The name is absent

t = 1,2,3... a battle takes place between the players in which A (resp. B)
expends effort a_t (resp. b_t). A victory by player A (B) in state i at time
t moves the war to state i — 1 (i + 1) at time t + 1. The state in period

t + 1 is therefore j (t + 1) = m_A + n_Bt — n_At, where n_At and n_Bt denote

respectively, the number of battle victories that A and B have accumulated
by the end of period t. This continues as long as the war stays in some
interior state j ∈ M^mt ≡ {1,2,..., (m — 1)}. The war ends when one of
the players achieves final victory by driving the state to his favored terminal
state, j = 0 and j = m, for player A and B respectively. A prize (for final
victory) of size Z_a is awarded to A if the terminal state j = 0 is reached
and, alternatively, a prize of Z_b is awarded to B if the terminal state j = m
is reached. Without loss of generality we assume that Z_a ≥ Z_b. Figure 1
depicts the set of states.

φ states j

Figure 1:

Player A’s (B’s) period t payoff ^_A(a_t,j(t)) (%_b(b_t,j(t))) is assumed to
equal Z_a (Z_b) if player A (B) is awarded the prize in that period, and
—α_t (—b_t) if t is a period in which effort is expended.⁶ We assume that
each player maximizes the expected discounted sum of his per-period payoffs.
Throughout we assume that 0 < δ < 1 denotes the common, time invariant,
discount factor.⁷

The assumption that the cost of effort is simply measured by the effort
itself is for notational simplicity only. Since a player’s preference over income
streams is invariant with respect to a positive affine transformation of utility,
if player A (B) has a constant unit cost of effort c_a (c_b) we may normalize
utility by dividing by c_a (c_b) to obtain a new utility function representing
the same preferences in which the unit cost of effort is 1 but player A (B) has

⁶Since the per-period payoffs do not depend directly on time, we have dropped a time
index.

⁷It is straightforward to extend our results to cases in which players have different,
time invariant discount factors 5_a and ⅛ .

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