Evolutionary Games and Ecosystems (12)

Free Market for Games with Natural Mixed Nash Equilibria

Mix-it-up

If you have followed us closely to this point, you might be thinking "Aha! game theory is a kind of oppressive totalitarian ideology". Of couse it is not. But if you have felt so, that is because we have been mostly dealing with game table which results in oppresive masure for improvements. We shall discuss the matter of ideology later on. Here let us just consider a game table given bellow.

This rule is essentially different from "prisoner's dilemma" and "collaborative game" in that your preferable choice depends on what choice your opponent makes. There are two point which is at the corossing end of your mmovement and your opponent's; red one and the blue one. They give two different Nash equilibria. To make profit, you should make the opposite choice to your opponent. But, of course, since the situation is the same for your opponent, you have no way to know what hand he will play. You are at the loss, and maybe want to resort to rolling dice to mix your two strategies. By trial and error, both parties eventually settle on a best rate of mixture. Of course this is nothing but the mixed Nash equilibrium we have already learnt before. Generally speaking, when there are two or more Nash equilibrium, there always is a mixed Nash equilibrium which often yield better payoff to both parties compared to the two pure equilibria. There is another type of games which has mixed Nash equilibria. The game table is given by

For good results, this one should be played by both parties playing the same hands. Since there is no preknowledge of your opponent's hand, you again play probablistically. Both parties eventually find out the best combination which will give the mixed Nash equilibrium as before. The process will be described by suitable stochstic process dynamics, and, if sumulated by replicator dynamics, eventually by Lotka-Volterra equation.

Invisible Hand of Gold Color

Therefore, when the rule of the game allows "reasonably profittable" mixed Nash equilibrium, there is no need for further intervention from the outside of the game (such as introduction of altruistic moral), and players' pursuit of self-interest is guided by 'invisible hand of game-theoretical god", so to speak. In hindsight, the free-market economics started by Samuel Adams, sorry, Adam Smith was based on this type of game. The extension of our first game to the N players is known as "minority game", and is known to be a rasonably realistic model of stock exchange markets, thus has been studied rather extensively in last decade. The N player extension of our second game should be called "majority game", and this could be of use for the research of the dynamics of elections and party politics.