Normal Form Games

(This article is the third in a series introducing Game Theory. To start at the beginning, click Economics Squared on the right sidebar.)

In this article, we start with normal form games, games which are represented in matrix form.  Let’s start by redefining what a game is.

A game must have

–          a number of players

–          set of strategies for each player

–          outcomes that depend on their choice of strategies

–          players preference of outcomes (rationality)

For now, we will focus mainly on games with 2 players.

Our first example is a 2-player zero sum game.  In this special case, the sum of utilities gained or lost in each outcome is zero.  In other words, if an outcome rewards one person a certain amount of utility, the other person loses the same utility.

In terms of practicality, zero sum games are not very interesting.  That is because in applications of game theory, situations are not always win-lose scenarios.  For example, two rival companies may decide to create a price floor (a minimum price) on their products.  Though they are still rivals, both companies can win as neither company has to slash prices now to deal with competition from the other.  We will discuss zero sum games later as we look at mixed strategies and fair division.

Other normal form games fall under the category of a non-zero sum game.  Non-zero sum games are games where the sum of utilities gained or lost in each outcome is not necessarily zero.  In other words, win-win or lose-lose situations may occur.

In this example (taken from the previous article), we see that at Rose Strategy A x Colin Strategy A, and Rose Strategy B x Colin Strategy B, both players gain positive utility.  Likewise, a lose-lose scenario occurs when Rose plays strategy A, and Colin plays Strategy B.

Now let’s look at some special cases.  There are many non-zero sum games, but these three games often catch my (and game theorists) attention…   

Our first special case is the game of chicken, a favorite among Hollywood characters.  To play chicken, two individuals in cars facing each other must drive towards each other at full speed.  The first one to swerve (thereby avoiding collision) is the chicken.  Of course, if none of them swerve, they collide and injuries result.  Ever see the 1955 classic, Rebel Without a Cause?  They play a variation of chicken there, when they drive their cars towards a cliff.

The matrix for chicken is generally like this:

Of course, the utilities can be changed (depending on the hospital bill, the cost of a collision may be more than -50 utility!).  But the main idea behind chicken is intact.  Both players would like to “win” and neither wants to end up in the -50 , -50 outcome.

We call the game of chicken an anti-coordination game.  It is better for both players to pick different strategies than the same ones.

A relevant application of the game of chicken is the Cold War nuclear arms buildup.

Had the two countries not built up their nuclear arms, there would be no threat of a nuclear holocaust, and money would be directed elsewhere, such as in education and public health.  Yet, history tells us we landed on the -200 , -200 spot.  Almost $5.5 trillion spent on a US nuclear arsenal makes this the worst game of chicken ever!  (It is quite ridiculous how much we spend on nuclear proliferation.  The Brookings Institution has a nice study here).

So how do we “win” chicken?  Two words: cooperation and threats.  We’ll discuss cooperation, threats and other strategies in a future article. (UPDATE: Brendan Greeley recently published a BusinessWeek article on the game theory in Obama’s debt ceiling deal.  It really is a game of chicken!  See it here.) 

Our next game is a simple one played by many: the battle of the sexes.

A husband and wife are looking for a place to go together on a Saturday night.  The husband prefers a baseball game and the wife prefers the opera, but both enjoy their spouse’s event as well.  However, neither will enjoy any event unless the two are there together.  Both are separated, forgot which event they agreed upon and have no way of communicating.  Will they spend their evening together?

Try a few games with a friend!  This is an example of a coordination game.  Both are better off if they go somewhere together.  Since neither knows which strategy the other chooses, it is up to the players to coordinate their strategies.  This game is often studied as a repeated game, a game played many times.  We’ll see the battle of the sexes later on, as we discuss cooperation and threats.

Our final example of a normal form game is the most famous: the prisoner’s dilemma.  This game deserves some special attention, as it is very applicable in the financial world.  Expect to see it in the next article!  Stay tuned…

<< Previous Article : Utility and Game Matrices <<

>> Next Article: Prisoner’s Dilemma: Part 1 >>