The value of commitment in contests and tournaments when observation is costly

Abstract

We study the value of commitment in sequential contests when the follower faces small costs to observe the leader's effort. We show that the value of commitment vanishes entirely in this class of games. By contrast, in sequential tournaments-games where, at a cost, the follower can observe the effectiveness of the leader's effort-the value of commitment is preserved completely provided that the observation costs are sufficiently small.

Commitment and observability in a contracting environment

“…This result is related to Morgan and Várdy (2007), who analyze a leaderfollower game with continuum action sets and strictly concave payo¤ functions. They show that the subgame perfect equilibrium outcome cannot be approximated in the game with costly observation.…”

“…1 by augmenting the payo¤ to action I by z; where is a small positive number, and z is the realization of a random variable which is distributed with density f on support [0; 1]. Assume f (:) > m > 0 on its support, 12 and that only the incumbent observes the realization of z, so that we have a private payo¤ shock as in Harsanyi (1973).…”

“…Given a price p; the incumbent's payo¤ from investing is given by v + z p. The marginal type of incumbent, who is indi¤erent between investing and not, is given by z = p v . Thus the payo¤ of the supplier from a price p equals 12 The lower bound on f (:) is not essential for our argument, but simpli…es its exposition.…”

Envelope theorems for locally differentiable open-loop Stackelberg equilibria of finite horizon differential games

“…Examples include applications as varied as pollution control (Long, 1992), wireless networks (Jiang and Ruan, 2008), optimal routing on transit networks (Yang et al, 2007), international trade in renewable resources between a single buyer and competitive sellers (Batabyal and Beladi, 2006), models of reputation (Ely et al, 2008), law (Nie, 2005), retailer-wholesaler supply chain interactions (He et al, 2007;Xie and Ai, 2006), sequential tournaments (Morgan and Vá rdy, 2007), models of espionage (Solan and Yariv, 2004), optimal investment and financing (Jørgensen et al, 1989), and duopolistic competition (Dockner and Jørgensen, 1984;Cremer and Riordan, 1985;Fujiwara, 2006). It is with an eye towards applicability that the ensuing envelope results are presented, for they allow one to infer intrinsic properties of the class of differential games admitting locally differentiable Stackelberg equilibria with an open-loop information structure, without requiring one to explicitly solve such games in closed form.…”

Bidding to Commit

“…The advantages of commitment are well known in economics, and much theoretical work has been done on commitment, including first-mover advantages (e.g., Bagwell, 1995;Frank, 1 Department of Psychology, University of Guelph, Guelph, Ontario, Canada 1988; Hoffman, Yoeli, & Nowak, 2015;Kambe, 1999;Morgan & Várdy, 2007;Schelling, 1960Schelling, , 2001. However, despite the wealth of theoretical work, there has been a surprising dearth of experimental work on how humans actually behave.…”