The name is absent

a prize value Z_a∕c_a (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, a_t ∈ [0,K] and b_t ∈ [0,K], for A and B, respectively,
where K ≥ Z_a.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, Z_b and δ, we say that player A is advantaged in state j
if 5⁷' Z_a > 5^m-∙⁷'Z_b , and B is advantaged if TZ_a ≤ 5^m-∙⁷'Z_b. We define
j₀ = min{j ∈ M^mt ∣5⁷' Z_a ≤ 5^m-∙⁷'Z_b } where this is non-empty, and j₀ = m
otherwise: player B is advantaged for j ∈ M^mt such that j ≥ j₀ and A is
advantaged otherwise.

If m = 2 and m_A = 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 = m_A = 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 8Z_a and 8Z_b for A and B, respectively, and the payoffs
in the unique equilibrium of this game (which are in nondegenerate mixed
strategies) are δ(Z_a — Z_b) 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 m_A and the full history of effort choices, denoted
as (a_t-ι, b_t-ι) ≡ ((α₁,..., α_t-1), (b₁,..., b_t-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) = m_A. Hence, we will summarize the
history at time t along any path which has not yet hit a terminal state by
h^t = (a_t-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 H^t. 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
Z_a in some period is strictly dominated by a choice of effort of zero in this and all future
periods.

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