Wednesday, October 5, 2011

Game Theory: Part IV: Non-Zero-Sum Games: Chicken

Up until now, we have only studied games in which players have directly opposed interests. But the real world does not work like that. There are times where we share interests with others, and times when we have the option of sharing interests with others.

The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.
 Let us move to the realm of non-zero-sum games. If a zero-sum game is one in which any gain by one player is balanced by an equal loss by another, a non-zero-sum game must therefore be one in which this is not the case. All outcomes in a non-zero-sum game are written as (x,y), where x is what the first player (in the games I've been showing you, the player whose options are written down the left) wins, and y is what the second player wins.

Let's start with a classic: the game of "Chicken!". Two people, usually teenagers, get into their cars. They put their cars on the same side of the same road, and drive at one another. When one moves to dodge the other, the friends of both players who have gathered to watch shout "Chicken!" (for those not familiar with the term, an accusation of cowardice). The diagram looks like this:


Stay on course
Dodge (Draw,Draw) ("Chicken!", Win)
Stay on course(Win, "Chicken!")(Crash, Crash)
where Win>Draw>"Chicken!">Crash (this is an ordering of outcomes, according to the utility function of each player).
So, let's look at what is going on here. This game has two of something called a Nash equilibrium: an outcome where, if both players were shown the other's decision, neither would gain an advantage by choosing to do something different. These are the outcomes where, when one player dodges, the other stays on course. The dodging player would cause a crash by choosing to stay on course, and the staying player would lose their win by choosing to dodge.
But, in game theory, it is generally assumed that neither player has access to their opponent's choices before making their own. Avoiding a crash thus comes down to reading one's opponent, and guessing their willingness to stay on course in comparison to one's own. This will be discussed in more detail later.
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