I’m good at some things, and foreign policy isn’t one of them. I have no idea if Syria actually used chemical weapons, how the U.S. should respond to any potential attack or the probability of attacks spilling over into larger conflicts. I don’t understand Russia’s incentives or how they will react to any acts of aggression. However, I have created an excel book that calculates mixed strategy equilibria, and I will use this as a game theory example.

As a study, game theory uses mathematical models to predict the actions of rational decision-makers. The world is complex and noisy. If we want to know anything, we must make simplifying assumptions. People aren’t actually rational, but we can measure where they fail and model imperfect agents. A simpler first-pass is to assume they rationally maximize “utility” and assign utility to all possible outcomes.

As a simple model of the situation, let’s assume that the U.S. and Russia can either Waver or Stand Firm. Let’s assume they make this decision simultaneously and once. Their payoff is defined by the following table:

As an example on how to read this, if both countries waver, then the U.S. will receive -20 utility and Russia will receive 40 utility. If the U.S. stands firm and Russia wavers, then Russia will receive a payoff of 0 and the U.S. will receive a payoff of 30.

All of these values are editable in the excel file attached below. These are subjective estimates that I admittedly just made up. A bit of reasoning on the choices I made.

- I’ve assumed the U.S. wants to give Iran a strong example of what happens to countries that use WMD’s of any sort. This means the U.S. would rather stand firm than waver, assuming Russia would not escalate the conflict.
- I’ve read speculation claiming Russia has an economic interest in keeping Assad in power, as his government has made it more difficult for Qatar to sell natural gas to Europe.
- To make the solution more interesting, I made these payoffs asymmetrical.
- I’ve made the outcome of both countries going to war large and negative under the assumption that even a proxy war in Syria would be very bad for both sides.

The first thing my workbook does is check to see if any strategies weakly dominate the other. A strategy weakly dominates another strategy if a strategy is always as good or better than the alternatives regardless of how the opponent plays. In this game, each country would rather stand firm if the other wavers or waver if the other stands firm, so no strategies dominate.

Next, the workbook looks for Nash equilibria. A Nash equilibrium is a situation where neither player would unilaterally change his position. In this case, there are two Nash equilibria at the top right and bottom left. In these boxes, neither player can improve their outcome by changing their strategy. The top right equilibrium is better for the U.S. and the bottom left is better for Russia. But which equilibrium will we likely end up in?

In this case, probably neither. The next thing my workbook does is, if no strategies weakly dominate each other, then it will solve for the mixed Nash equilibrium. A mixed strategy is when a player chooses a probability distribution across all possible strategies. In the mixed Nash equilibrium, each player chooses a distribution of strategies that leaves the opponent indifferent between their strategies. Feel free to play with the payoffs and watch as the mixed solution changes.

In this case, the U.S. will waver from conflict roughly 96 percent of the time and Russia will waver roughly 91 percent of the time. In equilibrium, Russia and the U.S. will fight over Syria less than half a percent of the time.

The final sheet of the workbook analyzes the outcome for both players, including the expected value of the outcome. Based on this simplistic analysis, it looks like the U.S. will not bomb Syria and, even if it does, it is unlikely Russia will escalate the conflict. This is a good thing for the world’s total utility.

This analysis could be furthered by having the U.S. and Russia play several rounds where they choose to Waver or stand Firm. Any improved utility estimates are welcomed.

Here’s a link to the excel book that generated these reports. Of course, this workbook can be used in other contexts such as pricing decisions in a duopoly, figuring out how to make your roommate do the dishes, analyzing the read option, modeling evolutionary equilibria or solving your Game Theory homework.