Throughout this module, we’ve gone through different situations from every game plan with prescribed advice. However, not every position can be neatly classified into categories with tidy rules. Sometimes, we need to use judgement to evaluate cube action. Building sound judgement takes experience and analysis, but here is a small shortcut to get you going. It’s called O’Hagan’s Rule, first described in our article on Market losers. Coined by 10x Giant of Backgammon John O’Hagan, the rule states that:
If you can find 9 market losing sequences out of 36, and you’re doing all right in the other 27, then you probably have a good double.
Below is the simplest example of O’Hagan’s rule I could come up with:
Gary was forced to leave us a shot that, if hit, will guarantee us a win. Otherwise, the race is dead even, and we have a 50/50 game to play out. The hit is certainly a market loser, and happens when we roll a 2 on any die (11 times out of 36). The rest of the time, we’re doing all right. According to O’Hagan, this is a clear double. Importantly, this position highlights that, whenever a hit would result in losing your market, a direct shot will always provide a sufficient number of market-losing sequences.1
Below is a more involved in-game position:
This position comes from a mutual-holding game, and doesn’t fit into the mold of any of the other articles in this module. Can O’Hagan’s rule help us here? Let’s analyze this position and see!
Starting with Gary’s homeboard, he has a 4-pt board, and we hold the golden point anchor. Despite Gary’s evident board strength, we aren’t really being primed in thanks to our advanced anchor. On our side of the board, we have a relatively weaker 3-point board, somewhat priming Gary’s 2-pt anchor. We also have 13 checkers in the zone with moderate flexibility (spares on the 6- and 8-pts, plus checkers to play on the 10-pt). There are pros and cons to the position, but overall it feels like we’re ahead. Should we double yet?
Well, what are our market losers in this position? First, if we roll a large double (66 or 33), then we’ll free our backcheckers, resulting in something that looks like a holding game. We already know that a 2-pt anchor doesn’t usually provide enough contact to warrant a take in a holding game, so that probably makes for 2 market losers.
What other sequences can lose our market? With a bit of luck, we might be able to fill the 4-pt, equalizing our board strength and forming a broken 5-prime. If that happens, Gary’s escaping chances become slim due to the prime and for fear of a blitz attack. He’ll also soon be forced to give up his outfield control.2 Any subsequent hit from this moment on would become fatal. These all add up to a game Gary will no longer want to play. How often does this scenario play out? We can comfortably make the 4-pt when we roll a combination of {2, 4, 6}, 22 or 66.3 This adds 22, 24, 26, 46 = 7 rolls to our total. We’re up to 9 market losers! Is this enough to double?
It turns out not to be quite enough. Why is that? This position calls attention to what happens if you’re not “doing all right in the other 27” sequences, which is the oft-forgotten second half of O’Hagan’s rule. We probably don’t double right away if Gary rolls 22 when we don’t fill in the 4-pt, or a 55 after any of our rolls? We therefore need to subtract these bad sequences from our total. Let’s round up, and subtract 2, leaving us 7 net market losers. Indeed, it’s not the number of market losers we need to count, but the net market losers.
Notice how hard this is to get right: you need to consider all of your rolls, the good ones, the bad ones; all of your opponents’ rolls that might negate your good rolls, all of your opponents’ rolls that might turn average rolls into bad ones, etc. It’s challenging, but fun to practice, and feels really rewarding to get right.
Since we’re so close to the threshold, let’s adjust the position ever so slightly to give us just a handful more market-losing sequences to see O’Hagan’s rule in action:
Here are 2 variations of the original position. In the first, we moved one checker 6/5, which creates a new builder for the 4-pt. Now, in addition to combinations of {2, 4, 6}, we can also use 1 to fill in the 4-pt, which adds 11, 12, 14, 16 to our good rolls. Adding to the 7 net market losing sequences we counted previously, we now have 14, and a good double! The second position breaks Gary’s midpoint by playing 13/10 (12/15 from our vantage point), giving us 5 fly shots, of which 3 are new numbers (53, 44), adding up to 10 total net market losers. That’s above 9, also making this a proper double.
One thing about backgammon that continues to fascinate me is how moving one checker a handful of pips can sometimes change the nature of a position enough to dramatically affect the right decision. O’Hagan’s rule is one piece of the puzzle to help discern why this happens in cube decisions. The rule turns out to be surprisingly universal! After reading this lesson, I encourage you to go back through the doubling module and see how it applies in every situation. I guarantee you’ll learn something you didn’t the first time around.
- A video of O’Hagan discussing his rule.
- As for Gary, if missed (25 of 36 times), he has a 50% winning chance, which combine to way more than a 25% (12.5/36 > 9/36), and therefore a take. ↩︎
- How would you play 64 as white, for example? ↩︎
- We don’t count 44 since it leaves a direct shot. ↩︎
Next lesson: Score-based cube actions
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