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.