for (Za,Zb) such that ^) ∈ (1, (1 — 5) 1) and the all-pay auction is more
efficient for ^1) ∈ ((1 — 5) l. ∞). In particular, for any given continuous
joint distribution of (Za,Zb). for sufficiently large 5 the tug-or-war is more
efficient than the all-pay auction.
4 Conclusions
We studied the strategic behavior of players who compete in a series of single
battles. A prize is allocated as a function of the sequence of battle successes.
A sufficient lead in the number of battle victories is needed to win the fi-
nal prize. We showed that there is a unique subgame perfect equilibrium
in Markov strategies and we characterized this equilibrium. Contest effort
concentrates on at most two states. Such states are characterized by three
factors: the ’distance’ which the two contestants need to win the overall
contest, the relative strength or dominance of contestants, and the discount
factor. The critical distance that determines the tipping states in which the
contest effort is focussed turns out to be a function of the contestants’ rel-
ative strengths (or, equivalently, in the relative valuations of the prize from
final victory) and the discount factor. The larger one player’s dominance in
strength, the higher must be this player’s distance to final victory, compared
to the other player’s distance.
Many animal species and economic institutions have developed mech-
anisms such as hierarchies, or other organizational structures to govern the
allocation of prizes, such as preferential food access and the right to reproduce
in the biological context, or prized jobs and contracts in the organizational
context. Behavior in these mechanisms could be interpreted as a conflict
that consists of a series of battles, or repeated opportunities to struggle. Our
results help explain why these structures may have evolved. The tug-of-war
delays the allocation of a given prize, compared to a single stage conflict, but
can considerably increase the efficiency of allocation of the prize and reduce
the overall resources that are dissipated among the group of players.
5 Appendix
Consider a tug-of-war with m > 3 and j0 ∈ {2,...,m — 1} with the property
that δj0Za < δm~joZb and 5°-1Za > 5m^-γ1 Zb. Then the Markov perfect
23