The optimal strategy for each of these cases is found by plugging the virtual payoffs into any VP program that computes a max-EV strategy. I don't know if any commercial VP programs will allow non-integer payoffs to be entered, but my own program supports this, and it is very useful for studying alternate strategies. If the commercial programs don't allow this, you might try multiplying all payoffs by 100 or 1000 and rounding to approximate the virtual payoffs.
For 9/6 JoB there is a simpler trick for finding the best-shot(royal) strategy. Changing the royal payoff to 1038, while leaving all other
payoffs at the actual values, just happens to give the correct RoRBR strategy. The same strategy results from any royal payoff between 1022 and 1043. However, this kind of trick doesn't work for 8/5 JoB or for 10/7 DB, and I haven't tried it for other games.
Here a quick summary of what each of these strategies is trying to do:
Min-RoR minimizes overall risk-of-ruin, which is equivalent to maximizing the probability that the player will play forever without ever going broke. The 50/50 bankroll for this strategy is 1661 units, which gives the player a 50% chance of playing forever without going
broke. The max-EV strategy gives a 50/50 bankroll of 1670 units.
The bs_RF column is best-shot(royal) strategy, which maximizes the probability of hitting a royal before going broke. This strategy has a 50/50 bankroll of 720 units, which gives the player a 50% chance of hitting a royal before going broke. This bankroll is smaller than the 50/50 bankroll for Min-RoR strategy, because the task of hitting a royal is easier than the task of playing forever. The max-EV strategy gives a 50/50 bankroll 731 units when trying to hit a royal before going broke.
The mc_RF column is min-cost(royal) strategy, which minimizes the average loss that occurs between royal flushes. When the royal payoff if 975.99 units, royals occur just often enough to pay their own cost, making the game exactly breakeven. By comparison, the average loss between royals when playing the Max-EV strategy is 984.299 units.
The Min-cost column is the strategy that maximizes the average number of units returned per unit consumed by losses. This views the game as if you were playing at a casino on the border between two countries and using the VP game to exchange currency. If the VP machine accepted only U.S. dollars but made all payoffs in Canadian dollars, then this strategy would give the player the best possible exchange rate. For this game, each dollar paid back to the player costs only 98.297651 dollars that are paid to the machine in lost wagers, on average. This strategy can also be viewed as a 'best winning streak' strategy. If you start with one unit and collect your net winnings, but re-play only the original unit and keep playing until you eventually lose, then this strategy maximizes your average payoff from these winning streaks. By comparison, the Max-EV strategy has an average cost of 98.297656 units lost per unit returned to the player. This strategy is almost identical to Max-EV for this game, and this tends to hold true for most games.
The Max-Royal strategy tries to hit royals at all costs. This strategy would discard a dealt straight-flush in order to try for a royal
instead. This would be the right play to make on the last few hands of a tournament, if your only hope for winning a prize is to hit a royal flush. Otherwise, this strategy is a great way to burn through a bankroll in a hurry.
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