Putting numbers to the results of the game

Now that we know the concept of Pareto optimality we want to apply it to the problem of Anna and Ben in the kitchen. To do this, we need to be able to compare each possible outcome to each other outcome and check if there is Pareto dominance. We cannot do this using the flow diagram. We will need a more complete representation of preferences. You have already seen it in the videos on the kitchen problem. We will be using numbers!

Ordinal preferences

Let’s stop for a moment and think about what preferences are. So far we were describing them using flow diagrams like the one below:

If we look at the dashed column in the table above we can say that if Anna cleans, Ben prefers not to clean. The outcome in which he does not clean and she cleans is better for him, than the outcome in which he cleans and she cleans. Analogously in the next column we read that the outcome in which he does not clean and Anna does not clean is better for Ben, than the outcome in which he cleans and she does not clean. If we take these two pieces of information out of the table we can present them on two diagrams:

It is easy to see now that a preference is simply an order in which the more preferred options are put after the less preferred options. The order in our example is incomplete. For example we do not know whether Ben prefers the outcome in which he does not clean and Anna does not clean or the one in which he does not clean and she cleans. We can reconstruct this information using the description from the kitchen video:

The best option for Ben is when he doesn’t clean and Anna does. The kitchen is still clean and Ben can enjoy his free time. The worst option for Ben is when it’s the other way round: Ben does all the cleaning and Anna doesn’t. The kitchen is clean, but Ben is really frustrated. When both Anna and Ben clean it is almost as good as when only Anna cleans, because the kitchen is clean and they can share the work. This leaves us with the fourth scenario when nobody cleans up. The kitchen is very dirty, but Ben avoids the frustration, so this option is better for him than cleaning by himself.

Using this description we can order the outcomes from the worst to the best one for Ben. The result is presented in the diagram below. Please check if it agrees with the description above.

Now our order is complete. Preferences which can be described in diagrams like this are called ordinal preferences. For convenience, the order is typically represented using numbers rather than arrows. The numbers should be getting larger as we proceed to the more preferred outcomes. Usually the simplest thing is to use consecutive natural numbers like in the figure below. This way the number tells us what is the position of a given outcome in the order of preferences for a given player. In game theory such numerical representation of preferences is called payoffs. The figure below lists Ben’s payoffs in our example.

Payoff Matrix

If we represent preferences using numbers it is easy to put them back in a table. This way we can gather information on preferences of all players in one place. The table will be called the payoff matrix.

To create the payoff matrix for our example we will first order the outcomes from the worst to the best one as seen by Anna and assign numbers representing this ordering.

Next we will put the numbers in the table. The blue numbers represent the position of the given outcome in Ben’s ordering. The green number represents the position of the given outcome in Anna’s ordering. The payoff of the player whose choices are in rows is typically listed first. This is why the blue payoffs are shown first in all the cells below.

Please note that we can still draw a flow diagram for this table and the information it conveys will agree with the information represented by the numbers. The blue arrows show the better option for Ben so they go from the smaller to the larger blue payoff. The green arrows show the better option for Anna so they go from the smaller to the larger green payoff.

We can also still use the flow diagram to identify the only equilibrium in this game. Do you remember how to do this? What are the payoffs in this equilibrium?