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Game Theory

by Ryan Wang

Basics of Game Theory

Suppose that the police suspect two people of wanting robbing the bank. Let us call the two suspects A and B. Let us assume that A and B are in fact not partners in crime and feel no attachment towards one another. Each are only concerned about their own self-interests. The police want to convict both of attempted robbery, but they only have enough evidence to convict the suspects of trespassing. The police devise a clever scheme to try to convict both. The two suspects are taken for questioning separately. If both suspects stay silent, both only serve a 1 year sentence for trespassing. If both rat each other out and admit guilt, both serve for 2 years. Finally, if one admits guilt but the other stays silent, the one who admits guilt leaves free as a reward for cooperating while the one who stays silent is punished by serving 3 years. 

Now try to put yourself inside the shoes of suspect A. If you cooperate, B could either cooperate or betray, leading to sentences of 1 or 3 years respectively. If you betray, B could again either cooperate or betray, leading to sentences of 0 or 2 years respectively. Since you know nothing about suspect B’s motives, you would assume that his chances of betraying or cooperating are equal. Thus, the expected value of cooperating is 2 years, while the expected value of betraying is 1 year. So the rational decision here is to betray. Suspect B is most likely thinking the same thing, so the expected result is a double betrayal. This scenario is known as the prisoner’s dilemma, and this result is called the Nash Equilibrium, named after the legendary Princeton mathematician John Nash.


To generalize things, we can use a payout table and measure results in units of utility, which is simply a measure of value. If you can imagine the numbers now as being the amount of money you gain instead of the number of years, you’ll be serving in prison. Again, the basic principle stands. The expected utility of defecting is less than at of cooperating.

Variations of Game Theory

So the classic example of the prisoner’s dilemma is cool, but it’s rather bland. To find a more interesting variation, we can look towards an infinite game. Let us use the same payout table as above. Now, let the numbers stand for thousands of dollars. Assume that this game is now played once every month, with the same stranger, let’s call him X, for the rest of your life. This game has now been transformed into an infinite game, and cooperation is now allowed. So, what would the strategy be here? Logically, you would want to cooperate with X. So long as you continue to cooperate with X, you both would reach the optimal result or getting 3 dollars every day. X is thinking this as well, but he goes a step further. X agrees to cooperate, but is secretly planning to betray. The next day, X betrays and gets 5 while you leave with nothing. What would you do? The logical sequence here is to betray X for the rest of the game as retaliation, or until X apologies and you go back to the nash equilibrium of dual cooperation. So we can now settle on an optimal strategy for this game. If X betrays at any point, betray forever afterwards. Otherwise, cooperate. This strategy is known as the Grim Trigger, and is the best way to ensure cooperation between you and X. 


But, there’s more. Money today is worth more than money tomorrow due to factors such as inflation, expected inflation over the long term, and our human impulse for instant gratification. To factor this in, we have to consider another variable, lowercase delta. Delta is the value that we place on money in the future. Delta fluctuates between 0 and 1. The closer to 0, the less value is put on money tomorrow, and vice versa. For example, if the value of money today is twice as much as value of money tomorrow for you (maybe you’re living in Venezuela in 2016), then your delta value would be ½, and it would make sense to betray at some point (the calculations for figuring out exactly where that point is is extremely messy and besides the point for this article). Or, if you believe that an astroid is 100% going to wipe out all humanity on earth as we know it tomorrow. Then it would also make sense to betray today, since your delta value is 0 (money tomorrow is worth 0% of money today). 

This is just one variation of the prisoner’s dilemma, there are an infinite amount of other possible senarios. The finale of the British TV show Golden Balls is a perfect example of a one-off prisoner’s dilemma with cooperation. OPEC is a great example of an infinite prisoner’s dilemma in real life, and the list goes on and on. Game theory is just a way of thinking, and once the basics are understood, it can be applied to any situation. 

What Game Theory Can Teach Us About War

Game theory is extremely useful when it comes to international relations. To take a recent example, game theory can help us understand why Russia decided to invade Ukraine, and help us determine if China will invade Taiwan. The decision tree above is a map of 4 possible outcomes. The numbers represent utility values. These value can and should be adjusted, and in reality are probably much different than in the illustration, but this is an oversimplified scenario. Russia can shown aggression, and the west can respond either by backing down or showing aggression of their own. If the west shows aggression, Russia can then respond by backing down or by showing even more aggression, which would likely lead to war. From the perspective of the West, if appears that against the possibility of Russian aggression, the optimal strategy would be to back down, as the expected utility would be higher. 


Or is it? Imagine that the West makes some sort of appeasement to Russia in order to prevent war. That would simply empower Russia to make more demands by threatening war. This is called an imbalanced game. When the West backs down to Russia’s aggression, it is playing a finite game, or a one-off, something that will only happen once. The purpose of a finite game is to win. Russia is playing an infinite game. The purpose of an infinite game is to continue playing. A finite game works towards a “something”, and infinite game works towards a “not something”. The imbalance occurs because both sides are playing by a different set of rules. Perhaps a better example of finite vs infinite is not the West vs Russia, but rather Russia vs Ukraine itself. In Ukraine, Russia is fighting a finite war. Ukraine is fighting an infinite one. Ukraine is fighting for their lives, and would fight forever if necessary. 


So it appears that it would be in the West’s best interest to deter Russia from showing aggression in the first place. It could do this by increasing the utility of Russia taking the passive approach, perhaps making some economic concessions in order to achieve a promise of non-aggression against Ukraine. 


Of course, keep in mind that this is an extremely dumbed down and wildly inaccurate verison of the problem. In reality, there are often hundreds, if not thousands of factors and variables that affect Russia’s ultimate decision. If it were this simple, the war would either be over already or the world would be destroyed. Rather, this aims to serve as a demonstration of the wide scope of game theory, and how it can be applied to so much in this world. Game theory teaches us an important lesson about humanity: The best outcome is often unreachable if people don’t cooperate. 

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