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Best of Alan Krigman

Gaming Guru

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Why Different Flavors of Pai Gow Taste the Same

31 July 2001

Pai gow class table games involve a Banker and one or more Players. Each receives a hand, and splits it into a high and a low set. To win, both sets must exceed their counterparts in the opposing hand. Topping one and being toppled on the other pushes.

Edge at pai gow arises through three mechanisms. 1) The casino charges a commission, normally 5 percent, on bets won by patrons. 2) If corresponding high or low sets in opponents' hands have identical rank, they are considered "copies" and are resolved in favor of the Banker. 3) In established versions of pai gow, participants can set their hands in diverse ways; the skill with which they do so influences their chances of winning.

Although pai gow settings for most circumstances are no-brainers, some can be tricky. No single strategy is optimum. Partly because approaches must be gauged relative to those used by opponents. Also because the algorithm that yields the greatest expectation depends on whether a participant is acting as Player or Banker.

A Player would have to know how the Banker, who might be the casino or another patron, will decide. A Banker faces the added dilemma that each opposing Player might use a different scheme. And, although casinos serving as either Bankers or Players follow predetermined "house ways," these differ among establishments.

Such inherent subtleties, along with variations in skill of participants and the distinct ranking structures of alternate versions of pai gow, might suggest that chances of Banker or Player winning, losing, or pushing would vary all over the lot. Not so. In fact, the probabilities and the resulting edge even in radically divergent variations of pai gow are remarkably uniform. In the traditional Asian "tiles" implementation, when Banker and Player both use one of the recognized nearly-optimal strategies, chances of resolution are 30.532 percent for Banker, 29.624 percent for Player, and 39.844 percent for a push. In the "poker" rendition, chances with nearly-optimal play are 32.6 percent for Banker, 31.4 percent for Player, and 36.0 percent for push.

This uniformity will hold for new pai gow games if and when they emerge. To understand why, consider that in the majority of situations, splitting initial hands into high and low sets is straightforward once a solid citizen knows such elementary rules for ranking as a six beats a four or triplets beat pairs. And, in many of the remaining cases, the penalty for a sub-optimal split is small. With this condition, the four elements formed from two opposing hands fall into a series from lowest to highest -- regardless of the specifics of the game. Ignoring "copies," which are comparatively rare, the elements can be ranked simply as levels 1 through 4. The accompanying chart gives the only six possible combinations of the four elements, and indicates how the hands would be resolved for each arrangement.

Player Banker Resolution
1-2 3-4 Banker wins (3B over 1P, 4B over 2P)
1-3 2-4 Banker wins (2B over 1P, 4B over 3P)
1-4 2-3 Push (2B over 1P, 4P over 3B)
2-3 1-4 Push (2P over 1B, 4B over 3P)
2-4 1-3 Player wins (2P over 1B, 4P over 3B)
3-4 1-2 Player wins (3P over 1B, 4P over 2B)

The chart shows Banker win, Player win, and push each in two of the six equally-likely arrangements. And two out of six is 33.33 percent. This figure serves as a base for a generic pai gow game.

Incorporating influences of copies and setting strategies for specific versions of the game nudges actual expected results away from 33.33 percent for each outcome. Copies tend to favor Banker at the expense of Player without affecting chances of pushes. Optimal setting strategies tend to raise the chances of pushes, decreasing both Banker and Player wins. These effects are evident from the values previously cited for the tiles and poker games.

Strategy changes at pai gow illustrate a fundamental but widely overlooked gambling principal. In games of repeated trials, you can achieve the same goals by losing less as by winning more, and pushes are often key to manipulating the offset.

The poet, Sumner A Ingmark, captured the conundrum in this acclaimed couplet:

When you tally results of your betting,
It's not what you're grossing, but netting.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.