Game Theory and Cooperation

Game theory is a branch of mathematics that deals with decision making when there are multiple decision-makers, called “players”, and outcomes depend on the decisions of other players. Game theory is based on the metaphor of a game, in which the rules of play and the outcomes are well defined. The game metaphor is a very useful abstraction. Game theory can be applied to economics, evolutionary theory, politics, warfare and other domains. It is especially important for understanding how society works.

The Prisoner’s Dilemma

The prisoner’s dilemma is an important concept in game theory. It shows the benefits and challenges of creating cooperation between selfish individuals. The prisoner’s dilemma is named after a thought experiment in which the police try to get two prisoners to confess and implicate each other in the crime. Unfortunately, that thought experiment is somewhat complicated and can be confusing. Rather than telling the story of the two prisoners, I will explain the prisoner’s dilemma in abstract terms.

The preceding table represents a typical prisoner’s dilemma. There are two players, A and B. Each has two choices: cooperate or defect. The scores are numbers that could represent dollars, megajoules, offspring or any other measure of value.

Let’s consider the decision problem from A’s perspective. If B cooperates, then A gets 1 for cooperating and 2 for defecting. If B defects, then A gets −2 for cooperating and −1 for defecting. In both cases, A improves his score by 1 if he defects. So, A should defect.

The same is true for B. Each player is better off defecting, regardless of what the other player does. So, both players will defect, and both will get −1.

If both players cooperated, they would each get 1. Both players would prefer that outcome to the one in which both defect and get −1. But each has an individual incentive to defect. So, unless there is some way to coordinate their actions, they will both defect and get −1.

The prisoner’s dilemma illustrates the problem of creating cooperation between selfish individuals.

Here is an example of a prisoner’s dilemma situation. Suppose that you are on a backpacking trip in a foreign country. Your backpack is heavy, and you would like to leave it while you walk around the city. You see a man sitting on the sidewalk, begging for spare change. You consider paying him to watch your backpack for an hour. The exchange would benefit both of you, but there is a problem. If you pay him up front, he might just take the money and run. If you offer to pay him afterward, he can’t be sure that you will pay him when you get back. The risk of defection prevents cooperation.

Here is another example. This time it is between two societies, rather than two individuals. Consider two neighboring countries, A and B. Each has the option of building an army that could be used either to invade its neighbor or to defend itself from invasion. There is a cost to building and maintaining an army, but there is a greater benefit to invading an undefended country. If B does not build an army, then A can benefit by building an army and invading B. Likewise, if A does not build an army, then B can benefit by building an army and invading A. If either country builds an army, then the other country will need an army to defend itself from invasion. Thus, both countries must invest in building and maintaining armies, just to maintain the status quo. Peaceful coexistence without armies would be better for both countries, but each is forced to build an army, because the other also has that option.

Cooperation between selfish players requires some coordination mechanism to prevent defection.

The Iterated Prisoner’s Dilemma

An iterated prisoner’s dilemma is simply a prisoner’s dilemma game that is repeated over time between the same players. The table above is the payoff matrix for one iteration. In addition to the immediate payoff, players also receive the expectation of future cooperation. F(A) and F(B) represent the long-term benefits of friendship with A and B, respectively.

Cooperation can emerge between selfish players if they interact regularly. The incentive to cooperate is the expectation of future reciprocity. Once bootstrapped, a virtuous cycle can emerge. This is called “tit for tat”.

Tit-for-tat only works if there isn’t a huge benefit to defecting on the first move. A country can’t disarm unilaterally and then wait to see if its neighbor disarms too. A tit-for-tat relationship usually begins with exchanges of small favors. As trust develops, the exchanges increase in size. Although the decisions are simultaneous in the standard prisoner’s dilemma game, tit-for-tat often involves exchanging favors at different points in time. Your friend helps you out today, and you help him out tomorrow. The basic principle — reciprocity — is the same.

Tit-for-tat doesn’t work between strangers, or when interactions are anonymous. It depends on the players knowing each other, keeping track of past interactions, and expecting future interactions.

Tit-for-tat tends to make people cluster into groups. People who associate tend to cooperate, and people who cooperate tend to associate. Thus, tit-for-tat causes people to cluster into small groups that are inwardly cooperative and outwardly competitive. Tit-for-tat doesn’t scale up to large groups. It is limited to the number of people that you can frequently interact with and keep track of mentally, which is about 100. Large groups require some other mechanism of social cohesion.

The Tragedy of the Commons

The tragedy of the commons is similar to the prisoner’s dilemma, but it involves the relationship between an individual and a collective, rather than two individuals.

The table above shows the costs and benefits of an action to both the individual and the collective. In this example, the individual receives a benefit of 1 for doing some action, and the collective benefit is also 1. This means that the individual receives all the benefit of his action, because the collective benefit equals the individual benefit, and he is part of the collective. The collective cost of his action is 2, but he only pays a fraction of that cost, 2/N, where N is the number of people in the collective.

The total cost exceeds the total benefit, but if N > 2, each individual has an incentive to do the action, because his individual benefit is greater than his individual cost. So, every individual will do the action, and every individual will receive −1. Everyone will be harmed by everyone doing the action.

Imagine a crowded room in which everyone speaks loudly to be heard over the noise of everyone else speaking loudly. Each individual would prefer everyone to speak quietly, but every individual has an incentive to speak loudly. So, they all speak loudly. That is a simple example of a tragedy of the commons.

Here is another example. Suppose that a group of people lives next to a lake. The lake could provide them with nice, clean drinking water. It could also be used for washing dishes and clothing. For an individual, washing his own clothes and dishes in the lake has little impact on him. He could wash his stuff in one place, and then walk a few feet away and dip his pot in the lake to get fresh water. But if everyone uses the lake for washing, the water will become undrinkable. It would be better for everyone to only use the lake for drinking water. However, every individual has an incentive to pollute the lake.

Scale matters in this example, as it does with tit-for-tat. Suppose that the lake is small, and the number of people is also small. When N is small, each individual pays a higher fraction of the cost himself, and he also has an incentive to police the actions of others, because their individual actions have a greater impact on him. When the group is small, the individual’s actions will be remembered by others, and he will want a reputation as a cooperator. Those incentives can make it better to cooperate than defect. But as the number of people increases, the incremental effect of an individual’s actions on any other individual, including himself, becomes negligible. There will be no incentive to curb his actions or police the actions of others.

The free-rider problem is similar to the tragedy of the commons, but it involves the absence of an incentive to be productive, rather than the absence of an incentive to not be destructive. For example, the welfare state allows individuals to benefit at the expense of others, without making a positive contribution to society. Another example is when a society allows people to opt out of military service, but receive the protection of the military. Free riders get a “free ride” at the expense of others.

There are two types of collectivism: top-down and bottom-up. In top-down collectivism, the benefits of individual action are collectivized, while the costs are paid by individuals. Communism is an example of top-down collectivism. In bottom-up collectivism, the costs of individual action are collectivized, while the benefits are received by individuals. The tragedy of the commons is an example of bottom-up collectivism. In top-down collectivism, there is no incentive to be productive. In bottom-up collectivism, there is no incentive to not be destructive. For society to work, there must be individual incentives to be productive and not destructive.

Using Coercion to Solve Problems of Cooperation

The prisoner’s dilemma and the tragedy of the commons are problems of cooperation. To create cooperation, we must solve problems of cooperation.

In small groups, tit-for-tat solves problems of cooperation. To create cooperation on a large scale, it must be imposed on individuals by coercion. The state is a mechanism for solving problems of cooperation. It makes cooperation possible between strangers by punishing defection.

The following table shows how the expectation of punishment solves the prisoner’s dilemma.

E(P) represents the expected penalty that will be imposed on a player for defecting. It is an expectation, because the player might not be caught and punished. Suppose that E(P) = 2. In that case, there is an incentive to cooperate, even in a single game with no potential for tit-for-tat.

To be effective, the punishment must be significantly worse than the potential benefit of the crime, because there is a chance of not getting caught, and thus not paying the penalty. The expected penalty is lower than the actual penalty if caught. For example, the punishment for theft should be more than simply replacing the stolen item.

The tragedy of the commons is solved by the state in a similar way.

The state uses punishment to create incentives that align the interests of the individual with the interests of the collective. If E(P) ≥ 1, the individual has no incentive to commit the socially harmful action.

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The necessity of the state raises two important questions:

• How is state power created?
• What prevents it from being abused?

I’ll leave those questions for another essay. They are too big to address here.

Given that human nature is selfish, modern society is almost a miracle. I can go to the store and buy groceries from strangers. When I walk home on crowded streets, no one hits me over the head and steals my groceries. We take these things for granted, partly because we view them as how things should be. But it is no small task to explain how they are possible.

Society is not based on altruism. It is based on cooperation between selfish individuals. Game theory formalizes problems of cooperation, and explains how we can solve them. Understanding game theory is necessary for understanding society.