Maths Illustrated
John Nash, Russell Crowe’s character in A Beautiful Mind, was awarded a Nobel prize for
his work on non-cooperative games. The example below illustrates Dr. Nash’s equilibrium
theorem for Haverford students and fans of Mr. Crowe. (Lynne Butler, Haverford College)
A two-player game: At the beginning of their friendship, Russell and Kevin enjoy chance
meetings at the local pub or frozen pond. On evenings they meet at the pub, both enjoy
drinking Fosters but Russell more so (say 4 smiles for Russell and 2 smiles for Kevin). On
evenings they meet at the pond, both enjoy playing hockey but Kevin more so (say 1 smile
for Russell and 3 smiles for Kevin). At this early stage in their friendship, Russell and Kevin
do not cooperate when planning their evenings; sometimes one finds himself at the pub and
the other at the pond. Neither enjoys himself without the other’s company. The payoffs to
each player are recorded in the tables R and K below.
R=Payoffs to Russell
R@pub
R@pond
K@pub K@pond
4
0
0
1
K=Payoffs to Kevin
R@pub
R@pond
K@pub K@pond
2
0
0
3
Equilibrium strategies: Russell plays this game by deciding what fraction of evenings he
spends at the pub and what fraction of evenings he spends at the pond. For example, Russell
might go to the pub 3/5’s of the time and the pond 2/5’s of the time. How much enjoyment
Russell gets pursuing this strategy (p = (3/5, 2/5)) depends on what Kevin decides to do. If
Kevin decides to go to the pub 1/5 of the time and the pond 4/5’s of the time (q = (1/5, 4/5)),
then the payoff to Russell
pRq is (3/5)(4)(1/5) + (2/5)(1)(4/5) = 20/25 = 4/5,
and the payoff to Kevin
pKq is (3/5)(2)(1/5) + (2/5)(3)(4/5) = 30/25 = 6/5.
This pair of strategies (p = (3/5, 2/5) and q = (1/5, 4/5)) is called a Nash equilibrium because
neither player can increase his expected payoff by unilaterally changing his strategy. So, for
example, if Russell changes to p = (1, 0) while Kevin continues with q = (1/5, 4/5), then
the payoff to Russell pRq is (1)(4)(1/5)+(0)(1)(4/5)=4/5, the same as before. Likewise, if
Kevin changes to q = (0, 1) while Russell continues with p = (3/5, 2/5), then the payoff to
Kevin pKq is (3/5)(2)(0)+(2/5)(3)(1)=6/5, the same as before. Neither man is motivated
to change his strategy. The other Nash equilibria for this game are when both men go to
the pub every evening and when both men go to the pond every evening. Notice that these
Nash equilibria are better for both men.
Further reading: John Nash’s elegant proof that any n-player non-cooperative game has
a Nash equilibrium may be found in the two-page paper “Equilibrium Points in n-Person
Games”, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), pp. 48–49. A good undergraduate text is
Game Theory by Guillermo Owen, published by Academic Press, Inc.