(This article is the second in a series introducing Game Theory. To start at the beginning, click Economics Squared on the right sidebar.)
Utility
It is hard to dive into game theory without first mentioning a bit about utility and game matrices. Let’s begin with the concept of utility. Ever hear the expression, “money isn’t everything”? It is quite true. Money isn’t everything… but utility is!
Utility is an abstract measure of worth. Though we usually think of worth in terms of money, utility measures worth more broadly. It takes into consideration all types of value, whether it be monetary, happiness, or something else.
Utility is best illustrated with an example. Suppose you buy an ice cream sundae and eat it. What types of outcomes may result? You might be down $3, yet be 500 calories “richer”. On a hot day, the ice cream may have made you happier. Perhaps the first bite that marked the end of your diet, causing you to feel guilt.
Utility takes into consideration all of these results. It assigns relative values to each outcome, such as -79 for the $3 paid, +18 for the happiness felt, and -22 for the guilt. Of course, an object’s utility is different for each person. Summing these values will give us the utility of an action. Utility is a great way of determining rationality. We mentioned maximizing utility earlier in the introductory article. What we mean by this is that if the utility is greater than zero, rational beings will always perform the action, as they are relatively better off than before. In face of two outcomes, rational beings will choose the one that gives them the greater utility.
Keep in mind that utility can measure just about anything. Whether it be an action (eating the ice cream) or an object (the sundae), utility puts a value onto it, making it much easier for mathematicians and economists to analyze.
Game Matrix
When we look at a game, the decisions that players make create an outcome that we can assign a value to using utility. There can be multiple strategies (decisions) that players can make, outcomes that vary on which players choose which strategies, and potentially different utilities of outcomes for each players…a jumbled mess.
The way game theorists neatly store information is with a table called a matrix. Here is an example of a game played by 2 players, Colin and Rose.
| Colin Strategy A | Colin Strategy B | |
| Rose Strategy A | 5 , 3 | -4 , -3 |
| Rose Strategy B | -3 , 2 | 1 , 6 |
Matrices conveniently display two-player games. Each column and row represents a different strategy that the player can take, and the outcomes of certain combinations of these strategies are within the table. Reading these matrices is pretty intuitive, as each box on the matrix corresponds to an intersection of two strategies. The first number in a cell is the utility gained/lost by Rose, the second being the result for Colin.
For example, if Rose chooses strategy A and Colin chooses strategy B, the outcome of these strategies is the box on the 1st row, 2nd column. The value in that box is (-4,-3). This means that if these strategies are chosen, Rose will experience -4 utility, while Colin will experience -3 utility.
Whew! Though utility and matrices may not be very exciting for now, they will be useful later on as we use them to discuss strategies and games. Stay tuned…
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