SIMPLE GAME THEORY
FROM: David K. Levine, Department of Economics, UCLA [http://levine.sscnet.ucla.edu/general/whatis.htm]
One way to describe a game is by listing the players (or individuals) participating in
the game, and for each player, listing the alternative choices (called actions or
strategies) available to that player. In the case of a two-player game, the actions of the
first player form the rows, and the actions of the second player the columns, of a matrix.
The entries in the matrix are two numbers representing the utility or payoff to the first
and second player respectively. A very famous game is the Prisoner's Dilemma game. In this
game the two players are partners in a crime who have been captured by the police. Each
suspect is placed in a separate cell, and offered the opportunity to confess to the crime.
The game can be represented by the following matrix of payoffs
|
Not Confess |
Confess |
Not Confess |
5,5 |
0,10 |
Confess |
10,0 |
1,1 |
Note that higher numbers are better (more utility). If neither suspect confesses, they
go free, and split the proceeds of their crime which we represent by 5 units of utility
for each suspect. However, if one prisoner confesses and the other does not, the prisoner
who confesses testifies against the other in exchange for going free and gets the entire
10 units of utility, while the prisoner who did not confess goes to prison and gets
nothing. If both prisoners confess, then both are given a reduced term, but both are
convicted, which we represent by giving each 1 unit of utility: better than having the
other prisoner confess, but not so good as going free.
This game has fascinated game theorists for a variety of reasons. First, it is a simple
representation of a variety of important situations. For example, instead of confess/not
confess we could label the strategies "contribute to the common good" or
"behave selfishly." This captures a variety of situations economists describe as
public goods problems. An example is the construction of a bridge. It is best for everyone
if the bridge is built, but best for each individual if someone else builds the bridge.
This is sometimes refered to in economics as an externality. Similarly this game could
describe the alternative of two firms competing in the same market, and instead of
confess/not confess we could label the strategies "set a high price" and
"set a low price." Naturally is is best for both firms if they both set high
prices, but best for each individual firm to set a low price while the opposition sets a
high price.
A second feature of this game, is that it is self-evident how an intelligent individual
should behave. No matter what a suspect believes his partner is going to do, is is always
best to confess. If the partner in the other cell is not confessing, it is possible to get
10 instead of 5. If the partner in the other cell is confessing, it is possible to get 1
instead of 0. Yet the pursuit of individually sensible behavior results in each player
getting only 1 unit of utility, much less than the 5 units each that they would get if
neither confessed. This conflict between the pursuit of individual goals and the common
good is at the heart of many game theoretic problems.
A third feature of this game is that it changes in a very significant way if the game
is repeated, or if the players will interact with each other again in the future. Suppose
for example that after this game is over, and the suspects either are freed or are
released from jail they will commit another crime and the game will be played again. In
this case in the first period the suspects may reason that they should not confess because
if they do not their partner will not confess in the second game. Strictly speaking, this
conclusion is not valid, since in the second game both suspects will confess no matter
what happened in the first game. However, repetition opens up the possibility of being
rewarded or punished in the future for current behavior, and game theorists have provided
a number of theories to explain the obvious intuition that if the game is repeated often
enough, the suspects ought to cooperate.
If you wish to learn more about game theory, there a variety of on the topic. © David K. Levine