For the Good of All

United We Stand

For united we stand
Divided we fall
And if our backs should ever be against the wall
We'll be together, together, you and I

Working together is not just the moral strategy, it is the richer strategy.

John Nash: If we all go for the blonde and block each other, not a single one of us is going to get her. So then we go for her friends, but they will all give us the cold shoulder because no on likes to be second choice. But what if none of us goes for the blonde? We won't get in each other's way and we won't insult the other girls. It's the only way to win. It's the only way we all get laid.

This quote from the 2002 Oscar Best Picture, "A Beautiful Mind”, is perhaps not the most elegant example of a Nash Equilibrium in Game Theory, but it does get the point across.  The scene takes place in a bar where John Nash and his friends are trying to pick up women.  If each friend acts without regard to what is best for everyone, then nobody will win. A User Optimal solution, getting the blonde, is not the System Optimal solution, making a pickup.  If each friend agrees not to pursue their own User Optimal solution, then the System Optimal solution is more likely to be achieved.

A key aspect in Game Theory is that games will be repeated, i.e. there will be a future.  If you want to find someone with whom to play a game, they have to feel that the game is fair, that you will not cheat, and that you are not misrepresenting yourself as being a worse player than you are.  It is why there are rules for the game and rankings, handicaps for players.  The price of cheating or misrepresenting yourself, hustling, is that you may not ever play another game.  If you believe in a future then you want to play another game, allow for growth.  If you believe in a future, then not pursuing the User Optimal solution may be the best strategy, for both yourself and others in the long run.

A classic example is the Ultimatum Game, where  Player 1 receives $100 to share with player 2.  The amount that Player 1 can offer to Player 2 can vary from $99 to $1.  If Player 2 accepts the offer, both players get to keep the money.  If Player 2 does not accept the offer, neither player gets to keep any of the money. The User Optimal solution is to give only $1 to the other player and keep $99 for yourself.  It was expected that this offer would always be accepted, because then each player would be richer. Player 1 by $99 and Player 2 by $1. But in practice Player 2 would not accept an offer of less than $30.  It seems that the other player expected the game to be played again and expected to offer at least $30 if the roles were reversed. When the Player 1 offered only $1, he indicated to Player 2  that he did not expect to play again, in other words the User Optimal  strategy places no value on future winnings. 

The User Optimal strategy is to offer only $1.  The System Optimal strategy appears to be an offer of $30.  If there is a second game, with roles reverse, and in that game Player 2 also follows a User Optimal strategy and that offer was rejected, then the result is that neither player has any money.   In the second game, with roles reversed, if Player 2 offered Player 1 $30 and the offer was accepted, then after two games Player 1 would have $30 and Player 2 would have $70, for a system total of $100. If the offer in both games was $30, the System Optimal Strategy, and was accepted each time, after two games each player would have $100, for a system total of $200..  

If both Players always pursue a User Optimal strategy, no one wins, ,e.g. no one gets the blonde. If both players follow a System Optimal strategy, in every game, both players and society would be richer.