Evolutionary Games and Ecosystems (10)

Advanced Sociology of Altruism : Game of Knights & Peasants

In Anarchy, Kights and Peasants have Equal Payoffs

Now, as the finish-up, we consider a society consisting of large number of people, who ramdomely paired and play games whose rule is given by the following table. The left-most colume is your hand and top raw your opponent's. The game is fair, so the opponent's point is given by a table which is given by transposing raw and column of this table. The characters a, b, R, f are some given real numbers which we shall specify later. According to the game table, you can choose from three strategies

* "Do Nothing" is as it says: Do not participate in the game, and always get zero point. then why is this thing here? Because player with this strategy can later participate by choosing something else at some point. So we need to keep this strategy. In physics, this is known as "vacuum". If you take out this one, the table would be reduced to that of prisoner's dilemma.
* "Peasant" earns points b per turn by producing good with his own labor. When he meets another peasant, his income get reduced by a for overcrowding. When he meets a kight, a large amount of wealth R (bigger than n) is taken away. Resisting is pointless because of the sward and chainmail of the kight.
* "Kight" needs to consume d per turn to maintain his armory. He can take away peasant's wealth R and make it his own with efficiency f. Meating another kight (or meating do-nothinger) will not do any good.

Every stategy has pros and cons. Which role would you play in this game?

Everybody is Equally Worse-off in Hobbsian State

You play this game repeatedly to maximize your own payoff. The best strategy depend on strategy of the others, so probablistic mixed strategy is employed to maximize the expected payoff assuming everybody else does the same. The resulying mixed Nash equilibrium is obtained by strategy vector (1-X-Y)(X)(Y) which is specified by
　　0 = x( b- ax - Ry)
　　0 = y( -d + fRx)
Interesting fact is that, at this equilibrium, your payoff doesnot change by changing your hand. This is so by the very nature of Nash equilibrium: If otherwise, you will choose a profittable strategy over others and the rest of the system will do the same and the system will head for some other state, which is in direct contradiction of the definition of equilibrium. In this sense, Nash equilibrium is a state in which no change in your strategy will improve your lot. Since do-nothinger's payoff is always zero, this means that you are end up with zero point with your best effort combined with the best effort of the others.