Symmetry, Marginal Utility, and the Cube
Danny Kleinman, 1980
Vision Laughs at Counting, Vol 2, © 1980 Danny Kleinman

## The Symmetry of Doubling and Taking

In backgammon, doubling and taking strategies must be considered separately as a rule. Your opponent’s psychology, more than anything else, should determine when you double. But when you should take depends primarily on your evaluation of the backgammon position and only marginally on your opponent’s psychology.

This changes when you are in the box in a chouette against opponents with a variety of styles and temperaments. You can no longer custom-tailor your doubles to a particular opponent’s loose-taking or fast-passing tendencies. If you figure that your opponents’ diffferent temperaments will average out, then you can revert to basing your doubles, like your takes, on your evaluation of the backgammon position.

Let us denote your take point by T and your doubling point by D. We can define T rather simply. It is the minimum chance of winning you require in order to take. Of course you must adjust your estimate of your winning chances in a particular game to take gammon possibilities into account. Ignoring psychology, we can establish T as 25% whenever owning the cube has no remaining usefulness to you. But if you may be able to use the cube again, you can assign a value of 5% or so to the cube and thus lower T to 20%.

We cannot define D quite so simply. D is that probability of winning at which you will certainly double. But when your winning chances approach D closely enough that they threaten to exceed D at your next turn, you will also double. Thus you will sometimes double slightly above D, sometimes slightly below. It will not be quite clear what your real value for D is.

An unbiased, symmetric view of backgammon positions implies a relationship between T and D: They must total 1. If you would have a hard time deciding whether to take from the other side, then you know you have a fine double. Thus 75% doubles follow from 25% takes, and 80% doubles correspond to 20% takes.

Two of the most common cube-handling errors still preserve this symmetry between taking and doubling. Players often differ in their assessments of certain kinds of backgammon positions. Suppose, for example, a particular player thinks back games especially favorable. He will tend to take doubles loosely when playing a back game. But he will also hesitate to turn the cube when his opponent plays a back game against him. If he is wrong, he makes errors in both situations. His assessment may be mistaken, but it is unbiased.

The second kind of error is strategic, betraying a misunderstanding of mathematics and the cube. Some players think a small advantage warrants a double. “I’m the favorite; why not double the stakes?” they think. These fast doublers become fast passers in moderately inferior positions. But their thinking is consistent and symmetric: “I’ll probably lose; why should I lose twice as much?”

Most players with biased views of backgammon positions double quickly and pass slowly. Sometimes, however, this stems not from any bias towards the position but from an awareness of their own superiority in playing. This superiority influences the chances of winning. Thus an unbiased expert may take a game that looks very poor because he feels he has a good chance to win it against this opponent.

## Steam Doubles

Many backgammon players make steam doubles when they are substantially behind for the session. Not only do they double early, but they also take many games they would pass if they were close to even for the session. This steaming implies an increasing marginal utility of money over the short run. An additional dollar lost doesn’t hurt as much as the failure to win an extra dollar.

Increasing marginal utility of money characterizes the gambler. A gambler prefers a 50% chance to win a dollar combined with a 50% chance to lose a dollar to the certainty of neither winning nor losing. In practice most gamblers prefer a 49% chance to win a dollar combined with a 51 chance to lose a dollar to the status quo. Sometimes gamblers will give away even greater vigorish in order to have action. This implies a sharply increasing marginal utility of money.

## Scoreboard Passes

Many backgammon players turn very conservative when they are substantially ahead for the session. Not only do they make scoreboard passes but they often fail to double in games they would gladly double if nearly even on the scoresheet. This implies a decreasing marginal utility of money over the short run. An additional dollar won doesn’t help as much as the preservation of a dollar already won.

Diminishing marginal utility:  When the perceived value or satisfaction gained declines with each additional unit acquired or consumed.

Decreasing marginal utility gives rise to the institution of insurance and is antithetical to gambling. This is close to the norm in our society. The diminishing marginal utility of just about everything is an assumption of classical economic theory.

A backgammon player with a diminishing marginal utility of money should play for as small a stake as possible, unless he has a marked advantage in skill.

Overcautious cube-handlers may try to rationalize their behavior by saying that they are willing to accept smaller mean expectations of winning in order to minimize the variance. This rationale makes sense only for players who have little exposure, for whom the single session and even the single game are significant. But for regular players who wager session after session on backgammon, conservatism saves very little variance over the long run. By a readiness to accept a big loss in a single game, you minimize the chances of having a big losing month.

## The High Price of Targets

As long as your marginal utility of money is anything but constant, you pay a price in the expected value of your winnings when you try to maximize your utility. From a mathematical point of view, a backgammon player can only act rationally when the stakes are sufficiently small in relation to his bankroll for each dollar won or lost to have equal value.

Typically, however, backgammon players display a strange and inconsistent mixture of the gambler’s temperament and the insurance buyer’s. They go back and forth between steam doubles and scoreboard passes. Their actions in any given game depend not just on the current position but on the results of all prior games in a rather arbitrary time period: the current session of backgammon.

Players who are minus on the scoresheet attach a special significance to getting plus, while players who are plus shun the prospect of going minus. The score of zero thus serves as a goal for each, a positive goal for the loser and a negative goal for the winner.

The single point zero is only a special case of a target range, (L, H), where L represents the low value and H the high value between which scores are felt to be satisfactory. Below L, a player feels he must get up to L even if it means taking poor gambles. Above H, a player feels he needn’t increase his score if it means taking gambles, even when the odds favor such gambles.

Below L, money has an increasing marginal utility, tapering off to a constant as L is reached. Above H, money has a decreasing marginal utility, tapering off to a constant as the score dwindles down to H. Only between L and H does the marginal utility of money remain constant. Only within the target range (L, H) does the player act rationally in the sense of trying to maximize his money expectation.

Typical backgammon player's utility curve for a single session. At different scores, he acts like a steamer, equity chaser, or Colonel Whiteflag.

We may define a caution quotient, K, as the ratio of utilities between a dollar lost and a dollar won. A value of K less than 1 represents increasing marginal utility of money; a value of K greater than 1 represents decreasing marginal utility of money. Of course K is 1 for constant marginal utility. Deviations from 1 depend on both the number of dollars at stake and the distance from the target range.

Each player’s value for his take point, T, and his doubling point, D, vary logically with his K at any given time. When the cube has lost any further usefulness, we can set

T =
 K 2 + 2K
and  D =
 1 + 2K 2 + 2K
.
When the cube adds the equivalent of a 5% extra winning chance, we can set
T =
 9K − 1 20 + 20K
and  D =
 11 + 21K 20 + 20K
.
Substituting the rational value of 1 for K, we get the familiar 25% and 20% take points, and the familiar 75% and 80% doubling points.

For K less than 1, both doubles and takes become looser. When K is only 12, T drops to 16 instead of 14 and D drops to 23 instead of 34. Similarly, for K greater than 1, both doubles and takes become tighter. When K is 2, T rises to 13 instead of 14 and D rises to 56 instead of 34.

Either way — steam take or scoreboard pass — the player whose marginal utility of money in one direction is twice what it is in the other direction may surrender equity up to 13 the stake by his eccentric taking policy. The steam double and scoreboard nondouble can be equally costly.

## Settlements

Backgammon players settle games precisely to avert big swings. This is antithetical to a desire to gamble, consonant with a desire to insure. For this reason, the steamer has an advantage in settlements but the scoreboard passer can lose out. The steamer will always refuse a fair settlement. He prefers to gamble it out by letting the dice decide, winning or losing the entire value of the cube. But the player whose value of K rises above 1 will accept an unfavorable settlement as a form of insurance. When K reaches 2, the insurance premium becomes 13 the value of the cube.

## A Coup against Colonel Whiteflag

Colonel Whiteflag goes by many names in backgammon clubs. Different players become transformed into Colonel Whiteflag under different score conditions. Some players become Colonel Whiteflag only while in the box or only when the cube reaches certain high levels. It isn’t always easy to recognize the Colonel when he puts in a surprise visit.

A man we had hitherto thought of as “Fearless Frank” was in the box in a 5-handed chouette. Two of the crew beavered when Frank turned the cube initially. Soon the game turned arround — just a little — and Frank got the cube back at 4 (8 for the two beaverers). Ultimately the game turned into a race, close until the very end.

Figure 1
Black on shake

From here, Frank figured to win 131 games in 216, giving him a positive equity of 46216 times the value of the cube (which was at this point 24 for him). Since white could not ever have a proper cube return, Frank had a redouble. Actually, any positive equity, no matter how small, would warrant a redouble for Frank.

Astonishingly Frank declined to turn the cube. Fearless Frank had suddenly become Colonel Whiteflag. By his inaction, he revealed that he was willing to relinquish more than 5 points for the privilege of not gambling. Fortunately for Colonel Whiteflag, he rolled doublets and won the game.

But it was equally fortunate for Colonel Whiteflag that all his opponents were gamblers. Otherwise someone could have executed a remarkable coup. If the Colonel was willing to pay over 5 points in order not to gamble with an additional 24, then perhaps he would also pay over 5 points to eliminate the 24-point gamble he was already taking. An astute crew member could have offered to settle for a wash. Even if Colonel Whiteflag somehow sensed the absurdity of such an offer, he might still have accepted an offer more favorable, such as 12 point from each crewmember, but 1 point from the two who had beavered.

In effect, making a settlement is the equivalent of undoubling. When you play Fearless Frank, you have only one way to win extra money from him because of his gambling attitude — by having him take the cube foolishly. But you can beat Colonel Whiteflag in two ways. You can take advantage not only of his reluctance to see the cube get higher, but of his eagerness to undouble as well.