MA331 - Setting a rotation

From Class, Monday:

You are putting together a pitching rotation that will be pitching in the Whirled Series. You want to put together a rotation that has the most expected wins in the seven games. Your four pitchers, labeled A,B,C,D, will pitch in this order: ABCDABC.

You are selecting the ability levels of your pitchers (are you going to try to have a consistent rotation or a star or two?). You will select the quality of each pitcher by listing the expected runs allowed for the pitchers subject to the following limitations:
I The sum of the runs, $A+B+C+D$, must be at least 13.
II No pitcher may have a runs total larger than 4.

Examples of valid rotations: 3.25,3.25,3.25,3.25 (sum = 13),
2.76, 3.24, 3.01, 3.99 (sum=13)

For Tuesday:

You are putting together a pitching rotation that will be pitching in the Whirled Series. You want to put together a rotation that has the most expected wins in the seven games. Your four pitchers, labeled A,B,C,D, will pitch in this order: ABCDABC.

You are selecting the ability levels of your pitchers (are you going to try to have a consistent rotation or a star or two?). You will select the quality of each pitcher by listing the expected runs allowed for the pitchers subject to the following limitations:
I The cost of a pitcher expected to allow R runs per game is 10/(R-1) million dollars.
II You can spend no more that \$20 million
III No pitcher can have a runs total less than 1.

Examples of valid rotations: 3,3,3,3 (cost = 4 times (10/2)=20)
2,4,3.5,4.75 (cost =10 + (3 + 1/ 3) + 4+ (2+2/ 3)=20)

Here's some Maple Code to match up pitching rotations (rot1,rot2). I've modified the code so that it shows the expected number of wins for the first rotation:
ws:= proc (rot1,rot2) local games,ser;
games:= [seq( rot2[k]^2/ (rot1[k]^2+rot2[k]^2) , k=1..4), seq( rot2[k]^2/ (rot1[k]^2+rot2[k]^2) , k=1..3)];
ser:= add(games[k],k=1..7);
games,ser; end;
Here are the results from the rotations passed in during class Monday: vs. Kissel
Balasundaram (1.76,3.64,3.75,3.85)
McCue (3.99,2.30,2.95,3.76) Won 3.502G
Barton (1.20, 3.93,3.93,3.94) Won 3.73G
Stuchel (2.75,2.75,3.51,3.99)		 McCue vs. Barton 3.54G
Sullivan (3,3.5,4,2.5) Won 3.55G
Chaille (3.713333, 3.713333,3.713333, 1.86)
Bradley (2.60,2.75,3.75,3.90) Won 3.52G
Whitaker (3,3,3,4) 		Sullivan vs. Bradley 3.73G Barton 3.75G vs. Bradley
Huster (2.5,2.5,4,4) Won 3.56G
Albert (3.5,2.65,2.85,4)
Freihaut (3.98,2.52,2.52,3.98)
Beverlin (2.5,2.5,4,4) Won 3.51G		 Huster vs. Beverlin Tie Huster/Beverlin 3.57G
Self
(2.74,3.37,2.99,3.9)
Schwarzmann
(3,3,3,4) Won 3.51G 		Schwarzmann vs. Shields 3.504G Shield vs. Kissel 3.51G
Shields Won 3.53G
(2.8,3,3.2,4)
Forsyth
(2.70,3.85,2.78,3.67)
Kissel Won 3.51G
(2.7,3.55,2.8,3.95)
Brosmer
(3.0,3.0,3.0,4.0) 		Kissel 3.51G vs. Wells
Wells Won 3.90G
(3.02,2.99,2.99,4)
Bye
(4.0,4.0,4.0,1.0)
The final round had Barton 3.73G over Huster/Beverlin.
In a round-robin tournament, pitting each rotation against all the others, so the average number of wins is 19*3.5=66.5;
Barton 70.87 Balasundaram 68.50 Freihaut 68.34 Beverlin 68.13
Huster 68.13
McCue 67.54 Albert 67.32 Stuchel 67.09 Bradley 67.06 Kissel 66.97
Shields 66.85 Wells 66.811 Brosmer 66.806
Schwarzmann 66.806
Whitaker 66.806
Self 66.55 Forsyth 66.26 Sullivan 62.46 Chaille 60.98 Bye 59.71
Notice that most of the "big winners" stacked the rotation so that the fourth pitcher was at or near 4 expected runs allowed. Conceding strength in spot D can be made up double in the other rotation spots. Another notable feature is that the strength of the rotation is not "transitive". i.e. if rotation 1 beats rotation 2, and rotyation 2 beats rotation 3, it is possible that rotation 3 could beat rotation 1. For example,
Balsundaram [1.76,3.64,3.75,3.85] vs. McCue [3.99,2.30,2.95,3.76] is won by McCue 3.502 to 3.498
        Game by game for McCue: (.1629,.7147,.6177,.5118, .1629,.7147,.6177)
McCue [3.99,2.30,2.95,3.76] vs. Huster [2.5,2.5,4.4] is won by Huster 3.527 to 3.473
        Game by game for Huster: (.7181,.4584, .3523, .4691, .7181,.4584, .3523)
Huster [2.5,2.5,4.4] vs. Balsundaram [1.76,3.64,3.75,3.85] is won by Balasundaram 3.562 to 3.438
        Game by game for Balasundaram (.6686,.3205,.5322,.5191, .6686,.3205,.5322)
McCue's rotation matched up nearly perfectly with Balasundram's. The sacrifice of games 1 and 5 were made up for in the other games. Against most of the rotations submitted, the overwhelming strength in games 1 and 5 made up for the slight disadvantage in the other games, leading to the second strongest overall performance in the round-robin tournament (behind a staff with a 1.20 ace leading off).
The average rotation submitted was (2.92266666,3.12566666,3.3366666,3.615). When this was pitted against the 20 rotations submitted, the expected wins were only 68.87 vs. 71.13 expected losses. Apparetly, unbalancing the rotation generally improved the series performance.
Using the cost function C=10/(R-1) to restrict the rotations, the following round robin tournament results we obtanied:
Chaille 69.35
[2.733,2.853,2.79,4.080322521] 		Laser 69.30
[2.75,2.75,2.75,4.5] 		Stuchel 69.02
[2.6,2.7,2.9,4.84] 		Bradley 68.78
[2.5,2.75,3,4.82]
Self 68.29
[3.54,2.65,2.65,3.54] 		Albert 68.284
[3.2,2.4,3.11,3.8] 		Birchall 68.278
[2.455,2.845,3.365,3.875] 		Huster 68.13
[2.4,3.2,3.11,3.8]
Forsyth 67.98
[3,3,3,3] 		Shields 67.77
[2.5,2.5,3.5,4.75] 		Kissel 67.71
[2.5,3.5,2.5,4.77] 		Wells 67.27
4,2.5,2.5,4]
Brosmer 67.15
[2.38,2.75,3.75,4] 		McCue 66.84
[3,2.5,3,5] 		Beverlin 65.38
[3.5,3.5,2.67,2.67]		 Freihaut 62.39
[5,2,3,5]
Sullivan 62.34
[2,3+2/3,4+5/33,4.25] 		Whitaker 62.15
[2,3,3.5,11] 		Barton 62.02
[2,4,4,4] 		Balasundaram 61.59
[2,3,4.5,6]

Again, the strength is not transitive. With more time to tihnk about how to set up the rotations, the spread between highest and lowest total was reduced.
An interesting pair of rotations to study are those submitted by Huster [2.4,3.2,3.11,3.8], and Albert [3.2,2.4,3.11,3.8] (do you think they worked together?). When pitted against each other, they tie. When matched against other rotations submitted, the sacrifice of game 1 seemed to be less damaging than the sacrifice of game 2. To see why, consider what happens in games 1 and 2 if they are opposing a rotation that starts (2,3,...). The probabilities of winning games 1 and 2 are:
Albert 2^2/(2^2+3.2^2)-->.2809 3^2/(3^2+2.4^2) --> .6098 .8907
Huster 2^2/(2^2+2.4^2)--> .4098 3^2/(3^2+3.2^2)--> .4678 .8776
The change in expected runs of 0.80 changes the probability of winning, but because we are dealing with a non-linear function mapping runs to wins, the effects will be different depending on where the change is made. The slope is largest near the middle and smaller near the ends. Thus, near the ends, the change of 0.80 will have a smaller effect on wins. Against this opponent, Albert has the 0.80 change nearer the middle of the chart.

Once again, the rotations with the most wins were those that sacrificed game 4 for strength in the other games. This time though, the stronger rotations had ABC of a more consistent quality.
This time, the average rotation was (2.8029,2.90323333,3.187325758,4.584766126), which has a cost of only $18.162 million dollars, so unsurprisingly, when matched with the 20 submitted rotations, did not do well. 68.34 to 71.66.
When considering the average cost, the average spent on the rotation spots was [$6.366M, $5.584M, $4.833M, $3.163M]. This gives an expected runs average of [2.57094, 2.7909, 3.0693, 4.16112]. This rotation actually does fairly well against the class, 72.40 to 67.60.

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