a terminal period t history. Denote the set of terminal period t histories by
Tt, and the set of (at_1, bt_1) generating elements of Tt by Tte.
If for an infinite sequence of effort choices, a = (α1,α2,...) and b =
(61,62,...) no terminal state is reached in finite time, we will call the cor-
responding history h∞ = (a, b, j) a non-terminal history and denote the set
of such histories as Hx .
Given these constructions, we define a behavior strategy σι for player
I ∈ {A, B} as a sequence of mappings σι(ht) : Ht → ∑[o,κ], that specifies for
every period t and non-terminal history ht an element of the set of probabil-
ity distributions over the feasible effort levels [0,K]. Each behavior strategy
profile σ = (σyσg) generates for each t a probability distribution over his-
tories in the set ∣Jτ<t Tτ U Ht. It also generates a probability distribution
over the set of all feasible paths of the game, ∣J^=1 Tτ U H x .
Since we assume that each player’s payoff for the tug-of-war is the expec-
ted discounted sum of his per-period payoffs, the payoff for player A from a
behavioral strategy profile σ is denoted un(σ) = Eσι∑i ∣∑ 1 ~Aîa;./î/ш ≡
Eσ(πn(a⅛-1, bj-1, jf)) where t is the hitting time at which a terminal state is
first reached.9 If a terminal state is never reached, t = ∞. Note that for a
given sequence of actions (aj_1 b^_1), t arises deterministically, according to
the non-random transition rule embodied in the all-pay auction, so that the
randomness of t is generated entirely by the non-degenerate nature of the
probability distributions chosen by the behavioral strategies. If ht+1 = (at
bt, jt+1) ∈ Tet+1 denotes a sequence of efforts that leads to a terminal state at
precisely period t = t + 1, then, the payoffs for A and B are
^A((at, bt, jt+1)) = <
t
-£ Si^
i=1
t
-£ ^1α
i=1
+δtZA if J (t + 1)=0
if J (t + 1) = m
(1)
9We adopt a notational convention throughout this paper that the action set available
to each player in a terminal state is the effort level zero, so that for any hitting time t,
a^ = ⅛ = 0. Hence, in these states πA(at,f) and ⅞(bt,j) include only the prize awarded
—t
to the victor, and we suppress the terms α^ and ⅛ in the notation Σ^=15t-1%A(αt,j(t)) ≡
πA(¾-i, bt_i, jt).