Hence, the considerable effort that is spent at the point at which the tug-of-
war becomes symmetric in terms of the prizes that are at stake for the two
contestants prevents the contestant who is lagging behind in terms of battle
victories from spending positive effort.
Discounting played two important roles in our analysis. First, discounting
leads to payoff functions that are continuous at infinity, allowing the applica-
tion of the one-stage deviation principle, which greatly facilitates our proofs.
Moreover, discounting is essential in giving a meaningful role to the distance
to the state with final victory. The following holds:
Proposition 4 For a given value of > 1, the tipping state j0 is an
increasing step function of δ. Moreover, as δ → 1, A wins the tug-of-war
without effort starting from any state j < m — 1.
Proof. The tipping state j0 is by definition the smallest state j for which
player B is advantaged: j0 = min{j ∈ Mmt ∣δ2Za ≤ δm~∙iZb } when this set is
non-empty, and j0 = m otherwise. For δ > 0, the inequality δ2 Za ≤ δm ^ Zb
is equivalent to δ2j ”'Za ≤ Zb. Since m ≥ 3, for δ sufficently close to
zero the inequality is clearly satisfied for j = m — 1, so that j0 is interior.
Moreover, since δ2j ”' ≥ 1 for j ≤ ψ ,it must be the case that j0 > A. As
δ → 1, the inequality is violated at all interior states, even at j = m — 1.
In this case, by definiton j0 = m, and from Proposition 2 player A wins
the war from any state j < m — 1. For any 0 < δ < 1, δ2j ”'Za ≤ Zb is
^R
log ɪ^
equivalent to 2j — m ≥ log , so that j0 is the smallest index j satisfying
the inequality. Since the left hand side of this inequality is positive, and
both the numerator and denominator of the right hand side are negative, as
δ increases, the right hand side monotonically increases, eventually diverging
to ∞ as δ → 1. Hence, as δ increases, the smallest index j satisfying the
inequality must increase in steps until it hits m. ■
As the discount factor increases, relative prize value or player strength
plays a greater role in the determination of the outcome than distance. For
any given value of ½A < 1, as δ increases the tipping state j0 moves in discrete
jumps towards m. Player A may suffer a greater distance disadvantage and
still win the prize with certainty.
19