The 4th Principle of Economics: People are stupid -- Cointossing Edition

Of course, Pr. Mankiw may claim that it's really "Rational People think at the Margins", but since most of us don't think very rationally, the principle really means that people are stupid (which sounds funnier than 'people are irrational').  I'm going to make this a recurring series.

People who don't play poker often ask me how I make so much money at poker (note: I'm not bragging here; many people think the $10k or so per year I make at poker is "a lot of money", but most poker pros make more than this in a month).  I don't mean to simplify the game too much, but the short answer is that to beat poker consistently you have to a) have an above-average intelligence (I'm not talking multiple standard deviations from the norm here), b) be willing to put in some time studying the game, and c) have a rational understanding of probability (you might think that c is included in a but I've met some smart people that don't understand probability at all).

The next logical point that comes up is that there is more and more information about poker all the times.  The 2+2 Forums will help you analyse practically any hand.  Coaching sites like LeggoPoker and DeucesCracked offer instructional videos and you can get a coach to teach you to play.  Poker books are nearly ubiquitous.  So, since the bad players keep going broke, and there are more and more good players (or the bad players are getting better), the games must be getting REALLY hard to beat, right?

In a word, no.  Thanks to the 4th Principle of Economics:  People are Stupid.

Take coin-tossing, for example.  Everyone knows the chance of winning a coinflip is 50% right?  I mean, there are no debates about this.  There are people in the world who are wildly misinformed about the probabilities of all kinds of events (winning the lottery, getting eaten by a shark, dying in an airplane crash, getting lung cancer from smoking, etc, etc ad infinitum ad nauseum), but winning a coin flip isn't one of them.  Nevertheless, you can get people to give you 53/47 odds on a coinflip (meaning I lose, I give you $.47, I win, you give me $.53).

The Sports Economist analyzes some trends on contracts for the coinflip at the start of the superbowl:

As anyone who has taken a statistics class knows, the expected value of the toss of a fair coin is 50/50 - a 50% chance of heads and a 50% chance of tails. So the expected value of a $10 contract on "heads on the Super Bowl coin toss" is 0.50*$10+0.50*0=$5, or 50 when expressed in the TradeSports price metric. TradeSports also makes price-volume data available for all contracts. There were 86 transactions, involving 275 contracts, in this particular market.

....


This distribution is interesting, to say the least. The mean price was 50.33 and the median 50.35. Why would anyone pay more than 50 for a contract with an expected value of 50? Of the 85 transactions, 52 of them took place at a price greater than 50. Those two trades at 52, which are the equivalent of betting $5.20 for a 50% chance of winning $10, both came minutes before the coin toss.

Yes, folks, that's right, half of the participants bought a 50% chance of winning $10 for $5.35 or more.  HALF.

Games will always be good.  They are good because people are pretty bad at estimating all kinds of probability -- I know that Ace-king of hearts is a favorite over pocket jacks on a TQ5 board with 2 hearts and 2 cards to come, but most players don't.  And I know that taking 2-to-1 odds is a really bad, bad, bad bet when I am a 6-to-1 dog, but most players don't.

But apparently, even if both my and my opponents both know the odds perfectly, at least half of the time I can get my opponent to take my bad bet anyway.  Good times.