Poker, Backgammon, and gamblers
Poker used to be about gambling, and the gambling lifestyle was not the same as today's tournament lifestyle. Today poker players will travel long distances for a shot at getting on TV. It used to be that poker players would travel long distances for a shot at playing in a good game, it didn't even have to be a poker game.
Back in the 1970's Norm Zadah published an article in the academic journal Operations ResearchNorman Zadeh (1977) Computation of optimal poker strategies, Operations Research 25, 541-561. giving a game theory solution to the first round of betting in draw poker. I was a graduate student (Mark Goodfriend) at the time, at Northwestern. Myself and another grad student who was also interested in gambling noticed an error in the article. Not a mathematical error, but an error in modeling.
He assumed that an optimal strategy would be to open with all hands H > H* where the relation > is based on a poker hand ranking. It turns out that's not true. Poker value before the draw isn't determined by poker hand ranking but by a probability distribution of possible final hand rankings. Sometimes a really strong draw is a better hand than a mediocre made hand.
It also turns out that in the case of draw poker, if the ante is very small relative to the bet size then the model works if you make the assumption he made, that with a very small antes it's never worth it to play a draw if only first round betting is considered (if you ignore implied odds).
Mark and I wrote up a comment on this, showing that his model doesn't find optimal first round strategies because he makes a faulty assumption about the mathematical form of the optimal strategy. Our main point was that poker math should be as much about poker as it is about math, maybe even more so. Back then operations research models of gambling got more attention within the operations research field than they do today, and there were a couple of sessions devoted exclusively to gambling.
Because of our presentation Mark and I ended up having some conversations with Tom Cover. Most of y'all probably never heard of him, but he teaches at Stanford and has done a bunch of research over the years in gambling related probability models.
I didn't actually play poker much back then, and the chance to talk to a well known math dweeb who did made an impression on me. During one of those conversations Tom mentioned that he'd pretty much given up poker for Backgammon. In the 60's and early 70's Tom had played a lot of poker, even flying to NYC once a month just to sit in on a private game hosted by Oswald Jacoby, who was pretty well known as a poker contributor (I don't mean that in a flattering way). Then when Tom discovered a Backgammon contributor across the bay at Berkley who not only never won he never gave up, he gave up on those transcontinental flights.
Gambling 30 years ago wasn't about winning tournaments and getting on TV, it was about finding a good game. These days getting on TV can pay off better than any gambling game is going to.