Hit frequency vs. paytables

August 20, 2007 11:25 PM
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When constructing a paytable for most games, there are two critical pieces of information that is required. The first is the frequency of the winning event and the second is the payout of that winning event.

By multiplying one to the other, and summing up these values for all possible winning events, we arrive at the payback of the wager. For many games, developing a paytable is rather simple because of this calculation.

It is a bit of trial and error, but you just keep plugging numbers into a simple spreadsheet until you get a payback you like and reasonable distribution for the paytable (you don’t want the rare hands paying less than the common hands just to get a good payback!).

Paytables for video poker are far more complex than this. Because the player makes a decision that can affect the final rank of the hand, this decision is impacted by the specific paytable in use. When I’ve done analysis work for inventors and their games are video poker based, I’m always asked what the frequency of a particular hand is, in order to determine the impact of raising or lowering that hand’s payout.

If only it was that easy. Every time there is a change to the payout of one hand, it causes the frequency of that hand to change. Since those hands must become something else, it also affects the frequency of many of the other hands.

Of course, minor changes in the paytable will not likely cause drastic changes in the frequency of a hand. One should not assume, however, that the paytable could be changed in any imaginable way with only these minor impacts. Let’s take a look at an example.

One game, with one of the most dramatic changes to the paytable is a game called All American Poker. It only pays 1 credit for a two pair, but pays 8 for a straight, flush or full house. This may seem rather odd given the different rankings of these hands.

The thought process may have been that in full-pay jacks or better, all three of these hands occur in rather similar frequency, and not in the order you would expect. A full house is actually the most common of these hands (1 in 87 hands), followed by a straight (1 in 89 hands) and lastly, a flush (1 in 91 hands).

These numbers are a direct result of the paytable in use for full-pay jacks or better. In All American Poker, with each paying 8, we find a very different distribution. The frequency of the full house actually decreases slightly to 1 in 92 hands. The flush increases to 1 in 64 hands, and the straight to 1 in 55 hands.

Normally, when we add 1 unit to the straight/flush/full house payouts, we know this adds 1.1% to the payback. Here, this calculation would lead to a rather faulty conclusion (and maybe that explains the 100.7% payback on this game).

A straight will now occur about 1.8% of the time, so the impact of adding the 4 units to this payout is 7.2%. Even this doesn’t tell the whole story, as there are additional 0.7% of the hands that will collect more than they would have. The problem is that we don’t know if these hands would have been losers, high pairs, two pairs or trips, so it is hard to quickly calculate the impact to the payback.

 To confuse the situation even more is the fact that royal flushes will become more rare, while straight flushes will become more common. The real key point here is that if the frequencies of final hands are changing so dramatically, then the strategy must be changing rather significantly too. If you play jacks or better strategy on All American Poker, you’re not going to achieve the 100.7% payback it offers.