The boy who had to bring a wolf

A game theory model that explains Vladimir Putin's actions

Plenty of real-world scenarios can be explained using game theory, a branch of mathematics which studies strategic interactions between rational agents. In many of these scenarios it’s easy for me to see what different rational agents would do, but perhaps that’s due to my background—which allows me to run algorithms in my head—and for other people it might not be so easy.

For example, I can easily see why an artificial intelligence playing poker inevitably learns to bluff without any guidance. It’s because given a scenario, bluffing is in the best interest of any rational agent, and artificial intelligences are rational agents. 2+2=4.

Let’s consider a simple scenario: a badger that doesn’t want you to get close. If you stay away from the badger, you are doing what the badger wants, and it doesn’t need to do anything. However, if you get close, or worse: starting poking it, the badger will have to respond in a threatening manner and will show you its teeth. If you ignore this threat, the badger will bite you.

It’s obvious that the badger doesn’t want to bite you, it wants you to consider it a threat (which it is). If you act as if the badger is harmless, then you are leaving the badger no choice but to bite you.

To me this is as crystal clear as 2+2=4. The actions of the badger are not just subjectively reasonable, it’s a mathematical fact that they are a rational response to your actions, just like bluffing in poker, or the result of an addition.

The same mathematical fact explains the actions of Vladimir Putin in Ukraine (at this moment in time we are six days into the invasion of Ukraine). Unfortunately humans get easily carried away by emotions and few people are thinking clearly at this moment.

So, in an effort to take away all the emotion from the situation, I’m going to construct a simple computer simulation to model different scenarios, and different rational agents. The end result is that Vladimir Putin had no choice but to conduct an act of aggression against Ukraine. This isn’t an opinion, it’s a mathematical fact, and any other rational agent would have done the same, in the same situation.

It should be clear that this is in no way an endorsement of any act. Whether or not it’s moral for a badger to bite is a different question, and it’s out of the scope of game theory.

The boy who cried wolf

The Boy Who Cried Wolf is one of the most popular fables in existence, I probably don’t need to explain it, but just for completeness: a shepherd boy repeatedly tricks nearby villagers into thinking a wolf is attacking his town's flock, when a wolf actually does appear and the boy again calls for help, the villagers believe it’s another false alarm and the sheep are eaten by the wolf.

The moral of the story is obvious: don’t squander your reputation or people will stop believing you.

To model this we are going to need numbers. The result of the simulation is completely dependent on the numbers, but we have to start somewhere, so we will pick some arbitrarily.

The most important variable is the wolf. Every day there will be a random chance of a wolf appearing, and we’ll say it’s 10%. Second is the level of trust the town has on the boy, we start with 100%, but each time the boy lies, he loses 50%, so after two lies nobody believes him. Next, the boy needs some incentives, the cost for losing the flock will be -100 (units don’t matter).

At this point if we consider 3 days, the result of the simulation hovers around -2.7 which is consistent with the math. (Code in GitHub).

In 3 days there are 8 combinations of wolf/no-wolf events, but the only way for the boy to lose value is if the town doesn’t believe him and there’s a wolf. In order for the town to stop believing the boy, 2 false alarms need to happen, therefore the boy needs to raise the alarm two times when there is no wolf. If the probability of a wolf happening is 10%, then the probability of two no-wolf days is 90% ⋅ 90%, or 81%. Then, a wolf needs to appear, which is a 10% chance, therefore 8.1%.

If the probability of the flock getting eaten is 8.1%, and the value lost is -100, then the boy in average loses -8.1, divided by 3 days we get: -2.7.

This means it isn’t rational for the boy to lie, since he has nothing to gain, and something to lose.

If we can calculate the expected value with math, then why not just use math? The main reason is that it takes significantly longer to calculate. It takes no effort for me to change the code of the simulation, and simply see what happens. Another reason is that it’s easy to get it wrong, and harder to prove I didn’t get it wrong. With code anyone with basic programming skills can verify the code themselves, or even write it from scratch. The final reason is that I have much more experience programming than doing math.

The jester boy

There’s one problem with the previous simulation: the boy gets nothing from fooling the town. That doesn’t match the fable. In the fable the boy gets enjoyment from fooling the town, so we need to give some value to that, let’s say 5.

If we update the code of the simulation to reflect that, now the expected value is 0.59. This means now it’s rational for the boy to lie, since even though he will lose the flock now and then, the enjoyment he gets compensate the potential losses.

The same result can be verified using math—although the result won’t necessarily match the simulation, since the simulation cannot be exact. Feel free to do that calculation if you don’t believe me.

However, this result depends completely on the numbers we use. If instead of losing -100 the boy loses -1000, then the expected value is -23.68, and now it isn’t rational to lie, since now the enjoyment of fooling the town isn’t enough to compensate for the losses.

Game theory doesn’t just let us understand why some rational agent would do some action, it also allows us find out ways to prevent that.

From the point of view of the town it isn’t in their best interest for the jester boy to raise false alarms. An easy way to fix the problem is to change incentives. If every time the boy raises a false alarm the town makes him pay a certain amount, then that counters the enjoyment, and the expected value returns to -2.70. This effectively transforms the jester boy to the neutral boy.

By simply changing the incentives, the behavior of the rational agent can be molded to some desired state.

Incredulous town

In order for our simulation to be closer to the badger situation, we need to change the expected value. Let’s say now the objective of the boy is to maximize the town’s trust.

In this case it shouldn’t be hard to guess what’s the best strategy for the boy, but just for completeness: a boy that constantly cries wolf gets an expected value of 22.15%, while an honest boy gets 100%. Obviously it makes no sense to lie, even a single time, since that reduces the level of trust of the town, which takes him further away from his goal.

What happens if the town doesn’t have any trust in the boy? If we start with a trust level of 0%, then the boy has no incentive to do anything. Be honest, lie, it doesn’t matter, since the town isn't going to react in any way, even if a wolf is really coming.

To fix the scenario, we can add another clause: if the boy raises a true alarm, then he gains 50% trust. Now it’s rational again for the boy to raise an alarm if a wolf is coming, and his expected value is 9.36%. A lying boy on the other hand gets 5.00%, so that’s not the best strategy (obviously).

But in fact there’s a better strategy: bring a wolf. If all the boy cares about is the level of trust of the town, bringing a wolf and raising the alarm will ensure the level of trust gets increased. The expected value of this handler boy is now 50%. Why wait for a wolf to come when you already know what will increase the level of trust?

This scenario is now on par with the badger scenario. The badger doesn’t need to wait for a bite to come, he can just decide to bite. An incredulous human who doesn’t believe the badger will bite him is akin to the incredulous town. All the badger has to do to make the human believe it, is bite him, and all the boy has to do to make the incredulous town believe him, is bring a wolf.

If the town believed the boy in the first place, the boy would have no need to bring a wolf. Similarly, if the human trusted that the badger would bite him, the badger would have no need to bite him.

In other words: the incredulous human is giving the badger no choice but to bite.

Once again, this is not a matter of opinion, it’s a mathematical fact, as certain as 2+2=4.

This should be obvious. The only tool the badger has to make the human believe it can bite, is bite. It’s the human’s low level of belief that caused the bite.

At this point there isn’t any math or probability involved. If the level of belief falls bellow a level the badger deems acceptable, it will bite. Period.

Threat theory

All that is needed to understand the actions of the badger in different scenarios is to understand the incentives of the badger, and the tools it has to influence those incentives.

The conclusion—which to me is obvious—is simple:

A threat only works if other agents believe it will be carried out. If other agents don’t want the threat to be carried out, they need to behave as if it will be.

In other words: if you don’t want to be bitten by the badger, don’t poke it.

And again: this isn’t an opinion, it’s a fact.

Vladimir Putin

Vladimir Putin isn’t a badger, but he is in the same situation. The only reason anyone wouldn’t be able to see it is either emotional fuzziness, or lack of empathy.

Putin wants to secure deals for Russia, and for those deals to be carried out. For that he needs Russia to be seen as a strong country. Anyone who has attempted any negotiation knows that without leverage you cannot get anything. A contract is worthless if there are no penalties for breaching it. It’s the “or else” that forces countries to action or inaction.

Have you heard the phrase “you and whose army”?

The stronger the image of Russia, the bigger Putin’s leverage, and the better deals he can secure for Russia.

Again, this has nothing to do with morality, this is the reality of world politics: strong countries get away with more than weaker countries. That’s just the way it is.

Of course, “strength” can be measured in many different ways. A country with a strong economy may be able to get better deals by offering trade in exchange, for example. But one way is undoubtedly by the size of its military.

Like the badger, Putin wants to maximize the perceived strength of Russia.

But the perceived strength is not necessarily the same as the actual strength. The actual strength of a country could be overestimated, or underestimated.

If USA gauges the strength of Russia to be 50% of what it actually is, Putin certainly wouldn’t be content, because beliefs translate into actions, and the kind of actions that affect Russia might not be in accordance with its actual strength.

What happens if USA believes the strength of Russia to be 25% of what it actually is? Actually, it doesn’t matter what USA believes, only how it acts. What happens if USA acts as if the strength of Russia is 25% of what it actually is?

Can Putin invite Biden to a public debate? Can he commission a study? Can he challenge Biden to a karate match?

There’s only one thing Putin can do, and that is to demonstrate Russia’s strength. Putin has to bite.

USA really left Russia no choice.

Of course, people will argue that Russia had other means to achieve its goals, like diplomacy. However, diplomacy has already been tried and USA gave Russia nothing. USA under Biden has acted as if Russia’s strength was 0%.

Is it moral for Russia to invade Ukraine? Is it moral for USA to threat Russia as if it was nothing? These are questions for moral philosophers.

The mathematical fact is that USA’s actions caused Russia’s actions, and they won’t stop until USA starts treating Russia seriously. By ignoring the badger’s teeth, the badger had no choice but to bite.

By setting the gauge at 0%, USA has left Russia with two options: be docile (0%), or bite (100%). If on the other hand USA had given Russia the smallest of concessions and set the gauge at 10%, maybe Russia would have opted to be docile (10%). But at 0%? No way.

Donald Trump was tough on Russia, but not tough to the point of leaving them no options. So was Barack Obama.

Anyone who doesn’t see this isn’t thinking clearly. Unfortunately people are advocating actions that are the opposite of what will resolve the conflict. If other nations keep deriding Russia, they will leave Putin with an impossible choice: either give up and accept that Russia will be treated like nothing for decades to come, or launch a nuclear attack.

The rational choice for USA is obvious: simply give some concessions to Russia, as any rational agent would do with any country with teeth. But World War III won’t be started by rational decisions.