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
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.
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.
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.