(Nash, Auman, Shelton, Schelling, Hasranyi & Shapely)
Game theory is first formulated by Von Nueman and Morgenstern. Later on John Nash, Richard Shelton and John Hasranyi further developed the game theory and they were awarded the Nobel Prize jointly 1994. The Nobel Prize was given again in 2005 to Schelling and Aumann for their analysis of “Conflict and Cooperation in Game Theory”. One way of remembering the five Nobel Economists above is to use the name NASH as an acronym. The first letter ‘N’ stand for Nash, ‘A’ stands for Auman, ‘S’ for Shelten and Schelling and ‘H’ for Hasranyi.
Game theory considers situations, in which there are two or more Players, each of whose actions influence the outcome of certain event. Depending on which event actually occurs, the Players receive various payments (pay-offs).
A famous example cited in Game theory is the Prisoner’s dilemma case. The story goes like this. Two individuals are known to be guilty together of a small theft case and a big Crime; there is evidence to convict them for a small theft but not for a big crime. They are kept in separate cells by the Police. A Public Prosecutor talks to them separately. He tells each individual separately the Rules. If one prisoner confesses and the other does not the person confessing will go free without any punishment. But the prisoner who does not confess get prison term for 6 years. If they both confess, they will be convicted for the Crime but in view of their good behaviour in confessing, the prisoners will get prison term for 3 years only instead for 6 years. If neither confess, then there will be no evidence to convict them for the crime. As such they get short prison time of 1 year each for the theft case.
We present below the punishment matrix (which is opposite to pay off). Least punishment means high payoff
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Prisoner - 2 |
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Confess |
Not-confess |
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Prisoner – 1 |
Confess Not-Confess |
(3, 3 ) ( 6, 0) |
(0, 6) ( 1, 1 ) |
Considering Prisoner-1, confess strategy dominates not-confess strategy. Taking the first row and first number in the cells, 3 and 0 in the confess row and compare them with the 6 and 1 in not-confess strategy. 3<6 and 0< 1. The prison terms are less in both cases (pay-offs are higher) in the confess category. As such prisoner-1 prefers to confess. Similarly for the second prisoner, taking the second numbers in the cells, column wise 3 and 0 in the confess category and compare them with 6 and 1 in not-confess category. Confess strategy dominates the not-confess strategy. So prisoner-2 prefers to play confess strategy. So both the prisoners confess and get 3 year prison each. This type of dominant equilibrium is termed Nash equilibrium.
The prisoners’ dilemma illustrates that self interest prevails over group interest and makes the prisoners confess. But it they cooperate with each other and both adopt the strategy of not-confessing, they reduce the prison term to one year each.
Nash Equilibrium is applied to Cournot Duopoly problem. Each firm chooses its best strategy to the other firm’s best strategy. In the Cournot equilibrium, which is discussed in the chapter on Markets. Each Duopolist produces an amount that maximizes his profits given the output of its competitor. Neither Duopolist has any incentive to change its output. It is termed Cournot-Nash equilibrium. It is a non-cooperative game. Games involving joint action and are contractual are termed cooperative games.
Hasranyi made significant contribution to games of incomplete information and Shelton made contributions to Extensive Games. Shelling made significant contributions to Strategies of Conflict in game theory and Aumann made valuable contributions to randomized strategies and Correlated equilibrium in game theory. For detailed treatment of game theory, readers may consult the excellent books on game theory mentioned in the Appendix.
In the first para of this Chapter, we mention that the letter ‘S’ in Nash stands for Shelton and Shelling. Now the letter ‘S’ stands for Shapely too. Shapely’s contribution are mostly in co-operative Game Theory. He used a value named after him as Shapely Value to the cost allocation problem in designing markets. We listed Shapely in this Chapter while his co-sharer of the 2012 Nobel prize, Alvin Roath is listed in Chapter 15 on Information Theory.