a prize value Za∕ca (Zb∕cb). Therefore, our model with asymmetric prizes
can be interpreted as one with both asymmetric prizes and fighting abilities.
Each single battle in the tug-of-war is a simultaneous move all-pay auction
with complete information. A player’s action in each period in which the state
is interior is his effort, at ∈ [0,K] and bt ∈ [0,K], for A and B, respectively,
where K ≥ Za.8 The player who spends the higher effort in a period wins
the battle. We choose a deterministic tie-breaking rule for the case in which
both players choose the same effort, by which the ’’advantaged” player wins.
Given m, Za, Zb and δ, we say that player A is advantaged in state j
if 57' Za > 5m-∙7'Zb , and B is advantaged if TZa ≤ 5m-∙7'Zb. We define
j0 = min{j ∈ Mmt ∣57' Za ≤ 5m-∙7'Zb } where this is non-empty, and j0 = m
otherwise: player B is advantaged for j ∈ Mmt such that j ≥ j0 and A is
advantaged otherwise.
If m = 2 and mA = 1, the tug-of-war reduces to the well-known case of
the standard all-pay auction with complete information at time t = 1, as in
Hillman and Riley (1989), Ellingsen (1991) or Baye, Kovenock and deVries
(1996). In this case, one single battle takes place at state j = mA = 1.
The process moves from this state in period 1 to j = 0 or to j = 2 at the
beginning of period 2, and the prize is handed over to A or B, respectively.
Accordingly, the contest at period t = 1 in state j = 1 is over a prize that has
a present value of 8Za and 8Zb for A and B, respectively, and the payoffs
in the unique equilibrium of this game (which are in nondegenerate mixed
strategies) are δ(Za — Zb) for A and zero for B. In what follows, we consider
the case with m > 2.
For each period t, if a terminal state has not yet been reached by the
beginning of the period, players simultaneously choose efforts with common
knowledge of the initial state mA and the full history of effort choices, denoted
as (at-ι, bt-ι) ≡ ((α1,..., αt-1), (b1,..., bt-ι)). Players also know the current
state j(t) of the war and the state in any past period j(τ), τ < t. We define
jt = (j(1),j(2),...j(t)), where j(1) = mA. Hence, we will summarize the
history at time t along any path which has not yet hit a terminal state by
ht = (at-ι, bt-ι, jt). We will call such a path a non-terminal period t history
and will denote the set of such histories by Ht. A history of the game that
generates a path that reaches a terminal state at precisely period t is termed
8This upper limit makes the set of possible effort choices compact, but does not lead
to a restriction that could be binding in any equilibrium, as an effort choice larger than
Za in some period is strictly dominated by a choice of effort of zero in this and all future
periods.