# Fearon (1994) - Rationalist Explanations For War

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Has article title | Rationalist Explanations For War |

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- This page is referenced in BPP Field Exam Papers

## Reference(s)

Fearon, J. (1994), Rationalist Explanations for War, International Organization 49, 379-414. pdf

## Abstract

The central puzzle about war, and also the main reason we study it, is that wars are costly but nonetheless wars recur. Scholars have attempted to resolve the puzzle with three types of argument. First, one can argue that people (and state leaders in particular) are sometimes or always irrational. They are subject to biases and pathologies that lead them to neglect the costs of war or to misunderstand how their actions will produce it. Second, one can argue that the leaders who order war enjoy its benefits but do not pay the costs, which are suffered by soldiers and citizens. Third, one can argue that even rational leaders who consider the risks and costs of war may end up fighting nonetheless. This article focuses on arguments of the third sort, which I will call rationalist explanations.

## Background

Fearon claims that:

- The central puzzle is that war is costly and risky, so states should prefer negotiated settlement to the gamble of war
- Leaders should reach ex-ante bargains to prevent war
- A rationalist arguement should explaing why both sides should prefer to fight

The paper provides five arguements for war:

- Anarchy - there is no supranational authority, but this doesn't prevent side payments.
- Expected benefits outweigh expected costs
- Rational preventative war - could be solved by inter-temporal bargaining, but there may be commitment problems.
- Rational miscalculation due to lack of information
- Rational miscalculation concerning relative strengths

The first three should lead to ex-ante settlements. The last two should be able to be prevented using communication. The paper proposes three mechanisms:

- Private information with incentives to misrepresent information - this prevents communication.
- Committment problems - one or both parties will renege
- Issue indivisibilities - some issues do not admit compromise. Side payments or randomization offer possible solutions

To address the puzzle of war it is important to consider both **ex-ante** and **ex-post** efficiency. If war is costly then it must be ex-post inefficient; the same final outcome could have occured without the conflict and the costs (or lesser costs). This is true even if there are benefits, unless fighting is a consumption good.

## Bargaining Preferred to War

Consider two states: [math]A\,[/math] and [math]B\,[/math]

The states have preferences over the issues (say territory) modelled as the interval [math]X = [0,1]\,[/math]. [math]A\,[/math] prefers outcomes ([math]x \in X\,[/math]) close to 1, [math]B\,[/math] prefers outcomes close to 0.

Utilities are [math]u_A(x)\,[/math] and [math]u_B(1-x)\,[/math], where [math]u_i(\cdot)\,[/math] is continuous, increasing and weakly concave (i.e. risk neutral or risk averse). WLOG set [math]u_i(1) =1\,[/math] and [math]u_i(0) = 0\,[/math].

War is modelled as gamble. A prevails with probability [math]p \in [0,1]\,[/math], and the winner chooses the outcome.

- [math]\mathbb{E}u_A = p u_A(1) + (1-p)u_A(0) - c_A = p - c_A\,[/math]

Where [math]c_A \gt 0\,[/math] is [math]A\,[/math]'s cost of war (particularly relative to any benefits). Likewise:

- [math]\mathbb{E}u_B = (1-p) u_B(1) + pu_B(0) - c_B = 1-p - c_B\,[/math]

There exists a negotiated settlement that is pareto optimal:

- [math]u_A(x) \gt p- c_A\,[/math]

and

- [math]u_B(1-x) \gt 1- p- c_B\,[/math]

For example in the risk neutral case [math]u_A(x) = x\,[/math] and [math]u_B(1-x) = 1-x\,[/math], [math]x \gt (p - c_A)\,[/math] and [math]x \lt (p +c_B)\,[/math] solves this set of equations therefore:

[math]x \in (p - c_A, p +c_B)\,[/math]

is strictly preferred to fighting. Risk aversion will increase the range. This range exists because bargaining is ex-post inefficient.

Proof: Choose an [math]\epsilon[/math] such that [math]0\lt \epsilon\lt \min\{c_{A},c_{B}\}[/math]. Let [math]a=\max\{0,p-\epsilon\}[/math], [math]b=\min\{p+\epsilon,1\}[/math]. Consider [math]x'\in[a,b][/math] by weak concavity, [math]u_{A}(x')\geq x'[/math]. Further [math]x'\gt p-c_{A}[/math], so it is better than war, because [math]x'\geq a \geq p-\epsilon \geq p- c_{A}[/math]. Can make same argument for B.

There are three assumptions needed:

- That [math]p\,[/math] be common knowledge - though even if it isn't rational leaders must know that there is a common [math]p\,[/math], and therefore that settlement is preferable to war.
- The leaders are risk neutral or risk averse. This is a reasonable assumption: A risk-acceptant leader is analogous to a compulsive gambler, that has the expected outcome of eliminating both the state and the regime.
- A continous range of settlement exists, to allow for feasible outcomes that lie in the range. But even without this, continuous side-payments would offer a solution, or randomization/alternation (c.f time-sharing!) would allow for negotiated outcomes. Though sharing a throne might be problematic.

## Private Information and Incentives to Misrepresent

Lack of information is not sufficient, as states could communicate. There must also be incentives to misrepresent information.

The new game is set up as follows:

- Suppose there is a status quo [math]q \in X\,[/math].
- State [math]A\,[/math] makes a unilateral choice of outcome [math]x \in X\,[/math] (where [math]x\,[/math] may equal [math]q\,[/math])
- Then state [math]B\,[/math] can either acquiesce or go to war (as before).
- Without private information, state [math]A\,[/math] chooses to push state [math]B\,[/math] back to its reservation level [math]p+c_B\,[/math] and [math]B\,[/math] acquiesces.
- With private information about either [math]p\,[/math] or [math]c_B\,[/math] state [math]A\,[/math] faces a trade-off: the more territory it grabs the more likely the war.
- In equilibrium state [math]A\,[/math] makes the trade-off and runs a positive risk of war.

But why could state [math]A\,[/math] not simply have asked state [math]B\,[/math] for the information as so avoided the (inefficient) war? Because there are strategic reasons to misrepresent!

### Incentives to Misrepresent in Bargaining

The model is now extended to give [math]B\,[/math] a foreign policy announcement (prior to [math]A\,[/math]'s unilateral choice) denoted [math]f\,[/math].

Then if the announcement has no direct effect on either side's payoffs, it is cheap talk and irrelevant. A will make the same demands irrespective of [math]f\,[/math] and the ex-ante risk of war remains the same as before. To 'fix' this, suppose that [math]f\,[/math] is costly to make. Specifically:

- The signal must be costly in such a way that a state with a lesser resolve or capability would not wish to send it.

## War as a Consequence of Committment Problems

### Preemptive War and Offensive Advantages

There may be the following advantages to preemptive war (i.e. a first mover advantage):

- The probability of winning may be higher as an attacker rather than as a defender
- The costs of fighting may be lower for an attacker - this doesn't lead to a committment problem, but does change the bargaining range
- Offensive advantages might change the variance of outcomes, that is total victory or loss may result rather than stalemate or small changes - this can reduce the expected utility for both sides thus increasing the bargaining range.

Only the first 'advantage' leads to a commitment problem.

Let [math]p_f\,[/math] denote the probability of [math]A\,[/math] winning with [math]A\,[/math] striking first, [math]p_s\,[/math] with [math]A\,[/math] striking second, and [math]p\,[/math] with simultaneous strikes, such that:

- [math]p_f \gt p \gt p_s\,[/math]

A peaceful resolution is possible iff neither side has an incentive to defect, therefore:

- [math]x \gt p_f - c_A\,[/math]

- [math]1-x \gt 1 - p_s - c_B\,[/math]

which imply the following must hold to have stability (i.e. no defection and conflict):

- [math]p_f - c_A \lt p_s + c_B \,[/math]

As [math]p_f\,[/math] increases and [math]p_s\,[/math] decreases, on either side of [math]p\,[/math], the range that supports this condition shrinks and then ceases to exist. If:

- [math]p_f - p_s \gt + c_B + c_A\,[/math]

Then no credible peaceful solution (i.e. one that the players can commit to without an outside enforcer) exists. The lack of the outside enforcer is the anarchy explanation for war, and we see its role in this explanation, though it, itself, if not sufficient. Note that even if this condition doesn't hold, then first mover advantages narrow the bargaining range.

### Preemptive War as a Committment Problem

Preemptive war arguements are dynamic, so the bargaining model is now modified to have and infinite number of periods, [math]t=1,2,\ldots\,[/math]. Assume that in each period [math]A\,[/math] can revise the status quo with a unilateral choice of [math]x\,[/math], which [math]B\,[/math] can either accept (acquiesce) or go to war over. The probability of winning the war is [math]p_t\,[/math] in each period and once a war is won, the winner chooses their preferred outcome which will remain in place for all time. There is a discount factor [math]\delta \in (0,1)\,[/math] and players are risk neutral. Note that war remains an inefficient outcome in this game.

If the states go to war in period [math]t\,[/math] the expected payoffs are:

- [math]\mathbb{E}u_A = \frac{p_t}{1-\delta} - c_A\,[/math]

- [math]\mathbb{E}u_B = \frac{1-p_t}{1-\delta} - c_B\,[/math]

It is easy to show that there exist some [math]x\,[/math] such that both states in every period [math]t\,[/math] would rather not go to war.

However, [math]A\,[/math] may not be able to commit itself to a future foreign policy that makes [math]B\,[/math] prefer not to attack at some point. Suppose in [math]t=1\,[/math] [math]p_t = p1\,[/math], whereas in [math]t \ge 2\,[/math], [math]p_t = p_2\,[/math], such that [math]p_2 \gt p_1\,[/math]. Without an outside enforcer, the unique SPNE is to choose:

- [math]x_t = p_2 + c_B(1-\delta) \; \forall t \ge 2\,[/math]

In the first period, [math]B\,[/math] is choosing between accepting [math]x_1\,[/math] or going to war. Accepting [math]x_1\,[/math] gives a payoff of:

- [math]1 - x_1 + \delta \frac{1-x_2}{1-\delta}\,[/math]

If [math]A\,[/math] gives [math]B\,[/math] everything (i.e. sets [math]x_1=0\,[/math]) this gives [math]B\,[/math] thier largest possible payoff:

- [math]1 + \delta \frac{1-x_2}{1-\delta}\,[/math]

But this can be less the payoff to attacking in the first period:

- [math]\underbrace{ \frac{1-p_1}{1-\delta} - c_B }_{\mbox{Attack in }t=1} \lt \underbrace{1 + \delta \frac{1-(p_2 + c_B(1-\delta))}{1-\delta}}_{\mbox{Accept }x_1=0}\,[/math]

- [math]\therefore \delta p_2 - p_1 \gt c_b (1-\delta)^2\,[/math]

Essentially, if [math]B\,[/math]'s expected decline is too large relative to its costs of war, then [math]A\,[/math]'s inability to commit makes preemptive war rational for [math]B\,[/math]. Note also that if [math]B\,[/math] could commit to fight in the second period, then [math]B\,[/math]'s bargaining power would not fall, and preemptive war would be unnecessary. Furthermore, there is no private information here. This model, taken literally, suggests that rising powers should transfer away their military strength to prevent preemptive war against themselves.

### Strategic Territory

One the topic of preemptive attacks, or of concessions, it is important to note that not all territory is, in the real world, created equal. Some territory may confer an advantage on the holder, and so it may be worth attacking to hold it (endogenously changing the probability of winning), or refusing to concede it (because the opponent may then be unable to commit to not attacking further).