Prisoner’s Dilemma
An illustration of Nash Equilibrium
Art and Bob are both suspects in a crime, and they
are both offered the following deal if they
confess…
Art’s Strategies
Deny
Bob’s
Strategies
Confess
Confess
3 yrs.
3 yrs.
1 yr.
10 yrs.
Deny
10 yrs.
1 yr.
2 yrs.
2 yrs.
Consider Art’s options…
1. If Bob denies and Art denies, then Art
will get two years. Art is better off
confessing and getting one year.
2. If Bob confesses and Art denies, then
Art will get ten years, so Art is much better
off confessing and taking three years.
Consider Bob’s options…
1. If Art denies and Bob denies, then Bob
will get two years. Bob is better off
confessing and getting one year.
2. If Art confesses and Bob denies, then
Bob will get ten years, so Bob is much
better to confess and take three years.
Thus, both parties will rationally choose to confess, and take three years – even though they
could have been better off denying. Each party does this because, considering the possible
options of the other party, they always found the better option was to confess. When neither
party has an incentive to change their strategy, they are in “Nash Equilibrium.”
Nash Equilibrium
John F. Nash, 1928 -
Russell Crow portrays John Nash in A Beautiful Mind
NASH EQUILIBRIUM An important concept in game theory, a
Nash equilibrium occurs when each player is pursuing their
best possible strategy in the full knowledge of the other players’
strategies. A Nash equilibrium is reached when nobody has any
incentive to change their strategy. It is named after John Nash,
a mathematician and Nobel prize-winning economist.
The Implications: The Duopolist’s Dilemma
Acme and Belco are the only firms operating within a given market. Thus, they are operating in
a duopoly. Both firms could increase their profits if they were to collude with each other in an
effort to restrict output. However, each firm could make even more money if they agreed to
collude, but then cheated and secretly increased their output. Thus, each firm is faced with the
following payoff matrix…
Consider Acme’s options…
Acme’s Strategies
Comply
Belco’s
Strategies
Cheat
Confess
Cheat Comply
$0
$0
-1.0m
4.5m
4.5m
-1.0m
2.0m
2.0m
1. If Belco complies and Acme complies, then
Acme will earn 2 million. Acme is better off
cheating and earning 4.5 million.
2. If Belco cheats and Acme complies, then
Acme will lose 1 million, so Acme is again better
off to cheat and earn zero profit.
Consider Belco’s options…
1. If Acme complies and Belco complies, then
Belco will earn 2 million. Belco is better off
cheating and earning 4.5 million.
2. If Acme cheats and Belco complies, then
Belco will lose 1 million. Belco is again better off
to cheat and earn zero profit.
Thus, both parties will rationally choose to cheat, and earn zero profits – even though they could have been
better off by complying with a collusive deal to restrict output. They will compete as fiercely as pure competitors continually lowering their prices and increasing production until they reach a point where they are breaking even.
This is where they will find themselves in Nash Equilibrium. They will have no motivation to move from this point.