By Mari N. Jensen

Rather than being a path to new research areas, teaching students who dislike your subject area sounds like the quick road to teacher burn-out.

But for Richard B. Thompson, creating a new introductory-level math course also fueled his research. As a result, Thompson and his colleague A. Larry Wright, both associate professors in the mathematics department, have found a new Nash equilibrium, one that provided a strategy for auction bidding. Nash equilibria are mathematical results that indicate stable strategies in many types of competitive situations, including auctions.

The study of such equilibria won mathematician John Nash, made famous by the film “A Beautiful Mind,” a share of the 1994 Nobel Prize in economics. Thompson’s path to a new Nash equilibrium started with two disliked math courses taken by pre-business students aspiring to enter UA’s Eller College of Management at the beginning of their junior years. Enrollment in the classes is about 900 students per semester. The required two-semester sequence, finite mathematics and brief calculus, were taught in the mathematics department and perceived by students to be used to weed the mathematically challenged out. The situation was not a happy one for the pre-business students or the mathematics professors.

To revise the courses, in 1997 Thompson teamed up with Chris Lamoureux, head of the finance department. The two professors revamped the courses to tailor them for business students and dubbed the courses Business Math I and II. The revised courses were taught for the first time during the 1998-99 school year.

“It used to be what we taught was scaled-down traditional math classes,” Lamoureux says. “There was nothing unique about the fact that it was business students sitting in the classroom. Business Math I and II came about because what we wanted was to recognize that business students are not what we call ‘little math students.’ Business students didn’t like math. They wanted to avoid math.”

The solution was to show business students that they needed math, rather than just tell them so.

The new classes revolve around real business problems rather than abstract mathematical ideas. Instead of starting with mathematical concepts, the students are presented with business problems that need solving.

“We found major business projects that are clearly real and that the students see as real business,” Thompson says. “We introduce the mathematics as tools ­ something that will help you make a better business decision.” He adds, “Business makes decisions in the face of uncertainty, and mathematics helps you make those decisions.”

Moreover, the course is structured to resemble how problems are tackled in the business world, he says. Students work in teams, use Excel software for calculations and report their solutions to the group using PowerPoint presentations. In fact, the course textbook isn’t a book at all: it’s an electronic text, using PowerPoint with computer animations, streaming video, links to Excel, and Internet sites.

Lamoureux says that students tuned out the older version of the class because it seemed the teacher was speaking an irrelevant foreign language. These new courses, however, create in the students what he calls “the thirst for the mathematician.”

“They come and say, ‘Now I need you to teach me this because I need to know this to figure out this very important business problem. Now, mathematician, teach me this.”

The courses, which tackle two problems each semester, don’t skimp on presenting complex mathematical concepts. After completing the sequence, the students’ expanded toolkit includes Monte Carlo and bootstrapping simulations, exponential models that indicate the time value of money, sensitivity analysis and an understanding of probability and expected value.

The last problem the students tackle involves what are known in the economics world as “first-price, sealed-bid auctions.” The government uses such auctions to sell off-shore oil leases.

Unlike the old-fashioned “I’ve got five dollars, do I hear ten?” auctions, in a first-price, sealed-bid auction, the bidders don’t know what price their competitors have bid and have only one shot to name a fee high enough to win.

Companies that bid on oil leases must figure out how much the lease is worth. The ideal bid would beat all others, but not exceed the value of the lease. However, the actual value of the lease is only known in retrospect, so each company’s estimate, the basis for its bid price, will have a degree of uncertainty. Moreover, to beat competitors’ bids, the winning company has to bid the highest price.

Thompson says, “On average, the companies’ geologists are correct, but for a given set of drilling rights, some estimate too high and some estimate too low. The auction method picks the company whose geologist made the worst overestimate, so the winner is guaranteed to pay too much.” The phenomenon is called “the winner’s curse.” Before this situation was understood, the winning oil companies were paying too much for leases. Thompson says, “They lost their shirts. The companies literally quit bidding ­ they couldn’t afford to win.”

But now oil companies can use historical information about oil-lease auctions to estimate the winner’s curse and reduce their bids by that amount, Thompson says. The students in Business Math learn how to do so: they review historical information on estimates from 18 oil companies for 22 specific leases. The students also know what each leased tract turned out to be worth. As a result, the students can use that information to find out how much each oil company’s geologist erred in estimating the value of each of the 22 leases.

The students then run computer simulations of thousands of auctions to find out the average amount the companies would have overbid if each company had bid its estimate.

For the specific set of historical information the students use, the overbid is $23. 9 million. Thompson says, “That’s now our estimate for the winner’s curse, if everyone bid their estimate.”

By doing the calculations the students have now learned a company facing a first-price sealed-bid auction could compute the dollar figure for the winner’s curse, whack that amount off the geologist’s estimate of the value and then bid the new, lower figure. If the company then wins the auction, it won’t have overpaid for the lease.

But the winning bid needs to be only one cent more than the second-highest bid. By calculating what mathematicians call “the expected value of the difference” between the highest and the second-highest bid, a company could take an additional amount off its bid and still win. The students also learn how to use computer simulations to calculate that additional reduction, which is known as “the winner’s blessing.”

Using the mathematical formulas for computing the winner’s curse and winner’s blessing would require several additional years of mathematics and statistics courses, Thompson says. By using computer simulations instead, he and Lamoureux developed a means to introduce freshman-level students to quite sophisticated statistical concepts.

And in the process of developing and teaching the class, Thompson discovered a new Nash equilibrium for bidding in first-price sealed-bid auctions.

The Nash equilibrium is a plan which, if pursued by all bidders, means that no single bidder could do any better by deviating from the plan.

Under a given set of circumstances, “There is a unique, specific, estimate reduction that is stable,” Thompson says. “If all other companies use this reduction, then my company’s best response is to make the same reduction. That common estimate reduction is the Nash equilibrium.”

Although Thompson initially found the equilibrium from his work with the kinds of computer simulations that the students use in class, such simulations cannot prove to a mathematician’s satisfaction that the solution actually exists and is unique. That required that Wright and Thompson show the result analytically, by doing what laypeople would call writing an equation or formula. The team found the formula in summer 2003. Wright presented the work at an international mathematics meeting in March 2004.

Having a formula is handy, Thompson says. For example, by plugging different numbers into the formula, one can quickly answer the question, “How will the Nash equilibrium change if the number of bidders changes?” Getting the answer using the formula takes 0.01 seconds, he says, whereas it might take three hours to develop and run a simulation from scratch to get the answer.

“Knowing what we know now,” he says, “We can do it either way.”

Thompson brings it all back to teaching.

“The most interesting thing to me is that we have freshman working with concepts that won a Nobel Prize within the last 10 years,” Thompson says. “And by teaching this class, I discovered a new Nash equilibrium in 2001. I was literally working on the class material when the light bulb came on. So it went right in that semester.”