Problems with Applications of Basic Game Theory
Game theory is studied to try to accurately predict the actions of players (specifically two players in the games we have been studying) given a game model, but often game theory runs into problems in real-world applications. John Nash contributed the famous “Nash Equilibrium” to game theory to explain which choices players will make and why. Basically, a Nash Equilibrium is reached when no player can change their strategy and increase their resulting payoffs. In the film “A Beautiful Mind”, which is based on Nash’s life, a scene attempts to capture the ideas of game theory with a scene about Nash and a few of his friends in a bar attempting to figure out the best way to approach a beautiful blonde:
While each of Nash’s friends wants to approach the blonde themselves because “in competition, individual ambition serves the common good”, Nash disagrees. He says that the way that each of them get the optimal payoff is by ignoring the blonde so as to be sure that each of them can at least get one of her brunette friends. While this is an interesting approach to the game, it is not using Nash Equilibrium. We can analyze this game as a simple two player game that we have been using in class. Let’s call the two players Nash and his friend. Now, if Nash and his friend both go after the blonde first, they will “block each other” (as Nash puts it), and neither will get her. Nor will either be able to get one of the brunettes because nobody wants to be a second choice. Let’s say that in this case, they both receive a payoff of 0. If one of them goes after the blonde, and the other goes after one of the brunettes, the one will get the blonde and be ecstatic, while the other will get a brunette and be satisfied, but not ecstatic. Let’s say the one that gets the blonde gets a payoff of 10, and the one that gets the brunette gets a payoff of 4 (because he is slightly mad that his friend got the blonde). Finally, if they both go for a brunette, they will both be satisfied, but not ecstatic. Let’s say that this results in payoffs of 5 for both Nash and his friend. The resulting game looks like this:
One can see that both Nash and his friend choosing to go for a brunette cannot be a Nash Equilibrium, because if either player knew that the other player were going to go for a brunette, the player could increase his payoff by just going for the blonde. In fact, there are two pure Nash Equilibrium in this game, where Nash goes for the blonde and his friend goes for a brunette, and where Nash goes for a brunette and his friend goes for the blonde. So, this scene does not use Nash Equilibrium in analyzing the game at hand but rather brings up one of the possible flaws of using game theory in real applications: if the players have time to communicate with each other (and are friends), they can try to maximize the payoff for each player, rather than go for what is individually best for them.
But there are other situations where game theory in the real world fails as well. For example, the New York Times had their readers participate in an interesting game, which can be seen here. In case you can’t get access to the article (they may force you to log in), I will describe the game briefly. Each reader is asked to pick a number between 0 and 100 that represents their guess as to what two-thirds of the average guess will be (rounded to the nearest whole number, presumably). To make this clearer, let’s say that there are three total readers who participate and they pick 2, 3, and 4. The average of the three guesses is 3, and two-thirds of that is 2. So the player who picked 2 would win.
So let’s get into a bit of game theory analysis of this game. It is not quite as simple as the two-player games that we have been studying in class since there are actually approximately 59,878 other players, each trying to choose the optimal strategy. Your first thought may be “Well, if everyone else picks at random, then the average will be 50, so I should just pick two-thirds of 50 (which is 33 rounded down), and I will have the best chance of winning”, but this logic is faulty because it doesn’t assume that the other players are trying to win as well. It assumes that the other players are just choosing randomly when, in fact, they are also thinking what the optimal strategy would be. So, if everyone thought the same thing as you, then they will all try to pick 33. So, if you pick two-thirds of 33 (which is 22), then would this be the optimal strategy? No! Again, you have to take into account that the other players are thinking like you are. They would also see this pattern and think to pick 22 as well. But then all players will think about picking two-thirds of 22 (which is 15 rounded up), and then two-thirds of 15 (which is 10), and so on. Eventually, using this thinking, every player should see that the optimal strategy is actually to choose 0, and there is no way to go any lower. We can actually see that every player picking 0 is a Nash Equilibrium. Assuming winning is a payoff of 1 and everything else is a payoff of 0, let’s assume currently every reader uses the strategy of choosing 0. Since 0 is two-thirds of 0, every player wins and gets their optimal payoff. In addition, if any single player changes their choice to a different number, their choice will no longer be two-thirds of the average, and they will receive a lower payoff. So, in this state, no player has any incentive to switch.
So, I played the game and chose 0, expecting to be congratulated on my victory. To my dismay, however, the New York Times Article informed me that while I did choose the optimal strategy from a game theory standpoint, I did not win. In fact, the average number chosen was 28, meaning that the winning number chosen was 19. Where did our game theory analysis go wrong? When we used game theory to analyze this game, we assumed that all other players were equally adept in analyzing the game to reach the optimal play. Unfortunately, in the real world not everyone has the experience in game theory that Networks teaches you. Other people just can’t think that far in advance. As a result, some people only went one or two steps into the game theory analysis (where picking 33 or 22 was the optimal strategy) and stopped there, thinking they had outsmarted the other players. Enough people thought this way to raise the average and cause those who chose the theoretically sound strategy of choosing 0 to lose the game. This is just another example of why its always good to be wary when using game theory in practice.
While I’ve pointed out a couple of examples where using game theory to analyze games and choose an optimal strategy isn’t always good at predicting what the players actually choose, this is a far cry from saying that game theory can’t be useful in the real world. There are some fantastic and brilliant uses of game theory in economics and business, for example. And then there’s this awesome analysis from a previous year of how game theory was used in a popular game show. The takeaway from this post should just to be cautious when utilizing game theory in real applications, because there ARE things that can go wrong.
Sources:
http://www.nytimes.com/interactive/2015/08/13/upshot/are-you-smarter-than-other-new-york-times-readers.html
http://www.nytimes.com/2015/05/25/science/explaining-a-cornerstone-of-game-theory-john-nashs-equilibrium.html
https://blogs.cornell.edu/info2040/2012/09/21/split-or-steal-an-analysis-using-game-theory/