Introduction to Game Theory

In this article we are going to talk about game theory.? A game at its most general is any process you go through to achieve some goal.? We usually talk about mathematical games which have defined rules.? Game theory is important in economics, mathematics, politics, and much more.? This article’s purpose is to give an introduction to what game theory is to help you appreciate some of the ways of thinking behind it.

Game theory provides a way for rational players to win a game.? A rational player is a player who always makes the best possible decision available to them.? Of course, in real life players aren’t purely rational.? For example, if you are playing a casual game with a friend you might let them win to be nice, or you might do an unconventional move to make it more interesting.? If though, you were playing for millions of dollars, you had years to make decisions and you had a team of advisors coaching you, then you would be much more likely to make the best, most rational decisions.? In general, players are more likely to behave rationally when the stakes are higher, when the games last longer, and when more people are making the decision.? For example, a country deciding to make a trade deal has all three, so it will most likely behave rationally.? Two traits that are common in rational behavior, but seem very unnatural are lack of selflessness, and lack of revenge.? This rational absence of a desire for revenge is very strange to people.? For example, sometimes when playing an AI in a game the AI will do something in the game that hurts you and then, immediately after, offer to trade which seems sort of surprising.? This is surprising because we don’t think like a computes, and the computer is not thinking of getting revenge, but only of maximizing their personal gain.? This can explain a phenomenon that can seem strange when learning about history.? After being at war, two countries sometimes will very quickly start trading.? This is because on an international level, the benefits of trade are very strong, and the countries will often act rationally and put aside bad feelings for the very high reward.? Rational beings such as computers can seem very unhuman, and some of their behaviors such as the lack of selflessness in places like science fiction can be thought of as evil, but most countries would prefer dealing with other countries that will behave rationally since it is much more stable and safe, to interact with an entity that will act predictably.

Let’s look at a couple of different types of games.? Any one player game is essentially a puzzle.? For example, solitaire involves one player trying to find a solution.? A two player game where both players are against each other like chess is often called a zero sum game because only one player can win.? It is also called a combinatorial game because the difficulty arises in the large number of possibilities.? In a game where there are only two teams the dynamic is essentially the same except each team has multiple people thinking through the scenario.? In a multiplayer game with more than one player/team things get really interesting and you have some strange and seemingly paradoxical behavior.? You can also have randomness in some games which is called imperfect information and mathematically can be thought of as introducing another player “nature”.??

One of the simplest games is called the prisoner's dilemma.? This example is extremely important in many fields.? The basic prisoners dilemma involves two prisoners.? Two prisoners are being interrogated, and are not allowed to communicate with each other.? They both have information that would provide serious evidence against the other.? If they both give up this information they will get a lighter sentence for giving up information.? If neither of them give up the information then they will both have to stay and be interrogated for only one more month, but if one of them gives information on the other without the other giving any then the first will go free, and the other will be charged without the lighter sentence.? In other words if one gives information on his own they will get the best outcome and the other will get the worst outcome, if they both stay quiet they will both get the second best outcome, and if the both tell, they will both get the second to worst outcome? Clearly, the best course of action is for both prisoners is to stay quiet.? The paradox is that from the perspective of an individual prisoner, it makes the most sense whichever the other prisoner chooses, to give up information.? This game is one of the most studied in game theory.

Another interesting problem to discuss is if there is a best strategy for winning games in general.? There are a couple strategies one can use, but one strategy that has the most success and also agrees with many peoples personal morals is the fair response strategy.? This is sometimes called tit for tat, and basically says that you will be friendly by default, but if somebody attacks you, you will attack them back.? This strategy can prove successful when the prisoner’s dilemma is run multiple times.? It also can be applied to other games and even situations in real life.? When employed, this strategy not only tends to be successful, but also agrees with many people's personal morals and beliefs. ? For example many people agreed it was necessary to fight the axis powers to save lives in the long run.? In situations though where it is possible they will strive for peace.? This is found in many countrie’s philosophies regarding international diplomacy and is the basis for ideas such as the Powell Doctrine. ? This strategy also is applied in situations such as when companies negotiate for millions of dollars, which is much less serious than international war and peace, but much higher stakes than most people have in day to day life.? There are other strategies as well that sometimes show success, but some of them require communication which is not allowed in the prisoner's dilemma problem.

One of the most important real life examples of this playing out was during the cold war.? In this case both the U.S. and the U.S.S.R had thermonuclear weapons and launching them would mean the end of life as we know it.? The stakes were as high as most likely they possibly could even be, both countries had years to ponder the situation, and many people thinking about, advising, and planning the best choices, so they most certainly were playing rationally.? The situation was basically the same as the prisoner's dilemma.? One question is why then the two countries didn’t take the most likely route of Mutually Assured Destruction.? The question of why they didn’t has been and is currently relevant.? Understanding it might be a key to solving problems today and making a better future.