Evolutionary Games and Ecosystems (3)

Game Theory and Nash Equilibrium

Pique Dame

"I am a gambler playing the card of life..". So sings Herman in Tchaikovsky's opera. He naturally keep winnig until the final big game, in which he draw the queen of spade which ruins him. Now, let us suppose you are Herman and play a card game against your friend/enemy prince Yeletky im pursuit of The Happiness. Both you and prince have two cards, marked as Angel and Devil, each of you showing one of the card together, and get gold Ruble coins according to the table shown bellow.

In both tables, left most column represents your hand, and top raw your opponent's. The table in the left shows your payoff Ruble. Let's call this Your Game Table. We assume that the game is fair and your oppenents would be getting the same treat, thus the his payoff should be given by the transoped one, which is the table in the right. According to this table, both players obtain 1 Ruble by going "Angel", but one player can benefit betraying the other with "Devil". The betrayed players suffer a lot. He will not repeat the mistake, thus also will go for "Devil" from the next time. In the end, both players, if they are "rational" (meaning properly calculating), will go for "Devil" and will stay poor getting nothing from the game.

The state which results from the persuit of both parties' own interest, the one shown in red, is called Nash Equilibrium. Some might recognize the name Nash as the hero of the well publicized movie "Beautiful Mind". The game we named "Angels and Devils" are also known by the name Prisoner's Dilemma. Technically, you can move up-and-down in your table to seek your benefit, and your opponent can move left-and-right in his table. nash equilibrium is the common end of those movement. In this particular table, you would be better of moving upward (Devil) in both case of your opponent playing Angel or Devil. With the same reason, your opponent moves left and choose Devil. So Devil-Devil becomes the Nash equilibrium.

Compassion and Pre-arrangement

Suppose, for now, that the brains of both players (or at least, its profit measuring part) are sudenly exchanged with some reason. In fact, this type of "irrational" devotion for others often occurs in real life. Think of love, for example. So you try to maximize your opponent's payoff, and your oppenent does same for you. This is to say, you look at opponents table and regard it as your own, and the same for the opponent. The result is easy to see, and both players will go for "Angel" and keep getting 1 Ruble each time they play.

So, in this case, altruistic game play out-performs egoistic (namely, usual) game play. But suppose that both players are seasoned veterans of life, and donot satisfy with this beautiful story of friendship. They try to squeeze the maximum profit and eventually find out that going for "Devil" ocasionary would produce better collective results, because, in this particular game table, the profit of betrayal exceeds loss of being betrayed. Trial and error would result in them finding theright mixture of random betrayal, which is one in four in this case. Both parties on average obtain 1.125 Ruble per play, giving 1/8 more payoff compared to pure friendship of 100% angels.

To prove that would require some space. So we change the chapter.