and, respectively,
|
t t -£ 5i-1αi | |
|
~b((at, bt, jt+ι)) = < |
i=1 |
|
-£ 5i-1ai < i=1 |
+δtZβ if j (t + 1)= m
if j (t + 1)=0.
(2)
If for an infinite sequence of effort choices, a = (α1,α2,...) and b = (b1,b2,...)
no terminal state is reached in finite time, payoffs are
∞ ∞
KA((a, b, j)) = - X δt~1α>t and ~b((a, b, j)) = - X δt~‰
For a given behavior strategy profile σ = (aA,aB) each player’s payoff in the
tug-of-war can be derived from calculating the expected sum of discounted
per period payoffs generated by the probability distribution over histories in
the set U∞=ι Tτ U H∞. Moreover, for any t and ht ∈ Ht, one may define each
player’s expected discounted value of future per-period payoffs (discounted
back to time t) conditional on the history ht by deriving the conditional
distribution induced byσ∣ht over S∞=t+1 Tτ U H∞. We shall refer to this as
a player’s continuation value conditional on ht and denote it by vi(σ ∣ht) =
Eσ∣ht(Eis=tSs^t^A(as,j(s))). Note that this has netted out any expenditures
accrued on the history ht.
Since the players’ objective functions are additively separable in the per-
period (time invariant) payoffs and transitions probabilities depend only upon
the current state and actions, continuation payoffs from any sequence of
current and future action profiles depend on past histories only through the
current state j. It therefore seems natural to restrict attention to Markov
strategies that depend only on the current state j and examine the set of
Markov perfect equilibria. Indeed, this partition of histories is that obtained
from the more formal analysis of the determination of the Markov partition in
Maskin and Tirole (2001). For any t, we may partition past (non-terminal)
histories in Ht by the period t state j(t), inducing a partition Ht(∙), and
define the collection of partitions, H(∙) ≡ {Ht(∙)}∞=1. It can be demonstrated
that in our game the vector of collections (Ha(∙),Hb(∙)) = (H(∙),H(∙))
is the unique maximally coarse consistent collection (the Markov collection
of partitions) in the sense of Maskin and Tirole (2001, p. 201). For any