Mathematics Awareness Week 1996

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Computer Undercut:
A Decision Making Game

Computer Undercut: A Decision Making Game

The 1996 Mathematics Awareness Week theme, "Mathematics and Decision Making," is depicted in a game, "Undercut," developed by Douglas Hofstadter in his book Metamagical Themas, and used by Jonathan Choate, mathematics department chair at The Groton School in Groton, MA. Members of Choate's advanced topics in mathematics class helped with this project, including Ted Chase, Molly Gregg, Alex MeVay, Thad Pollock, Cabot Henderson, Peter Nkongho, Ben Lyons and Amaurie Laurincent. The game is a variation on "prisoners' dilemma," one of the classic examples of the branch of mathematics known as game theory, which differs from statistics and probability in that two or more "players" have different goals or objectives.

Choate, as part of his advanced topics course, has developed Computer Undercut for use in middle and high school mathematics classes to illustrate such concepts as selection of optimal strategies, bargaining and negotiation, costs or benefits,and equilibrium outcomes. A key element is the opportunity to base current choices on previous ones.

Choate offers suggestions for teachers on how to include Computer Undercut in their classes; he also explains how to arrive at a "best strategy" using the idea of expected value.

Computer Undercut is played by two players (or teams) that alternate moves. In the first move player A secretly chooses one number between 1 and 5. Player B also selects a number. If one player's number is one less than the other than that player receives the sum of the two numbers chosen, otherwise they each receive the number they chose. For example, if player A chose 5 and player B 4, player B would receive 9 points. Play continues alternately until one player accumulates a pre-determined number of points (40 seems to work well) and wins the game. Winners are determined by playing best of three series.

Download the Game

Related Topic: Spatial Prisoners' Dilemma

See "Nice Guys Sometimes Finish First," a recent article which details 3-dimensional prisoners' dilemma, in the Mathematica in Education and Research Journal.

Background Note

Modern game theory is generally thought to have begun with the publication of "Theory of Games and Economic Behavior" by mathematician John von Neumann and economist Osar Morgenstern. One summary of game theory is in "For All Practical Purposes: Introduction to Contemporary Mathematics," W.H. Freeman, 1988, Chapter 11, Game Theory: The Mathematics of Competition. There is also an accompanying video. Another excellent resource is William Poundstone, "The Prisoners' Dilemma," Oxford University Press.
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