Why calculate 'risk' and 'reward'?
When calculating (unadjusted) take-points, it is common to make a 'risk-reward' calculation. For example, suppose that are receive a double when we are 5 away/5 away. Consulting a match equity table, we see that:
- If we drop, we will go to 5 away/4 away, so will have 42.3% match winning chances.
- If we take and win, we will go to 3 away/5 away, so will have 64.8% match winning chances.
- If we take and lose, we will go to 5 away/3 away, so will have 35.2% matching winning chances.
Players often say that, by taking, they are 'risking' 42.3% - 35.2% in order to gain a 'reward' of 64.8% - 42.3%. Then they calculate the (unadjusted) take-point using risk/(risk + reward), i.e.
(42.3% - 35.2%)/(42.3% - 35.2% + 64.8% - 42.3%) = 24.0%
I have never understood why you would do this. Instead, I just calculate:
(42.3% - 35.2%)/(64.8% - 35.2%)
Of course, this is the exact same expression (since the 42.3% terms in the denominator cancel) but involves less computation.
Can anyone explain why the 'risk/reward' approach is so popular?
5 Replies
It's because if you know the risk and reward, you don't have to do the division sometimes. For instance, if you recognize that the reward is slightly over 3 times the risk, you know the take point is slightly under 25 and that's good enough 99% of the time. (But I suck at division so do what works for you.)
I agree with Z’s suggestion. I try to avoid doing math over the board as much as possible (and mostly try to memorize instead). But in the rare situations where I do calculate one (e.g., something important that I don’t have memorized, like a big last-roll cube in a long match), I almost never go beyond calculating the numerator and denominator and saying “ok I’m getting a little over a little over 3:1” or “ok I’m getting a little less than 2:1” or similar.
I agree that your approach probably requires fewer steps if you are calculating from scratch and are going to actually do the division.
Another variable to think about is “mental RAM”. For common calculations that come up OTB (eg race cubes) I find it helpful to plan out my order of operations to minimize the amount of numbers/intermediate results that need to be juggled. For example, when I am deciding whether to double in a race, I count my pips, calculate the point of last take, store that number (either mentally or on fingers), then feel free to forget my own pip count while I count the opponent’s pips and compare it to the point of last take. I find that the “risk/gain” approach for calculating takepoints helps me remember the intermediate results, but that’s probably just because I’ve practiced it more so do whatever works for you.
Thanks for the responses. I agree that one will want to use heuristics to approximate the fraction. However, you can do this under either method. Under my method, you could say: "the numerator is about 42% - 35% = 7%, and the denominator is about 65% - 35% = 30%, so the take point is a bit lower than 1/4".
I should stress that the *only* difference in the methods is how you calculate the denominator. In the risk-reward approach, you compute 42.3% - 35.2% = 7.1% (risk), then 64.8% - 42.3% = 22.5% (reward), then 7.1% + 22.5% = 29.6% (risk + reward). Under the simpler method, you just compute 64.8% - 35.2% = 29.6%. Of course, you can use less precise match equities, but this is true under either method.
I don’t know the history, but Kit Woolsey is at least one of the more prominent authors to write about this early on (eg in “How to Play Tournament Backgammon”, and the popular “Five Point Match” article linked below) and he seems to have a preference for using odds rather percentages and the risk/gain terminology (e.g., “2 to 1” risk to gain). I don’t know who invented it, but it is at least plausible that Kit’s writing has contributed to making those the standard terms.
Yeah, that makes sense. So to sum up, if you aren't going to do the division, in both methods you have to do two subtractions and one comparison/estimate. But if you want to do the division, your method saves you from having to an additional addition. So I agree that's more streamlined assuming you don't mind thinking in probabilities (e.g. "1 in 4") as opposed to odds (eg. "3 to 1").
Maybe people think in risk/reward odds because when learning the math it seems more logical that way? I probably will keep doing what I'm doing but I rarely do the division anyways.
