Remember the movie A Beautiful Mind? Directed by Ron Howard, it starred Russell Crowe, who played a brilliant but schizophrenic mathematician, John Forbes Nash, Jr. Though at heart a mathematician, Forbes received the Nobel prize in Economic Sciences in 1994. There is no Nobel prize for mathematics.
Why economics? Because his mathematics of partial differential equations combined with game theory explains so much of why people act the way they do, and industry structures form the way they do.
Core to this is the concept of a particular equilibrium that is now called the Nash Equilibrium.
John Forbes Nash, Jr.
A group of players are in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others. That is, each player plays his or her own optimal strategy with that knowledge that the others are playing their optimal strategy too, and what that strategy is. If no player has an incentive to change, it's an equilibrium. A game may have one, many, or no equilibriums. According to Mathworld, cited above, "in a Nash equilibrium, no player has an incentive to deviate from the strategy chosen, since no player can choose a better strategy given the choices of the other players."
Let's take a famous example, the prisoner's dilemma. Two criminals cooperated in a crime, and are arrested and placed in solitary confinement. The prosecutor can charge each for a minor offense, but he is thinking strategically. Each is offered the same plea deal:
- If both confess the crime, each of them serves two years in prison
- If one confesses but the other denies the crime, the confessor will be set free but the other will serve five years in prison
- If both deny the crime, each will only serve a six-month sentence
No matter what the other does, it is better to rat on your accomplice. There is one equilibrium for this -- where they both squeal. This is true even though both are worse off than if they remained silent.
Now, let's consider stag hunting. A hunter can hunt a stag or a hare. Stags are much more valuable than hares, so why not hunt stags? Stag hunting, in this game theory exercise, requires cooperation from another hunter to be successful, but hare hunting doesn't. This game has two equilibriums. Both hunters either hunt stags, or hares, but never mixed. If both are hunting stags, why change? Of course, this is an equilibrium, because both are highly rewarded with the prized stags, and any change causes them both to lose. If both are hunting hares, one hunter changing would deny himself the hare, but he would not get the stag. Each hunter loses if they alone change, so they won't. That's the second equilibrium. One hunting stags, and the other hares, is not stable. Sooner or later one will adopt the hunting of the other, and the game then snaps to that equilibrium. Both of the above examples assume there is no communication between the players.
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