When I introduced my goodness-of-fit test against pure luck last month, the objection was raised that it didn’t take into account the skill of opponents, that is, a win against better opponents should count more than a win against not-so-good opponents. Ryan doesn’t provide us a breakdown of wins according to quality of opponent, but if you keep track of how often you play at a particular level of table, you can make a rough quality-of-opponent adjustment to your test against pure luck score in the manner described below.

If outcome of a game is due to pure luck (as if each player was randomly assigned a number from 1 to 7 and whoever drew the 1 was first, whoever drew the 2 was second, and so on), the Elo scores would be based purely on luck. Those with higher Elo scores would advance to a higher table, but the opponents at the higher table would also have arrived there as a result of pure luck. Therefore, for all games played at that table, your distribution of wins would resolve to 14-14-14-14-14-14-14, just as at the lower table. By extension of the reasoning, your pure luck distribution would be 14-14-14-14-14-14-14 at all levels of table at which you play, the only difference being that the number of games you play at the higher tables would be less because you would be eligible to play there less often.

The Quality of Opponent Adjustment (QOOA) is a comparison of how many times you play at a particular level of table compared to how many times you would play at that level if your Elo score was a matter of pure luck.

At my last check, there were a total of 42 tables, 24 of which have the zero minimum, 5 of which have the 1500 minimum, 8 of which have the 1600 minimum, and 5 of which have the 1800 minimum. One might think that there ought to be as many tables above 1500 as below 1500, since 1500 is supposed to represent a mean or median value of play, but the likely reason for there being more lower-level tables is that people don’t always play at the highest level of table that they are eligible for (you might find some 1500-er’s or 1600-er’s at a zero minimum table, for example). So I will take the current distribution of tables as the datum against which to calibrate the QOOA. In percentage terms, the distribution is 57%-12%-19%-12% going from lowest to highest.

To calculate your QOOA, divide each of your four percentages of actual attendance at a given level of table by the percentages I just listed, take the sum, and divide by 4. For example, if your distribution of games at the four levels of table is 50-20-30-0, your QOOA would be (50/57 + 20/12 + 30/19 + 0/12)/4 or 1.03. If your distribution is 70-10-20-0, your QOOA would be (70/57 + 10/12 + 20/19 + 0/12)/4 or 0.78. Multiply your test against pure luck (TAPL) score by the QOOA to get a score that is adjusted for quality of opponent.

Now someone may ask the question: could I not improve my QOOA by never playing at the 1600 level tables again and playing only at the 1500 level tables, since the denominator for the latter is smaller? If my distribution is 70-30-0-0 instead of 70-10-20-0, my QOOA would be 0.93 instead of 0.78. The problem with this reasoning is that there are not enough 1500 tables to meet the extra demand. Ryan would have to create more 1500 level tables for all the 1600-er’s who decide to “slum� down to that level and the datum would have to be recomputed. Furthermore, the quality of opponents at the 1500 level tables would increase and I would have to earn my 0.93 or I would wind up playing at the zero minimum tables.