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 . A and B have utility functions and respectively, where are increasing and concave. Intuitively, A wants the state of the world to be close to , while B wants it to be close to . 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 respectively and there is a probability with which A wins the war and a probability 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 and likewise B will always set .

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

  • A's expected payoff is
  • B's expected payoff is

By assumption and are concave, and therefore we know that

Therefore, if there is a war, A's expected payoff is strictly less than and B's expected payoff is strictly less than .

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 . This compromise gives A a utility of and gives B a utility of , 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 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 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 and 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 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 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.

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A very nice read. Something to note: typically in the arms race that can precede a war the side that is behind will invest in unconventional weapons. For example Germany's U-boat investments.  They didn't know for sure if submersibles were a good tactic or not, but they did know for sure that their ordinary battleships would be outnumbered even if they committed the full navy budget to them. So they took a punt on submarines.

This adds more ambiguity. If the underdog has invested much of their budget into a new-fangled weapon that has never seen real action before then they introduce additional uncertainty into the two sides likelihoods of victory.

It’s more than that, there is no uncertainty about probabilities in the https://en.m.wikipedia.org/wiki/Chicken_(game) and yet, there can be a conflict.

You might want to fight even if it’s more expensive than compromising, because you are playing a repeated game. You don’t want to send a signal that you are a kind of person who compromises with an aggressor.

Two points about this:

  1. In repeated versions of the hawk-dove game, it's difficult to get war in an equilibrium unless there are some players who have very high discount rates for the future. There's always the implicit off-equilibrium threat of resisting if someone deviates and tries to go to war, but such an off-equilibrium threat doesn't need to be observed in order to be effective.

    Maybe the best example of this is monetary policy: the US doesn't have a hyperinflation because people understand the Federal Reserve's commitment to contract monetary policy enormously if that happened. However, in fact we never observe the Federal Reserve's commitment to fight hyperinflation be tested by the US dollar becoming worthless overnight.

    You can say that in the real world these commitments will always be tested, but the reason they are tested is usually because of disagreement about how credible the commitments are, not because there are some madmen going around who wage war for fun.

  2. What I've outlined in the post is not the only way in which war can happen, since the world is complicated and a big phenomenon like war is going to defy any simple explanation. I think it's better to read the post as saying "ambiguity causes conflict", as is stated in the title, rather than "all conflict is caused by ambiguity", which would be false.

Perhaps an interesting question here might be just what one can assess Putin's discount rate as.

I think with Putin the question is more about what his objective is rather than what his discount rate is. It seems to me that Putin is motivated by his desire to see the Russian Empire restored in some form or another, and he cares less about indicators such as Russia's productivity, economic potential, soft power, et cetera.

He has repeatedly said that he sees the fall of the Russian Empire (in the form of the dissolution of the USSR) as a disaster for Russia. In this, there are definite parallels between Putin and Hitler: both of them were driven by nationalism and a desire to see what they saw as people belonging to their nation and their national territory returned to their country. For Hitler, these were the German-speaking populations in the old Austro-Hungarian Empire and the territories that Germany had lost in the Treaty of Versailles. For Putin, they are the territories that Russia lost when the USSR collapsed.

That's why Ukraine is more important to him than it might've been to us if we were in his position, and why he's willing to bear such large costs in order to ensure Ukraine doesn't drift away from the Russian sphere of influence and enter NATO, which Putin sees as a military alliance hostile to Russia.

Edit: If you think I'm wrong, you can write a reply to the comment along with (or instead of) downvoting it. I'd like to know exactly where I'm wrong so that I can correct my model of the situation.

I think your observations on motivation/goals are correct -- though also admit I'm not qualified to be sure about those claims. They are, it seems, rather widely shared/suggested.

But I don't think that has really changed for, what?, maybe 20 years now. So is the fact we see Putin escalating things now to what most in the world seem to think are insane levels maybe says his discount rate (or his time horizon, had not considered that before) has changed I would think. That's following on your statement in point 1.

If the game theory is saying we get war with high discount rates - or that the probabilities of such outcomes greatly increase -  then what might we think Putin's rate changed to? Why? And, perhaps, what other implication would that change suggest?

I had not thought about it before starting this comment but wondering what the ramifications might be in the game when one player suddenly is given a much shorter time frame in the game. Does that perhaps produce the same outcomes as the high discount rate parameter in the game model?

I've published a postmortem on my prediction about the war in Ukraine in this post. Posting it here as a comment so people can find it.

I'm not sure I'd frame it as ambiguity causing the conflict.  The conflict is about underlying different desires and beliefs.  The ambiguity makes the lines of power less obvious, and allows for more negotiation and variety of resolution, at the cost of sometimes having to test the limits.

It's well-known that uncertainty of outcome is the only reason to fight, but that is somewhat different from the intentional ambiguity of position that has both costs and benefits.

I'm not sure I'd frame it as ambiguity causing the conflict. The conflict is about underlying different desires and beliefs. The ambiguity makes the lines of power less obvious, and allows for more negotiation and variety of resolution, at the cost of sometimes having to test the limits.

I think here you're using the word "conflict" with a different meaning than the one I'm using in the post. There's no reason for us to go to war just because we have different interests, and I'm using "conflict" to mean a physical conflict, so either war or something that is of a similar character but on a smaller scale.

It's well-known that uncertainty of outcome is the only reason to fight.

This is not what I'm saying. The outcome can be uncertain, but that is by itself not a reason to fight. What causes the fight is disagreement. If the outcome was uncertain but the parties agreed on the probability of it going one way or another, that wouldn't be a reason to fight.