a casino is a funny place

a casino is a strange place

worries about the foundations of probability

Continuing on from a previous week’s post, and from a conversation I had this week with a mate who had commented on it … We know, as a matter of historical fact, that the theory of probability was originally developed by analysing games of chance^[1]. I think this might be an original sin, because when you think about it, a casino is a very strange environment for thinking about chance and risk.

· You always know how much you have at stake

· You always know how much you can win

· There are regulations and systems to protect you from cheaters

· You are not allowed to cheat yourself

· The probability of winning, in the sense of the long term expectation, is either known or can be calculated

Not all of these things are always completely true of every game (if you bet on backgammon with a doubling cube, you might not know the stakes at every point, if you bet on shove ha’penny it’s not obvious how to calculate the odds). But they’re the paradigm which nearly everything in the world of gambling approaches. And absolutely none of them are regularly satisfied in real-world conditions of decision under encertainty.

As I said in the post before, optimisation rules deduced as part of this model shouldn’t be given a dignified name like “rationality”. And when human behaviour in embodied, contextual situations diverges from the rules of probability and expected utility, this shouldn’t be called “irrationality” without quite a lot of analysis and work aimed at showing that it’s not the probability/expected value model that is at fault.

Of course, the most obvious piece of evidence that I’m right is that everyone knows that “rationality” often gives absolutely stupid answers when you apply it to problems which involve trust and cooperation. Game theorists do lots of hard work in trying to reconcile the two, but the game is fundamentally a bogey; the point of the Prisoner’s Dilemma, the Tragedy of the Commons and all that is that rationality ain’t always very rational.

Could it have been any other way? Maybe. There is another historical line which could perhaps have also been used. The word “risk” originally comes from “rhizikon”, Byzantine Greek meaning “a sharp rock or reef”, later “any hazard to shipping”, later “whatever it is that they buy and sell in Lloyd’s Coffee House and the like”. The word “average” comes from an Arabic word “awariya” meaning “lost merchandise”; the “Law of General Average” is the principle that when stuff has to be chucked overboard in a storm the losses will be shared out fairly between the merchants so that the crew are can grab the heaviest objects as they see fit without worrying about who owns what. If mathematicians had first got into the area by thinking about maritime risk rather than games of chance, I wonder what might have developed differently?

^[1] Although I guess that this depends what you mean by “the theory of probability”; Kolmogorov wasn’t very interested in games of chance but measure theory is a whole other bee in a different bonnet for me.