Equations in Real Life
Professors Jim Hoste and David Bachman empower students to calculate risk through a unique application of game theory to real-life scenarios inside the casino and out.
If you are lucky enough to stumble into Associate Professor of Mathematics David Bachman's class on the right day, at the right time, you may think you have accidentally entered a casino rather than a college campus. The cards, the chips, the makeshift gaming tables—the only things missing are lingering clouds of cigarette smoke and the ever-present clanging of the slot machines. But what does a spirited game of Texas Hold ’Em have to do with academics?
Everything. Pitzer's courses such as The Mathematics of Games and Gambling, The Mathematics of Poker, and Pencil and Paper Games set out to break new ground in introducing non-math-oriented students to an innovative way of applying math concepts to the liberal arts.
“Games are a natural place to find a lot of mathematics, statistics and probability,” Bachman explained, “and people find them lighthearted and fun.” The goal in studying games is to expose students to serious mathematics, not fluff. Many topics in mathematics have a deeper meaning and relationship to other areas of study—not just math and science.
“However,” added Jim Hoste, professor of mathematics, “in no way are we promoting gambling or encouraging students to be irresponsible with what they learn in class. Instead, we give math beginners a place to further explore the connection that math has to the entire Pitzer curriculum.”
Hoste began teaching The Mathematics of Games and Gambling in 1991. “We learned how to calculate various probabilities associated with rolling dice and dealing cards and used this to analyze all the usual casino games,” he recalled. “From there we turned our attention to game theory.”
Today, the course covers combinations, permutations, probability, expected value, Markov chains, graph theory, as well as game theory. Throughout the class, students learn to analyze games such as keno, roulette, craps, poker, bridge, and backgammon— always applying the concepts of probability to real-life situations. For many students, this knowledge provides a solid basis for statistics courses, as well as for using game theory in the social sciences.
When Bachman arrived at Pitzer four years ago, he came up with the idea of teaching the Pencil and Paper Games course. “Jim's class is all about games where chance is a factor,” he said. “My class was going to complement his because I was only looking at games where there are no chance elements. Two years later, I thought I'd try the poker class.”
While there is some overlap with Hoste's class, Bachman's poker class focuses on how to use game theory to make decisions, and Hoste's class emphasizes using probability to deal with chance in a game. Here is how the two differ: Mathematicians group games into several different categories. There are games of chance, in which probability plays a big role. That would be Hoste's The Mathematics of Games and Gambling class. There are “games of perfect information,” which are the games Bachman's Pencil and Paper Games class addresses. “In these games there is no randomness, and you always know both your own options, as well as your opponent's,” Bachman explained.
Then there is a third category, called “games of imperfect information,” which The Mathematics of Poker covers. These are games that also do not involve probability, but you may not know your opponent's options.
A fourth class of games include “single player” games such as puzzles, solitaire or Rubik’s Cube. Hoste is developing a new class on this subject that he will be teaching in the fall.
As a branch of applied mathematics, game theory attempts to mathematically capture behavior in strategic situations, where an individual's success in making decisions depends on the choices of others. But how does understanding game theory in mathematics play into the College's mission of providing an academically rigorous, interdisciplinary liberal arts education? Economics and philosophy students can use it to develop theories of ethical behavior, and to understand what constitutes good behavior; while political science scholars can apply game theory concepts to discussions of fair division, political economy, public choice, positive political theory and social choice theory.
If you have never studied game theory, the concept can be somewhat difficult to envision. To better understand it, Bachman offers this example of a game theory problem. Jim and Dave are each given a king and a queen card. They secretly choose one of their cards, and simultaneously reveal their choice. If they choose differently, then Dave must pay Jim $3. If they both choose kings, then Jim pays Dave $4. If they both choose queens, then Jim pays Dave $2.
It seems like this is a fair game, since both players would make an average of $3 every time they win, if they were to play completely randomly. What you discover through game theory is that Jim has a strategy by which he has an eight-cent advantage over Dave.
David Lee '08, a Political Studies major, readily admits that he has no inclination toward studying math. However, taking Bachman's The Mathematics of Poker changed all of that. “I frequently played poker with friends, but didn’t understand the math concepts behind it. The class was fantastic.”
While Lee admits that the class hasn't made him a better player per se, it is the first time that he's taken a math class and thought “Wow, what we're doing is applicable to real-world situations.”
“I regret not taking more math classes at Pitzer,” he said. “I now feel confident that I can handle the concepts. By making math interesting, it allows the subject to become more manageable.”
Dan Mitchell '08, also a Political Studies major, found that while math can be taught in several different ways, he was attracted to the real-life application to the types of problems solved in class. It's the type of correlation of subject matter to practicality that he was used to finding in his other liberal arts-focused classes, but not something that he expected to see in a math class.
“In calculus, we solved equations with x, y and g variables; in The Mathematics of Poker, we see equations with the variables being the number of cards played, the number of chips on the table. All those variables have meaning. I really enjoyed applying the mathematical formulas to what we were learning,” Mitchell said.
First-year student Ian St. Lawrence, an English major, initially decided to take Hoste's The Mathematics of Games and Gambling because applying mathematics to gaming scenarios sounded like fun. “We started off with dice and ended with black jack and game theory,” St. Lawrence said. “Professor Hoste enjoys what he teaches. He showed us a trick to multiply numbers by twelve and he really got into it. The class was more practically based than studying the derivative of slopes or finding the area under a shape. I'm now interested in taking other not-strictly-math classes, such as the Geometry of Nature.”
For Hoste and Bachman, comments like this mean they have accomplished their mission to show the relevance of math in everyday life and in a liberal arts education. “Our goal with these classes is to empower students to make good decisions,” Hoste said. “Although you don't know the outcome, what are the chances something will happen? We want to get students thinking quantitatively to assess the various risks in their lives.”
Professor Bachman poses this game theory question in The Mathematics of Poker class: Jim and Dave play a simple game of poker. Both players ante $1. Jim is then given a king (both players know this), and Dave is given either an ace, with probability of one-quarter, or a queen, with probability of three-quarters (this card is hidden from Jim). Dave may then either bet $2, or fold. If Dave bets, then Jim may either call $2, or fold.
It's not hard to see that Dave should always bet when he gets an ace. The question is, should he bluff and bet when he has the queen as well? Game theory gives the answer: Dave can maximize his winnings by bluffing precisely one-sixth of the time when he gets a queen. And when Jim is faced with a bet, his best strategy is to call half of the time, and fold the other half.