27 November 2010

Game Theory for Noobs

Game theory is not so much about Playstation as it is a science about strategy. The purpose of game theory is to determine the most rational course of action. Rational in this sense means the course of action that results in the highest amount of possible gain, or in the lowest amount of possible loss.

Christina Aguilera Christina Aquilera Hollywood Star
Christina Aguilera. She recently received her own Hollywood star. Not sure why, but it must mean she knows a thing or two about strategy.

John Nash is frequently associated with game theory. His tragic bout of schizophrenia is often abused to ridicule game theorists and to discredit the science of game theory, as is the case in the BBC documentary The Trap. However, John Nash was a fairly latecomer to the field as John von Neumann established the science in collaboration with Oskar Morgenstern with their book, Theory of Games and Economic Behaviour. This book reported a means to quantify the qualitative notions of expected utility, so in a way von Neumann and Morgenstern gave us mathematics for hope just like Bayes gave us the mathematics for faith with Bayesian analysis.

Nash Bridges
Nash Bridges. Not really relevant here.

The Nash equilibrium

Nash's contribution is the Nash equilibrium. Nash equilibrium results when all players involved know the strategies available to themselves and the other players, but no player has anything to gain by switching strategies. This appears superficially similar to the African philosophy of Ubuntu, which is an ethical concept based on the premise that “I am what I am because of who we all are”.

Thus, my strategies are limited by your strategies and your strategies are limited by my strategic choices, but neither of us have anything to gain by changing our current courses of action. If a Nash equilibrium is available, the most rational course of action is to cooperate and not to change your strategy. A Nash equilibrium is however not possible in all kinds of games. In these games, one can maximise gains by not cooperating (that is, by “cheating” or defaulting).

“Seven deadly sins, seven ways to win”

Strangely scientific lyrics from Iron Maiden in Moonchild. Seven of the games in which cheating or defaulting may be beneficial to you and detrimental - fatal, even - to other players as identified in Len Fisher's excellent book Rock, Paper, Scissors are:

  1. The Prisoner's Dilemma

  2. The Tragedy of the Commons

  3. The Free Rider Problem

  4. Chicken

  5. The Volunteer's Dilemma

  6. Battle of the Sexes

  7. Stag Hunt

In the next couple of posts, I shall deal with each of these and elaborate in my usual sombre tone.

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