Why matches?

Backgammon has been enjoyed for centuries as a great game, and often as a gambling game.  However, not everyone approves of gambling.  Fortunately, in the last century, a tournament format of backgammon became popular.  Whether or not there is an entry fee, a participating player is assured a certain number of matches and, during a winning streak, can play more matches. 

Match play offers unique advantages over cash games.  Foremost, since pairings in a tournament start out random, the average player may be lucky enough to compete against a world class player on equal terms during a match.  This thrill is a memorable experience for anyone, and an unexpected victory becomes another thrill and a future story, shamelessly embellished.  There are very few sports or pastimes where an amateur can challenge a top opponent on equal terms. 

Another advantage of match play is the social aspect.  A monthly tournament often develops friendships over the years.  Since a backgammon match can last many games, a few games, or even a single game, there is plenty of time to socialize and commiserate and discuss.  Most players view the entry fee as payment for the locale (a restaurant or pub), for operating expenses of the organizer, and for the annual trophy, plus the occasional treat of a first or second place finish paying off.  Any backgammon player can enjoy membership in a regular tournament. 

Match play introduces a new level of skill in the use of the doubling cube.  The expert player welcomes this opportunity for extra skill, but an enthusiastic amateur need not shrink from the cube subtlety.  Unfortunately, current literature and discussions about the cube during match play is awash with mathspeak, which can discourage any player from the simple pleasure of tournament play.  This essay intends to rectify that situation.  No one should miss out on the fun of match play because of distractions about cube issues. 

Unlike cash game backgammon, match play must adjust the decision of whether to take or pass an offered cube to the current match score.  Many players make this decision more complicated than the vital decisions of playing the checkers and spotting the doubling opportunities.  Hence the distraction and the detraction.  Why be so precise in adjusting the cube decisions to match score?   Instead, every player should focus on the best play and know when cube action is needed.  Simplify the match cube process and sharpen the checker play and cube awareness – that is one ticket to improvement, and to better enjoyment of match backgammon.

Why three-point matches?

Three-point matches capture the intricacy of match play in short story form.  Only about two dozen winning pathways are available for Black, and the same two dozen for White.  Playing three-point matches provides frequent practice to cement match play ideas.  Three-pointers also have cube silence and cube actions, times when the cube cannot or can be used.  These binary poles of the doubling cube give rise to a transition from positional thinking early in the match to score management later in the match.  Practicing three-pointers hones these distinctions, and provides strategic harbors during a short match. 

The fulcrum of the transition from positional to score-based play is the Crawford game.  When Black reaches a score of two before White does, the next game only is played without the cube.  It is Black’s one chance to finish the match without White’s desperate and dangerous automatic double to overtake Black’s match lead.  Should White win the Crawford game, but not yet the match, then White is free to cube relentlessly.  The Crawford game is one example of a silent cube situation.

Longer matches of five-points, seven-points, nine or greater scores often evolve into the functional equivalent of three-point matches.  So 3-pointer practice helps in many longer matches during later stages. 

Redoubles are limited.  The moments of mutually-assured-destruction with a redouble are limited to two instances during the match score of three-pointers.  This restriction controls added complexity to a pair of situations.

Another valuable reason to focus on three-point matches is the ease with which match data at various scores can be accurately approximated.  No need to memorize a bunch of numbers to clarify cube actions at each match score in three-point matches.  The approximation is not quite accurate enough at higher match scores.  But for three-point matches, four-points, and many five-pointers, it is.

What is match equity?  Why does match equity matter? 

The main difference between cash games and match games is the objective: to win the match.  Unlike a cash game session, a cube position can be a take of a double at one match score but a pass of the double at another match score.  Hence the probability of winning the match is the final arbitrator of each and every cube decision and checker play.  Each position still has probabilities of winning or losing the game, either single games or gammons.  But these game-winning chances are the servants of the chances of winning the match at various match scores while the match progresses.  At positions within a game, estimating the match chances of a position given the experience and intuition developed by cash game play can be quite complicated.  However, between games in the match, when the checkers are being set up and one player has just improved her match score, the chance of winning the match from the opening configuration can be calculated beforehand.  This match equity before a new game is then known from tables.  These numbers say when a player in the match can pass an offered double.  Such data are collated into a match equity table (MET).  The current gold standard of MET was developed by Kazaross and Rockwell, with a slight modification as Kazaross XG2.  The main table applies before and at the Crawford game.  Simpler data applies after the Crawford game.

For three-point matches, a simplification to the tables requires no memorization.  The same approximation works nicely with four-point equivalents and some five-pointers. 

Before the simplified table is shown, the jargon has two identical ways of referring to the current match score: the direct way and the to-go way. For example, a match score is Black one and White zero in a three-point match. This is the direct way. Same example, Black is 2-away while White is 3-away is the to-go way.

In all discussions except cube actions this essay will use the direct score wording.  But during cube actions the match equity will use away score wording. 

My simplified table for three-point matches (called baronMET) and its value at each score is:

For her: ( his away # ) / ( her away # plus his away # )

For him: ( her away # ) / ( her away # plus his away # )

The baronMET is:

This table is a very good simplification of the KR-MET for three-point matches (and four-point matches, and most later games of five-point matches).  Let’s be crystal clear – baronMET is not a new MET derived separately but a simplified mnemonic to the existing K-XG2-MET.

The essay restricts the match to three-pointers. 

Each value in baronMET is the probability of winning the match for him or for her at the start of the game with the current away score, including the Crawford game.  After the Crawford game, it is simple enough to discuss individually the isolated pathways to victory. 

When does the MET tell the whole story?  And when not?

First, a brief review of the MET shows that winning a single game results in different increases in match-winning chances at different match scores.  In the first game, should Black double White and White pass, then Black’s match-winning chances increase from 1 / 2  at the start of the match to 3 / 5 at the start of the second game; White’s match chances decrease from 1 / 2 to 2 / 5 at the start of the second game.  During game two, should Black double and White again pass, Black’s match-winning chances increase to 3 / 4 and White’s decrease to 1 / 4 at the start of the third game, the Crawford.  Black’s match chances increase by 10% after the first dropped double, and increase by a further 15% after the second dropped double. 

The MET tells the whole story when assessing whether to pass a double.  Thankfully, the needed match-winning chances are easily recalled from the away scores.  When considering a take of the cube, the player now can focus instead on the valuable hard work:  an estimate of the game-winning chances, including estimates of gammon possibilities for both players.  For simplicity, triple games are ignored (but counted as gammons).   Once a best estimate of the game-winning chances is made, the scenarios that lead to the end of the match are constructed, and the probability of winning the match by dropping the cube is compared to the match-winning chances from the match equity table.  Worse chances, pass.  Better chances, take.  Of course, the situation must be designated as before, at, or after the Crawford game.

A game position is described as fractions for Black (W,S,G) and for White (w,s,g) where G and g are gammons wins, S and s are single-game wins, and finally W and w are game victories of any kind:  Clearly W = S + G  and  w = s + g  and someone always wins every game, hence the identity 1 = W + w.

What about when to double?  The MET can help decide when to double, but the table cannot establish the decision precisely in all situations, primarily due to cube ownership that varies with board position.  Cube ownership means only one player can redouble.

What is all this talk about Take Points?

During long matches, a statistic called the Take Point is often used to help decide whether to pass an offered cube.   The Take Point applies the same lengthy calculations at each score, but for short matches it is numerically cumbersome, needlessly so.  The above plan – construct scenarios that lead to the end of the match — replaces the Take Point computations for three-point matches.

Before the Crawford game there are four meaningful take decisions that depend on both the game chances of the position and the current score of the match.  Not all occur within a specific match.

• Black doubles in the first game.  White decides whether to pass the cube.
• White doubles the first game.  Black takes.  Later, during the game, Black redoubles.  White decides whether to pass the cube.
• In the second game with score Black one and White zero, Black doubles.  White decides whether to pass the cube.
• In the second game with score Black zero and White one, Black doubles.  White decides whether to pass the cube.

Of course, the colours Black and White can be swapped.  There are other situations during a three-point match where a player would take the cube, but these situations are automatic and have little direct relationship to the game-winning chances.  These situations have everything to do with match-winning chances.

What happens after the Crawford game?

In a three-point match the Crawford game is the fulcrum between cubes primarily based on game chances beforehand and exclusively based on match score afterwards.  The Crawford game occurs immediately after the first player gains a score of two in a three-point match: for example, score Black 2* where the asterisk denotes the Crawford game, and neither player can use the doubling cube during this game.  The cube goes silent.  There will always be an endgame and a bear off during a Crawford game.  Gammons still count, thus Black (the match Leader) tries to avoid being gammoned.  For Black to win a gammon is unnecessary, but acceptable – any Black win finishes the match.  When White is specifically 2-away during the Crawford game, a White gammon victory also wins the match for White. 

After the Crawford game the score Black two and White two is called double match point (DMP) and whoever wins this game wins the match.  This game too is cube silent.  The DMP game is played until the endgame and the bear off remove the last checker from the board. 

When White (the match Trailer) wins the Crawford game, White’s score will either be White one or White two – White odd or White even.  Black’s score is still two.  White will double immediately in the game following the Crawford, either before the second roll or the third roll of the dice depending on who wins the opening toss. 

Suppose the score is Black two and White one after the Crawford game.  White doubles.  If Black takes, then this game decides the match.  If Black drops, then the following new game decides the match with equal chances beforehand.  In other words, only one game remains in the match and Black (the match Leader) gets to decide whether it will be the first or the second post-Crawford game.   At White’s very early cube, if Black has any tangible advantage, however small, Black should take the cube in the first post-Crawford game.  As a mnemonic, the score White one is White Odd and Black can choose to take or drOp with little cost – the “free drop”.  Either way, the match has only one game left. 

An aside:  In longer matches post-Crawford, such as 5-pointers with score Black 4 and White 2 or White Even, then Black has a single ”must takE” assuming Black avoids gammon.  Black thus plays two games to try to finish the match, so any success in the first game after the Crawford is gravy.

Because of cube silence or desperation cubes that depend only on match score, the checker play becomes old-school for Black (the match Leader) at and after the Crawford game.  Black aims to win a single game during a quiet and boring bear off and tries to avoid being gammoned.  Simple game plans and low volatility positions prevail for Black.  White of course wants the opposite.  White wants high volatility games producing gammon situations and lots of complexity and contact.

It is legal to double in the double-match-point game, but meaningless.  The cube will always be accepted.  Etiquette leaves the cube untouched.

When to double?

The match-winning chances (abbreviated MWC) strongly influence when to double, but are not the only factor.  During any game the value of owning the cube changes.  An example familiar to every player occurs very late in some bearoffs where the cube is sometimes worthless before the last roll.  During the bearoff the possibility of cashing the game after a lucky streak varies considerably, and so does the value of cube ownership.  A half century ago Oswald Jacoby and John Crawford published a bearoff position known as Jacoby’s paradox where the effect of cube ownership is clearly and undeniably illustrated. 

Since the game-winning chances (abbreviated GWC) are subordinate to the match-winning chances during a match, cube ownership is even more subtle. 

Any advice as to when to double is still valuable.  Here are some ideas to think about.  Examples will appear in later posts.

In medicine a doctor swears the Hippocratic oath: To do no harm.  In match backgammon there is a selfish do-no-harm version of the oath.  Do not double if your opponent’s pass (or take) leaves your MWC less than the chances at the current score. 

This next piece of advice is the match equivalent of playing on for the gammon during a cash game, where the position is too good to cube.  In matches the window of MWC in each game varies with score.  A joker exchange of your good luck followed immediately by your opponent’s bad luck can vault the position into a “too good” to cube.  This circumstance inaugurates the great descent: a series of rolls in a very strong position that may or may not dip below the MWC and trigger Double/Pass, cashing this match game. 

Perhaps surprisingly, once every couple hundred three-point matches, a unicorn triple game will finish the match in the first game without use of the doubling cube.  The scenario is an early joker exchange transforming the cube from no double to the great descent, followed by a miraculous backgammon during the bearoff. 

In short matches, gammons for both White or Black can be demoted to the equivalent of mere single wins or promoted to a knighthood.  This situation is known as the gammon price.  An obsession of estimating and appreciating gammons is therefore part of match play. 

Redoubles in three-point matches are confined to two situations: in the first game or in some second games.  Although redouble opportunities are infrequent, they are still vital.  The reason is the rewhip:  a Trailer who makes the redouble succeeds in demoting all gammons to single game wins or, stated another way, all game wins are a match victory, single games or otherwise, for both White and Black.  Thus previous progress in the match is forgotten, usually to the chagrin of the Leader.  The current game position alone determines the match.  The rewhip must be factored into many doubling decisions.

During a short match the fluctuating balance between score-based doubles and position-based doubles must be watched and understood. 

Lastly, there is something to be said about the terms “probably” versus “in the bank”.  Passes become known MWC that are in the bank.  Nearly equal MWC situations, where the game continues, allow the spectre of human error to muddy the waters.  Whether the stronger or weaker player, gamesmanship can sometimes be used to advantage, particularly during very close cube decisions.

What are some checker play subtleties?

This essay is primarily about cube play in three-point matches.  However, checker plays also can be tailored to a match score.  In the opening, where some positions have several close choices, a style of opening is sometimes favored based on the score.  Ahead or behind in a match suggests preferred game plans.  Not every player steers the play to his advantage.  The good players do.

Cube silence means endgame and bearoff skills are quite important as checker plays.  Study and master bearoffs.  Keep alert to endgame simplifications when match simplifications are advantageous.

In short matches blitzes are often the straw that breaks the camel’s back and wins the match.  Become obsessed with the tactics of blitzes.  Study them.  Master them.  Know which scores favour the bold. 

In short matches some anchors wait forever.  Depending on the score backgames and deep holding games remain an eternal thorn in the opponent’s side during a key game.  Unlike cash game play sometimes your gammon losses in a match are irrelevant. 

The opposite side of the coin, know when saving gammons is paramount and at which scores. 

When trailing in the match try to steer the game into volatile situations and maintain an interest in the specifics of market losers.  Even in matches the concepts of market losers still apply.  Sometimes in match play try to make your own luck by embracing volatility.

Lastly, are there any crazy doubles that are correct?  Leave this situational question unanswered for now.

Practice positions?

In future posts about the cube in three-point matches, there will be many examples of positions posed as problems.  Some of these situations can be saved as interesting benchmark positions.  There will also be practice with the simple baronMET calculations, as promised. 

What to study next in match play?

After the rudiments are mastered a retrospective will suggest bigger ideas to study and understand. 

For example:

• Understand the great descent where it is too good to cube.
• The details and improvement of blitzing skills.
• Practice positional estimates of gammon chances.
• When running is a favoured game plan.
• Stepping stones to longer match scores.
• Watching bot versus bot battles in three-pointers.
• The personality of each cube at all match scores.

Here ends the outline for future posts about the cube in three-point matches.