 Little Professor Baseball
We do the arithmetic, so you can play baseball.

Here at Little Professor Baseball, we not only do the arithmetic, we also tell you how it's done. So sharpen your pencil and pull up a chair. Here's where the little professor gives the (ball)game away. Math Warning: this is no walk in the (ball)park.

On the Mathematical and Statistical Foundations of Baseball Simulation
by The Little Professor

An appropriate place to begin is with a simple criterion for simulations that are intended to be accurate simulations.

The First Principle of Accurate Simulation
The expected outcome of a simulated event matches the average outcome of that event as measured over the intended time frame, typically an entire season or career.

Part I: Batting
Batting by itself is the simplest aspect of the game to model, though judging from Mark Cooper's Baseball Games, even accuracy for an average at bat was slow in emerging. Perhaps that was because modern statistical thinking was evolving and entering the public discourse along with the game of baseball itself. Conceptually, it is simplest to begin by modeling a single batter.

Batting Average Over a Season: A First Example
The first example is of a very simple simulation of a batter's batting average over a season that satisfies the first principle of accurate simulation. Take Manny Ramirez of Boston, who in 2002 batted .349 in 436 at bats. An accurate model would be to give him a hit chance of .349 for a hit in each at bat. The simulation of an at bat involves generating a random number between 0.0 and 1.0 with a uniform distribution and determining if it is less than or equal to .349. If it is, the simulation produces a hit; otherwise, it produces an out. Simulating an entire season involves simulating 436 at bats. The expected value of simulating 436 at bats with a per at-bat likelihood of .349 is simply .349 * 436 = 152, exactly the number of hits Ramirez had in 2002. Thus this simulation satisfies the first principle of accuracy.

Exercise: Computer Simulation: Histograms
Simulate Manny Ramirez's batting average by generating the outcomes of 436 at bats individually and computing the resulting batting average. Was the result close to the expected value of .349? Repeat 1000 times and plot a histogram of the results with bins defined to 3 decimal places. (That is, compute the number of times the simulation predicted a .000 average, a .001 average, a .002 average, ..., a .349 average, a .348 average, ..., a .999 average and a 1.000 average. What's the likelihood that Ramirez will beat 1.000 or 0.000 in 436 at bats?

If the granularity of the model were not an individual at bat, but an entire season, then there are any number of models that satisfy the first principle. For instance, the entire season could be simulated with one roll with a .349 chance of producing a 1.000 batting average and a .651 chance of producing a 0.000 batting average. The expected outcome is the same, because .349 * 1.00 + .651 * 0.00 = .349. The critical flaw with this second model is that it assigns roughly a 35 percent chance of Ramirez batting 1.000 for a season, which is patently absurd. It also assigns no chance to his batting his actual average .349, or likely alternatives such as .347 or .359. In statistical terms, the model has the right expected mean, but the wrong expected variance. What this means is that if 10 seasons are simulated with each model, the expected variance of the first model is much much lower than the second model. And somehow, that seems closer to reality, leading to a second criterion for accuracy.

The Second Principle of Accurate Simulation
The variance of the model should be equal to the actual variance.

Unfortunately, it's not clear what the "right" variance is for a model, because a season is only played once. Under the assumption that each at bat is independent and has a chance of a hit equal to the season average, then the expected variance will be correct. This seems a reasonable assumption to make, as illustrated in the following exercise.

Exercise: Computer Simulation: Histograms
Consider a batter with a .298 average who comes to the plate 5, 10, 25, 50, 100, 200 or 400 times during a season. For each number of at bats, plot the result of 1000 simulations. Do the distributions look like bell curves? They should, because they're drawn from normal distributions. How do the histrograms differ based on the number of events simulated? What is the expected variance for each number of at bats?

At Bats: Extending the number of outcomes
Although the batting average example only had one outcome, a similar average-based model may be extended to multiple outcomes. For instance, Bernie Williams batted 612 times, had 204 hits, 37 doubles, 2 triples and 19 home runs; he also walked an additional 83 times, for a total of 695 plate appearances. A plate appearance may be simply modeled by assigning the possible outcomes the following likelihoods.

```
204/695 1B
37/695 2B
2/695 3B
19/695 HR
83/695 BB
```
Note that the remaining 350 outcomes were outs. This distribution can be converted into a tabular format suitable for printing on a card by indicating the numbers in a cumulative fashion.
```
Bernie Williams (2002 Season)

OUT  000
BB   350 (= 0 + 350)
1B   433 (= 350 + 83)
2B   637 (= 433 + 204)
3B   674 (= 637 + 37)
HR   676 (= 674 + 2)
```
To use this distribution, a number between 0 and 694 inclusive (for a total of 695 = 694 - 0 + 1 possible outcomes) is generated at random from a uniform distribution. Then the best outcome (for the batter) is chosen that is greater than or equal to the number generated. In that way, the number next to the outcome is the minimal number required to produce that outcome. For example, an outcome from the random number generator of 643 would represent a double, whereas 312 would be an out and 676 a home run. With this arrangement of outcomes, a higher random number is better for the batter. This feature is one of the distinguishing features of Little Professor Baseball's rolling system. In addition, the outcome is easier to determine than in other game,s because an exact number need not be looked up, just a range.

Exercise: More Outcomes
Bernie Williams struck out 97 times. Generate a new table for Bernie Williams that indicates the difference beween strikeouts and hit outs.

The main problem with the last representation is that it requires an event on the 0 to 694 scale to be generated, which is not particularly straightforward with dice. By normalizing to a 000 to 999 scale, each player can use an ordered sequence of three ten-outcome dice, one representing each digit in the outcome. The canonical dice for Little Professor Baseball are black, grey and white, with the black being the hundreds position, the grey the tens and the white the ones. Your colors and orders may vary. To determine the normalized table, just multiply each number by 1000/695, producing the following result.

```
Bernie Williams (2002 Season)

OUT  000
BB   504
1B   623
2B   917
3B   970
HR   973
```
Note that the numbers were rounded to the nearest integer, which introduces some arithmetic accuracy error. A three-digit decimal representation of the original outcomes introduces the same degree of rounding error. Also note that the worst outcome will always have a number of 0, so it may be removed from the table without loss of information.

Generate a team for Ethan Allen's All-Star Baseball. The set of outcomes is: strikeout, ground ball, fly ball, base on balls, single, double, triple and home run. Outcomes are determined by spinner, so the 360 degrees of the circle needs to be broken into areas proportional to the likelihood of various outcomes.

Take a break and play a game. You deserved it. If you've done all the exercises, you are hereby awarded an honorary B.B.S.Sc., otherwise known as a Bachelor's of Baseball Simulation Science.

Part II: Pitcher vs. Hitter
Simulating pitching versus hitting introduces considerably more complexity into the modeling of individual players. For simplicity throughout, the target of the simulation is a hitter facing a pitcher for a single at-bat. As such, individual pitches are not modeled, nor is pitch count/fatigue, situation or lefty versus righty differentials. Statistics will be drawn from season averages. Note that exactly the same techniques as are used to model an at-bat could be used to model a pitch, an inning, or even a whole game.

Pitcher versus Hitter: On-base versus out
Historically, games have taken two tacks to determining the balance of contribution to the outcome made by a pitcher facing a hitter. The Strat-o-Matic route assumes that an initial roll determines whether to read the outcome from the hitter or pitcher's cards. The APBA method uses a double indirection of a 36-outcome roll of two labeled six-sided dice that is then mapped to an outcome from 1-42 on a batter's card, of which outcomes 1-11 are determined by the pitcher's grade A through D, and outcomes 12-42 are determined by the cumulative fielding points of the defense. Without computing the mapping distribution, it cannot be determined to what extent the outcome will be determined by the pitcher's, batter's, or team's fielding statistics. Strat-O-Matic is more straightforward; a six-sided dice is rolled and a 1-3 is read from the hitter's card and a 4-6 from the pitcher's. Two six-sided dice are then rolled and added to determine the outcome (modulo so-called splits, which are employed by both APBA and Strat-O-Matic, and are discussed in part three of this paper). Of course, if the dice are labeled, all three may be rolled at once.

For simplicity, the first model follows Strat-O-Matic and assigns equal likelihoods to reading from the pitcher's or hitter's statistics. Any method may be used from flipping a coin to a game of scissors, rock and paper. This measure is merely a stopgap until the more sophisticated method underlying Little Professor Baseball is introduced in the final part of this paper.

Assuming that each batter and pitcher faces an overall average set of opponents over the season, accuracy requires that the result of a pitcher facing a sequence of average batters to have an expected value of the pitcher's actual statistics. Similarly, the expected outcome of a batter facing an average lineup of pitchers should be the batter's season statistics.

The average hitter statistics can be computed by simply adding all the at bats for the entire season together. Note that this is also equal to the average pitcher statistics. Again, a granularity choice must be made as to whether the averages are drawn from the American or National League or from both. Little Professor Baseball took the leagues separately in computing the statistics, though there is residual error due to interleague play as outlined in the following exercise. Thus accuracy requires a National League pitcher facing an average assortment of National League batters will produce his actual averages.

Exercise: Effects of League Variation
With the designated hitter rules, the average hitter and pitcher statistics are considerably different. Compare the 2000 American League Statistics with the 2000 National League Statistics. How much error would be introduced into if averages for both leagues are used rather than for the individual leagues? How much of that variation might be due to designated hitters? Did you notice that the National League Pitchers hits allowed and runs allowed is not equal to the National League Batters in terms of runs and hits? That's thanks to interleague play, so the only proper adjustment is the one mentioned in the exercise on adjusting statistics for the opposition. How much error is introduced because a team does not actually play itself, but the stats for each team is included?

To begin slowly, reconsider the case where an at bat has two outcomes, hit and out. Thus the target averages being modeled are the batting average of a batter and the percentage of outs by a pitcher (disregarding walks altogether). To introduce some real numbers, in 1970, before inter-league play and designated hitters complicated matters, in the National League, batters eked out 17151 hits in 66465 at bats, for a league cumulative batting and hits-allowed average of .259. Johnny Bench, playing for the Big Red Machine, beat the averages considerably by hitting .293 in 605 at bats. Against Tom Seaver, of the New York Mets, batters only managed 230 hits in 1102 appearances (note that innings pitched * 3 + bases on balls is the number of at bats against a pitcher), for an allowed average of .209.

The obvious thing to try is to set the cards up according to the player averages. Recall that the pitcher's card is used 50% of the time and the batter's card the remaining 50% of the time. The expected value of Johnny Bench facing an average array of pitchers will be the same as reading half the resuls from the average and half from Johnny's statistics. But this yields the rather disappointing:

```     (1/2)*.259 + (1/2)*.293 = .276
```
For simplicity, the average player's card is assumed to provide the league average chance of a hit: 25.9%. Working backward from the desired result, Johnny's card must be adjusted to a value such that:
```     Johnny Bench's Average =
1/2 * League Average + 1/2 * Johnny Bench's Card
```
This is the fundamental formula of opposition. A little quick algebra, and:
```     1/2 * Johnny Bench's Card
= Johnny Bench's Average - 1/2 * League Average

Johnny Bench's Card
= 2 * Johnny Bench's Average - League Average
```
Plugging in the actual numbers yields:
```     Johnny Bench's Card
= 2 * .293 - .259 = .327
```
Pitching works the same way, so that:
```  Tom Seaver's Card
= 2 * Tom Seaver's Average - League Average
= 2 * .209 - .259 = .159
```
This adjustment for blending with the average accounts for the apparent skew seen on cards in other games. The statistics need to be "juiced" when you read off a player's card in order for the averages to work out.

Exercise: Generalizing the Fundamental Formula
What adjustments would have to be made to the fundamental formula in order to account for having 75% of the outcome determined by the pitcher and only 25% by the hitter? Will the expected outcome of a hitter facing a pitcher be different?

There are several desirable consequences of the present model. First, if Johnny Bench faces average pitchers, his expected average is equal to his actual average of .293. On the other hand, if he faces weaker pitchers, then his average will be higher, whereas if he faced stronger pitchers it would be lower. Second, if an entire season is simulated an at bat at a time, using the actual hitter and an average pitcher, the expected results for each batter and the entire league will be accurate. This also follows for pitching. It also works if pitchers are faced off against batters by being selected according to their number of plate appearances.

Exercise: Prove It and Extend It
Prove the assertions in the preceding paragraph. Generalize them to show that if each hitter faced an average lineup of pitchers and each pitcher faced an average linup of batters that the result of simulating an entire season with the actual pitcher versus batter matchups would have the right expected values for each player and for the cumulative averages.

Why is it not guaranteed that if the actual pitcher versus batter events are simulated that the expected average is the actual league average? How could it be adjusted so that it would be accurate? If batters or pitchers do not face a representatively average set of opponents, how might their statistics be adjusted to account for the caliber of opponents? Did you get the limit construction? How much is your answer like the way Google ranks are determined? How about the way BCS ranks are determined for college football teams? Is the BCS second-order approximation reasonable?

Exercise: Playing Across Generations
Why is it impossible to compare across generations? Would it be fun to play them against each other with their Little Professor cards anyway? How could you justify enjoying such a game if you had to.

Pitcher versus Hitter: Multiple Outcomes
The same way as the simple batter-only model was extended to multiple outcomes, the pitcher versus hitter cards can be extended to multiple outcomes by evaluating each outcome the same way as was done for hits versus outs. Thus all percentagle values on a card will be double the player's percentage minus the average percentage. Returning to the 1970 National League, the total stats, in the format typically reported in league averages and on baseball cards, were:

```     National League (1970)    Johnny Bench (1970)
AB: 66465                 AB: 605
H: 17151                  H: 177
2B:  2743                 2B:  35
3B:   554                 3B:   4
HR:  1683                 HR:  45
BB:  6919                 BB:  54
SO: 11417                 SO: 102

1B: 12171                 1B:  93
PA: 73384                 PA: 659
HO: 37897                 HO: 326
```
Note that singles can be recovered by:
```     1B = H - 2B - 3B - HR
```
The total number of plate apperances (otherwise known as batters faced for pitchers) is given by:
```     PLATE APPEARANCES = AB + BB
```
Finally, the number of non strikeout outs is the difference:
```     HIT OUT = AB - H - SO
```
Recasting the above tables as percentages and ordering them from worst to best outcome for a hitter produces the following table:
```            NL     JB      2*JB-NL
SO:   .156    .155    .154
HO:   .516    .495    .474
BB:   .094    .082    .070
1B:   .166    .141    .116
2B:   .037    .053    .069
3B:   .008    .006    .004
HR:   .023    .068    .113
```
Applying the fundamental formula requires each stat on Johnny Bench's card to be twice Johnny's percentage minus the league's percentage, which reading from the third column and converting to cumulative dice rolls yields Johnny's card and the league average pitcher/hitter card:
```
Johnny Bench (1970)     Average Pitcher/Hitter (1970)
SO: 000                 SO: 000
HO: 154                 HO: 156
BB: 628                 BB: 672
1B: 698                 1B: 766
2B: 814                 2B: 932
3B: 883                 3B: 969
HR: 887                 HR: 977
```
But what about Tom Seaver? The immediate problem is that so many additional game statistics (W/L, ERA, etc.) are provided for pitchers that their hits are rarely broken into singles, doubles, and triples. For instance, the reported statistics for Tom Seaver in 1970 are:
```     Tom Seaver (1970)
IP: 290.7 (290 2/3)
SO: 283
BB:  83
HR:  21
H: 230

AB: 1102 (IP * 3 + H)
PA: 1185 (AB + BB)
HO:  589 (PA - BB - H - SO)
1B + 2B + 3B: 209 (H - HR)
```
Because the ratio of singles to doubles to triples is unknown, it must be approximated somehow. A simple approximation involves using the league ratios of singles:doubles:triple, which are: 12171:2743:554, or normalized to percentages, .787:.177:.036. This ensures that the distributions are right on average for the batters, and that the expected number of hits (the known statistic) is right for the pitcher. Multiplying these results through Tom's 209 hits yields:
```     1B: 164
2B:  37
3B:   9
```
This allows the calculation of a card according to the percentages:
```            NL     TS      2*TS-NL
SO:   .156    .239    .322
HO:   .516    .496    .476
BB:   .094    .070    .046
1B:   .166    .138    .110
2B:   .037    .031    .025
3B:   .008    .008    .008
HR:   .023    .018    .013
```
If the resulting total does not sum to 1.000, it should be normalized in some way so that it does; in htis case, .001 was subtracted from Seaver's hit outs. This yields the following card for Tom Seaver in 1970.
```
Tom Seaver (1970)
HR: 000
3B: 013
2B: 021
1B: 046
BB: 156
HO: 202
SO: 678
```
Anticipating the actual presentation order of Little Professor Baseball, outcomes are presented on the card in the opposite order for pitchers so that they, like batters, will want to roll high.

Exercise: Generate a Card
Generate your own cards for a batter, pitcher and league average for a year other than 1970. Generate an average pitcher card with the same statistics as the average batter card, but in the reverse order of outcomes to match Tom Seaver's card.

Exercise: Total Outcomes
If Johnny Bench faces Tom Seaver, what is the likelihood of each outcome? What if Johnny Bench faces an average pitcher or Tom Seaver faces an average batter?

Exercise: More Cards
Generate your own cards for a batter, pitcher and league average for a year other than 1970.

Exercise: Estimating Hit Ratios
Could the ratio of home runs to hits be used to assign a better estimate of the distribution of hits to pitchers? How would you prove that extra-base hits and home runs were correlated?

There is one complication which has yet to be addressed. What if double the player's percentage minus the average percentage is negative? Unfortunately, this actually occurs quite often. For instance, 2 percent of the plate appearances in the National League in 1970 resulted in home runs. Some players didn't hit any. Therefore, 2 * 0.0 - .023 = -.023.

The real bummer is that it's hard to compensate for this effect assuming that fifty percent of the outcomes are on the batter and fifty percent on the pitcher. If a pitcher gave up any percentage of home runs, and there is any chance of a home run on the pither's statistics, then there is no hope of accomodating a weak hitter directly. Drastic measures may be taken and a weak-hitter designation given, which amounts in home runs being converted back into doubles, and the doubles category similarly downgraded. This step was taken in Strat-O-Matic, but not in the basic game of Little Professor Baseball. Triples could also be downgraded to doubles in order to satisfy the first principle. If there is still underflow, then doubles must be downgraded to singles and so on. For instance, a player who doubled only .010 plate appearances and never tripled or homered must be designated so that pitcher outcomes of triples or home runs are downgraded to doubles (.31 total) and that a percentage of doubles are re-rolled and possibly downgraded to singles and so on.

Why might smoothing be useful to deal with rare outcomes like triples? What is the expected variance in number of triples for a batter with the league chance of .008 per plate appearance of hitting a triple with 500 at bats?

Allow for the possibility of errors, with an accuracy criterion given by having a player (or team's) expected number of errors equal that of the actual season being modeled. Hint: Set aside some of the hit out likelihood on the pitcher and catcher cards for errors. Calculate the likelihood of an error based on team fielding reserving the other outcomes for hit outs. If modeling individuals, pro-rate the likelihood of error based on players in the field. Although this provides an accurate model of the distribution of errors, where does it fall down as a model of fielding? How could that be accomodated? Second Hint: See either APBA or Strat-O-Matic.

If you've read this far, take a break and play a couple games. Now you really deserve it. If you've done all the exercises, you are hereby rewarded an honorary M.B.S.Sc., aka the Master's of Baseball Simulation Science. If you've done the advanced exercise, consider it your master's thesis.

Part III: The Rolling System
In the rolling system just described, as in Strat-O-Matic, two rolls are required to generate the outcome of an at bat. The first roll determines which player's statistics are used to determine the outcome, and the second roll generates an outcome from that player's card. One of the charms of Little Professor Baseball is that both players roll simultaneously, and the outcome is read off of the card of the player with the higher roll; if both players roll the same result, they roll again.

Compensating for the High Roller Bias
The problem introduced by reading the result off the card of the player who rolled highest is that higher rolls are more likely to result, introducing a bias into the outcome. Luckily, this bias is easily calculated by considering the events consisting of the two rolls and their likelihoods. First consider the game as it stands. There are 1000 outcomes from 000 to 999, and each is equally likely. But only half the outcomes will be read from each card, so each number on the card represents 1/2000 of the probability mass.

In the situation where the player's card with the highest roll is chosen, a roll of 000 will never be used, so instead of 1/2000, the probability mass is 0. At the opposite extreme, a roll of 999 by a player occurs 1/1000 times, but will be used 999/1000 of the times that it is rolled, for a total of 999/1,000,000, or roughly double the probability mass of the same outcome in the original game. In general, a roll of N will be assigned N/1,000,0000 of the probability mass, because the likelihood of a roll of N is 1/1000, and if it is rolled, it stands an N/1000 chance of surviving as the high roll, beating every outcome from 0 to N-1 inclusive.

Previously a range of outcomes between N and M inclusive would constitue (M-N+1)/2000 of the probability mass. With the highest roll wins model, it will now constitute:

```     M/1,000,000 + (M+1)/1,000,000 + ... + (N-1)/1,000,000
```
of the probability mass, because this is the sum of the probabilities of each outcome in the range. Note that this can be reduced to a closed expression by:
```     M/1,000,000 + (M+1)/1,000,000 + ... + (N-1)/1,000,000
= (M + (M+1) + (M+2) + ... + (N-1))/1,000,000
= ((1 + 2 + ... + (N-1)) - (1 + 2 + ... + (M-1)))/1,000,000
= (N*(N-1)/2 - M*(M-1)/2)/1,000,000
= (N * (N-1) - M * (M-1)) / 2,000,000
```
The tricky step is the third one, where the sum of numbers between `M` and `N-1` is replaced with the difference between the sum of the numbers between `1` and `N-1` and the sum of the numbers beteen `1` and `M-1`. The result is the fundamental formula of high-roll-wins calculations.

With this observation, cards can be generated straightway for Johnny Bench, Tom Seaver, and the average 1970 pitcher/hitter. Recall the statistics for Johnny Bench, Tom Seaver, and the National League.

```            NL     JB      2*JB-NL    TS      2*TS-NL
SO:   .156    .155    .154       .239    .322
HO:   .516    .495    .474       .496    .476
BB:   .094    .082    .070       .070    .046
1B:   .166    .141    .116       .138    .110
2B:   .037    .053    .069       .031    .025
3B:   .008    .006    .004       .008    .008
HR:   .023    .068    .113       .018    .013
```
Note that in terms of batting average, good pitchers differ more from the league averages, and in terms of power, good batters diverge more. To start, Johnny Bench need a .154 chance of strikeouts, where the strikeout range begins at `000`. Suppose the strikeout range ends at the number `N`. Then the total probability mass will be `(N*(N-1) - 0*(0-1))/2,000,000`. Recal that only half of Johnny's number is required, yielding the following equation:
```   .154/2 = N*(N-1) - 0*(0-1)/2,000,000
.077 = N*(N-1)/2,000,000
154000 = N*(N-1)
0 = N*(N-1) - 154000
0 = N*N - N - 154000
```
This is a simple quadratic equation of the following form, where the unknown variable `N` is replaced by `X` to make the discussion line up with the standard presentation of the quadratic equation.
```     A*X2 + B*X + C = 0
```
with `A=1`, `B=-1` and `C=-154000`. Every quadratic equation has two roots (which might be the same, and which might be imaginary), given by the following formula:
```     X = (-B + (B2 - 4*A*C)(1/2))/(2*A)
```
A quick search on the web finds a nice quadratic equation calculator, which yields roots of 393 and -392. This fills in the number next to HO on Johnny's card.
```
Johnny Bench (1970)
SO: 000
HO: 393
```
As expected, the number is much higher than the one on the original card where a determination is first made of whose card to read the result from. Continuing the calculation, note that Johnny has a hit out percentage of .474. The starting number is `393` this time, yielding the following equation:
```     (N*(N-1) - (393)(393-1))/2,000,000 = .474/2
(N*(N-1) - 154056) = .474*1,000,000
N2 - N -1102056 = 0
```
which has a positive solution `793`. Continuing in this fashion produces the final card for Johnny Bench.
```
Johnny Bench (1970)
SO: 000
HO: 393
BB: 793
1B: 836
2B: 903
3B: 940
HR: 942
```
Note that by the range for home runs is now smaller just as the range for strikeouts and hitouts is higher. This is to be expected given the non-linear effect of the fundamental formula of high-roll wins. Also note that the result is slightly different than seen in The Basic Game, becuase the basic game calculated statistics against the combined American and National League averages.

Exercise: Average Batters and Pitchers
Create the cards for Tom Seaver, the average batter, and the average pitcher for 1970, given the statistics reported above.

Ties Go to whom?
The only thing left unaccounted for theoretically is ties.

Exercise: Likelihood of Ties
What is the chance that two players rolling `000` to `999` both roll the same thing?

Because a retry will result in the same ratio of outcome likelihoods as the original set of non-tie outcomes, in the limit, the results can simply be normalized back to 1.0. Proving this goes beyond the simple algebra assumed so far, so the result is left as an exercise (involving calculus).

Exercise: Ties on Rolls Converge
Show that doing ties over results in the correct result in the limit.

Alternatively, ties could go to the batter or to the pitcher, and the results re-normalized to take this into account.

Exercise: Ties Go to the Batter
Assume that ties go to the batter. How much error would be introduced if the same cards were used? Would ties going to the batter be to the batter's advantage? How could the fundamental formula be adjusted to take into account ties going to the batter?

Exercise: Pitcher Fatigue
To account for pitcher fatigue, a number could be subtracted from the pitcher's rolls based on the number of batters faced, which is a reasonable proxy for pitch count due to the high correlation. For sake of argument, assume `50` is subtracted from Tom Seaver's rolls. What is the likelihood of each outcome in that situation? Is it the same as if `50` is added to each of Johnny Bench's rolls? Is the result intuitively reasonable? If you can find the real statistics for a season, calculate whether any straight sum would be appropriate. If not, could there be a non-linear adjustment for fatigue?

Bases for Dice
One benefit of the new math is that a generation of American children should be able to conver the statistics used in Little Professor Baseball from base 10 to base 6. The advantage of this is that it is far easier to come by 6-sided dice than 10-sided ones. The manufacture of the traditional 20-sided dice with two sets of 0 to 9 has ceased. This is a shame, because a 20-sided die is an icosahedron, and as every geometer knows, an icosahedron is a platonic solid, with 20 equilateral triangle sides (The others are tetrahedrons [4 equilateral triangle sides], cubes [6 square sides], octahedrons [8 equilateral triangle sides] and dodecahedrons [12 pentagonal sides]). Icosahedrons roll very nicely, whereas the abominable 10-siders roll more like (American) footballs.

Exercise: Base 6
As an exercise, comnvert Bernie Williams's 2002 card from decimal, as above, to base 6. You should assume there are 4 six-sided dice ordered to create a resulting sequence of heximal digits ( (not to be confused with hexidecimal, which is base 16 and only understood by computer nerds). Could the same thing be done using base 2 and coin flips? What about base 52 and drawing from an ordinary deck of cards with replacement?

How could outcomes be assigned if dice are summed? How many six-sided dice would be required for the same arithmetic precision as Little Professor Baseball? How many dice would be needed if higher rolls are better outcomes for the roller? What's the lowest likelihood that can be estimated directly from a player with 600 at bats?

Exercise: Splits
If only two six sided dice are rolled, then the finest granularity of outcome is 1/36, or roughly 3 percent. This is clearly inadequate for modeling hits (see the league statistics). In order to compensate, both APBA and Strat-O-Matic resort to the notion of some initial rolls requiring follow-up rolls. These follow-up rolls then provide further granularity for outcomes. Which system do you prefer? Could Little Professor Baseball be redone with splits? APBA uses two consecutive six-sided die outcomes (1/36) plus some splits which are followed by a second identical roll (1/36). Strat-O-Matic uses a sum approach, having outcomes 2-12 corresponding to the sum of two six-sided dice; for splits a 20-sided die is rolled. Order the systems in terms of the granularity of their representations: APBA, Little Professor, Strat-O-Matic.

Previous Work
The most obvious influences on Little Professor Baseball are the baseball board games. Mark Cooper's coffee-table book, Baseball Games, reproduces the boards and boxes of a range of editions from the beginning of the game to the modern era.

The first game that modeled individual players was Clifford A. Van Beek's National Pastime which was patented in 1925, marketed in 1931, then discontinued due to the depression. In 1941, Cadaco issued the long-running Ethan Allen's All-Star Baseball. APBA Baseball, which was first released publicly in 1951, remains a popular choice, extending National Pastime to pitching and fielding, although the models of such are crudely quantized into four pitching grades and total team fielding of three grades. Like several early games starting with Parlor Baseball in 1878, the outcomes in APBA were determined situationally. APBA is particularly amusing for the narrative nature of its outcomes, despite their lack of statistical justification. Beginning in 1961, Strat-O-Matic Baseball took the art of baseball simulation to a higher level by modeling pitching, hitting, running, fielding and eventually endurance, health, stadiums, weather conditions at a very detailed level. If you know of other early games that modeled individual players or even the outcome of average at bats, the professor would love to hear from you.

Exercise: Literature Review
Provide a more detailed literature review. Extra credit for annotated bibliographies.

Take a break and play as many games as you like. You've earned it. If you've done all the exercises, you are hereby rewarded an honorary Ph.D. in B.S., aka the Philosophy Doctorate in Baseball Simulation. Congratulations. Send the little professor your thesis. Please.

References

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