Prize the Power of the Push at Blackjack

The sand in the gambling gears results partly from punting palaces being places where people are prodded, subtly in some cases and blatantly in others, to trust what they want to be true and not what they see or know is. But phenomena more fundamental are afoot. Rigorous analysis occasionally yields surprising if not outright counterintuitive conclusions. Among other things, this explains why many blackjack veterans never bother to learn the nuances of "Basic Strategy" before risking hard-earned money on a game they don't quite understand, deliberately flout the rules because they haven't a clue as to how they were derived and think their gut will serve them better, or follow the strait and narrow because they're "supposed to" but are uneasy doing so.

Basic Strategy comprises the decisions that maximize the "expectation" for each set of player totals and dealer upcards. This implies the fattest profit when players are favored, and the leanest loss when the house is in the catbird seat. It's calculated by finding the chance of winning times the payoff and subtracting from this the chance of losing times the wager. For convenience in comparing alternate situations, values are usually given per dollar bet or expressed as percentages.

The root of many Basic Strategy head-scratchers lies in the role played by pushes -- hands where players and dealers have equal nonbreakers so bets neither win nor lose. An example might be a player's 12 versus two-up. Basic Strategy is to hit. Many solid citizens stand. Some because they selectively recall hitting and breaking. Others after reasoning that four cards (the 10s) break and lose immediately, four leave totals less than 17 (ace through four) which won't improve their position, and only five help but still may not win unless the dealer busts, so why not stay in contention rather than take what seems like a futile longshot?

A 12 versus two-up is an underdog no matter how it's handled. The expectation is to lose roughly $0.29 on the dollar (29 percent) by standing and $0.25 on the dollar (25 percent) by hitting. "The book" says to hit because it's several percent less onerous.

How do these numbers originate? The expectation associated with standing is relatively straightforward to picture. On the average, a dealer with two-up who must hit anything under 17 and stand otherwise will bust 35.4 percent of the time; this is when a player with 12 wins. A total of 12 can't push, so the player loses on the remaining 64.6 percent of all cases. Expectation is the difference, 35.4 - 64.6 or -29.2 percent.

Expectation for hitting the 12 can be found by calculating a) the chances players break, end below 17, or finish at 17, 18, 19, 20, or 21; b) the likelihoods the dealer will bust or finish at equal, higher, or lower levels from 17 through 21; and c) the probabilities of the combinations. The accompanying table gives the resolutions. The data show players' overall chances as 34.8 percent of a win, 4.9 percent of a push, and 60.3 percent of a loss. Subtracting 34.8 - 60.3 yields -25.5 percent expectation.

Chances of alternate resolutions of
Player 12 vs two-up when hit in blackjack

Players who push after hitting 12 against two-up deem themselves lucky to have escaped free from an adverse situation. They rarely appreciate just how vital pushes like those are to the game. The card player's Coleridge, Sumner A Ingmark, put it like this:

When weak hands, played well, finish even,
A bettor learns what to believe in.