t = 1,2,3... a battle takes place between the players in which A (resp. B)
expends effort at (resp. bt). 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) = mA + nBt — nAt, where nAt and nBt 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 ∈ Mmt ≡ {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 Za is awarded to A if the terminal state j = 0 is reached
and, alternatively, a prize of Zb is awarded to B if the terminal state j = m
is reached. Without loss of generality we assume that Za ≥ Zb. Figure 1
depicts the set of states.
φ states j
01
mA-1 mA mA+1
m-1 m
Figure 1:
Player A’s (B’s) period t payoff ^A(at,j(t)) (%b(bt,j(t))) is assumed to
equal Za (Zb) if player A (B) is awarded the prize in that period, and
—αt (—bt) 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 ca (cb) we may normalize
utility by dividing by ca (cb) 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 5a and ⅛ .