time t, the current state j(t) therefore constitutes what they call the payoff-
relevant history. Since our game is stationary, we may partition the set
of all finite non-terminal histories by the same state variables, j ∈ Mmt ≡
{1, 2, ...(m —1)}, removing any dependence of the partition on the time t. We
label this partition {j(t) = i}i∈Mint. This is the stationary partition defined
by Maskin-Tirole, (2001, p. 203).
In the continuation, we restrict attention to (stationary) Markov strategies
measurable with respect to the payoff relevant history determined by the sta-
tionary partition {j(t) = i}ieM '. A stationary Markov strategy σι for player
I ∈ {A, B} is a mapping σι(j) : Mmt → Σ[0,κ], that specifies for every interior
state j a probability distribution over the set of feasible effort levels [0,A-].
If in the continuation game starting in period t and state j, σ = (σj,σ^) is
played, then the continuation value for player i at t is denoted as vi(σ |j ) and
can be calculated as the discounted sum of future expected period payoffs in
a well-defined manner similar to that described above.
In this context we are interested in deriving the set of Markov perfect
equilibria; that is a pair of Markov strategies that constitute mutually best
responses for all feasible histories. In Propositions 1-3 below we demonstrate
that the tug-of-war has a unique Markov perfect equilibrium for any com-
bination of mA,m,ZA,ZB and S.
Before stating these propositions, it is useful to derive some simple proper-
ties that must hold in any Markov perfect equilibrium of our model. Suppose
σ* = (σ*A,σ*β) is a Markov perfect equilibrium and denote player i’s continu-
ation value in state j under σ* by vi(σ* |j) = vi(j). Subgame perfection and
stationarity imply that competition in any state j, j ∈ {1, 2,...m — 1}, may
be viewed as an all-pay auction with prize zA(j) = SvA(j — 1) — SvA(j + 1) for
player A and zb(j) = Svb(j + 1) — Svb(j — 1) for player B. In equilibrium,
the continuation value to player I of being in state j at time t is equal to the
sum of the value of conceeding the prize without a fight (and thereby moving
one state away from the player’s desired terminal state) and the value of
engaging in an all-pay auction with prizes zA(j) = Sva(j — 1) — Sva(j + 1)
for player A and zb(j) = Svb(j + 1) — Svb(j — 1) for player B. An imme-
diate consequence of the characterization of the unique equilibrium in the
two-player all-pay auction with complete information (see Hillman and Riley
(1989) and Baye, Kovenock, and De Vries (1996)) is that local stategies are
uniquely determined and the continuation value for the two players in any
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