# Ambiguity causes conflict

Consider the following model of conflict:

Two parties, say A and B, have a dispute about some issue. The state of the world can be represented as a real number in the interval \\( s \in [0, 1] \\). A and B have utility functions \\( u_A(1-s) \\) and \\( u_B(s) \\) respectively, where \\( u_A, u_B \\) are increasing and concave. Intuitively, A wants the state of the world \\( s \\) to be close to \\( 0 \\), while B wants it to be close to \\( 1 \\). Concavity is equivalent to A and B being risk-averse.

A and B also have the ability to go to war. If they do so, they pay some nonzero utility costs \\( c_A, c_B > 0 \\) respectively and there is a probability \\( p \\) with which A wins the war and a probability \\( 1 - p \\) that B wins instead. The winner can set the state of the world as they want, so if A goes to war and wins then A will always set \\( s = 0 \\) and likewise B will always set \\( s = 1 \\).

Now, let's calculate the expected utility of both sides if they go to war.

- A's expected payoff is \\( p u_A(1) + (1-p) u_A(0) - c_A \\)
- B's expected payoff is \\( p u_B(0) + (1-p) u_B(1) - c_B \\)

By assumption \\( u_A \\) and \\( u_B \\) are concave, and therefore we know that

\\[ p u_A(1) + (1-p) u_A(0) \leq u_A(p), \, \, \, p u_B(0) + (1-p) u_B(1) \leq u_B(1-p) \\]

Therefore, if there is a war, A's expected payoff is strictly less than \\( u_A(p) \\) and B's expected payoff is strictly less than \\( u_B(1-p) \\).

What if instead of going to war A and B decide to compromise? The obvious point of compromise is the one that linearly interpolates between the endpoints of the interval according to the balance of power, so \\( s = 1-p \\). This compromise gives A a utility of \\( u_A(p) \\) and gives B a utility of \\( u_B(1-p) \\), and so it makes both A and B strictly better off than a situation in which they go to war.

# Assumptions

This model is well known in the rationalist international relations literature, and much like the [Ricardian equivalence](https://en.wikipedia.org/wiki/Ricardian_equivalence) theorem of macroeconomics, it derives a very strong conclusion (namely, war should never happen) under a set of assumptions. Let's list those assumptions explicitly:

1. The dispute in question is *divisible*, in other words, it can be accurately modeled by a continuum such as \\( [0, 1] \\) rather than a discrete space with only a few options. This assumption is not crucial, since even if disputes weren't divisible, it's possible to get around this fact by using lotteries to compromise.

2. The parties involved in the dispute are risk-averse. This assumption is important, since if A and B have extreme utility functions such as indicator functions supported on the points \\( 0 \\) and \\( 1 \\) respectively, then as long as war is not too costly they will always choose to go to war. Notice that when it comes to states or other large organizations going to war, principal-agent problems in which leaders have convex preferences whereas the organizations led by them have concave preferences can also lead to a failure of this assumption.

3. The parties agree about the balance of power in the situation, in the sense that they agree on a probability distribution over the outcomes of the war. [Aumann's agreement theorem](https://en.wikipedia.org/wiki/Aumann%27s_agreement_theorem) gives some theoretical support to this assumption, but empirically people disagree with each other all the time even when they are communicating in good faith. Since in practice deception is an essential part of warfare, we should expect Aumann agreement to be even more difficult to establish in our setting.

4. Wars are costly overall, even if they benefit the winners. This is a plausible assumption, but there are also mechanisms through which some degree of warfare could lead to macro level improvements: perhaps the competition brought about by war is an effective force in regulating [mazedom](https://www.lesswrong.com/tag/moral-mazes) in large organizations, for example, in which case two organizations that go to war could both obtain some benefits from the conflict. Again, principal-agent problems could also play a role in leading to a violation of this assumption.

Which of these assumptions is most likely to be false in practice? I maintain that most wars are caused by a failure of the third assumption: disagreement about the balance of power. There are also wars in which a failure of the second assumption plays a key role, but I think these situations are rarer and most wars don't fit into this category. In practice I think the first and fourth assumptions are almost never violated.

One of the important ways in which the third assumption can be violated is if there are more than two parties, but the relationships or commitments between some of the parties are ambiguous. In this case, different parties will interpret these commitments differently, leading them to a different assessment about the overall balance of power in any specific dispute. In short: *ambiguity causes conflict*.

# Examples

Here is a short list of wars in which I believe such disagreements played a key part:

## The First World War

An important reason behind the outbreak of the war was the United Kingdom's deliberate policy of keeping their relationship with France, the Entente Cordiale, ambiguous. Britain wanted to avoid making explicit continental commitments, and their reluctance to do so was interpreted by German leaders as a sign that the United Kingdom would most likely not intervene in a continental war. The French, in contrast, saw the Entente Cordiale as an actual alliance and expected support from across the Channel if they ever found themselves at war with Germany.

In the end, the United Kingdom entered the First World War under the pretext of Germany's violation of the neutrality of Belgium. It's difficult to know to what extent this was genuine - for example, the Entente would go on to violate the neutrality of Greece in 1916, much in the same way Germany had violated the neutrality of Belgium. However, it's a fact that the British armed forces had been in talks with their French counterparts for many years, and had even made detailed and secret plans of sending the British Expeditionary Force to France in the event of a war with Germany.

The German leadership was aware of the risk they were taking by invading Belgium, but they saw British intervention as a lower probability event than the French did, and this may have been the crucial disagreement that tipped the scales towards war.

## The Second World War

This is the same story as the First World War, except now the ambiguous relationship is between France and Poland instead of Britain and France. While Britain and France had formally guaranteed Polish independence earlier in 1939, Hitler doubted that they would actually honor this guarantee given how the Czechoslovak crisis of the previous year had been resolved. Moreover, Hitler also judged the French military doctrine to be purely defensive, and thought a serious French offensive into German territory was improbable even if France and Germany ended up at war.

In contrast, the Poles not only believed that they would have support from Britain and France; but they also believed French assurances that in the event of a German attack on Poland, France would launch a major offensive on the Western front and force Germany to split its army into two in order to fight off the French attack. The Poles counted on this support to such an extent that they stationed a substantial number of their units close to the German border where they knew the units would be encircled and forced to surrender. The aim was to drive up Polish losses in the first days of the war so as to expedite British and French intervention.

Hitler underestimated the willingness of Britain and France to go to war over Poland, but the Polish leadership also overestimated the willingness of their allies to physically support them against the German invasion by launching an offensive. Hitler attacked Poland with most of the German army, a decision that caused considerable anxiety to many of his generals due to how it exposed Germany to a potential attack from the West. In the end, the only attack that came was the tiny, inconsequential and symbolic Saar Offensive, a way for the French to claim that they had technically fulfilled their promise to Poland by "launching an offensive".

There are many other disagreements that fueled the Second World War: another important one is how Hitler held American military and industrial capabilities in low regard, and this led him to the grave mistake of declaring war on the United States. It seems obvious now that Hitler would have been better served by cutting off his relations with Japan in response to the attack on Pearl Harbor, as the alliance between Japan and Germany didn't commit Germany to assisting Japan in an offensive war. If Germany and the UK had agreed on how US participation in the war would alter the balance of power, they might have reached a compromise instead of continuing the war as they did.

## The Gulf War

The motives behind Saddam's invasion of Kuwait have baffled many observers at the time and continue to baffle people to this day. I do think that the ambiguity of the relationship between Kuwait and the US played a key part in the invasion, but it's not what I will discuss in this section.

Instead, I'll talk about the Shiite uprising against Saddam's rule that happened after Saddam's forces were routed by the US-led coalition in Kuwait. Throughout the Gulf War, American aircraft dropped leaflets all over southern Iraq, encouraging the Shiite Arabs in the region to rise up against Saddam's Sunni regime and overthrow him. The Shiites, expecting US support in their uprising, duly obliged.

However, the US had no intention of actually supporting the Shiite rebels, as they didn't want to get entangled in a potentially lengthy regime change project in Iraq at this point in time. The coalition forces stopped their advance after kicking the Iraqi army out of Kuwait, and the Arabs were left to fight alone against the Iraqi army. The uprising failed and the Iraqi regime massacred around fifty thousand rebels and sympathizers.

In this case, had the US been clearer about their commitments or the lack thereof to the Shiites they encouraged to rebel against Saddam's regime, the uprising and the massacres that took place in reprisal could have been avoided. 

## Russo-Ukrainian War

This is the event that motivated me to write this post, but I think the general message is more important than how well this specific event fits the mold described in the post. That said, since the war between Russia and Ukraine motivated me to write the post, I should probably say a few words about it as well.

In my opinion, the key ambiguity here was about the relationship between Ukraine and Western countries, including both the US and the European Union. Ukraine's ability to resist a Russian invasion by itself is even less than Poland's ability to resist a German invasion in 1939, and I don't think anyone doubted this. Like it was with Poland in 1939, I believe the Ukrainian leadership was emboldened by their relationship with the West, and mistakenly believed that they would receive more support from the West in the event of a Russian attack. Russia, on the other hand, believed that the Western commitment to Ukraine was weaker than Ukraine thought it was. The disagreement couldn't be resolved through diplomacy, which led to the eventual war.

If the US and the EU had committed to the defense of Ukraine in clear terms, Russia would not have invaded Ukraine. On the other hand, if the US and the EU had been clear that they would offer no support to Ukraine whatsoever, the Ukrainian government would have been compelled to reach a compromise with Russia. In either case war would have been avoided.

The only commitment Western countries have made in clear terms is that they will send no ground troops to fight in Ukraine, and there's much more that they could have done to support the Ukrainian government without breaching that line. They haven't done so, and I think in that they have shown Putin's judgment of the situation to be accurate.

# Whence ambiguity?

I see two primary drivers of ambiguity in the policies of large organizations:

1. Credibility comes with short-run costs and long-run benefits. Agents with short time horizons have difficulty establishing it because they are always tempted to break their promises for short-term benefits. This prevents others from trusting their promises, leaving their relationships and commitments ambiguous.

2. Organizations have turnover in personnel, and they often also have decentralized decision-making structures in which the weight of different sub-units of the organization can change in unpredictable ways. This means that it's not easy to establish a tradition of credible commitments even with the best intentions, and as different groups within the organization push for different policies, the overall stance of the organization can end up being ambiguous.

There are most likely other reasons that don't occur to me, so if you can think of anything let me know in the comments.

# Conclusion

I'll end the post here as I think the examples I've presented should be sufficient to make my point. I'm curious to hear what people have to say about this, as it's an idea I haven't seen discussed anywhere as a potential driver of conflict in general.

I'm fine with any discussion in the comments so long as it's not obvious spam, so if you think my portrayal or description of one of my examples is inaccurate you're free to tell me how wrong I am.