Attaching values to 3 Card Poker

Oct 24, 2005 3:11 AM

Last week, I discussed how the concept of Expected Value (EV) is used by blackjack players to guide their decision to hit, stick, double down and split. Because of the almost unlimited possibilities with blackjack, it may be difficult for some to understand how the math works. After all, for our blackjack problem, I used a computer simulation to run a million hands with each strategy and chose the one that gave us the highest EV. I’m sure there’s a skeptic or two who thinks 1 million hands is too few or that the whole process is flawed because they’ll never play a million hands of blackjack.

So, today I’ll discuss how expected value was used to determine the optimal strategy for a game like Three Card Poker. By know, many of you know the strategy for this game. If you’re dealt a Q-6-4 or better, you play. If dealt a Q-6-3 or less, you fold. Why is this?

For each of the 22,100 unique hands the player can be dealt, there are 18,424 unique hands that the dealer can be dealt. So, for a hand like Q-6-4 or Q-6-3, we simply look at all of the possible hands that the dealer can have and sum up the results. When we do this, we get these results (see table).

In Three Card Poker, if we fold, we lose our original ante wager. Thus, if we are looking at the 18,424 possible dealer hands, we would lose 18,424 units. If we Play, we risk twice this many or 36,848. If by playing we have at least 18,424 returned to us, then our loss will be less by playing than by folding. From the table above, we see that we cross this threshold right between these two hands. Thus, a Q-6-4 warrants being played, while a Q-6-3 should be folded.

If we were to put a gigantic table together of all the 22,100 unique player hands, and add up the total units returned for hands that are worthy of playing, and add up total units wagered for each hand (If we play, we count two units for each of the 18,424 hands. If we fold, we count one unit for each of the 18,424 hands.), we’ll find the overall payback of Three Card Poker. We’ll also need to adjust the total coins returned by the amount of the ante bonus for those hands paying one. For ante/play, we’ll find the total payback to be 97.98%.

What happens if we decide to "simplify" our strategy and play all hands of queen high or better? If we do this, then, instead of adding 0 to our total units returned and 18,424 units to our total units wagered amounts, we’d use the numbers we’d get if we played. The net result will be that we will wager more and we will have more returned. But, just as the Q-6-3 example shows above, the net result will be that the number of units lost will increase. It is simply NOT mathematically possible to have the number of units lost increase, and expect to somehow INCREASE our overall payback of the game. This is why by its very definition; expert strategy maximizes the overall payback of the game.

In the case of Three Card Poker, I used a program that literally dealt out every possible hand combination (22,100 times 18,424), which is just over 407 million hands. Programs that analyze video poker basically do the exact same thing.