Make your own free website on

The Math Learning Portfolio Casino: Test Your Luck

Probability Laws
Four Probability Properties
Pascal's Wager and Bernoulli Trials
John Forbes Nash and Game Theory
Chance Games
Math in Music
Demonstration of Goals

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 Carnegie-Mellon 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 “Non-cooperative 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 Lopez-Harrison de Lard. 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 insulin-coma 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. Hungarian-American 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 “win-lose” 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 – non-cooperative (strategic) games, and cooperative (coalitional) games. John Nash claimed that all cooperative games could be reduced to some form of non-cooperative 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.





Enter Industry


Stick to Toilet Paper


Enter Industry






Stick to Toilet Paper






The upper-right and lower-left 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 lower-right 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 10-year sentence and you go free. If they confess and you remain silent, you get the full 10-year 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 “non-zero-sum” 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).


Real-Life 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

-          Plea-Bargaining: plea-bargaining 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.:

  1. Surgical air strike (A): wipes out all the missiles already installed, possibly followed by invasion of Cuba
  2. 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:

  1. Maintenance (M) of missiles
  2. 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



-          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)



-          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 toilet-papering 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 rock-paper-scissors, 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?



-          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!