The name is absent



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
KZa.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 TZa5m-7'Zb. We define
j
0 = min{j Mmt 57' Za5m-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(ZaZb) for A and zero for B. In what follows, we consider
the case with
m2.

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 m
A 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
j
t = (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.



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