup.…”