Faculty Scholarship

Professor Koether and Professor Pendergrass

Dr. Robb T. Koether, Professor of Mathematics and Computer Science, Dr. Marcus H. Pendergrass, Visiting Assistant Professor of Mathematics

PROFESSOR of Mathematics and Computer Science Robb T. Koether, with his former colleague John S. Osoinach, Jr., and a current colleague, Visiting Assistant Professor of Mathematics Marcus H. Pendergrass, continues to work on variations of the Lying Oracle problem, previously treated in “Outwitting the Lying Oracle,” which was published in Mathematics Magazine. In the basic problem, a player is placing wagers on the outcome of a coin-toss. An all-knowing oracle announces how the coin will land, but there is a chance that the oracle is lying. What is the player’s best strategy?

Koether also analyzed a variant of the familiar game of Scissors-Paper-Stone. In this variant, the game begins when a referee tosses a biased coin. If the coin lands heads, then player #1 is told to choose either scissors or paper; if it lands tails, he is told to choose either paper or stone. Player #2 does not know how the coin landed. What is player #2’s best strategy? Koether discovered that if the probability of the coin landing heads is between 1/3 and 2/3, then player #2 should play as though there were no coin. However, if the probability of heads is less than 1/3, then his best strategy is to assume that the coin landed tails, and if the probability is greater than 2/3, then he should assume that the coin landed heads. In either case, he should play according to that assumption.

The interest of Pendergrass in the mathematical field of game theory was developed after attending a colloquium on the Lying Oracle game led by Koether. One generalization formulated by Pendergrass is called “The Tourist Game,” in which the “Guide” is leading the “Tourist” through a “maze,” or “directed graph,” and the Tourist is trying to guess where the Guide will go next. The Tourist places a wager on the guess and receives a payoff if the guess is correct.

While the game is seemingly different from the Lying Oracle game, on a certain type of maze they are in fact the same. Pendergrass has derived the optimal strategies for the Tourist Game and investigated the way in which the game progresses when the players are using the optimal strategies. The results have, in turn, shed new light on the Lying Oracle game.

That work and related work have resulted in several presentations at regional and national conferences.

In addition to his work on game theory, Pendergrass, who came to the College in 2005, drew on his prior experience as a scientist in the communications industry to co-author a chapter on “UWB Propagation Channels” in UWB Communication Systems: A Comprehensive Overview (Hindawi Publishing Company, 2006). UWB, an abbreviation for “ultrawideband,” is an emerging radio technology in which an ultrawideband radio transmits information on many frequencies at the same time, in contrast to the single frequencies of the more familiar AM or FM radios. Pendergrass has developed mathematical models that describe the changes that UWB signals undergo as they propagate through the environment.

In addition to that theoretical work, Pendergrass holds nine patents in UWB communications, most of which center on modulation and coding techniques to enable reliable communication.

Koether came to the College in 1981 with a B.S. degree from the University of Richmond, and M.S. and Ph.D. degrees from the University of Oklahoma. Pendergrass received his B.A., M.A., and Ph.D. from the University of Alabama in Huntsville.

BEYOND THE Classroom FOR THE Classroom
Hampden-Sydney College Faculty Scholarship 2005-2008
A report by the Office of the Dean of the Faculty