Keywords: winner-take-all, all-pay auction, tipping, multi-stage
contest, dynamic game, preemption, conflict, dominance.
JEL classification numbers: D72, D74
1 Introduction
Final success or failure in a conflict is often the result of the outcomes of
a series of potential battles. An illustrative example is the decision making
process in many organizations. Resources, jobs and other goods that involve
rents to individuals inside the organization are frequently allocated in a pro-
cess that has multiple decision stages. For instance, hiring decisions often
involve a contest between candidates in which a hiring committee makes a
decision and forwards this decision to another committee. This committee
approves to the initial decision and forwards the case further until a final
decision stage is reached, or may return the case to the previous committee.
Candidates could expend effort trying to influence the decision process in
each stage, but if at all, typically serious efforts are expended by the candid-
ates only in early stages of the decision process. Such multi-layered decision
processes obviously cause delay in decision making and this can be seen as
a cost. We will argue here that, compared to a single stage decision process
in which the rival players spend effort in a single stage all-pay auction, the
multi-stage decision process can be advantageous as it may improve allocat-
ive efficiency and reduce effort that is expended by rival contestants in the
conflict.
In more general terms we describe the multi-stage contest as a tug-of-war.
As a modeling device, the tug-of-war has a large number of applications in
diverse areas of science, including political science, economics, astronomy,
history and biology.1 It consists of a (possibly infinite) sequence of battles
1 To give a few examples: In politics, Whitford (2005) describes the struggle between the
president and legislature about the control of agencies as a tug-of-war. Yoo (2001) refers
to the relations between the US and North-Korea and Organski and Lust-Okar (1997) to
the struggle about the status of Jerusalem as cases of tug-of-war. According to Runciman
(1987), at the time of the Crusades, when various local rulers frequently attacked one
another, they sometimes succeeded in conquering a city or a fortification, only to lose
this, or another, part of their territory to the same, or another, rival ruler later on. The
conflict between two rival rulers can be seen as a sequence of battles. They start at some
status quo in which each rules over a number of territories with fortified areas. They fight
each other in battles, and each battle is concerned with one fortress or territory. In the