Social Games

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 beginning with 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 puzzle explores 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 profit of 3. If Alice decides to lower prices while Bob doesn’t, then Alice will enjoy a profit of 4 while Bob gets a profit of 0, because nobody will buy from him. At that point, Bob will lower his price to receive at least a profit 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 profits will drop to 1 each.

This arrangement can be expressed in the following table, where Bob’s profit is shown on the left in each pair and Alice’s profit is on the right. So, the upper right corner, for example, represents the state in which Bob charges a high price (and receives a profit of 0) and Alice charges a low price (and receives a profit of 4).

Open table as spreadsheet

	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 in this case is the bottom right corner. Neither Bob nor Alice will unilaterally raise prices. If Bob raised prices (thus moving the state to the upper right corner), then his profit would decrease to 0. If Alice raised prices, she would move to the lower left corner, thus decreasing her profits 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 does not always lead to socially beneficial consequences. 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 yield a social benefit or harm.

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 value of -5 (e.g., 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:

Open table as spreadsheet

	Alice Honest	Alice Cheats
Bob Honest	3, 3	0, 3.1
Bob Cheats	3.1, 0	0.4, 0.4

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

Warm-Up

Suppose you can increase punishments (i.e., make them more negative). Can you do so in a way to make the upper left a Nash equilibrium while keeping the bottom right a Nash equilibrium as well? Assume you cannot change the probability of being caught.

Solution to Warm-Up

Suppose the punishment becomes -8. Alice’s gain in the upper right corner state becomes (4 × 0.9) + ((-8) × 0.1) = 2.8. The gains at the bottom right become (1 × 0.9) + ((-8) × 0.1) = 0.1. This yields the matrix:

Open table as spreadsheet

	Alice Honest	Alice Cheats
Bob Honest	3, 3	0, 2.8
Bob Cheats	2.8, 0	0.1, 0.1

So, honest-honest is a Nash equilibrium, because neither Bob nor Alice has an incentive to cheat to leave this state. On the other hand, the lower right hand corner is also a Nash equilibrium. If both Bob and Alice cheat, then they each still receive a tiny expected gain. If one then becomes honest, that person’s gain is eliminated.

Here are some challenges for you.

1. How would punishment have to change to eliminate cheats-cheats as a Nash equilibrium?

As civilizations have advanced, societies have evolved the notion of punishments that fit the crime. This principle might limit the extent to which overly severe punishments are considered acceptable.

2. If you are a public policy maker and you want to limit the punishment for cheating to -5, how much must you increase the probability of catching criminals to maintain the upper left as the only Nash equilibrium?

In modern societies, inequalities exist so that the benefits of an honest society may benefit one party more than another. Say that Bob gets 5 whereas Alice gets only 2 if both are honest. Assuming for the moment that the probability of being caught is 0, this gives us the following game matrix:

Open table as spreadsheet

	Alice Honest	Alice Cheats
Bob Honest	5, 2	0, 4
Bob Cheats	4, 0	1, 1

The total social good of the upper left corner has increased at the cost of inequality. One side effect, however, is that incentives to dishonesty now change.

3. Assuming a 10 percent likelihood of being caught, is there a punishment value that would cause the only Nash equilibrium to be the upper right state (Alice commits crime but Bob does not)?

4. Still assuming a 10 percent likelihood of being caught, what must the punishment be to force Alice to stay honest?

Let’s close this puzzle on a non-punishment note. Let’s return, in fact, to the competing manufacturers Bob and Alice. Recall their game matrix.

Open table as spreadsheet

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

The invisible hand guides them to lower prices, but neither likes this. They see that if they cooperated, their total profit would be 3 + 3 = 6 instead of 1 + 1 = 2 as in the competing scenario. For that reason, they might consider merging their two enterprises into BobAlice and thereby achieving a total profit of 6 instead of 2. That is, when Bob looks at Alice, he doesn’t value her company at the profit of 1 that she is currently achieving, but at a profit of 5, the additional profit she could bring to a potential BobAlice relative to Bob in a competing world. Of course, this increased profit to Bob and Alice comes at an approximately equal cost to the consumer.

Next time you hear someone expound about inequality, crime, enlightened self-interest, and antitrust laws, bring out your game matrices. You may find more wisdom in them than in the blah-blah.