# Simulating Flow

One of the features that I’ve added to the new version of the simulator is the ability to simulate the flow of a tournament in which rounds overlap. This post will introduce this feature, and in the next I’ll report what I think is an important simulation result that using the feature made possible.

Here’s a single match from a new kind of analyzed bracket that the flow simulation makes possible. In a working bracket, the names of the two teams would occupy the top and bottom lines, but here the bracket reports several numbers, in various colors, instead. The numbers that relate to the individual player of team on a line are one that line, or just below it, and numbers that relate to the match itself are centered between the two lines. In this example, the numbers come from an simulation actual simulation that was run 1,000,000 times.

The blue D2 indicates that this line is filled by a drop from a higher bracket – the second match in the fourth, or “D”, round. The upper line is attached to a previous match in the same bracket, and so has no drop marker.

The black numbers, 0.175 and 0.730, represent the skill levels of each player, expressed as Z scores. Thus, the player on the top line is 0.175 standard deviations better than the average skill level for the population. The player on the bottom line is considerably better, at 0.730 standard deviations better than average. This large difference one of several things about this match that suggest that this is rather a peculiar match.

The green numbers represent the average prize money equity for the occupant of the line. For most analyzed brackets, I’ll base this on the assumption that the total prize fund is \$100, which means that the green numbers can also be interpreted as showing a percentage of the prize fund. In most of my previous analyzed brackets, the analogous number was the percentage of overall wins that could be expected from that line, but the new simulator allows the simulation to consider a number of different payouts, which means that it can cope with formats like consolation tourneys, where some parts of the bracket have no route to overall victory. This technique for doing this was discussed in conjunction with the revised fairness (C) measure. Note that the two players, though in the same place in the bracket, have quite different expectations – the D2 drop is, on average, a better player, and so can expect to win more money.

The red 6.5% repeats shows that there is a small possibility that this match is a rematch between two players or teams that have already met earlier in the tourney. With properly drawn drops, most matches, even in lower brackets, should avoid repeats all together, and in those cases there is no repeats line on the analyzed bracket. In the later rounds of a lower bracket, it is impossible to completely avoid repeats, but well-allocated drops will tend to reduce their frequency.

The numbers that are left are the ones that simulate the flow. The grey 4:03 indicates that this match begins, on average, four hours and three minutes after the tourney started at the notional hour of twelve noon. The red numbers below the lines on the right side show the average time, in minutes, between the end of the last match for the player on the line and the start of this match. Thus, the player on the top line would, on average need to wait an hour and 49 minutes for the D2 drop to be determined and the match to begin. This is, as you might imagine, an unusually bad case, showing that this format, or at least this match, has a severe problem with flow.

The calculation of the times for individual matches is based on a number of factors, most of which are parameterized so that I can attempt to simulate the actual flow of various kinds of event. Each match has a notional length, which varies according to three factors. First, each hypothetical player has a normally-distributed playing speed which stays with the player for the duration of the iteration, so that a player who is slow in one match will also tend to be slow in later matches – here I’m trying to simulate the tendency of a single slow player to retard the progress of a whole section of the bracket. There’s also a factor that varies the time for individual games, with an adjustable variance. And each match in the table can be made faster or slower, on average, by specifying is as best of three, best of five, or whatever, which also adds to the variability for the match as a whole.

In the next post, I’ll post some analyzed brackets applying flow tracking to a couple of particular tourney formats, including the one that includes the peculiar match shown above.