Non Zero-Sum Games

The scenario on the previous page is quite an interesting one for mathematicians who might be intrigued by the various mathematical possibilities that could stem from such games. However, as a representation of reality, it is somewhat removed from many of the ordinary interactions that happen in daily life and in economic activity. What is more realistic, therefore, is what are called non zero-sum games.

In non-zero-sum games there is no optimal solution like that of zero-sum games. A non zero-sum game occurs where the sum of the gains and losses to each player does not total zero - it is possible therefore for both players to gain or to lose. We can see such an example in the theory of comparative advantage. There are benefits to both sides from trading if each focuses its resources on specialising in the product in which it has a comparative advantage. The important thing to remember with these games is that there is no one 'best' solution.

International trade - an example of a situation where parties to a bargain can both become better off - a non zero-sum game. David Ricardo recognised this in the 18th Century but did not recognise it as a game as such. Copyright: Kevin Walsh, from stock.xchng.

The Prisoners' Dilemma

Non-zero sum games are more complex to analyse as a result. The classic case of the non zero-sum game is the so-called Prisoners' Dilemma. The dilemma is based around the following scenario. Two people are arrested and placed in separate rooms on a charge of (let us say) robbery. Each is then questioned and faces the following options. They can either confess to the crime or stay silent. They are each fully aware of the consequences of their action which is represented in the matrix below.

If I confess and you do choose to stay silent, I go free but you get 20 years in jail.
If I confess and you also confess, we both get 5 years in jail
If I stay silent but you confess, I get 20 years in jail but you go free
If I stay silent and so do you, we each get 1 year in jail

There is now a risk element in playing the game; there is no optimal solution. If I choose to stay silent there is a 50% chance that I could end up with a paltry sentence of just a year but equally I run the risk of going to jail for 20 years. If I choose to confess I have a 50% chance of going free or receiving 5 years in jail.

In this game, it is assumed that there cannot be any cooperation between the two players. Cooperation would imply that the best option is for us both to stay silent and take the lesser sentence of 1 year. Without the option of cooperating, I have to think about what I would least like - that seems to be the idea of spending 20 years in jail. I don't want to spend any time in jail but 5 years is preferable to 20 years but not as preferable as only 1 year.

Given this logic, therefore, my best option is to confess. I have a 50% chance of going free or facing 5 years in jail. If I chose to stay silent there is a 50% chance I could end up in jail for 20 years!

Want to spend 20 years in here or would 1 year be a better option? You had better think about what the best solution for you is in such a dilemma! Copyright: Nick Winchester, from stock.xchng.

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