Since Adam Smith observed that the "invisible hand" of the free market would force self-interested manufacturers to offer low prices to consumers, governments and politics have never been the same. It took almost two more centuries, however, to achieve a proper mathematical analysis of the consequences of selfish behavior in Morgenstern's and Von Neumann's game theory and the work of John Nash. (You may recall Nash as the mathematician who was the subject of the movie A Beautiful Mind.)

This month we will explore game theory as it applies to social goods. Let's start with the invisible hand.

Bob and Alice are competitive manufacturers. If they fix their prices at a high level, then they will share the market and each will receive a benefit of 3. If Alice decides to lower prices while Bob doesn't, then Alice will enjoy a benefit of 4 while Bob gets a benefit of 0, because nobody will buy from him. At that point, Bob will lower his price to receive at least a benefit of 1. Similarly, Bob receives 4 and Alice 0 if the roles are reversed. Simply following their self-interest, both will lower their prices and their benefits will drop to 1 each.

This arrangement can be expressed in the following table.

 

  Alice High Alice Low
Bob High 3, 3 0, 4
Bob Low 4, 0 1, 1

John Nash defined the concept of an equilibrium state, since known as a Nash equilibrium, in which no party has an interest in deviating from that state. The only Nash equilibrium is the bottom right corner in this case. Neither Bob nor Alice will unilaterally raise prices. If Bob raised prices (thus moving the state to the upper right corner), then his benefit would decrease to 0. If Alice raised prices, she would move to the lower left corner, also decreasing her benefits to 0.

This is the invisible hand at work. For the consumers, Bob and Alice's competition leads to lower prices, a social good. Virtually every modern economy gives evidence of this.

Unfortunately, selfishness is not always a good thing. Suppose that instead of representing price choices of competitive manufacturers, the table represented choices about honesty or social responsibility. That is, the "High" row represents states in which Bob acts honestly. By contrast, the "Low" row represents a situation in which Bob cheats (say, steals, pollutes, or bribes lawmakers). The upper right corner portrays a state in which Alice cheats but Bob doesn't. As you can see, the cheater then benefits. If both cheat, then their benefits go down. Selfishness leads to social loss. Corrupt nations, high crime zones, and brawling families give much evidence of this.

Game theory is neutral. The same game matrix and the same Nash equilibrium can lead to a good or bad state. Selfishness can be good or bad.

Now, let us say that it is your job to design public policy. You are confronted with the matrix representing the selfish benefits of cheating and you want to change it somehow. So you establish a police force and criminal justice system that makes it 10 percent likely that cheaters will be caught and given a reward/punishment value of -5 (i.e., jail time).

In that case, Alice's benefit in the upper right corner has a 90-percent probability of being 4 and a 10-percent probability of being -5. Thus, her expected benefit is (4 * 0.9) + ((-5) * 0.1) or 3.1. In terms of expectations, the matrix becomes:

 

  Alice High Alice Low
Bob High 3, 3 0, 3.1
Bob Low 3.1, 0 0.4, 0.4

Now, the benefits of cheating when starting from the high-high state are much reduced. Further, if the punishment increases or the probability of being caught increases, then high-high may become a Nash equilibrium.