Do payoffs always track the odds overcome to win a bet?

Payoffs on casino wagers generally track the odds against winning them. At double-zero roulette, for instance, odds are 37-to-1 on a single-spot straight-up for a 35-to-1 payoff, 35-to-3 (11.67-to-1) on a three-number row for an 11-to-1 payoff, and 26-to-12 (2.17-to-1) on a 12-number column for a 2-to-1 payoff. The offset between the figures in each pair is what provides the house with the statistical advantage or edge which lets it eke out a profit from its loyal patrons’ action.

Offhand, you might suppose a form of odds tracking would also hold for slot machines and other games offering a range of payoffs linked to the specific resolutions of each coup. Perhaps with payoffs proportional rather than equal to the odds. This isn’t necessarily, or even ordinarily, true.

Say you venture $1 on a simplified side bet having three possible winning outcomes with odds of 20-to-1, 10-to-1, and 2.5-to-1. Further, imagine the wager is “full-pay” – the house has no edge – and will pay in dollars and cents rather than only whole dollars. As payoffs, the numero-noodniks on the seventh floor propose $7.07 for the 20-to-1 hit, $2.63 for the 10-to-1 win, and a push for the 2.5-to-1 upshot; everything else loses $1. Games like this usually display results as “returns” – payoffs plus the original bets back – so the schedule would posted as $8.07. $3.63, and $1.

If these payoffs were in proportion to the associated odds, success at 20-to-1 would pay 2X that at 10-to-1 and 8X that at 2.5-to-1; the 10-to-1 gain would pay 4X that at 2.5-to-1. They don’t. The bosses want this. So they ask the math mavens for payoffs that meet this criterion and still keep edge at zero. The new amounts are $4.48 at 20-to-1, $2.24 at 10-to-1, and $0.56 at 2.5-to-1.

Most actual slot and video poker machines have payoffs far out of proportion to the odds up and down the ladder of levels. The accompanying table presents the odds for an appropriate expert strategy and the corresponding values per dollar bet of an actual full pay jacks-or-better video poker game. In addition, it indicates the earnings on winning hands for the same machine and playing strategy, when payoffs are in proportion to the odds rounded off to the nearest penny. Remember that the posted returns would be $1 more – the payoffs plus the original bets.

Hand                        Odds      Actual Payoff  Proportional Payoff
                                         per $1 bet           per $1 bet
Royal flush           39802.54-to-1           799              2545.00
Straight flush         8379.25-to-1            89               535.78
4 of a kind             422.59-to-1            24                27.02
Full house               85.92-to-1             8                 5.49
Flush                    88.36-to-1             5                 5.65
Straight                 89.59-to-1             3                 5.73
3 of a kind              12.44-to-1             2                 0.80
Two pair                  6.74-to-1             1                 0.43
Jacks or better           3.68-to-1             0                 0.24
Lose                      0.83-to-1            -1                -1.00

The data show that solid citizens are short-changed for the big bucks when they beat the highly adverse odds against straight flushes and royals. As compensation, they’re overpaid for lower-valued frequently-occurring hands such as the high pair, two pair, and three-of-a-kind. In the long run, it averages out the same way. In the short term, where volatility dominates, actual payoffs tend to be more conducive to a pleasant gambling experience because the greater amounts at relatively high frequencies tend to keep players in the fray longer. Lots of folks receive a little something, as opposed to a few players cleaning up and everyone else quickly biting the dust.

The data reveal another factor which should interest video poker buffs. Following the implied optimum strategy, straights are slightly less likely than flushes, and flushes tougher to get than full houses. The payoffs go inversely – the most for full houses and the least for straights. The proportional values show that players should get nearly the same amount for all three hands, the highest for straights, decreasing slightly for flushes, and dropping a bit more for full houses. The amounts in the actual game are selected to be consistent with live poker and are arbitrary. The apparent disparity arises because the hold-discard strategy that gives the highest expectation at video poker deviates from that in live games – the objective of the former being to achieve the highest hand ranking on a list and of the latter to outmaneuver or beat the other players.

But, then, you didn’t really think the skills involved in video and live poker were the same. Er, did you? Anyhow, remember what the poet, Sumner A Ingmark, asked punters to ponder: