between two contestants who accumulate stage victories, and in which the
contestant who first accumulates a sufficiently larger number of such victories
than his rival is awarded the prize for final victory.2
To our knowledge, Harris and Vickers (1987) were the first to look form-
ally at the tug-of-war. They analyse an R&D race as a tug-of-war in which
each single battle is determined as the outcome of a contest with noise. Such
exogenous noise makes the problem less tractable and has so far ruled out a
fully analytic description of the equilibrium. Budd, Harris and Vickers (1993)
apply a somewhat more complicated stochastic differential game approach to
a dynamic duopoly, seen as a tug-of-war involving a continuum of advertizing
or R&D battles that determine the firms’relative market positions. Using a
complementary pair of asymptotic expansions for extreme parameter values
and numerical simulations elsewhere, they isolate a number of effects that
govern the process. Several of these appear in our analysis which, unlike
their framework, derives an analytical solution for the unique Markov per-
fect equilibrium. Morever, our analysis explicitly solves for equilibrium for
both symmetric and asymmetric environments.
The term ‘tug-of-war’ has also been used in biology. In the context of
within-group conflict among animals, subjects could struggle repeatedly.3
For instance, the formation of hierarchies and their dynamic evolution oc-
curs in repeated battle contests. As Hemelrijk (2000) describes for several
examples, individuals may try to acquire a high rank, but the differentiation
sequence of successes and failures, the fortresses or territories are destroyed or reallocated,
and the conflict continues until one of the rulers has lost all his fortresses or territories
and is thus finally defeated. If battle success alternates more or less evenly, then such a
contest can go on for a very long time, possibly even forever. The end comes only when
one of the rulers has been more successful than his rival sufficiently often.
2 According to Wikipedia the term tug-of-war refers to a rope pulling contest in which
two contestants (or groups) pull a rope in different directions until one of the sides pulled
the rope (and the opponent group) across a certain limit. In more abstract terms, the
contest consists of a series of battles, where a battle victory of one player makes both
move one unit towards the winner’s preferred terminal state, and where one contestant
wins the war if the difference between the winner’s number of such battle victories exceeds
the other contestant’s number of battle victories by some absolute number.
3The term also refers to contests between different species. Ehrenberg and McGrath
(2004) refer to the interaction of microtubule motors, Larsson, Beignon and Bhardwaj
(2004) and Zhou et al. (2004) refer to the interaction between viruses and the dendritic
cells or other parts of the immune system as tugs of war. Tibbetts and Reeve (2000)
consider the role of the amount of reproductive sharing within a group for the likelihood
of within-group conflict among the social wasp Polistes dominulus.