Mathematically, time is running out to shore up Ukraine
Mathematics has been used in warfare for millennia. Besides his contributions to pure mathematics, Archimedes, working 200 years before…
Mathematics has been used in warfare for millennia. Besides his contributions to pure mathematics, Archimedes, working 200 years before Christ, was perhaps the first “military contractor” to provide mathematical and technological assistance to the state. His giant claw could overturn Roman ships while his catapult designs were so impressive the Romans captured one and copied it. He is also well known for his invention of a “death ray” that could use mirrors and the Sun’s rays to burn ships out at sea. Modern attempts to replicate this feat, including a well known challenge by President Obama to the Mythbusters team to attempt it, failed. The math was good but the physics was not. Reflected sunlight is simply too diffuse to direct over long distances.
Mathematics can certainly help military affairs when applied to navigation, weapons, targeting, and estimates of logistics (food and weapons, for example). Military strategy, however, has tended to rely on either experience or military theories that may have little empirical backing.
The continued popularity of Sun-Tzu’s Art of War is telling in that a general who lived in the 5th century BC could, based on experience, give advice that still rings true today (and has equally bad consequences when ignored).
In the current Ukraine war, there is no shortage of arm chair diplomatic and military strategists, many of whom don’t even have the experience of reading Sun-Tzu, let alone understanding the complexities of modern military engagements. Opinions about what to do range widely from demanding no-fly zones to imploring that any direct contact between NATO and Russian forces would automatically lead to nuclear Armageddon. Leaders who, ostensibly, have skin in the game are almost unanimously in favor of waging an overt proxy war, but savvy brinksmanship may require more.
In a previous article, I suggested that game theory implies that in this game of brinksmanship, NATO members should hold their cards closer to their chest, lest the Russians decide they can push their luck. The only person in leadership I found who agreed with me was Joe Manchin. I’m not sure he’s heard of game theory. Perhaps he’s a poker player.
We got here through the conventional thinking that a Russian invasion of the whole of Ukraine was “unthinkable”, and we are forced now into considering likewise unthinkable things. Yet, far from being unthinkable, the current situation (in hindsight) was inevitable because of the asymmetry in the global power structure. This asymmetry enables one superpower to make gains through war while the opposing superpower is hamstrung. The global system is only stable as long as certain actions are seen as “unthinkable”. As soon as a leader like Putin thinks them, instability results. This instability will continue long after Putin is gone. It is inherent in the mathematics of global power.
In this article, I want to talk more concretely about what the long game looks like from a game theoretic perspective and why the situation is so urgent from a strategic point of view.
The Ukraine war is a game of escalation with three entities, Russia, Ukraine, and NATO+some other allies. I’ll call these last “the West” even though you could include Japan and some other Eastern nations in this group.
You can argue about who made the first move in this game, was it Ukraine by rejecting its pro-Russian leaders in 2014? Was it NATO in making overtures to Ukraine but failing to follow through on them or provide assurances? I won’t quibble, and it doesn’t matter now. We will start with Russia making the first move by invading Ukraine.
Each player in this game can make three types of moves: concede, deny, and escalate. A concede move implies a de-escalation, pulling out, giving in, etc. A deny move means meeting force with proportional resistance. Escalate means meeting force with a higher level of force.
Each move has both a potential reward and a potential cost. To win the game, the goal of the player must be to maximize their own reward while minimizing the reward to the other player.
Ukraine and the West have closely aligned interests and hence any reward to the West can be considered a reward to Ukraine and vice versa. The magnitude of the reward may be different, i.e., the stakes are much higher for Ukraine than the West. But, we can construe them to be one player.
If someone else were in charge of Russia, we could consider this a game with the potential for cooperation, which would make it more complex, but, since it is Putin, this is a zero sum game. A zero sum game means that any move that results in a gain for one player, is automatically a proportionate loss to the other.
Russia has a considerable menu of escalations it could undertake such as taking a scorched earth approach to the war (substantial bombing with many civilian casualties), attacking neighboring states (including border skirmishes), or even using tactical nukes.
The West also has a menu of escalations from sanctions to military assistance to interventions.
What is “likely” for each actor to do is where mathematicians and arm chair strategists diverge. Mathematicians have as their main criteria reward maximization, which assumes that the two players will make moves that they believe gives them an advantage. This is the criteria of “rationality” but it is a low level definition called instrumentalist rationality. An instrumentalist approach to rational decision making does away with questions about information bias or variances in personality. It simply says that a person will make choices that they believe are in their own interests.
An irrational person, then, is simply someone who knows what their interests are and yet does something else. For example, if I am offered a piece of chocolate cake and I like cake and would like to eat a piece if offered, yet I take the cake and throw it away, not because I don’t like cake, am full, on a diet, or think it is poisoned, but because I just “felt like it” then I am acting irrationally.
Many a military campaign failed, not from a lack of rationality, but because the rewards of a particular action were misapprehended or the long game was not appreciated.
This is clear in that when Russia chose to invade Ukraine, they expected Ukraine to concede, but instead Ukraine chose to deny. The West likewise responded with intense pressure on the Russian financial system as well as rapid arms shipments to Ukraine. (Short-sightedness was not only on the Russian side here or those weapons would have already been in Ukraine as well as a decent air defense system.) Threats Russia considered not credible were actually very credible.
If Russia had maintained their invasion as is, Ukraine and the West would sustain a limited conflict, possibly to a stalemate, but, with the move in Russia’s court, Russia appears to have chosen to escalate (e.g., hiring Syrian mercenaries and escalating artillery and bombing hitting civilians), trusting that, by regrouping, they would gain the upper hand. This move, which is far from complete, requires that Ukraine and the West quickly turn their initial denial of the invasion into an escalation before it is too late.
Based on a game theoretic analysis, if Russia completes their escalation, a deny action by Ukraine and the West will no longer succeed. Two options result: (1) the Ukrainian army is driven from their cities and forced into rural areas, avoiding the Russian army and carrying out a guerilla war, or (2) Russia decides not to hold the entire country (for obvious reasons) and Ukraine concedes to its demands not to join NATO or the EU and to let go of Crimea and its Eastern regions.
A subsequent escalation by Ukraine and the West, however, could lead to all out conflict because the Russians will be entrenched. The West won’t allow that to happen, so Russia will win on their terms.
On the other hand, an escalation on the part of Ukraine and the West now could lead to an unwillingness on Russia’s part to escalate things further. This is based on the assumption that the Russians would lose (fail to get their demands met) if they chose to deny the escalated conflict, but that they would be unwilling to further escalate and risk all out conflict. Indeed, if the West makes the right choices now, their escalation will require Russia to risk all out war, thus, forcing Russia into a checkmate posture between all out war and concession.
So how do we escalate things now? Well, we are in a bind there. Short-sightedness on both sides has both limited the West’s options and given us a window of opportunity.
Let’s first look at the option that the Ukrainians favor: the no-fly zone. A no-fly zone is a bad idea because it invites the Russians to provoke us and escalate the conflict out of control. We would have to fly sorties over Ukraine, and, if Russians flew against us, which they will, we would have to pull the trigger. It is ineffective at best and catastrophic at worst. We have been here before in Iraq of the 1990s. The Iraqis repeatedly challenged our no-fly zone and were repeatedly shot down. We can’t treat the Russians like the Iraqis since unlike Iraq the Russians actually have Weapons of Mass Destruction. This is also what Kennedy did in the Cuban Missile Crisis by blockading Cuba. He got lucky, but Kennedy was ready to push the nuclear button. This is brinksmanship at its worst, and a repeat of the CMC is the opposite of what needs to happen. We need to force the Russians to have to pull the trigger against the West, preferably against NATO nations which are outside the warzone.
A better option is to provide Ukrainians with more heavy weaponry (such as the MIG-29s from Poland, heavier field artillery, portable targeting jammers and decoys) as well as staging supply bases in border nations. As long as border nations are not actively firing weapons across the border or flying sorties from there, anything that Russia did in retaliation such as firing missiles into Poland would be an escalation on their part, not a response in kind. This is preferable to a no-fly zone. (It is also one of the tactics that the Taliban used against the United States in Afghanistan, staging from Pakistan.) It effectively forces the Russians into the same position a no-fly zone would force the West into.
The theme in all escalations is that they must force the Russians into a bind of having to escalate against the West. In a desire to avoid all out conflict, the Russians will be unable to counter these. These may be weaker than direct support, and, while a no-fly zone may be a proven tactic, it would be a strategic blunder. If we keep up sustained pressure, they will eventually have to back down and negotiate a more reasonable settlement.
The Russians will attempt to shut the borders. Unfortunately, we cannot airlift or drive weapons across the border with impunity. The next step will be to create more covert means of getting materials into Ukraine. This may rely on tunnels, light aircraft, disguised vehicles, or even submarines. (There is really no shortage of techniques for smuggling.)
For another mathematical take on this crisis, a 2015 analysis of the Nash equilibrium of the Ukraine situation examined eight scenarios. We are in the 8th in which Russia demands control of Ukraine. Their research suggested that the current balance in which the West matches a Russian invasion with military assistance to Ukraine was near equilibrium. That means that we could be here for a while, but the devil is in the details and the equilibrium is meta-stable at best. When the equilibrium collapses, we want it to be in our favor.
Right now, at this moment, Russia’s first move has completed. There is no turning back. Ukraine and the West’s response is currently incomplete and needs to escalate to a point where the Russian invasion collapses. We are running out of time for that to happen.
Quackenbush, Stephen L., and Frank C. Zagare. “Game theory: Modeling interstate conflict.” Making sense of IR theory (2006): 98–114.
Ericson, Richard E., and Lester A. Zeager. “Ukraine crisis 2014: A study of Russian-Western strategic interaction.” Peace economics, peace science and public policy 21.2 (2015): 153–190.