There's more to play-or-fold and dealer qualification than meets the eye

In some casino table games, rounds are decided by comparing players' and dealers' hands. If the hands have equal probabilities, additional provisions are needed or edge would be zero and neither bosses nor bettors would have average – statistically-expected – wins or losses. Several methods are used to introduce unbalance and give the house an edge. A relatively new genre of games uses an approach based on a requirement that dealers must "qualify" by being above a designated minimum or players get a default rather than a full payoff for having the better hand. This feature is coupled with a feature letting players fold after seeing their hands or making secondary wagers and moving to the showdown phase. The edge depends both on the ranks at which dealers qualify and those at or below which players elect to fold.

Players who do fold lose their primary bets and exit the round. When they make the secondary bets, one of four results is possible. 1) The dealer doesn't qualify – players win their primary bets and push on their secondary wagers. 2) The dealer qualifies and the two hands have equal rank – players push on both bets. 3) The dealer qualifies and has the higher-ranking hand – players lose both bets. 4) The dealer qualifies and has the lower ranking hand – players win both bets.

The principle and the complex nature of the relationship between qualification level, folding strategy, and edge can be illustrated by a hypothetical game involving a deck of 20 cards comprising two each having values from 1 to 10. Players and dealers get one card apiece. The primary and secondary bets are equal, and both pay even money.

Pretend the dealer qualifies with a rank of 1 or better (i.e. plays every hand – none fail to qualify). Players who fold with anything from 1 to 1 through 5 then have an edge. Their advantage is highest folding with 1 through 3. Folding from 1 through 6 to 1 through 10 gives the edge to the house. This is highest when players fold from 1 through 10 – every hand.

If the dealer in this hypothetical model qualifies at 2, players still get an advantage folding with 1 and 2, 1 through 3, and 1 through 4. It's highest when the player folds with1 through 3. The house has the edge for all other folding strategies. Likewise, players can get an edge by optimal folding when dealers qualify at 7, 8, 9, or 10. The accompanying table shows the levels involved.

Qualification level	player has an edge folding at
1                       1, 1 & 2, 1 thru 3*, 1 thru 4, 1 thru 5
2                       1 & 2, 1 thru 3*, 1 thru 4
7                       1*
8                       1*, 1 & 2
9                       1*, 1 & 2, 1 thru 3
10                      1*, 1 & 2, 1 thru 3, 1 thru 4

The house always has an edge if the minimum for dealer qualification is 3, 4, 5, or 6. The joints aren't about to knowingly let patrons have an advantage, so the prototype game would be offered at one of these levels. The choice among them would be made so solid citizens who've done their homework can use a folding strategy yielding a house edge high enough for the casino to expect a profit in the long run, but not so inordinate that everyone who plays busts out and never returns.

For the proposed fictitious game, at a qualification level of 3, players who fold on 1 through 3 would be fighting an edge of only about half a percent. This would be too low for the casino to cover its operating costs, so chalk off that option. At a qualification level of 4, the smallest house edge would be 8 percent when players folded on 1 through 3. For qualification at 5, the smallest house edge would be 12 percent and would hold when players folded on 1. And for qualification at 6, the smallest house edge would be 7 percent, also when players folded on 1. Were these the alternatives, the casino bean counters would likely select qualification at 6; an optimum edge of 7 percent is excessive for players putting real dough at risk, but it's the best of the lot in this example. A game you might actually find on a casino floor would have a wider variety of hands, possible qualification levels, and folding strategies – leading to combinations that would give the house an edge in the 3 to 5 percent range against proficient players.

Intuitively, given the casino's choice of qualification level, you might guess that the optimum playing strategy would be to fold with anything below this value. The thinking would be that a qualifying hand would beat anything of lower rank, so folding lets players accept a loss of one rather than chancing two units. Decent qualitative reasoning, but not necessarily correct. This rule of thumb indeed applies for qualification at 4, where the optimum folding levels are 1 through 3. However, if the casino elected to have qualification at 5 or 6, players would be facing the lowest edge folding in either instance at 1. Counterintuitive? Yes, but that shouldn't seem strange to betting buffs who carouse in casinos where the very prospect of success is counterintuitive. As the poet laureate of the casino scene so aptly noted: