Quote:
Originally Posted by David Sklansky
That seems right. Thank you. Could you now perhaps do the same calculation with one additional rule? There is still only one round of betting after the first card is dealt but when there is a showdown the active players get a second card which they add to their first one. Best total wins.
Also, I would like to see your solution when called pots are raked three dollars.
I believe this game could, in principle, be solved in a similar fashion as the original game. However, the equation system will be significantly more complex. I'll give it a try but it might take me a while. Using heuristic methods might give an (approximate) answer more quickly.
Quote:
Originally Posted by Kenji
I did come up with similar values using fictitious self play. @joker I am curious about what kind of multivariate polynomial equations you solved - any reference on this approach?
Let me explain it for three players for simplicity. I use variable names r, rc, rcc, and rfc for the thresholds above which BTN raises, SB calls, BB calls after SB called, BB calls after SB folds, respectively. At equilibrium we'll have r<rc<rcc and r<rfc. Also players are indifferent between folding and entering the game at the thresholds. Consider, for example, player SB at threshold rc conditioned on BTN raising before. SB wins the pot if BTNs card is below rc and if BB folds. The success probability of calling is therefore (rc-r)/(1-r)*rcc, which must equal the pot odds 14/(14+15+1+2) to make SB indifferent. Similar considerations for the other thresholds give the below four equations (which can be turned into polynomial equations by multiplying with the denominators).
rc*rfc=15/(15+1+2)
(rc-r)/(1-r)*rcc=14/(14+15+1+2)
(rcc-r)/(1-r)*(rcc-rc)/(1-rc)=13/(13+15+15+2)
(rfc-r)/(1-r)=13/(13+15+1+2)
One can solve this numerically using free online solvers. The system has multiple solutions but only one with all thresholds in [0,1] and r<rc<rcc and r<rfc. This must be an equilibrium.
For five player I wrote a small script to generate the equation system automatically and solved it with an built-in solver.
I am not aware of a reference for this approach but it seems elementary to me.
Concerning your approach based on fictitious play: Is there an existing software or library for this that can be flexibly configured or did you program it yourself? This approach might be easier and quicker to get approximate solutions than the approach above.