Tonight at the department party we were continuing the debate on the usefulness of game theory. Part of the argument was about whether the math has any utility, or if you could get just as much from verbal arguments about strategy as in Schelling’s work. Rakesh had some good positive examples in pricing and auctions which perhaps he’ll write up here if he has time. Here’s one that came to my mind just now. You may not consider poker “real world” but bear with me; I think it illustrates a general point.

Consider the simplified model of poker in von Neumann-Morgenstern. In one of the versions, there is a result that the unique equilibrium strategy is to bet with the best 30% and worst 10% of hands. Now, nothing nearly so simple is optimal in real poker, and furthermore all poker players, even the least mathematical, know that they should bluff sometimes. So what was gained by this exercise? Well, we have a qualitative recommendation to **bluff only with your very worst hands**. This is far from obvious, and I never thought of it before doing the mathematical exercise. True, I can translate it into a verbal argument, as follows: bad hands and medium hands become equal if you bet with them, since they both lose whenever (or almost whenever) you get called. But medium hands are significantly better than bad hands when you check, since they may win the pot in a showdown. So you bluff with bad hands, not of course because they are better to bluff with, but because they are worse to check with. Note that this logic only applies fully when you are last to speak in the last round of betting. In earlier rounds, “semi-bluffs” where you hope for a fold but have chances if called are a common part of good strategy, and are more common than pure bluffs.

Enough about poker; here is the broader point of the example. Solving the problem mathematically imposes a discipline on our reasoning process which forces us to discover an important qualitative insight we could easily have missed otherwise. True, I am sure some strong poker players came to this insight intuitively over the years without formal study of Bayesian games, but many players surely missed it. The examples Rakesh was discussing seem similar to me. Yes, once you hear certain insights described in words, you may decide we never needed the math. But this is much too facile, akin to thinking that every problem is easy once you’ve seen the answer. Any illuminating chain of reasoning can be missed as easily as found, and formal models can channel our reasoning in the right direction.

## 7 comments

October 17, 2010 at 10:30 am

michael websterI think you are right with this example. But did you mean to make this a general point: we should do the quantitative calculations in order to discover qualitative phenomena that we would have missed but for modelling?

October 17, 2010 at 1:20 pm

Jonathan WeinsteinYes, I was saying that this is a frequent benefit of modelling, and that we shouldn’t think any given model is useless just because it can’t be used in a precise quantitative way.

October 17, 2010 at 4:02 pm

michael websterI would go further an argue that a) since to do any serious calculations we need some form of cardinality measure for the preferences, but b) since it seems extremely unlikely that people satisfy VM or any of the variants, then c) we should be qualitative features.

Calculations are not solutions, or recommendations to action.

Here is an example of what I think is a very bad use of game theory.

http://www.psychologytoday.com/blog/brain-candy/201010/date-night-dilemma-solution-0

The calculations in the little unrealistic game are presented as solutions or advice for the strategic encounter.

When I used to teach game theory, I started off by saying:

Game theory assumes that labels don’t matter, but they do.

Game theory assumes that the payoffs are given, which they aren’t

And game theory assumes, that the players are superb calculators, which they aren’t.

But, if in this very simple world we uncover phenomena that likely exists when we add back the uncertainty of intentions, preferences, and calculations, which otherwise would remain hidden in this fog, then those are the phenomena worth studying.

October 18, 2010 at 11:37 am

EranThere are two separate questions here: the usefulness of math and usefulness of game theory.

Regarding math, for me it’s just a language to express insights and assumptions in an unambiguous way. Of course you can get qualitative insights from quantitative models even without precise calculations. This is so also in physics: you can get qualitative predictions about the impact of acceleration on a body in a train even without numerical values for precise predictions.

However, I don’t view your example, either in math or verbal form, as a satisfactory answer to Rubinstein’s usefulness challenge. At the risks of repeating myself and of saying things you agree with as if they were at odds with your argument, I will delve into the mud once more and then I promise to shut up for some time.

For the sake of our anonymous responders I should first emphasize that when I say that game theory is not useful for predictions and policy recommendation I don’t necessarily mean that it is useless to know game theory if policy recommendations is all you care about. Usefulness of studying or knowing a theory and usefulness of the theory itself are two different things, and I believe Jonathan’s example reflects this difference.

There are two problems with this example (and also with Eilons’ and to the extent that I understood them Rakesh’) as an answer to the applicability challenge:

First, the example doesn’t do justice to game theory. The theory has a lot to say about strategic decisions under uncertainty: the very definition of Bayesian games, the existence or nonexistence of equilibria and other solution concepts, the implications of higher order beliefs. The theory can be used to give interesting interpretations for conceptually complex ideas like probability and information. It can be used to explain social phenomena and to identify regularities in the the behavior of human beings and other living creatures. All this body of knowledge is irrelevant for your example. What is left is just the fact that working on Bayesian games, you are more aware of the components of strategic interactions (like the participants, their interests, information, possible actions, possible contingencies,) which are not the output but the starting point of game theory, and that framing a strategic situation using our models forces you to systematically pay attention to all of these components. The exact formulation of the Bayesian games model was also irrelevant here: you would probably get the same `disciplinary’ advantage if you used, say, Bruce de Mesquita’s model. From the perspective of game theory, the Bayesian game model might have other advantages, but that’s because imposing discipline is not the purpose of modeling. Similarly, you could dispense entirely of the other players, and analyze the situation a single agent’s optimization under uncertainty.

Second — knowing game theory turned up to be useful in this example, but that’s not what people refer to when they talk about applying other sciences (most notably physics but also a big chunk of social sciences and economics). In other sciences, by being useful one doesn’t mean `channel our reasoning’, `imposes a discipline’ or `point our attention to’, but that the output of some research is directly applicable (my stupid body in a train example is such an application). I think proponents of `application of game theory’ approach means this more conventional sense of `using’ and so it seems to me that even while the type of examples you provide ignore almost all of game theory, they still don’t save the applicability assertion.

October 18, 2010 at 3:37 pm

Jonathan WeinsteinHmmm, it seems as if we agree on most everything except what it means to apply a theory, or for a theory to be useful, or maybe even on what we mean by “game theory.” Probably the main lesson then is to use more precise language? If you would prefer to say “learn something from studying game theory” or “apply insights from game theory” in cases where I might go so far as to say “apply game theory,” you may well have a good point that this is more clear. I guess I figured that it goes without saying that you can’t apply game theory as you would apply physics, but if it doesn’t I’m all for more careful language.

Certainly, I could be much more refined and quantitative when applying physics than game theory. I suppose you argue it risks an undue presumption of parallelism to say “applying game theory”? You might be right.

If I understand right, you say that when I anticipate that I will fall forward when a train stops, I am applying physical theory, but when I use a game-theoretic analysis to get a vague idea of how to play poker, I am not applying game theory. These sound similar at first glance — I apply a quantitative theory in a qualitative way — but let me try to understand why you make a distinction.

In the physics case, the theory has both a key axiom about the physical world (objects in motion tend to stay in motion) and a logical consequence that not everyone might think of (the illusion of being thrust forward when the train stops.) In poker, I assume that each type of each player optimizes, and I get a logical consequence that not everyone might think of. How is this different? By the way, while I focused on just one player in the discussion, it was implicit that the other player has a cutoff strategy, which is actually an outcome of the analysis.

By the way, since the poker example dates to the birth of game theory, almost by definition you don’t need to know game theory to solve it. Nevertheless I do consider “equilibrium strategies in this simplified poker game are xyz” to be a theorem of game theory — one of the first.

October 18, 2010 at 3:47 pm

AnonymousThere are perhaps simpler tests for whether a theory is of any use.

(a) If you had two experts with differing `artistic’ viewpoints – one who wants to use the theory and one who wants to either use an alternate theory or an alternate way of solving problems; then is there some intrinsic sense in which the former would beat the latter, systematically? This is how most others, e.g., electrical engineers, measure the utility of a new conceptual framework.

(b) Is there some phenomenon regarding which one achieves a much better understanding, e.g., in terms of prediction of a hitherto unknown entity (e.g., Higgs boson) that is more or less unachievable without the theory?

The arguments put forth regarding utility so far seem too far from falsifiability to be actually useful.

October 18, 2010 at 4:14 pm

michael websterYou might find it useful to formulate your questions with respect to what is currently known about models in science:

http://plato.stanford.edu/entries/models-science/