L.A. Math: Romance, Crime, and Mathematics in the City of Angels
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CHAPTER 9. THE WINNING STREAK
1. One of the common problems confronting both a sports bettor and an investor is how to allocate funds to investments with differing degrees of risk. A detailed analysis of this problem, known as portfolio theory, won the economist Harry Markowitz the 1990 Nobel Prize in economics. Bettors generally solve this problem by deciding on a basic unit for each bet—such as $100. If their analysis shows that a particular bet is unusually attractive, they might make a larger bet. If the basic unit is $100, a $200 bet is known as a two-unit bet, a $300 bet as a three-unit bet.
CHAPTER 10. ONE LONG SEASON
1. When a betting line includes a half point, such as three and a half, a push is no longer possible, since the final result of a game is always a whole number. If the line is three and a half, the favorite covers if it wins by four or more.
2. An if bet is a second bet that is contingent on winning a first bet. Pete has bet $200 on the Knicks and has told his bookie that if the Knicks cover and the line on the Lakers is six points or better, the $200 that he won on the Knicks should then be bet on the Lakers.
CHAPTER 11. THE GREAT BASKETBALL FIX
1. Many bookmakers offer a separate line for the total number of points scored in a game. This is known as the totals line, and the bettor selects either over, betting that the total number of points scored by both teams will be more than the totals line, or under, betting that the total number of points scored by both teams will be less than the totals line. Winning bets are paid off at even money, losing ones pay off at 11–10, just like the points line.
2. The process of hedging is common in investment; it is an attempt to obtain a profit no matter what the outcome of the event. If a bookmaker offers a line such as minus five, the ideal situation for the bookmaker would be to have the same amount bet on the favorite and the dog. That way, unless the game is a push, he would win 110% of the money on the bets he or she won than he or she lost on the losing bets—guaranteeing a profit. However, sometimes a line attracts a preponderance of betting on one side. A bookmaker may offer a line and find out that four times as much is bet on the favorite as on the dog. In that case, the bookmaker can increase the line to attract more money on the dog.
Here’s a simple example.
All is well for the bookmaker as long as the favorite does not win by precisely six points. For instance, if the favorite wins by seven, the bookmaker loses $4,000 to those who bet on the favorite at the original line and wins $1,100 from those who bet on the dog at the original line. He also loses $1,000 to those who bet on the favorite at the modified line but wins $4,400 from those who bet on the dog at the modified line. This results in a net profit of $500 to the bookmaker.
However, catastrophe strikes the bookmaker if the game ends at six points, in the middle of those two lines. He wins $1,100 from those who bet on the dog at the original line and also $1,100 from those who bet on the favorite at the modified line. However, he loses $4,000 to those who bet on the favorite at the original line and another $4,000 to those who bet on the dog at the modified line, for a net loss of $5,800. In effect, the bookmaker is offering odds of $5,800 to $500 that the game will not end with the favorite winning by six. In this case, the bookmaker is said to have been middled. In the above example, if the modified line had been Favorite minus 8½ (which might have happened had 6½ not attracted enough betting on the dog), the bookmaker would be middled if the game ended with the favorite winning by six, seven, or eight. The larger the gap between the prices, the greater the chance that the bookmaker will be middled.
This is also a problem in the commodities futures markets, which operate in a fashion that is greatly similar to sports betting. In this case, the line is known as the futures price of the commodity, and the result of the “game” is the price of the commodity at a particular future point in time, known as the settlement price. The major difference is that the value of a winning or losing “bet” in the commodities futures market is not a fixed amount (such as a $100 bet) but the difference between the price at which the futures were bought and the price at which the commodity settles, much as the value of a stock purchase depends upon the difference between the price at which the stock is bought and the price at which it is sold.
INDEX
add-on interest, 168
algorithms: greedy, 230; nearest neighbor, 124, 229–30
amortization, 172
anti-symmetry, 178
arithmetic progressions, 156–60
Arrow’s Theorem, 223
average, 25–27, 112, 148–52, 192, 194–95, 202
balance, 172–73
base, 18, 147
binning, 205
binomial distribution formula, 210
Borda count, 220
Chinese Restaurant Principle, 185–88
common difference, 156–57
complements, 179
complexity, 229
compound interest, 55, 168–71
conditional probability formula, 201
discount, 145
down payment, 171
edge, 227
elements, 176
empty set, 179–80
even money, 196
event, 193; certain, 194; impossible, 193
expectation, 194–96
experiment, 190–91
favorite, 232
frequency distribution, 203
Fundamental Counting Principle, 182–83
future value, 168
Gauss trick, 159
graph, 226
independence (of choices), 186
independence of irrelevant alternatives, 223
insincere voting, 220
installment buying, 172–73
interest period, 167
intersection, 180–81
linear equations, 163
line betting, 232–33
logical equivalence, 141
maxmin, 214
mean, 205
minmax, 215
nondictatoriality, 223
normal distribution, 207
NP-complete, 228
odds, 196
outcome, 190–91
pari-mutuel betting, 231–32
payoffs, 213
percentage, 18, 55–56, 70, 144–47, 151, 167–68, 203–4, 211–12
plurality, 220
present value, 168
preservation of unanimous preferences, 223
principal, 167, 172
probability: empirical, 192; theoretical, 192
probability density function, 207
probability distribution, 204; continuous, 207; discrete, 207
probability function, 191
propositions, 138–41, 179
push, 233
random variable, 203
rate, 152, 162, 165, 167–70
Rate Principle, 162
recursive definition, 154–56
reflexivity, 178
runoff, 220
saddle points, 216
sample space, 191
sequences, 154–56
set-builder notation, 176
set difference, 183
sets, 175
simple interest, 55, 167–68
standard deviation, 206
strategies, 213; mixed, 216–17; pure, 216
subsets, 177–78
systems of two equations in two unknowns, 163–65
tautology, 139
Totals Principle, 162
transitivity: of inclusion, 178; of preference, 221–22
Traveling Salesman Problem, 227
truth tables, 139–41
underdog, 232
uniform probability space, 192
union, 180–81
universal sets, 178
vertex, 227
vigorish (or “vig”), 233
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