Monotone comparative statics in games with non-monotonic best-replies: Contests and Cournot oligopoly

Abstract

In this paper, we employ lattice-theoretic techniques to derive a number of comparative statics in a logit contest -a class of games for which best-replies are generically non-monotonic. Using the same approach, we obtain several comparative statics in a Cournot oligopoly model without imposing the usual assumptions that ensure the game is one of strategic substitutes. Our analysis illustrates how latticetheoretic techniques can deliver powerful insights in games with nonmonotonic best-replies.

Submodularity and supermodularity in contest games

“…A similar result can be obtained without an upper bound for the ϕ function, if instead both players' strategy sets were bounded from above, as also assumed byAcemoglu and Jensen (2013).6 In their Footnote 3,Gama and Rietzke (2019) state that the logic behind their negative result in the decisive case (i.e., when R = 0) also applies to a nondecisive case (i.e., when R > 0). However, as we show here, this is not necessarily valid for some ϕ functions.188 | KARAGÖZOĞLU ET AL.…”

“…It is worth noting that when R > 0, the generalized structure of the utility function assumed in (3) implies that both draw and loss yield a zero payoff to each player. This means that the draw corresponds to a case where both players lose, which can be considered a reasonable assumption in patent races and inducement contests, as also noted by Acemoglu and Jensen (2013) and Gama and Rietzke (2019). On the other hand, it can be argued that a positive draw prize appears in some sports competitions (in terms of tournament points collected) and territorial conflicts (where both sides keep their own territories in the case of a draw).…”

“…As argued by Gama and Rietzke (2019), such a rent-seeking contest is neither submodular nor supermodular when it is decisive (i.e., one of the two players will win the contest). This is because P x y P x y ( , ) + ( , ) = 1…”

“…Once again, as noted by Gama and Rietzke (2019), given that a lottery-form CSF is decisive, a submodular or supermodular game cannot be obtained in a rent-seeking contest when a standard cost function is considered. Yet, in the current subsection, we show that such a contest game can be supermodular for certain types of effort cost functions that allow for the possibility that the rival's effort influences one's own cost.…”

“…1 Later, Jensen (2016) improves upon those results, thereby extending the applications to a wider set of contest games. As another alternative approach, Gama and Rietzke (2019) employ lattice-theoretic techniques to obtain comparative static results in games with nonmonotonic best responses, including logit contests. All these complement the existence and comparative static results in contest games, such as Nti (1997), Szidarovszky and Okuguchi (1997), and Cornes and Hartley (2005).…”