Polimath

Today I got up, finishing a decision I started last night about how much to sleep before today.  I will choose my attire to fit the weather and strike the right tone in the classes I will teach. I will go to work and spend the day at work making optimal decisions about how to allocate my time and effort considering my immediate goals, teaching effectively and  preparing for an experiment, and longer term goals like getting along with my peers and building my tenure packet.  I will come home along a route that balances safety, convenience, fuel economy, and curiosity.  I will talk with my wife, play with my daughter, read to my son, all with an eye toward building both their individual lives and my relationships with them.  I may make a few allocation decisions about improving our house or saving for retirement.  I will decide whether to work out tomorrow morning, then  begin the decision about how much to sleep before tomorrow.

I don’t know if I’ll make the right decisions. I cannot know, even afterward, whether I did.  It turns out that it would take a very long time to check.  Constantinos Daskalakis has shown that it would take a computer as complex as the entire universe longer than the lifetime of the universe to solve for Nash equilibria of many common games, like how to invest for retirement. There all I am trying to do is optimize money.  When I read to my son I am doing something harder:  I am trying to influence his education, his confidence, his happiness, and my relationship with him, all without being so bored that I fall asleep.  Most of the decisions we make all day are too hard to solve.

Of course, I can simplify each of these decisions to make them tractable. Any simplifying assumptions I make are wrong, and therefore I am answering the wrong question.  I can make specific, certainly wrong assumptions, I can approximate a bunch of decision characteristics by combining them into one or more stochastic elements in a simpler game.  Either way, I am limiting my rationality, or really acknowledging that my rationality is bounded. When I make one of my decisions I am not answering the question before me but rather a different, simpler, but wrong question.

You might object.  “Once we give up rationality, we give up prediction.  There are no limits to how we can be irrational.”  How does the joke go?  “The difference between genius and stupidity is that genius has its limits.”  There is no limit on the answers we can get if we allow our models of human behavior to admit irrational behavior.  If you allow for heuristics, anything could happen, so we know nothing!

True, anything could happen, but we can still note that some things happen more than other things.  Data can guide us.  We can examine which ways of simplifying are likely and which are unlikely.  Assigning likelihood statements to different behaviors is the statistician’s strategy.  Some steps along this route are taken by quantal response theory. However, there is still much work to be done.

Acknowledging that when we model we always lose essential details, George Box said “All models are false but some models are useful.”  At some point we all have to decide to answer the wrong question, whether in the economy or just in deciding when to go to bed.  Otherwise we wouldn’t be able to get up in the morning.

A recent exchange at the excellent Cheap Talk focused on how the uselessness of the United States’ recent promise not to nuke other states who comply with the Nuclear Non-Proliferation Treaty (NPT).  Sandeep Baliga writes

This is an attempt to use a carrot and stick strategy to incentivize countries not to pursue nuclear weapons.  But is it any different from the old strategy of “ambiguity” where all options are left on the table and nothing is clarified?  Elementary game theory suggests the answer is “No”.

…

We are left with the conclusion that a game theoretic analysis of the Nuclear Posture Review says it seems little different from the old policy of ambiguity.

Baliga raises a simple, clear, and important question:  When we say we are not going to nuke you, what keeps us from doing it anyway?  “[T]he words of the [new policy] are just that – words.”  Baliga seems to imply that there is no reason to make such a promise.

Suppose that the assumptions of the “elementary game theory” employed are all correct.  They are standard assumptions and they have been employed correctly; therefore, our administration has wasted its time.

Let’s look at it from the other direction.  Whether or not the folks who make our policies have studied game theory, are bright enough to look back and notice which things tend to work and which things tend not to. We may have only 6+ decades of experience with nuclear politics, but we have many, many more years of experience with the role of cheap talk in diplomacy.  Apparently, our policymakers seem to think that cheap talk can work, at least enough to be worth the effort of making a statement.  Therefore, our administration did not waste its time.

Barring a logical error, one of three things must be true.

1. The assumptions of the game theoretic analysis are correct (or close enough to being so,) so the administration is wrong and can learn from the theory.
2. The assumptions of the game theoretic analysis are wrong (or not close enough,) the administration is correct, and theorists need to update their model.
3. The assumptions of the model are wrong, but the conclusion is correct and the administration is still wrong.

Setting aside case (3), on to the central question:  Is the theorist being positive, describing the world, or normative, telling us how it should be?  (Both are important, useful roles, and formal theorists can, and I believe should, speak up on normative issues whenever science can help anchor moral choices.)  In case (1) the theorist is describing the world, providing information.  In case (2) the theorist is incorrect, but still earnestly trying to describe the world.  In either case, he is being positive.  However, by asserting or implying that case (2) is not under consideration, he is taking a normative position:  not about the conclusion (that in contexts like this cheap talk is useless) but about the assumptions.  He is saying, for example, that it should be the case that we can ignore audience costs.  Unfortunately, this assumption and other similar ones turn out not to be tenable even in theory.  More importantly, I agree with the administration that promises like this can have a real, positive effect; given this, the assumptions of the model must not be correct.

In my view, it comes down to this:  When a formal theorist derives a behavioral prediction that does not coincide with what people actually do, maybe the model is correct and we can learn from the model, or maybe the model is wrong and the modeler should learn from the world.  Perhaps one’s goal (understanding policy or shaping it) should drive that decision.

If you are confident that your theory explains the relevant situation very well, go ahead and use it to make recommendations.  Just remember Cromwell’s Rule:

I beseech you, in the bowels of Christ, think it possible that you may be mistaken.