Since I had some time I pursued the I idea I posted above. Rather than dealing with real poker hands, I simplified to the standard [0,1] poker hands. Players are "dealt" a random number between 0 and 1 and high value wins.
In our experiment, if there are N total players at the table, all but one of them can discard their initial hand-value and draw another random hand-value if they so choose. The Nth player does not have the ability to discard/redraw. To simplify things further, we will assume that the N-1 players who can discard/redraw make their decisions simultaneously.
When N is small, it is possible to derive analytically the optimal equilibrium cutoff each of the first N-1 players should utilize (each of these N-1 players utilizes the same cutoff in equilibrium).
Then it is straightforward to derive the probability each of the N-1 players will win a hand (of course each of the N-1 has the same probability) and the probability that the Nth player will win a hand.
I derived the analytical solutions for N=2, 3, and 4. Beyond that is too much to work through the calculations manually so I programmed a simulation.
Before I present the numerical results, I want to present an interesting finding which, I think, generalizes to higher N.
It is straightforward to show that:
P1 = [1 + c - c^2]/2 for N=2, where P1 is the prob that Player 1 (the player who can discard/redraw) wins and c is the optimal cutoff Player 1 utilizes.
P1 = [2 + c - c^2 + c^3 - c^4]/6 for N=3.
P1 = [3 + c - c^2 + c^3 - c^4 + c^5 - c^6]/12 for N=4.
If this pattern holds, then:
P1 = [4 + c - c^2 + c^3 - c^4 + c^5 - c^6 + c^7 - c^8]/20 for N=5.
P1 = [5 + c - c^2 + c^3 - c^4 + c^5 - c^6 + c^7 - c^8 + c^9 - c^10]/30 for N=6.
Etc.
At this point these higher-N formulas are conjectures, but they do seem consistent with my simulations. If anybody has any thoughts on those formulas, feel free to share them. I wonder if this is a "well-known" result that I stumbled upon (assuming that the formulas are even correct).
The table below assumes those formulas are correct and continue all the way through the N=10 case.
Total Number of Players | Optimal Cutoff | _____ P1 _____ | _____PN _____ |
---|
2 | .50000 | .62500 | .37500 |
3 | .60583 | .38774 | .22452 |
4 | .67033 | .28041 | .15877 |
5 | .71454 | .21942 | .12231 |
6 | .74705 | .18015 | .09926 |
7 | .77214 | .15277 | .08341 |
8 | .79218 | .13259 | .07187 |
9 | .80860 | .11711 | .06309 |
10 | .82235 | .10487 | .05621 |
Of course, the optimal cutoff naturally rises as the number of players increases. The lone player who cannot discard/redraw naturally is at a significant disadvantage in this game.
Last edited by whosnext; 11-19-2021 at 08:50 PM.