The name is absent



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 = (α12,...) 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], 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) = Eσιi ∣∑ 1 ~Aîa;./î/ш ≡
Eσ(πn(a-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


tZA 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%At,j(t))
πA-i, bt_i, jt).



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