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Game of Theories: John Nash and the Nash Equilibrium


Justin Liu

By

April 20th, 2016


Game theory is used everywhere today, but few people could have guessed that John Nash’s theories could be so universally applied. Justin Liu reiterates the importance of theory in a modern context.


Ron Howard’s A Beautiful Mind paints John Nash as a brilliant yet troubled mathematical genius. The film portrays him as a man capable of the incapable, yet it fails to go into detail about what Nash theorised – and how extensively his theories describe the way the world works.

 

Nash’s economic theories are subsumed under an umbrella called game theory. Though this sounds divorced from the standard “demand and supply” mainstream economics, the fact is, economics is fundamentally concerned with decision making. Game theory explores how and why agents make one particular decision over another, rather than focusing on the end result.

 

Nash is renowned for his conceptualisation of the Nash equilibrium. In a nutshell, a Nash equilibrium illustrates a state where two agents have nothing to gain from changing their own decisions, if the other agent’s decision remains unchanged. This can be quite difficult to understand, but this phenomenon is actually quite widespread and can be observed in our everyday lives.

 

One of the reasons Earth has not been plunged into a nuclear winter is due to an effect known as nuclear deterrence. Because nuclear weapons pack so much destructive power, an attack using them is bound to cripple its intended target. If two countries have a nuclear weapon arsenal, any attack by either of the two countries on the other would result in the complete destruction of both countries as the defending country retaliates. As such, no country would want to start a nuclear war. No country would also want to disarm their own nuclear weapons, as that would put them at a distinct disadvantage to the other country that kept their weapons. This would create an incentive against disarmament because simply possessing nuclear weapons act as a strong deterrent.

 

A more complex variation of the Nash equilibrium, and probably the more well-known, is the Prisoner’s Dilemma. As an example, let’s imagine that two people – named Alice and Bob are arrested by the police for some minor crime. However, the police think they have committed a more major crime, but do not have enough evidence to convict either of them. The police separate Alice and Bob, and each of them are offered a deal: (1). Sell out your partner by giving us information that leads to their arrest, and if your partner remains silent, you are free to go while your friend receives three years in jail. (2). Remain silent. If both Alice and Bob remain silent, the police just don’t have enough evidence and they both receive reduced sentences of one year each. If both of them sell out each other, they will both receive two year sentences. Game theorists like Nash would represent this rambling story in what is known as a payoff table, shown below:

(Alice , Bob) Silence Sell out
Silence 1 , 1 3 , 0
Sell out 0 , 3 2 , 2 (Nash Equilibrium)

 

The schedule in the middle represents the number of years that Alice and Bob respectively have to serve in prison, depending on each other’s decisions. The immediate conclusion is to assume the best possible solution is to cooperate and remain silent, as this results in total prison term of only two years. However, upon closer inspection, from an individual standpoint, Alice and Bob always gain from selling out the other. For example, if Bob remains silent, Alice gains no prison time as a result of selling out Bob (down from one year). If Bob sells out Alice, Alice only receives a two-year sentence (compared to three years). This holds in reverse. Not only do they end up collectively worse off in terms of total years in prison, they are also worse off than if they had cooperated.

 

A more realistic scenario could also be found in the market for airplane tickets. Two airlines, JetStar, and Virgin have two options they could undertake in their business operations. They could choose to discount their fares to increase sales; or, they could choose to hold their fares steady. An updated payoff table for this market would look like this:

(Jetstar, Virgin) Maintain Discount
Maintain 100, 100 25 , 150
Discount 150 , 25 50 , 50 (Nash Equilibrium)

 

When both companies maintain their pricing, their revenue remains unchanged. However, when one organisation discounts its products, its revenue increases due to higher sales whilst the other company’s revenue decreased due to reduced sales. If both companies decide to discount their products, both of them experience a fall in revenue. It can also be assumed that once one company begins to start discounting their products, the other company will respond by discounting because there is little incentive to keep prices high while the competitor is discounting. This phenomenon explains much of the price competition that we see in everyday life (e.g. Coles and Woolworths).

 

The Nash Equilibrium continues to be used today and has moved away from economics into other disciplines, including evolutionary biology and military strategy. The Nash Equilibrium even describes much conflict that we experience today, whether it may be personal conflict with our friends, conflict between corporations or even international conflict. Ultimately, Nash’s equilibrium is a perfect example of the cross-disciplinary relevance of economics and the impact that one person’s idea can have on the world.

The views expressed within this article are those of the author and do not represent the views of the ESSA Committee or the Society's sponsors. Use of any content from this article should clearly attribute the work to the author and not to ESSA or its sponsors.

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