Softball.app Lineup Optimizer Gallery

Monte Carlo Exhaustive

Runs Monte Carlo simulations of games for all possible lineups. The optimizer then averages the runs scored acrosss all possible lineups and returns the lineup with the highest average runs scored.

Overview

Simulation flow. This flowchart refers to the parameter g which is a configurable value indicating the number of games to simulate. The stages with thick black outlines have sections w/ additional details below.

flowchart TD; A[Select a lineup from the lineup pool.]:::addtDoc-->B; B[Simulate a game, remember the number of runs scored.]:::addtDoc-->D D{Have we simulated more than *g* games?}--yes-->E D--no-->B E[Compute the mean score of all simulated games using this lineup.]-->F F{Is this lineup's mean game score better than all the others we have simulated?}--yes-->G F--no-->I G[Remember this lineup as bestLineup and its mean score as bestScore.]-->I I{Is the lineup pool depleted?}--yes-->J I--no-->A J[Done! Print the bestLineup and bestScore that we remembered earlier.] classDef addtDoc stroke:#333,stroke-width:4px;

The Lineup Pool

This optimizer simulates games for all possible lineups. The number of possible lineups for each lineup type is given by these equations where 'm' is the number of male batters and 'f' is the number of female batters:

Standard

numberOfLineups = (m+f)!

Alternating Gender

numberOfLineups = (m! + f!) * 2

No Consecutive Females

numberOfLineups = m! * f! * (\\binom{m}{f} + \\binom{m-1}{f-1})

Performance

Because of the factorial nature of these equations, adding even one more player to the lineup can make the optimizer take significantly longer to run.

Example simulation (7 innings, 10000 games)

# players in lineup	possible lineups	runtime (ms)	~runtime (human)
6	720	1958	2 seconds
7	5,040	15982	16 seconds
8	40,320	135255	2 minutes
9	362,880	832902	14 minutes
10	3,628,800	6613484	2 hours

Simulating a game

This flowchart refers to the parameter i which is a configurable value indicating the number of innings to simulate (typically 9 for baseball, 7 for softball, but can be anything). Same as before, the stage with the thick black outline has a section w/ additional details below. Simulate a game

flowchart TD; B[Set active batter to the first player in the lineup.]-->C C{Simulate a plate appearance for the active batter.}:::addtDoc--Result: 0-->D C--Result: 1,2,3,4-->E D[Increment the number of outs by one]-->F E["Advance the runners. Increase score if the plate appearance drove home (a) run(s)."]-->G F[>= 3 outs?]--no--> G F--yes-->I G[Set active batter to the next batter in the lineup.] G-->C I{Have we simulated more than *i* innings?}--yes-->J I--no--> K J[Done! Return the score for this simulated game.] K[Increment the number of innings simulated by one. Clear the bases. Clear the outs.]-->G classDef addtDoc stroke:#333,stroke-width:4px;

Simulate a Plate Appearance

Each plate appearance result (Out, SAC, E, BB, 1B, 2B, 3B, HRi, HRo) is mapped to a number indicating the number of bases awarded for that plate appearance. The mapping is illustrated in this table:

Result	Bases
Out, SAC*, E, K	0
1B, BB*	1
2B	2
3B	3
HRi, HRo	4

We can then use the frequency of each type of hit to build a distribution that reflects the way any given player is likely perform when they get a plate appearance. Whenever we need to simulate a hit for that player, we draw a random sample from that player's distribution.

An Example

Tim's historical at bats are as follows: Out,1B,2B,SAC,E,HRo,3B,1B,1B,Out,Out,2B,1B,Out,Out

First we translate those hits to number of bases using our mapping from the table above: 0,1,2,0,0,4,3,1,1,0,0,2,1,0,0

Then we determine the histogram and chance of each hit:

# of bases	# of times	% of plate appearances
0	7	47
1	4	27
2	2	13
3	1	7
4	1	7

And every time we simulate a plate appearance for Tim, we'll draw a random hit with that distribution. That is to say, for every simulated plate appearance, Tim has a 47% of getting out, 27% chance of getting a single, a 13% chance of getting a double, a 7% chance of getting a triple, and a 7% chance of getting a home run. Of course, other players will have their own distribution of hits to draw from based of their historical performance.

Other Notes

Things that are not accounted for in the simulation:

Double/triple plays
Stolen bases
Players who were on base advancing more bases than the hitter
Any pitching data

*We can debate about how walks or sacrifices should be counted. It probably depends on what flavor of the sport you are playing. IMHO sacrifices should be counted as outs in slowpitch softball and kickball, but not baseball or fastpitch. In any event, these mapping are configurable (or will be configurable soon). So you are welcome to impose your own philosophy.

Monte Carlo Adaptive

Employs the same approach as the Monte Carlo Exhaustive optimizer but instead of simulating a fixed number of games for each lineup, performs a variable number of simulated games. The exact number of games simulated for each lineup is determined by continuing to do simulations on a lineup until a statistical t-test determines that the expected run totals for two lineups are significantly different (by some configurable alpha value). The lineup with the lower mean is then rejected and the larger one remembered as the best so far.

Monte Carlo Annealing

A faster (time constrained), less accurate optimizer that doesn't test the entire search space of possible lineups. Instead, it employs simulated annealing to seaerch only a subset of possible lineups. Like the Monte Carlo Adaptive optimizer, this optimizer uses statistical t-tests to determine when a particular lineup is better or worse than another.

Expected Value

Calculates the expected runs scored mathematically up to a specified max number of batters. The number of batters is limited because there there are an infinite number of possibilities i.e. many teams could theoretical bat forever.

Dummy Test

Calculates the expected runs scored mathematically up to a specified max number of batters. The number of batters is limited because there there are an infinite number of possibilities i.e. many teams could theoretical bat forever.