Where does the money come from?

Here is a brief scene which may (or may not) have been found in the cutting room, clipped then stuck to the floor. A scene from that classic film: The Three Stooges Discuss Backgammon. Decades later, the screenplay went cold and decidedly devalued, only to morph online into a real comedy, entitled the baron talks backgammon.

Curly:  Yuck, yuck.  Hey, Moe. In backgammon, where does the money come from when you double?

Moe:  What you mean, where zit come from?  Look at the board. You’re winning this game, knucklehead, so why not play for more moolah?  (Moe stares gobsmacked at Curly)

Curly:  Yeah, just like you to double before your first roll.  (Curly snubs his nose at Moe)

Moe:  Why you little!  (Moe waves some oblique physical threat)

Larry:  Maybe what Curly says is: “Why do you double now? Right now?” as opposed to the roll that was or wait until the roll that will be. Double or wait? 

Moe:  That what yer asking, Curly?

Curly:  I … I …… I’m not telling.  Moe. You’re always mean to me.  (Curly makes obscene gestures as Moe looks away. Camera pans to the backgammon board, then fades to black.)

Okay, fine, yes it’s silly. Admittedly.  But. Pregnant pause. Is Larry’s question a good one? 

Why do we double now as opposed to before our last roll or wait until our next roll?  And, pentimento, where does the extra money come from?

To illuminate these questions we must take stock of those moments in a backgammon game when we have control.  Real control. There aren’t that many. We have no control over anyone’s dice.  We have no control over the opponent’s checker plays.  We have some control over our checker plays.  But, and the theme of this post, we have total control over our cube decisions. 

Now, control in backgammon is strange — it’s opposite is randomness. Not chaos. Nor resignation. But randomness. In backgammon, buffeted by random fluctuations, all control needs to be gentle, forgetful, and optimistic.

In mathspeak, backgammon exhibits the Markov property.  Specifically, the probability of winning from any backgammon position does not depend on the history of positions which led to the current position.  Of course, this position was created by passing through the previous positions; that fact is not in dispute.  It is the probability of proceeding to the next position that depends only on the current position, and not at all on the probability (or pathway or order or memory) of past positions. 

From a probability standpoint, the past has no memory in backgammon. Numerical analysis in backgammon begins with the chances in the current position only, and no position from the past – a huge simplification. 

By contrast, consider this decision in Texas holdem poker.  Compare the following two hands of poker:

• You have red pocket aces, you raise preflop.  All fold except one player who calls.  The flop comes 2-of-clubs, 3-of-clubs, king-of-hearts.  You bet.  Your opponent calls.  The 8-of-clubs comes on the turn.  You bet. Your opponent thinks.
• You have red pocket aces, you raise preflop.  All fold except one player who calls.  The flop comes 2-of-clubs, 3-of-clubs, 8-of-clubs.  You bet.  Your opponent calls.  The king-of-hearts comes on the turn.  You bet.  Your opponent thinks.

The poker situation at the turn seems identical, but the parlay to arrive at this situation is quite different.  While the opponent thinks, you realize that your opponent is more likely to have kings (a pair or a set) in the first case, and is more likely to have the club flush (or flush draw) in the second case.  Most card games do not exhibit the Markov property and therefore the entire history of a hand must be reviewed to make informed decisions.  In this sense, card games are harder than backgammon.

Thus backgammon has the Markov property. The cycle of control in a backgammon game is called the full ply.  All cube decisions begin by imagining how best to play your 36 dice rolls and then, for each of your rolls, spread out the pathways to imagine how your opponent best replies with her 36 rolls: an imagined total of 36×36=1296 future positions on the board (nine gross?).  If you double or redouble, imagine also those unhappy situations where your opponent immediately cubes you back.  That’s a lot of imagining for one human. Luckily, bots relish this kind of work. Note that the full ply takes your current cube decision (double or wait) into your next cube decision (double or wait), a transformation from one moment of total control to the next moment of control, a Markov step for cubes. 

Admittedly, many of the nine gross board positions are duplicates.  Still, for a simple discussion, view the bunch of next-roll positions as 1296 individual pathways of the full ply.  Behind the headlines, each pathway in a full ply has its own financial story.  A story like, poor kid strikes oil and becomes rich.  Or, tycoon gambles on stock market and loses it all. Package the financial stories into common bins. Apply the basic financial plot of a bin to the transition from your current position to your next position.  Start with the current position. Suppose you have access to the cube.  Since you are considering a double or redouble, you are clearly winning and expect to make money – a positive equity.  Suppose you double. On average, you find yourself in each of the 1296 next positions.  The financial stories partition into six classes:

• Lasting regrets. Occasionally $#@& happens.  You roll horribly. Your opponent laughs aloud, then redoubles. You curse aloud, then you pass the cube.  Label each of these rare events a lasting regret.  Regrettable since you just doubled into a position where you now find yourself losing.  Lasting because you just passed a cube, game and money gone.  If only you had waited and not turned the cube, then you could drop the cube for $1 and not $2. No probability here, just a dollar wasted.
• Sad reversals. Sometimes you roll poorly, the opponent immediately redoubles and you must take.  You are not happy. The game still continues, but you are currently losing.  This is a sad reversal.  Sad because you are an underdog even though you just escalated the stake twice.
• Setbacks. Often the game becomes close but you realize you’re no longer winning.  This is a setback.  Your opponent sees it’s too soon to redouble, but watches intently and talks tenderly to the cube, building a nest for it.
• Rugged marches. Often you are still winning, but somehow your position on the board seems anemic after the redouble.  This signals a rugged march.  You have lost some equity by surrendering the cube and weakening your position, but your equity is still positive, and now pays off at twice the stake.  To milk a win, you plan to outwit the opponent with brilliant checker play. 
• Earned luck. You rolled well and, after the opponent’s roll, your position has improved somewhat.  This is earned luck.  If you waited to double until this moment, the opponent would still take, grudgingly.  Note that earned luck positions have no financial influence on the decision whether to double at the start or the end of this pathway; the equity has doubled in both cases after the next roll. 
• Market losers. You rolled very well, the opponent not so good, and your position has greatly improved.  If you waited to double until this moment, the opponent would certainly pass the cube.  That didn’t happen.  Instead, you already doubled. The opponent accepted this cube and now must play the rest of the game from a bad position. And for twice the stake. This is called a market loser. Your opponent wishes you had delayed a cube offer until now so the opponent could drop and wriggle out of this expensive situation.  

For a redouble decision, the nine gross pathways cluster into six bins: lasting regrets; sad reversals; setbacks; rugged marches; earned luck; market losers.  The identical bins appear for any first cube decision when you choose to double.  When you wait to make a first double, the bins may populate differently, since a gammon does not count under the Jacoby rule and the opponent still has access to a centered cube.  Note that the opponent cannot pick and choose from the six bins.  Only future positions are binned, glints in a dad’s eye. These pathways live only in probability space; just one will arrive on the board. With a board position, the take decision is all-or-nothing, beforehand.  Marry in haste, repent at leisure.

The full ply is packaged as six populations in six exclusive classes.  If you wait to redouble, the populations mix in one fashion; if you redouble, they mix in another fashion.  Similarly for wait or first double. When you wait, the equity of your current position is defined as the average of the equities for all 1296 next positions, where an estimate of the equity of each next position is made with the cube held back.  When you double, the equity of your current position is the average of the equities over all 1296 next positions at twice the cube level. However, the cube is surrendered and will be used by your opponent, as appropriate.  This pair of calculations is what the bots do spectacularly; humans will never be able to match these calculations.  For the bot a simple comparison of the pair of equities decides the cube action.  For us, we must sweat. 

Where does the money come from?

As discussed, waiting until the next position to redouble from 2 to 4, the important class of market losers does not work in your favour.  Instead of continuing the game at 4 from the next position with the opponent’s situation a shambles, the opponent will pass a late cube.  The bin that nurtures your best-case scenarios — the market losers — will be capped at $2; the excess above $2 at a 4-cube is thereby forfeited, a contribution missing from the profits.  You have lost your market for this extra money.  Clearly, much of the extra money you gain by doubling now results from the opponent agreeing to escalate the stake before the position evolves into a market loser. 

When you wait in the current position to redouble at the next position, the bin of earned luck is redouble/take now and redouble/take next.  By the meaning of the term “correct double”, the current position is akin to those later positions in the class of earned luck.  Thus earned luck positions gain money by redoubling, either now or next. However, on average, little extra money accrues to earned luck by doubling next.  Small equity changes between earned luck bins (now and next) mean extra profits are small too. Big profits from a joyful surfing on the market losers beckon instead.

When you wait and the next position is in the rugged march bin, you gain slightly by not doubling the stakes because the current position, correctly an earned luck cousin, has decreased into a smaller lead of a rugged march position, so you make a bit less money.  Yet the cube is still at 2 and you still own it or have access to it. 

That’s half the story: on the plus side, only surfing the wave by doubling into the market losers offers extra windfall profit. 

Without enough market losers you have no business touching the cube.

The other half of the story is the rare yet painful situation where you are in fact losing the game at the next position – these situations obviously prefer waiting to double, thereby reducing losses while you retain cube access.  Specifically, in a setback at the next position, you are losing but the opponent has no cube action yet. The setback position loses equity, partly from your weaker position and partly when the opponent owns the cube.  Those few cases where the opponent immediately redoubles and you must take – the sad reversals – are one-quarter as expensive when you wait.   Finally, the very rare but deadly class of lasting regrets, where the opponent immediately redoubles and cashes the game, costs a full $2 plus a few vanishing miracle wins that your cube ownership would buy by waiting to redouble.  Note that a lasting regret resembles a reverse market loser on steroids; one lasting regret is worth more than 36 market losers.  Fortunately, they are rare. 

To simplify the conclusion, ignore small equity reductions from the rugged march and the setback bins, and hope for an empty lasting regret bin and a modest sad reversal bin, banking on their rarity.  Then, the main source of any extra money won by an immediate double comes from the windfall of continuing to play the market losers.  A small sum comes from twice the average equity in the earned luck bin after discounting the loss of cube ownership.  In short, the market losers pull the load.  To double, you must have market losers.  Double now, and the money comes mostly from playing the market losers to a sweet bearoff victory. 

A double packs most of the profits into the endgame play of the market losers.

This reasoning applies to all doubles, not just close to reference cube positions.  For first doubles of a centered cube, the mix of credited gammons may differ, double or wait, because of the Jacoby rule.  For redoubles, the power of cube ownership by your opponent amplifies any hard luck stories on your immediate roll.