Fearon's Rational Reason's for War: Notes on a Critique

James Fearon ‘Rationalist Explanations for War’ International Organization 49,3, Summer 1995

i. Main Concern
To explain war
3 types of explanation; 1) leader’s irrationality 2) leaders do not bear costs 3) rational leaders may still go to war
Fearon focuses on (3) because particular wars seem to have been started rationally; a rationalist explanation for war is essential for neorealism

ii. The Problem

States could reach a rational bargain without war based upon probability of winning or losing

Definition of A Rational Bargain

Fearon defines a rational bargain in the following way

2 states A and B, hold preferences over outcomes x∈X =[0,1]

A prefers outcomes closer to (1); B prefers outcomes closer to (0)

The preferences of A and B are represented in their utility function

Ua(x)= A’s utility for outcome x

Ub (1-x) is B’s utility for outcome x

These utility functions are continuous, increasing and weakly concave.

The Gains of War

A will win a war with probability p.

Ci=A,B= utility for the costs of war

Thus Ua for war= pUa (1)+(1-p) Ua (0)- Ca = pUa- Ca

For B
(1-p)Ub (1)+pUb(0)-Cb =(1- p)Ub- Cb

The Existence of a bargaining range

there exists a subset of X s.t. for each Ua (x) > pUa- Ca and Ub (1-x)>1-p- Cb.
the bargaining range, in which the outcomes for both A and B war.

So why do wars happen?

3 assumptions
i)‘there is some true probability p that one state would win’

ii) States are risk averse or risk neutral; there may have been risk acceptant leaders but ‘even if we admitted such a leader as rational, it seems doubtful that many have held such preferences.’

iii) issues are perfectly divisible on a continuous range [0,1] – possibility of linkages and side payments. Often violated by nationalistic attachment to territory

Rationalist Arguments
1) Anarchy
States go to war because there is no power to stop them.
Why? If states are rational then they know the costs of war and which one is likely to win, without reference to a central power anyway.
Does not address central question

(2) Benefits of war outweigh costs
Argument above shows that some ex ante bargain always exists that Pareto dominates war

(3) Preventive War
A declining power may attack arising power if it expects attack by that power in the future. But – doesn’t change bargaining solution

(4) Rational Miscalculation due to lack of information

States disagree about relative power – one side has private information. Why not share it?

iii. Fearon’s thesis
Private information + incentive to misrepresent => rationalist explanation for war
States have an incentive to misrepresent information in bargaining, ‘a rational state may choose to run a real risk of (inefficient) war in order to signal that it will fight if not given a good deal in bargaining.’

Commitment problem leads to preventive war

For periods t=1,2,…, A demands x. B acquiesces or fights a war; A wins with pt. Future payoffs discounted at d =>(0,1).

Uat=(pt/(1-d))-Ca and Ubt ((1-pt)/(1-d))-Cb.
No third party to ensure bargain will be kept. Let p1> p2. At t2 A demands xt= p2+Cb (1-d). So at t1 B gets war or 1- x1+d(1-x2)/(1-d). If A sets x1 =0, max Ub = 1+d(1- x2)/(1-d). But if d(p2)- p1 >(1-d)Cb then B will attack at t1.
iv. Criticisms
(1)Fearon identifies rationality with certain preferences rather than the form of those preferences. Risk averse leaders are not so rare as he says – their utility function could be convex rather than concave.
(2)‘Indivisibility’ encouraged by nationalism introduces an external explanation.
(4) Probability of withholding information should appear in calculations of relative power
(3) Fearon’s concept of the bargaining range depends upon risk aversion and neutrality– also represented as discount rates. If discount rates differ there is an incentive to risk lovers.

PROOF

The Nash Bargaining solution predicts that there is only one rational and feasible bargaining outcome


Claim #1;

There is an incentive for states to be risk lovers within Fearon’s bargaining range

I prove this by applying the Nash Bargaining Solution to the bargain between State A and State B in Fearon’s bargaining range. The Nash Bargaining Solution states that there is a unique solution to any bargaining game. In the 4 steps below I show that this solution will favour the more risk loving bargainer.

I. Assumptions

Let x∈X be an outcome in the bargaining range [p- Ca,, p+Cb].

II. Differing Utility Functions

Here I change Fearon’s assumptions by introducing different utility functions.

Let UA be a linear function UA=f(x).

To make B a risk lover, let UB be a convex function UB=yn=(1-x)n= (1- UA)n

III. The Nash Bargaining Solution

The Nash Bargaining Solution is the value of x that maximizes the product P of the
bargainers’ utilities.

Thus

P= UA(x) ∗ UB(1-x) = x(1-x)n

To find the maximum, we find the value of x that sets the first derivative of P(x) to 0.

dP(x)/dx=d(x)/dx ∗(1-x)n + (x)∗d[(1-x)n]/dx = 1 ∗ (1-x)n + x + n ∗ (-1) ∗ (1-x)n-1

= (1-x)n-1((1-x))-nx)= (1-x)n-1[1-x(n+1)].

This derivative is equal to 0 if x=1 or if 1-x(n+1)=0, i.e. x=1/(n+1).
For n>0 P(x) is positive when x<1 and 0 when x=1.

Therefore to maximize P set x=1/(n+1). This is the Nash Bargaining Solution.



IV. The Incentive to Risk Taking

In x=1/(n+1) x is the demand made by state A and n is the degree to which B is risk
averse or risk loving.

The larger n is, the more willing B is to take risks.

In x=1/(n+1) as n increases x decreases ( and consequently 1-x increases).

Therefore there is an incentive for states to be risk lovers even within Fearon’s bargaining range.