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Best of Alan Krigman
To Some Craps Players, the Value of Hedges Is Worth their Price15 March 1999
By Alan Krigman
Craps offers numerous ways to hedge bets -- to "protect" one wager with another. And dexterous dice devotees often cherish the chance to do so, frequently following the philosophy that hedges can never get too elaborate and the most baroque are the best.
The laws of probability decree that all hedges penalize players by raising house advantage relative to the edge on the main bet. For two reasons. (1) Bets used as hedges are normally longshots with high inherent house advantage. (2) When action is on both sides of the same result, casino commissions are based on the sum of the bets while player winnings depend on the difference.
But probability isn't the sole source of standards solid citizens use in making gambling, business, and other decisions. Utility also offers a metric, essentially accounting for the emotional rather than merely numerical worth of prizes and penalties.
Lotteries illustrate the upside. When jackpots are high, people grab what seem to be cheap tickets. Like a shot at $10 million for $1. Assume chances are one in 50 million, winner take all. Probability says a "fair" payoff on $1 would be $50 million. Utility says $10 million may merit a try, regardless of odds. Hedging shows how utility works on the downside. Say a craps player bets $40 "no four," paying $1 vigorish. Normally, he wins $20 and recovers his $40 when the dice show 1-6, 2-5, 3-4, 4-3, 5-2, or 6-1; he loses the $40 on 1-3, 2-2, or 3-1. Including the vigorish, this is six ways to clear $19 versus three to lose $41.
The player could hedge with a $6 "hard four." On a seven, he'd still win $20 and recover the $40 but lose the $6. On 2-2, he'd still lose the $40 but the hardways would pay $42 and return the $6. On 1-3 or 3-1, he'd lose $46. Including the vigorish, this is six ways to make $13, one to earn $1, and two to lose $47.
The hedge cut the amount the player can win and raised what he can lose. But rather than six paths to victory and three to defeat, he has seven roads to joy and two to sorrow. Math mavens mock this hedge because in nine statistically-correct decisions, the edge on $40 "no four" alone amounts to ($41x3 - $19x6) = $9 while on the combined bet it's ($47x2 - $1x1 - $13x6) = $15. Utility legitimatizes sacrificing $6 to reduce the likelihood of losing $41 for some individuals under appropriate circumstances.
Players inclined to hedge are reasoning qualitatively and no jumble of numbers is apt to change their minds. Still, some hedges make more sense than others. Although the arithmetic to compare options can get complex, you won't go too far afield forgetting the frills and choosing the simplest way to do a job.
What if you wish to "protect" your bet during the come-out roll? Most players who like this idea simply bet $3 "any craps." This cuts their loss on the combined bet to $4 on a craps, but naturals yield only $22 and all other rolls lose $3. A fancier alternative is to bet the $3 as "three-way craps." The possible net results on the combination are: two or 12 win $3, three loses $12, naturals pay $22, and all other rolls still lose $3.
Which is better? You could go around in circles comparing the apples and oranges of ways to win or lose different amounts. Based on edge, the plain-vanilla $3 any craps costs a theoretical $0.33 per come-out and the rococo three-way craps sets you back $0.39. The simpler way provides cheaper protection.
Hedges are never bargains when you pencil out the percentages. But numbers may not be the only or even the most crucial criteria to individuals making decisions in particular instances. Especially not in a universe governed by both law and chance. The pragmatic poet, Sumner A Ingmark, knew this when he penned:
The value structure may be lost.