## Sunday, September 16, 2007

### Free Ridership and Game Theory

My nighttime reading this week has been devoted to game theories –theories that looks how rational people make rational choices and the consequences that result.

One of the more interesting game theory situations is called the ‘prisoner’s dilemma’. This dilemma comes in many forms – one of which gives the dilemma its name. For my purposes, the form that is the most useful is one that captures the concept of what economists call ‘free ridership’.

In this post, which will be the first in a series, I will simply set up the free-rider problem, look at the standard (political) solution to that problem, and explain how the political solution can make things worse.

Constructing the Dilemma

To put the problem of free ridership in terms of a prisoner’s dilemma, let us assume that there is some project that people can invest in. However, there is no way to restrict the payoffs of this investment only to those who invest. The payoffs must be distributed evenly. For the sake of this argument, we will assume that the rate of return is 50%. That is to say, for every \$1.00 that society invests in this project, they will get \$1.50 rate of return.

Now, we have two citizens, each with \$1000 in investment capital.

To start off with, we have a voluntary situation. People are free to donate, or not donate, as they wish.

Citizen One is trying to decide whether to donate.

First, assume that Citizen Two donates \$1000.

Now, should I keep my \$1,000, or should I donate \$1000 as well.

If I also donate \$1,000, then there will be \$2,000 in the public treasury. This will produce \$3,000 in social good, which we will distribute evenly. That is to say, I will get \$1,500.

On the other hand, if I keep my \$1,000, then there will still be \$1000 in the public treasury. This will produce \$1,500 in social good, which will be divided evenly. The \$1,000 that I kept plus \$750 worth of social good means I get \$1,750.

Since \$1,750 is more than \$1,500, then the best thing for me to do is to donate nothing.

Second, assume that Citizen Two donates nothing.

If I donate \$1,000, this will produce \$1,500 in social good, which will be divided up evenly between us. That is to say, we each get \$750.

If I donate nothing and keep my \$1000, then there will be nothing in the social pot, so I will get nothing. However, I get to keep my \$1000.

Since \$1000 is more than nothing, if I assume that Citizen Two will donate nothing, the best thing for me to do is to donate nothing as well.

In fact, no matter what Citizen Two does, I am better off donating nothing. So, I am going to donate nothing.

Of course, Citizen Two goes through the same line of reasoning concluding that he, too, should donate nothing. They end up each with the \$1000 they started with. Whereas, if both of them would have contributed their \$1,000 and taken their share of the social wealth that resulted, they would have each had \$1,500. They each lost a chance for \$500 they could have very easily had.

Examples

This is a real-world problem because there are a number of institutions that provide benefits for the whole population, where it is not possible to provide the benefits for some people but not others.

Military protection, for example, is something that defends everybody. There is no way for the military to protect one house but not the one next to it. So, the benefits that the military provides to the population are spread equally. There is no way to provide benefits only to those who donate.

The police and court system has the same effect. When a potential rapist is locked away, all potential victims, and all people who care about those potential victims – obtain a benefit. We don’t even know who they would be, so we cannot charge only those who actually benefit.

The same is true for education. An individual can keep some of the benefits of his education private and offer it only to those who will pay for it. However, some of the benefits will leak out and get distributed to everybody. An educated population provides some benefits to everybody.

All of these are cases where an investment made by any given individual gets distributed to the whole population. As such, all of these are cases where, if we depended on voluntary contributions, each individual has an incentive to contribute nothing, and “free ride” off of those who do contribute.

The Political Solution

One way to deal with these types of situations is through a political solution. That is to say, the government taxes everybody \$1,000. It puts the money into a pot. It invests the money. In doing this, it provides everybody with \$1,500 of value instead of the \$1000 in value each person would have had.

Notice that each citizen has reason to support this type of ‘tax and spend’ policy with respect to those goods that have distributed benefits. Each person faces a situation where, “If the government were to take this money and invest it in these public goods, I will end up better off than if we relied on private contributions.”

However, please note that each citizen has an even stronger incentive for a different type of political action.

Citizen One thinks, “On the other hand, what if I can get the government to force Citizen Two to pay, but to give me an exemption. If I can pull this off, then I will get my \$1,750 in social benefit, which is more than I would get from a system that taxed everybody equally. Even if I have to invest \$100 to get a candidate elected who will exempt me from this tax, I still end up \$150 in the black.”

Meanwhile, Citizen Two is going through the same type of process, looking for a candidate that will support a tax on Citizen One while providing his constituent with exceptions and exemptions. In the mean time, society continues to suffer a loss of social welfare that would have otherwise been available.

However, at this point we run into another ‘game theory’ problem – the political auction.

What we need to do now is to determine the costs of winning or losing the election.

Citizen One needs to determine how much money to invest in the election. Now that there are competing candidates (each advocating that the other pays for these social goods while their constituent does not pay), Citizen One is faced with two options. If his candidate wins, he gets \$1750. If his opponent wins, then the Citizen One gets \$750.

Let us assume that Citizen One has already spent \$1000 to get his candidate elected. So has Citizen Two – and Citizen Two’s candidate is ahead in the polls. Citizen One still faces two options: \$1750 minus \$1000 in political costs for \$750 if his candidate wins, or \$750 minus \$1000 in political costs for a net loss of \$250 if his candidate loses. If, by spending another \$250, he can bring about a win, then he will end up with \$500 (as opposed to minus \$250 if he does nothing).

Of course, Citizen Two is thinking the same thing when his candidate is behind, seeing just as much reason to throw more and more money into these campaigns.

Eventually, both Citizen One and Citizen Two end up being worse off than they would have been if they simply refused to try to obtain a political solution. By trying for a political solution, they went from a situation where each had \$1000, to one in which they each had significantly less, because of the resources drained away in political fighting.

Summary

So, free rider problems create situations where the a population has an opportunity to realize some significant benefit, but cannot get people to contribute to that benefit. They have no reason to voluntarily contribute, because ‘free riders’ who live off of the benefits that others provide end up being better off than those who provide the benefit.

If we aim for a political solution that forces each person to make a contribution, we are at risk of setting off a political battle. In this battle, each candidate proposes an option that is less than optimal (their constituents benefit while the other candidate’s constituents pay). The resources that then get drained by this political fighting actually leaves people worse off than they would have been if they had not sought a political solution.

Tomorrow, I would like to look at the moral solution.

Curiosis said...

"The same is true for education. An individual can keep some of the benefits of his education private and offer it only to those who will pay for it. However, some of the benefits will leak out and get distributed to everybody. An educated population provides some benefits to everybody."

Compared with military and police protection as well as the court system, education seems out of place here.

If Bob graduates from high school, clearly Bob benefits, but the benefit to the rest of the populace seems negligable, if it could be measured at all.

Once could likewise argue that we would all be better off if every graduating senior were given \$10,000. This would stimulate the economy and give them all a better chance in life.

However, this is wealth redistribution, plain and simple. State-provided education is no different.

For example, I could make some substitutions in your last sentence.

A healthy population provides some benefits to everybody. Ergo, we need socialized medicine.

A happy population provides some benefits to everybody. So, we need the government to offer entertainment and mental health care.

Anonymous said...

If Bob graduates from high school, clearly Bob benefits, but the benefit to the rest of the populace seems negligable, if it could be measured at all.

That's an odd claim. You live in a country where every one gets one vote. You don't see the advantages of having an educated populace?

Not only that, your life would be reduced basically to subsistance hunting/farming without a society to live within (as would anyone who was isolated from all others). In light of the fact that the only way civilization continues is through the vast network of interdependance and specialization, you don't consider it a net benefit to everyone is every member is as educated, productive, and motivated as possible?

As for your substitutions - yes. The government should in fact provide some level of medical and mental health care - beyond the help it already provides through Medicare/Medicaid/VA benefits/insane asylums. And, while I don't think it should, the government already provides large amounts of entertainment. Most sports stadiums and arts are funded at least partial by the government.