The name is absent

a terminal period t history. Denote the set of terminal period t histories by
T^t, and the set of (a_t_1, b_t_1) generating elements of T^t by T^t_e.

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 H^x .

Given these constructions, we define a behavior strategy σ_ι for player
I ∈ {A, B} as a sequence of mappings σ_ι(h^t) : H^t → ∑[o,κ], that specifies for
every period t and non-terminal history h^t 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 ∣J_τ<_t T^τ U H^t. It also generates a probability distribution
over the set of all feasible paths of the game, ∣J^=₁ T^τ 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) = E_σι∑i ∣∑ ¹ ~_Aîa_;./î/ш ≡
E_σ(^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 h^t+1 = (a_t
^bt^{, j}t+1) ∈ T_e^t+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

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(b_t,j) include only the prize awarded
—t

to the victor, and we suppress the terms α^ and ⅛ in the notation Σ^=₁5^t-1%A(αt,j(t)) ≡
πA(¾-i, ^bt_i, ^jt).

<<   6   7   8   9   10   11   12   >>

More intriguing information