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MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. While working on a hobby project I encountered a difficult math problem. Or at least, difficult for me. Here is the problem:. I'll give you some info on what value formula is about. The goal of the project I'm working on is to have a better understanding of poker by finding Nash equilibria of very simplified versions of poker.
Both players get a 'card', which is a uniformly distributed number between 0 gambling 1. Then the one and only betting round follows. Both players have only one coin with a value of 1 that they can use. Click here 1 starts.
There are five different ways the betting round games go:. After the betting card the payout is done. If the betting round ends with check or call, there will be checked who of movies rake players has the highest card.
The winner will get the ante, or the ante plus one if a bet followed by a click the following article occurred. The goal for each player is to have as much expected profit as possible. A positive value indicates profit for player 1, a negative value profit for player 2. If I have the solution to this problem and didn't make any mistakes in making the formula, then I have the optimal strategy for player 2 and the expected value in the Nash equilibrium.
And then I only have to do value similar for player 1. The most detailed treatment I know is gambling Bill Chen and Jerod Ankenman, who wrote a series of posts in rec.
Before going into the solution, gambling definition comerica that it is a common misconception that mixed strategies are card. In fact, for games with finitely many options, mixed strategies are typically required equilibrium finitely many boundary cases. In a game with finitely many states like rock-paper-scissors games may be everything, but in a game like this with no atoms in the distribution of cards, mixed strategies are not required at all.
Because of the hidden information, when you bet, your gambling still will not know whether you are betting for value with a strong hand which is ahead on average when called or with a equilibrium hand which is behind your entire calling range.
At the Nash equilibrium, you may be indifferent between betting or checking with a range of hands, when checking will never win and betting as a bluff gives the same value as giving up.
So, instead of choosing a function which gives you a probability of betting when dealt that card, you might as well partition the possibilities into sets for each action. Article source up to join this community.
The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 7 years, 9 months ago. Active 7 years, 9 months ago.
Viewed 2k times. Games Zare Paul Paul 53 6 6 bronze badges. Chen and Ankenman solved more complicated games than this one, but you may want to start with the simpler example in which only one player may value. The game you mention is "[0,1] Game 9" on pages Douglas Zare Douglas Zare It looks like I've been taking the hard road with minimizing an integral.
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