To fix ideas consider two countries a and b that quarrel about a foreign policy issue6.
Assume this issue can be captured by a rent R. These countries each have to appoint first a politician to act on their behalf. These politicians then have to decide how much of a given budget Ba to spend in the contest7. We solve for a subgame perfect Nash
equilibrium by first solving the final stage contest game taking the acting politician’s 6See Paul and Wilwhite (1990) for a similar interpretation.
7For simplicity we will, without loss of generality, set out the primitives of the model from the per-
5.2. BASIC MODEL 69 types as given. Then we use these results when deriving the optimal delegation decision of a country. There is no asymmetric information in the model.
The citizens of the two states may have differing valuations of this rent. The valuation of the rent to citizen i is αi
aR, i.e. αia can be seen as the weight placed on the foreign
policy issue. αiis continuously distributed according to the distribution function f a(α)
within each group. The only restriction we put on the distribution functions is that they have to be bounded on (0, αa] , i.e. there exists a most radical typeαa .
An integral part of the model is the contest success function (CSF) g(ma;mb) that
determines the probability of winning the contest for a contestant dependent on the resources spent by him, ma , and the opponent, mb . To avoid technical difficulties
assumeg(0; 0) = 1/2.
To model the contest we use a Tullock style contest success function ma
ma+mb.Our results
would hold for all “constant returns to scale” contest success functions, i.e. functions of the form θma
θma+πmb that are homogenous of degree 0. See the Appendix 10.1 for an exposition
with a general constant returns to scale contest success function.
Skaperdas (1996) shows at least that the general structure Pha(ma)
hj(mj)) , with hj(mj)
being an increasing function, is the only structure that fulfills several desirable axioms: the contest success function satisfies the conditions on a probability distribution, the success probability is increasing in the own expenses, an anonymity property applies and independence of irrelevant actions, i.e. actions of non-participants, holds8.
To ease the exposition we focus on Tullock’s initially proposed function ma
ma+mb. As
we stick to risk neutrality throughout one can interpret this not only as the winning probability but also as being the share the group secures for itself.
An individual citizen i’s utility function in country a is given by
ui =αiaR ma ma+mb
+ (Ba−ma)
8For contest success functions of the more general form mk a
mk a+m
k
b we have the problem that we do not
get closed form solutions fork6= 1. However numerical examples give us some hope that our main results should remain qualitatively unchanged with increasing returns to scale (k >1) or decreasing returns to scale (k <1) contest success functions.
This states that utility is increasing in the (expected) rent and decreasing in the re- sources spent by the country in the contest. This cost −ma can be considered as the
foregone public good which is produced with a simple linear production function from the exogenously given budget Ba not spent in the contest9.
In our extensions section later on we will introduce heterogeneity in the cost of provision of the public good and analyze the effects.
We proceed from here by first deriving the equilibrium of the contest stage dependent on the politician’s types. Then we use our results to derive in the next section the citizens’ preferences over politician’s types.
In the contest stage the two agents i (for county a) and j (for county b) in charge maximize their utility by deciding upon ma and mb.
max ma ui = αiaR ma ma+mb + (Ba−ma) max mb uj = αjbR mb ma+mb + (Bb −mb)
From the two first order conditions of this problem we can solve for the reaction func- tions ma = p mbRαia−mb and mb = q maRαjb −ma
and the equilibrium values ofm∗
a and m∗b which are uniquely determined by
m∗ a =R (αi a) 2 αjb αi a+α j b 2 and m ∗ b =R αi a α j b 2 αi a+α j b 2.
They depend only on the politicians’ types and on the size of the rent under consider- ation10.
It is interesting and facilitates the intuition of our results later on to note already here how these equilibrium values form∗
aandm∗b behave in the limits with respect to the acting
9Alternatively think of the contest expenditure financed by an equal per-capita-tax. However our
model works as long as the share in financing can not be made dependent on the valuation.
10Note that forαi a=α
j
b = 1 , i.e. the situation analyzed by Tullock (1980) the values not surprisingly
boil down to his solution, namelym∗
a=m∗b = R
5.3. INDIVIDUAL PREFERENCES OVER TYPES 71