0 ≤ VA(j) ≤ δj Za (7)
0 ≤ vb (j ) ≤ δm- Zb (8)
and
va(j) + vb(j) ≤ max(δjZA,δm-jZb) (9)
We can now prove the following
Proposition 1 Consider a tug-of-war with m > 3. Suppose jo ∈ {2, ...m—1}
exists such that
δj°-1Za > δm-(70-1)ZB and δj° Za < δm-j°Zb . (10)
Then a unique Markov perfect equilibrium exists which is characterized as
follows:
For all interior states j ∈ {jo — 1,jo}, the equilibrium effort choices are
a(j) = b(j) = 0. Only at j0 — 1 and j0 does a battle with a positive probability
of stricty positive effort choices take place. Payoffs for A in the continuation
game at j are δZa for j < jo — 1, [δj°-1ZA — δm (j° 1) Zb] for j = jo — 1,
and 0 for j > jo; payoffs for B are δm-Zb for j > jo, (-^ [δm-j°Zb—δj°Za]
for j = jo and 0 for j ≤ jo — 1.
Proof. We consider existence here and relegate the proof of uniqueness
to the Appendix. We consider the following candidate equilibrium: For all
interior states j ∈ {jo — 1,jo}, the effort choices are a(j) = b(j) = 0. At jo — 1
and jo players choose efforts according to cumulative distribution functions
Fj and Gj for players A and B in states j as follows:
Fj°-1(a) =
—s a ⅛ for
δ λ9° J
(1-δ2) δBA
1 for
a ∈ [0, (¾]
. ^δB°a
a > (1-<S2)
(11)
12
More intriguing information
1. A Computational Model of Children's Semantic Memory2. Ahorro y crecimiento: alguna evidencia para la economía argentina, 1970-2004
3. Does Market Concentration Promote or Reduce New Product Introductions? Evidence from US Food Industry
4. Higher education funding reforms in England: the distributional effects and the shifting balance of costs
5. The Role of State Trading Enterprises and Their Impact on Agricultural Development and Economic Growth in Developing Countries
6. The name is absent
7. The name is absent
8. Electricity output in Spain: Economic analysis of the activity after liberalization
9. The name is absent
10. The name is absent