Spoiler: positive expected utility. That’s what it’s good for.

In The War Trap^[1], Bruce Bueno de Mesquita lays out a theory of war that beats all other theories. This is a great example of a model that is parsimonious, wrong, and useful. For every assumption Bueno de Mesquita makes, the reader goes, “What? That’s not how things work.”

Yet when we measure the things Bueno de Mesquita asks us to measure, and apply the equations he lays out, the result ends up closely matching reality. What’s better, his equations arrive at the same results that previous ad hoc theories did – and also explains observations those theories could not account for.

The work Bueno de Mesquita did on The War Trap is creative, intelligent, and inspiring.

We don’t know what a potential war will be like

Here’s the scene for the book: there’s an active international dispute, and one country has issued a threat of solving it with violence. We are faced with the same questions that troubled Tolstoy when he wrote War and Peace:

• Will the threat escalate into a full-on war?
• If so, which countries will join the war?
• How bloody will the war be?

Tolstoy’s main complaint was that when people write about wars after the fact, they spin stories that are just so. They say things like

Of course France would attack Spain as part of the Thirty Years’ War because X.

but then this X is generally something that’s only obvious in hindsight. Often X is also very loosely correlated to the outcome: whenever X happens the next time, France does not attack Spain. The of-course-because-X theory is very common but it’s a bad theory because it can explain everything and predict nothing.

Bueno de Mesquita’s theory is different.

We measure military alliances and fuel usage

Here’s the weird thing: Bueno de Mesquita’s calculations depend only on three things:

• For each country, the distance from its centre of power (usually the landmass which contains the capital^[2]) and the closest point on every other country. This affects how well a country can project its power onto another.
• For each country, we establish its at-home power through some broad metrics like population size, metal production, fuel consumption, and military expenditure. We assign each country a number in the range 0–1 based on how large their values of these measurements are compared to the total for all countries. This is a proxy for how well the country can fight on the battlefield.
• For each pair of countries, we compute the correlation between their sets of military alliances (ranging from -1 to 1). Note what this says: we don’t need to know what the alliances are for any country, just the correlations between pairs of countries. This is a proxy for how much countries agree with each other in policy choices.

The last one deserves an elaboration. What we really want to measure is how strongly countries feel for each other’s choices, because this determines how ready they are to use violence to impose their own choices on other countries. However, it’s hard to measure what countries think of each other across the world and a long span of time, so Bueno de Mesquita figures that countries generally enter into military alliances with other countries they agree with, and if two sets of alliances are completely disjoint, it’s likely the two countries disagree on a lot of other things in addition to who to ally.

It’s a brilliant operationalisation of a value that is difficult to measure directly.

Then we compute expected utilities

We then assume^[3] that winning a war means you get to impose your will on the loser, in effect forcing them to become your best friend. That is, the utility gain from winning a war (​) is computed as if you could force your opponent to adopt your exact set of alliances and no more or less. If you lose a war, the same thing happens except to you, and the utility loss (​) for you is computed as if you had to drop all your existing alliances and adopt your opponent’s.

The hypothetical probability of winning (​) a war depends on the power of the involved countries, including any third parties that join the war on either side.^[4] The uncertainty around the probability of winning, inherited from the uncertainty about third-party support, is why two countries going to war can have wildly different ideas of what their opponent can gain from fighting.

Once we figure out the probabilities and win/loss utilities using the measurements we have, we can compute how much, on average, any country would gain from declaring war on any other, as a regular mathematical expectation:

This can be a positive number – a profit – in which case war is a possibility, although by no means guaranteed. It can also be a negative number – a loss – in which case there is no rational grounds for entering a war. This expectation is most commonly negative (a random pair of countries has a 68 % chance of having negative utility of war against each other) which explains a part of why there is so much peace^[5], despite the anarchic nature of international affairs.

All of the above involves fairly basic plus and minus calculations where we plug in our measurements from before. The resulting number is the expected utility of going to war.

If we can assume people are rough expected utility maximisers, which they seem to be, we would then predict that wars have historically been initiated by countries that according to our calculations have positive expected utility of going to war. Indeed, that has been the case in the past 200 years: the initiator is the one with positive utility 85 % of the time.^[6]

Stop to appreciate how amazing that is! We have a very crappy way of measuring the terms of a very simple theory, and we can still predict with decent accuracy who initiates a conflict. We already have a theory that tells us who the initiator will be among a pair of countries, assuming that they end up in war. If only Tolstoy was around to witness this.

Predictions from expected utility calculus

What else can we learn from the theory?

1. Entering into alliances is traditionally viewed as something that strengthens a country’s position. The model, in contrast, predicts that entering into alliances can actually make a country weaker. By declaring its affiliation with some countries, a country is implicitly distancing itself from others. Any expression of opinion is a double-edged sword. The record of history confirms: it is not strange for a country to be attacked by their stronger neighbour only after they entered an alliance.
   
   Even a weaker country with no alliances of its own may initiate war against its stronger neighbour when they enter an unpopular alliance, counting on support from third parties who form an impromptu coalition. (The example Bueno de Mesquita uses here is one that would be very familiar to Tolstoy: the Crimean war, which was fought “not for Turkey, but against Russia.”)
2. The model predicts that in extreme cases, enemies can fight alongside each other against a common foe. All it takes is for the supporting country to have sufficiently lower utility for the common foe than the country they support. (This corresponds to the well-known expression my enemy’s enemy is my friend – except it gives us more details on exactly when that saying applies. Often it does not.)
3. Wars are more often fought between neighbours than across large distances. Some writers try to explain this as some sort of inherent tendency of neighbours to be more belligerent against each other (appealing to culture, historical territorial claims, etc.) The model explains this instead as a difficulty of projecting power across long distances, which makes the expected utility of war low. Much like the theory predicts, very powerful countries can and do fight long distance wars.

Those consequences of the theory are probably not surprising. They cover events that happen relatively frequently in history. But there are also more surprising predictions.

The surprising effects of good relations

We stated above that we don’t care about who is allied to whom. What makes it possible to ignore that? Allied countries are, it turns out, not inherently more likely to help each other just because they’re allied. What matters for an outsider to a war is whether they see some benefit of their own from entering the war. If they don’t, they won’t give their support. If they do, they will support enough to further their personal goals. It doesn’t matter whether they are allied to one of the belligerents or not.

In practise, allied countries often see it as beneficial to step in, but the model explains this as an attempt to preserve or spread their world view, rather than an obligation to their partner.^[7]

Here’s something else that struck me as particularly odd:

• Countries that really hate each other (because they have so incompatible views on the world) are less likely to go to war with each other, because their opinions of each other are so low that they can’t get much worse. It’s a relationship in which there is nothing left to save, and thus no point of a military intervention.
• On the other hand, countries that are really close and share a common world view are more likely to go to war with each other. The reason is that they can’t love each other more, so any change in their dispositions is likely to be negative. Those changes may be most easily preventable by swift military intervention. Both the US and Soviet ran military interventions in their allies’ countries at various points during the cold war, for example.

Apparently, this is also confirmed by history. Wars between allies have been more common than wars between very antithetical foes.

Predicting the severity of wars

We can go even further with the theory.

• If the defending country has a very negative expected utility compared to the initiator’s positive utility, the theory leads us to think the defending country will try to find ways to get out of the war, e.g. by yielding to the initiator’s demands without escalating the fight. This is the classic offer that can’t be refused. Thanks to Bueno de Mesquita, we know what the size of the threat must be relative to the concession desired.
• The above is true also during a war. If at any moment one of the countries have their expected utility go very negative, we would predict a peace deal follows shortly thereafter. Specifically, the peace deal will be in favour of the country that still has positive expected utility – i.e. has the BATNA^[8] of “keep fighting”.
• One might think from a quick reading of the theory that a war will only break out if both sides have positive expected utility of war, but this is not true. If the defender has negative expected utility near zero, it is still a rational alternative for the defender to resist and fight back when attacked, even if they would not initiate a war themselves.
  
  Intuitively, this situation means the initiator and defender both agree the defender will lose – but they disagree on the size of the concession the defender should need to make. The defender may need to resort to fighting to show the initiator that they are not as weak as the initiator assumed.

That is the basic logic of when war happens: there needs to be an initiator who thinks they can benefit from it, and then the war goes on for as long as the defender thinks they can negotiate a less costly deal in the future if they continue to fight today.

Following the same logic, we can guess when wars will be bloody: when the initiator and defender have very different ideas of what the outcome should be. In that situation, negotiating a settlement both agree to will be difficult, and it will take some time for their views on the appropriate settlement to converge. During this time, the fight goes on.

Can third parties stop the war? I.e. if a powerful third party says they will fight on behalf of e.g. the defender unless the initiator withdraws, will this deter the initiator? Not necessarily. The third country’s claim is not credible if the third country does not have positive expected utility of a war against the initiator.^[9]

It's impossible until the right person tries

As much as I’d like to explain all the juicy details, and list more consequences of the theory, there’s a certain limit to how much of a book can be squeezed into an article and I think that limit has been reached for now. The War Trap is one of the most interesting books I have read in a while. In part because it covers a serious and important topic, but in part because it is an excellent lesson in how to build quantitative theories.

Sometimes I get the feeling that every discovery a regular person can make has already been made. Sure, I could have come up with a simple algorithm like Dijkstra’s if I needed to – but that has already been done. Then there are the unanswered questions in, I don’t know, particle physics, which I’m sure still has discoverable answers but a regular person does not have the supercolliders to discover them. Then there are questions that are seemingly unanswerable, because they involve seemingly-chaotic systems to which no general laws apply, like wars.

Then Bueno de Mesquita comes along and comes up with a working theory of war using maths I know, plugging in data I have access to. He just had the right idea and connected all the pieces. I could have done that. And there are loads of those seemingly-unanswerable questions still hanging around, just waiting for someone to connect the dots in the right way and turn chaos into order. That gets me excited.

1. ^{^}

   The War Trap; Bueno de Mesquita; Yale University Press; 1983.

2. ^{^}

   This sort of thing matters a lot because otherwise in the early 1900s the distance from any country to Britain would be near zero.

3. ^{^}

   Again, incorrectly but usefully.

4. ^{^}

   Actaully, in The War Trap, Bueno de Mesquita had not yet figured out a parsimonious way of including risk tolerance, uncertainty, and the effect of third parties into the probability ​ so in the book the equations end up a little more complicated, including additional terms that are defined as binary cases that need to be considered separately and then added together to the main equation. I think that is rather immaterial to the usefulness of the model, so I’ll present the typographically simpler version here.

5. ^{^}

   If there are on the order of 200 countries in the world, there are around 4000 pairs of countries. The vast majority of these 4000 pairs are not at war with each other.

6. ^{^}

   Bueno de Mesquita doesn’t give this figure, but he gives and meaning the odds ratio of looking at the initiator of a war if they have positive utility is 5.4. Since the base odds are 1, this gives us 85 %.

7. ^{^}

   Bueno de Mesquita talks a little about how there are no actual ways of enforcing commitments in international relationships, because that’s what sovereignty means.

8. ^{^}

   Best alternative to negotiated agreement.

9. ^{^}

   If they don’t, they wouldn’t rationally enter the conflict anyway, so their posturing is just a trial balloon.