




John Forbes Nash and Game Theory




John Forbes Nash and Game Theory


John Forbes Nash: Life
Early
Life
John Forbes Nash was born June 13, 1928 in the small city of Bluefield, West Virginia to John Nash
Sr., an electrical engineer, and Margaret Virginia Martin, who had been a schoolteacher before she was married. He also had
a sister, Martha, who was two years his junior. His father had served in France as a lieutenant in the supply services during
the First World War. Nash’s parents had come to Bluefield so his father could work for Appalachian Electric Power Company.
He was interested in academics from an early age, learning from a picture encyclopaedia and other books as a child to supplement
his standard schooling. His early initiative regarding education developed as he grew. He performed numerous chemical and
electrical experiments in high school, and he took supplementary math courses at Bluefield College for a year prior to entering
university.
Secondary
Education
Nash attended Carnegie Institute of Technology (now
known as CarnegieMellon University) in Pittsburgh, on a full scholarship, with his major as chemical engineering. However,
after one semester of those studies, Nash shifted to chemistry instead. But the mathematics faculty, who recognized his genius,
encouraged him to change his major again – to mathematics – which he did. He graduated from Carnegie with a M.S.
as well as a B.S.
After
being offered fellowships to study at both Harvard and Princeton, Nash chose Princeton for his graduate studies. He began
his PhD when he was just twenty years old. It was at Princeton where his interest in game theory developed. His doctorate
was entitled “Noncooperative Games”, and graduated in 1950.
Personal
Life, and Mental Illness
Nash met a woman named Eleanor Stier, and in 1953, she gave birth to his son, John David Stier.
Nash refused to marry her, though she tried to persuade him. Then, during an academic sabbatical from MIT in 1957, Nash met
Alicia LopezHarrison de Lardé. She was originally from El Salvador,
and had graduated as a physics major from MIT. They were married in February that year. It was around the time his wife became
pregnant in 1959 that Nash’s “mental disturbances” began. She had him involuntarily hospitalized in a private
psychiatric hospital outside of Boston, Mclean Hospital, where he was diagnosed with paranoid schizophrenia. After he was
released, he left MIT and travelled to Europe with the intention of renouncing his U.S. Citizenship. His wife followed, and
had him deported back to the States. They returned to settle in Princeton. However, Nash’s illness only worsened. In
1961, again his wife had him committed to hospital (this time, Trenton State Hospital in New Jersey). Treatments for mental
illness were very primitive at that time in history Nash endured insulincoma therapy, which was risky and painful. Then,
he endured divorce – but Alicia housed him in Princeton (she described him as her “boarder”, and they were
like “two distantly related individuals living under one roof). They renewed their relationship in 1994, when Nash won
the Nobel Prize for Economics (which he shared with John C. Harsanyi and Reinhard Selten.
Portrayal
in “A Beautiful Mind”
Russell Crowe portrayed John Nash in the 2001 major motion picture “A Beautiful Mind”,
which was based on Sylvia Naser’s book of the same name. The film was nominated for various Academy Awards – including
Best Actor (Russell Crowe), Best Editing, Best Makeup, Best Music (original score) – and won several as well: Best Actress
in a Supporting Role (Jennifer Connelly), Best Director (Ron Howard), and Best Screenplay (based on previously published material).
It even won the Best Picture Oscar. However, there are important inaccuracies, and omissions, in this film that should be
taken note of. One major criticism was “over dramatization” of Nash’s life. There was also the omission
of his time spent in Europe. We heard nothing of when his wife divorced him, after the she could no longer tolerate his delusions
and behaviour. Many of the details of his mental illness were also flawed. PBS produced a documentary, “A Brilliant
Madness”, which intended to portray Nash’s life more accurately. It featured interviews with other mathematicians
and economists as well as Nash himself and members of his family.
Mathematics:
Nash and Game Theory
Game
Theory?
Game theory is a branch of mathematics that has a wide range of applications, including economics,
biology, political science and law, sports, psychology, and military strategy. Interactions with games (or “formalized
incentive structures”) are studied with the use of models. It concerns the behaviour of individuals in these “games”,
and most advantageous strategies. HungarianAmerican mathematician John von Neumann is credited as the father of game theory.
He published a book with Oskar Morgenstern called “Theory of Games and Economic Behaviour”. Neumann treated “winlose”
competitions, and Nash showed him a stable mathematical scenario where both, or neither, side would win. Strategies in game
theory can be “pure” (that is, to play a particular move) or “mixed” (randomly played). Game theory
can be (roughly) divided into two categories – noncooperative (strategic) games, and cooperative (coalitional) games.
John Nash claimed that all cooperative games could be reduced to some form of noncooperative game.
Nash
Equilibrium
Suppose that there are two (or more) players in a game. If they are each using a consistent strategy,
and any change of strategy would not benefit the player, they have reached a Nash equilibrium. In a Nash Equilibrium, any
change of strategy will result in a deterioration of the player’s fortune.
Economics
and Game Theory
Game theory
plays a huge role in economics – for instance, with risk aversion. Game shows are a good example of risk aversion. Often,
when someone is faced with the choice of taking a smaller amount for sure, or have a chance of winning a larger
amount, they will take the smaller, but certain, amount. The opposite behaviour is exhibited with the lottery – people
taking a chance with the odds stacked very high against them – which is risk seeking. Game theory is also important
to the corporate world. Let’s say that Cottonelle and Royale were both thinking of expanding into the paper towel industry,
and assign hypothetical numerical values to their benefit from it.
Cottonelle
Royale 
Enter Industry 
Stick to Toilet Paper 
Enter Industry 
10
10 
10
1 
Stick to Toilet Paper 
1
10 
5
5 
The upperright and lowerleft squares are both Nash Equilibria. Cottonelle or Royale has to convince the other
they are wholly committed to a strategy “no matter what”, so that the other will take the “stick to T.P.”
option. But, those Nash Equilibria are inefficient, since the lowerright square would give a total “pie” of 10,
which no other square would do.
Biology
and Game Theory
In 1973, John Maynard Smith and George R. Price introduced the evolutionary stable strategy (also
called the evolutionarily stable strategy). It can apply to genetically determined physical traits, such as the length of
an animal’s tale, and as well to behavioural traits, such as whether to challenge or retreat from an opponent (the strategy
may also be conditional – for instance “fight if opponent is smaller, retreat if opponent is larger”).
.
The Prisoner’s Dilemma
“The police arrest two suspects: you and another
person. The police have inadequate evidence for a conviction. But, having separated both of you, they visit each of you and
offer the same deal: if you confess and your accomplice remains silent, they get the full 10year sentence and you go free.
If they confess and you remain silent, you get the full 10year sentence and he goes free. If you both stay silent, all they
can do is give you both a few months sentence for a minor charge. If you both confess, you each get 6 years.”
The Prisoner’s Dilemma is a “nonzerosum” game where it is
assumed, and reasonably so, that each individual player intends to capitalize on their own advantage or benefit without concern
for the other players – meaning that their goal is to minimize their own jail sentence. That gives you two options:
cooperate with your accomplice and remain quiet, or betray them and confess. The outcome of either choice depends on what
your accomplice decides to do, but the prevailing strategy (since you do not know what your partner will do, or if you can
trust them) is confession. Even if you trust your accomplice to cooperate and stay silent, it’s in your best interest
to confess, so that you can go free right away. In addition, if your partner does cave as well and confesses, you still have
a reduced jail term. However, if this problem were considered in the best interest of the group, or both people, then
the smartest choice would be to both cooperate and stay quiet, getting a very small sentence.
The Prisoner’s Dilemma has one Nash Equilibrium – when both players
confess (“defect”). Although that option, as determined above, is not as good a choice as cooperating, both staying
silent is “unstable”, since one player could benefit greatly from defection while their partner (opponent) cooperates.
Therefore, “both cooperate” is not an equilibrium, but “both defect” is. It should be noted that mutual
defection is not only an equilibrium but also a “dominant strategy”. The difference between the two is that a
dominant strategy is your best move, regardless of what anyone else’s strategy is, whereas the Nash Equilibrium
says that it is your best move, given the strategies of everyone else. Douglas Hofstadter said that the Prisoner’s
Dilemma “payoff matrix” could be written in different ways, so long as it is based on this principle:
T > R > P > S, meaning T (temptation to defect – when you defect and
the other person cooperates) > R (reward for mutual cooperation) > P (punishment for mutual defection)
> S (“sucker’s payoff” – when you cooperate and the other person defects).
RealLife Examples of Prisoner’s Dilemma
 Arms race:
(also related to military strategy) between different states (two options, to either increase military spending or to form
an agreement to reduce weapons, both states incline towards increasing military expenditure since there is no guarantee the
other would keep to such an agreement)
 Tour de France,
or other cycling race: the two cyclists well ahead of the pack can work together through mutual cooperation by sharing the
front position, where there is no shelter from wind. Mutual defection, meaning neither cyclist made an effort to stay in front,
would result in the larger pack catching up. But often, one cyclist does the hard work (cooperation) at the front of the pack,
while the second cyclist follows behind (defection), while taking advantage of the slipstream, and ends up winning the race
 PleaBargaining:
pleabargaining is illegal in many countries because of the theoretical conclusion of the Prisoner’s Dilemma –
since it’s in the best interest of both people to confess and testify against the other, even if they’re both
innocent. Possibly worse is if a guilty party confesses against the innocent, who is of course unlikely to confess
Military Strategy: The Cuban Missile Crisis
The Soviet missile installation in Cuba, and the resulting American attempts to remove the
weapons, can be related to game theory.
Options for U.S.:
 Surgical air strike (A): wipes out all the
missiles already installed, possibly followed by invasion of Cuba
 Naval Blockade (B): a.k.a. “quarantine”,
prevents shipment of more missiles, but does not remove existing ones (followed by stronger action, i.e. diplomatic action,
so U.S.S.R. would withdraw missiles already installed)
Options for Soviets:
 Maintenance (M) of missiles
 Withdrawal (W) of missiles


Soviet Union 



Withdrawal (W) 
Maintenance (M) 
United States 
Blockade (B) 
Compromise (3,3) 
Soviet victory, U.S. defeat (2,4) 

Air Strike (A) 
U.S. victory, Soviet defeat (4,2) 
Nuclear War (1,1) 
Key: (x,y)
= (payoff to U.S., payoff to U.S.S.R.)
4 = best, 3 = second best, 2 = second worse, 1 = worst
Game Theory in Popular Culture
Movies
 The Hunt for the Red October: during the Cold War, the captain of a Soviet submarine tells
his fellow officers he has informed the Soviet government of their intention to defect to the U.S. (example of “burning
bridges”)
 Rebel Without a Cause: original game of chicken on film – first to jump from their car
as it drives towards the edge of a cliff is the chicken (two pure strategy equilibria, one mixed strategy equilibrium)
 The Good, the Bad, and the Ugly: Clint Eastwood’s character set up a situation where
each man evaluates his possible moves (with information given to them, which has been manipulated). But, really, Eastwood’s
character has already won the game.
 Murder by Numbers: example of Prisoner’s Dilemma
 Thirteen Days: showed game theory during Cuban Missile Crisis (effectively a game of
chicken)
Television
 The Weakest Link: contestants must help each other to make money, but have to beat each
other out in the end to keep the money for themselves, so it is important to know who you should vote off or keep on
 South Park: when the boys are arrested for toiletpapering a teacher’s house, they face a classic
Prisoner’s Dilemma (with detention instead of jail terms)
 The Simpsons: in an episode where Bart and Lisa compete in a game of rockpaperscissors,
Lisa predicts Bart’s use of “rock” (demonstrates the importance of unpredictable mixed strategies)
 Survivor: Who should you vote off – when is it smart to start voting off the strongest members?
Or to keep loyal to alliances?
Music
 Murray Street: Murray Street is a 2002 Sonic Youth album. It was described as “a product of cooperative game theory. Like all Sonic Youth albums, it
is a result of individuals, striving in a collectivist environment, for goals that are only understood once they are achieved”.











math is fun!






