This is cool. I was actually asked for the optimal strategy for player 1 in this game as an interview question for a quant research role; I didn't know anything about game/decision theory, other than the basics, and so I only got as far as determining what to do with an Ace and a King, and then figuring out that you should bluff with a certain frequency if you hold the ace.<p>Anyhow, I came up with another question while considering this problem. Player 1 should only bluff at the optimal rate if player 2 punishes them for deviating from this rate by changing the frequency that they call when holding the king. Otherwise, Player 1 gains more on average by bluffing with the queen more frequently. So, how should Player 2 figure out if Player 1 is actually bluffing at the optimal rate (and not higher)? It could happen, with exponentially small probability, that Player 1 decides whether to bluff randomly, and bluffs with the first 10,000 queens before ever folding with a queen. How does Player 2 distinguish this from Player 1 only ever bluffing with the queen?<p>But perhaps this is something that the Nash Theorem deals with? I don't know. Any comments from people who know more than me on this are welcome!
For the uninitiated like me it's confusing at first the following sentence when deriving the Nash equilibrum:<p>"If first player has a K, because of 1. and 2., he would only lose if he bets. So first player should pass if he has a K."<p>It actually means that the first player always loses on average if he bets (expected utility -1/2 if not mistaken). If he folds however the expected utility is clearly zero since half of the time wins and half of the time loses.<p>I have not reached yet the description of the algorithm.<p>I actually find somewhat paradoxical that the first player should fold with a K and bluff with a Q! I remember that I loved little results like this when years ago I first followed the game theory course in Coursera. It was a little dry but I completed it nevertheless since I found it quite fascinating. I do wonder what possible applications are for my working field, machine learning, to apply these kind of algorithms. I would love to have an excuse to devote some time to it.