A Mathematical
Model of Professional Major League Baseball
Baseball is an example of a process that
can be successfully simulated. The game is complex enough that a theoretical
analysis is difficult, but it is an orderly and well documented process.
Past data can be fed into a computer, and the computer can then "act out"
the game.
On the basis of two teams statistics (batting
and fielding averages, slugging percentages, steals, and so on), it is
possible to simulate different situations (hits, strike outs, fly out,
ground out, bunt, double steal, etc.). A player's statistics can be determined
and expressed as a probability of occurrence. Strat-o-matic, APBA and Pursue
the Pennant use DICE to simulate the probability of an appropriate outcome.
Quantum Baseball uses playing cards.
The MATHEMATICS of DICE GAMES
When a player's batting statistics are
expressed as a probability along a line segment, the portions along the
line would represent the percentages of times a batter flies out, grounds
out, strikes out, walks, singles, etc. ... Let us assume, for example,
that a player has a batting stats as shown below.
RESULT
PROBABILITY
Fly Out
0.110
Ground Out
0.210
Strike Out
0.220
Walk
0.140
Single
0.220
Double
0.050
Triple
0.025
Home Run
0.025
1.000
We can simulate this batter at the plate
by generating a random roll of dice. If one die is red and the other is
white, there are 36 possible combinations. There is only one double 1 and
one double 6 combination, hence 1/36 chance for each to occur. Percentages
wise this means, there is a 2.8% chance of occurrence. If a player hits
a home run only 2.5% of the time (example: 13 HR at 502 at bats), then
rolling dice, a double 1 roll could represent a home run, etc...
Other batters are treated in an analogous
manner. Base runners likewise have statistics, such as the number of bases
that they will advance in different situations ... so do pitchers. In this
way an entire game can be simulated. As a result, if played enough times,
the laws of probability will average out to yield the players past performance.
There are many other variety of dice baseball board games, however they
all fall short in the realism. Strat-o-matic ranks as the best dice game
with its 85% approximation of the real effects occurring in professional
baseball.
When the laws of probability are applied
to playing cards, more advanced mathematical techniques are required utilizing
probability matrices of Markov chains and eigenvalued problems. In addition,
some 400 hours of mind boggling number crunching was required to find a
basic consistent pattern, to unfold an accurate baseball game, enough to
satisfy the most demanding baseball fanatic. At the same time, the objective
for a good and easy card game was essential with relatively few rules.
From the point of view of John Von Neumann's
game theory, an advanced mathematical technique, Quantum Baseball is more
than a stoic Markov chain of probabilities. Quantum Baseball is more than
a Monte Carlo method of simulation, although these features were instrumental
in following the thread that lead to its modeling. According to Category
Theory, another advanced branch of mathematics, baseball is indeed a game
that is in the class of games identified as a Finite Non-deterministc Automaton.
However, in the actual play of this game,
no one ever is required to have any knowledge of mathematics, let alone
the advanced concepts used to find a mathematical model of professional
baseball. All one needs to know are some preliminary basic rules of baseball.
The rest unfolds as if the game was physically being played by real ball
players.
The MATHEMATICS of PLAYING CARDS
There are twenty four (24) different possible
initial states of the base paths when a batter comes to the plate (bases
empty, runner on 1st, runners on 2nd and 3rd, no outs, one out, ... etc.).
The player's batting statistics are subject to the conditions on the base
paths as well as his hitting capabilities. The EVENT of his batting performance,
which can be translated into a probability transition matrix, has an effect
on the situation on the field. A home run clears the bases and produces
a run(s). A double advances the runner(s) with some possibly scoring. A
line drive to short stop could become a double play or it can lead to either
a fielder's choice or a ground out. Some plays advance runners, others
do not.
The amazing similarity of the above equation
to both Schordinger's wave equation and Heisenberg's matrix equation led
to the conjecture that if the laws of nature in Quantum Baseball were to
be replaced by the rules of baseball, then the mathematics of the matrix
equation would describe an accurate mathematical model of professional
baseball. Hence the name, Quantum Baseball.
In the application of this mathematical
model to playing cards, incorporating every situation generated in baseball,
a deck of 60 playing cards are required. Such a deck would use a standard
deck of 52, plus four jokers and four "ones". This arrangement generates
an accuracy of 98% of the events of professional baseball. As a result
Quantum Baseball is unique amongst all baseball games. Since it is based
on a mathematical model, an equation, it is not a simulation.
This has never been accomplished before!
All previous attempts to obtain an accurate card game were met with little
or no realism. Quantum Baseball, not only captures the realism of baseball
that the top dice board games accomplish, but also, is the only game that
incorporates the complete drama and excitement of major league baseball
by developing the complete count (balls and strikes) on the batter. Dice
games do not. One merely throws the dice and the event happens! With Quantum
Baseball, you are in effect playing an actual game of baseball, pitch by
pitch, without the physical exertion.
My children and their friends, everyone,
not only learn the rules of the card game quickly, but they also recognize
the strategies of baseball. In all the excitement of play, they discover
the wonders of baseball.
