What game theory can and cannot do

Game theory offers a formal language to study strategic interaction. Is it possible to use game theory to predict the outcome of every possible strategic interaction in the real world? Of course not. There are, at least, two problems. One is that game theory when applied to some real-life strategic interaction, needs to be empirically well informed. The game theorist needs to know who the relevant decision makers (the players) are, what they can do (their strategies), and how they care about the various outcomes (their payoff function). The game theorist also needs to know what the relevant decision makers know about each other, what they know about each other’s motives, and what they know about what others know (the players’ information). Only if the game theorist knows all this, or has a good first-order approximation of all this, can they begin to try to come up with a prediction for the strategic interaction at hand.

For instance, it may be possible to use game theory to predict the outcome of the war in Ukraine for a very well-informed game theorist, but it is certainly not possible for me, as I wouldn’t even know where to begin modeling this conflict as an empirically reasonable game. If you would ask me what I believe the outcome of this conflict would be, I would first be asking you questions. And as you would probably not be able to answer most of them, I would not be able to give you a prediction. For a good game theoretic prediction, you would, therefore, need a game theorist teaming up with someone who has the appropriate empirical knowledge. That is why I, in this blog, tend to write about real-life strategic interaction, typically of a smallish nature, that I have personal experience with, so that I can be reasonably confident that I will make the appropriate assumptions when I build my game theoretic model. After all, I am no expert on any particular real-life strategic interaction, I am an expert (if at all) on how to formally model strategic interaction and how to mathematically “solve” these models.

But there is also a second problem. Suppose the game theorist does have a good idea of what the right game is for the real-life situation at hand. The game theorist may still struggle to come up with good predictions. Consider tic-tac-toe. You know, the game with crosses and circles. There is a 3 by 3 grid and players alternate one player placing crosses and the other circles in this grid. The first of the two players to have managed to place three of their symbols in a row, horizontally, vertically, or diagonally, wins (and the other player loses). If neither player can do so, it is a draw. Suppose both players want to win. With this knowledge, the game theorist has, one would think, all they need to be able to write down the appropriate game. Ok, one could argue, and in that case what I will write in a moment is just a version of the first problem, that the game theorist still doesn’t necessarily know how much the players understand the game. In fact, this is the problem that I wanted to come to. A game theorist would be able to come up with predictions for this game. This is a relatively simple game, with the feature that both players could, in principle, at least guarantee themselves a draw. So, Nash equilibrium, for instance, would predict that every such game ends in a draw.

I want to argue that, first, this is, of course, empirically wrong, yet, second, this is still very useful knowledge and helps us explain something about the world.

Ok, first, of course not every real-life tic-tac-toe game ends in a draw. I have often seen kids play this game and observed that sometimes one child and sometimes another wins this game. I have also involuntarily watched a grown man one row in front of me on the airplane repeatedly losing tic-tac-toe against the computer in the easy setting of the tic-tac-toe program on the back of the seat in front of him.

As a game theorist, you would really need to know a lot more than the players, strategies, and payoff functions, to be able to predict all these outcomes correctly every single time. You would probably need to know, what Jeeves (from the P.G. Wodehouse books – if you don’t know about this you are missing out), would call the “psychology of the individual.” I like this term because it stresses the term “individual.” It would not suffice to know something general about psychology, you need to know the individual, as obviously not everyone plays the game in the same way. In fact, you need to know even more, you need to know how the players feel at the moment and how motivated they are and what they know about and what they have learned about the game so far, and probably much more, to have any hope of coming up with a good prediction of their behavior every single time.

I believe that there are cases of real-life strategic interaction, such as some random kids playing tic-tac-toe, that are essentially unpredictable. This does not discourage me from using game theory. Why not? I still think the Nash equilibrium prediction (other ideas based on the elimination of dominated strategies would also come to the same prediction) is useful. It explains why we do not have high-prize competitive tic-tac-toe tournaments. If we had such tournaments, the incentives to win in these, given the high prizes, would be very high. People would have a strong incentive to learn to play this game well. Such learning will eventually, at least in simple games such as tic-tac-toe, lead to Nash equilibrium behavior. As this behavior is finally always the same it would be very boring. Nobody would come and watch a tic-tac-toe tournament where every game ends in a draw and the winner has to be decided by coin flip every single time.

While the Nash equilibrium prediction would, thus, be empirically wrong in many real-life settings and probably also in many potential lab experiments, it is still useful to understand something about our world.

2 comments

  1. A few people have correctly pointed out to me that there is actually a bit of strategy in the high jump and pole vaulting: the decision as to which height you are trying to beat. This is probably a decision you will make dependant on what you think others will do. The more gradual you increase the height the more you tire yourself out. The higher you make it, the more you risk not beating it. If you think all others cannot jump that high, you might choose a low height at least at first. If you think others can jump quite high, you might want to go for a higher height even to begin with.

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