Mexico (game) - Strategy and Odds

Strategy and Odds

There is a moderate amount of strategy involved in Mexico, thanks to the fact that the first player to roll determines the conditions of play on a given round by re-rolling up to twice if he or she isn’t satisfied with their initial result. On the other hand, if any subsequent player is satisfied with a particular roll, that player is not forced to roll as many times as the leading roller. Thus, any player can stop after the first roll if they like (or anytime before the maximum role allotment), even if his or her result isn’t the highest yet rolled, which follows from the fact that in this game one is only concerned to avoid rolling lowest in a given turn. Accordingly, the worst thing someone can do as lead roller is roll three times, only to end up with a mediocre result. Therefore, unless one has rolled something very easy to beat, and unless the field of opponents is extremely narrow (particularly in the case in which only one opponent remains) it is best to satisfy oneself with an average or even slightly below-average single result.

The odds against any particular non-double result are 17 to 1, and the odds against each double are 35 to 1. However, because of the game’s open rolling structure and idiosyncratic adding system, determining what is a good roll is somewhat counterintuitive, and the most important thing to know is the median result for a given roll (i.e., the result such that it is an even proposition whether the next roll is likely to beat it). The median first roll result, in terms of the likelihood of some other result beating it, is 54—not 62, although there are ten possible results above and below the latter—one of the game’s oddities. Thus, the main thing to be kept in mind on the first roll is where one stands in relation to 54. However, a good result on the first roll is entirely different from a good result on the second or third, with the median result becoming more and more difficult to attain on each reiteration: The median results for the second and third rolls are, respectively, 64 and 1-1 (double ones).

Naturally, if one knows the likelihood that they will beat their own last roll by rolling again, they also know the likelihood that someone following them will do the same. For example, if a player were to roll a 52 and then stop, that player would be looking at approximately a 61% probability that the next person will roll something equal to or better than that in one try. In other words, their next opponent would be a 3-to-2 favorite to beat or tie them. However, if the first player were to decide to roll again, he or she would then need to roll a 62 on their second turn, or a 64 on their third to maintain even this mediocre statistical position. Another result of 52 on a second roll would raise the likelihood of losing to or tying an opponent with two chances to roll to roughly 85%. This is of course a terrible position to be in, and the lead roller would almost certainly roll a third time, which would, in turn, give opponents more chances to beat him or her. If the lead roller were sufficiently hapless as to come up with yet another result of 52 on the third roll, an opponent would have approximately a 95% likelihood of beating or tying that result in three attempts, making our first roller a 19-to-1 underdog for that round - just as if that player had rolled for the lowest possible result, a 31, and then stopped after just one roll.

Below is a statistical breakdown of all possible rolls and their likelihood of being beaten (but not tied), according to the number of attempts an opponent has to roll against them. The median rolls, depending on how many rolls have been taken, are in boldface text.

Result (x)	Likelihood: Any 1 roll > x	Likelihood: Any 1 of 2 rolls > x	Likelihood: Any 1 of 3 rolls > x
31	94.44%	99.69%	99.98%
32	88.89%	98.78%	99.86%
41	83.33%	97.22%	99.54%
42	77.78%	95.06%	98.90%
43	72.22%	92.28%	97.86%
51	66.67%	88.89%	96.30%
52	61.11%	84.88%	94.12%
53	55.56%	80.25%	91.22%
54	50.00%	75.00%	87.50%
61	44.44%	69.14%	82.85%
62	38.89%	62.65%	77.18%
63	33.33%	55.56%	70.37%
64	27.78%	47.84%	62.33%
65	22.22%	39.51%	52.95%
Double 1	19.44%	35.11%	47.73%
Double 2	16.67%	30.56%	42.13%
Double 3	13.89%	25.85%	36.15%
Double 4	11.11%	20.99%	29.77%
Double 5	8.33%	15.97%	22.97%
Double 6	5.56%	10.80%	15.76%
21 ("Mexico")	0.00%	0.00%	0.00%

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