Pga’s can be spotted from their stats because they play each other more often than one would expect from probabilities. What I mean by a pga is a “pre-game affiliate” which could be a pre-game ally, a pre-game enemy, or a pre-game secret admirer. To further separate these three we would need stats on how often they attack each other. For now I will stick with the general category of pre-game affiliate.

Suppose you want to know if players x and y have some sort of affiliation. Suppose that player x has sat down at a table at which player y has not appeared. What would be the probability that player y would show up and sit at that table? It would be the number of empty seats left, divided by the number of players watching at all tables of that level (assuming they are all eligible to sit in), times the probability that player y is currently among the watchers. For each successive game played by player x (considering for the moment only games in which player y has not sat down before player x), the expected number of games that player y would show up (if he is watching) would depend on the number of empty seats and the number of watchers at each game, but for a reasonable number of games (say twenty), the law of averages would set in and we could say that the expected number of times that y would show up would be the probability that y is watching times the number of games times the average number of empty seats when x sits down divided by the average number of watchers. It turns out that the expected number of times that y shows up is governed by the Poisson distribution which for a reasonable number of games has a mean and variance similar to those of the normal distribution. So we can develop a statistic that determines whether player y shows up significantly more often than would be the case if the distribution of players at the tables were purely random. I will call this statistic the Passive Participation Coefficient (PPC) and its formula is

PPC = f * square root (wn/pe)

where f is the fraction of the games played by x in which y did not sit down before x but shows up later, w is the average number of watchers, n is the number of games played by x in which y did not sit down before (whether or not y shows up), p is the probability that y is watching, and e is the average number of empty seats. If the value of the PPC is 3 or more, player y has an affiliation with player x.

I use the word “passive” to describe this statistic because no assumption is made that player x is initiating any attempt to form an affiliation. This is merely player y’s affiliation to player x. To measure player x’s affiliation for player y, we would have to use different data. We would use all the games played by y in which x did not sit down earlier and calculate player y’s PPC. Player y’s PPC is player x’s APC, which means the Active Participation Coefficient (APC). The same number serves two functions. Player x’s PPC with player y is player y’s APC with player x. Since we are looking for intent, it is the APC that we use to determine whether a player is a pga.

Before you rush out and use this formula to try to nail some pga’s, keep a few points in mind. First, you don’t know if the affiliation is friendly or hostile. Second, the value of p is hard to come by. We know how many watchers there are, but we don’t know who they are. To identify watchers, you would have to have spies at all the tables, all the time, when either x or y are playing. Incidentally, if you are spying, you have to subtract yourself from the number of watchers! If you don’t have a good idea of what p is, you will have to give x and y the benefit of the doubt and make p equal to one. Of course that means a smaller APC and a lesser likelihood of your having found a pga. Third, you need a good number of games—at least twenty—to have a reasonably reliable number.

With the above caveats,

Happy hunting!