The Voter's Paradox
Extended Analysis
The most puzzling aspect of Social Choice theory[1] is that people cooperate much more than the theory suggests. There are several reasons why this is so, including the fact that people are not always rational -- particularly by the definition of "rational" used by the Social Choice theorists! Another obvious reason is that people act out of ignorance much of the time.
The situation of theory and practice not tracking may not be as bleak as it appears, however. The magnitude of the cost to the individual for cooperating has a far greater influence on the decision to cooperate or defect than the basic theory recognizes. Sure, a person cooperates by voting because the cost of voting is inconsequential to her. However, would this same person be so quick to cooperate if the ante were raised to a level that would seriously impact her personal welfare? Consider the example of a person that has been badly bitten by the neighbor's dog? Would this person sue for a large settlement if she were encouraged to do so by the claims of her lawyer that she would be successful? Of course suing is harmful to society and everyone suffers by having to pay higher insurance rates. So suing is defecting. Nevertheless, most people will sue.
Another example given elsewhere in this series of documents is the case where a person has a substantial amount of savings in a bank in which rumors are circulating that it could fail. Most people will withdraw their savings in a situation like this -- which of course causes the bank to fail.
I believe that this effect of magnitude of the cost to cooperate also contaminates many of the Social Dilemma (SD) experiments that researchers have conducted. A reward of a few bucks to a college student (the typical volunteer for such experiments) is just not meaningful. The reward needs to be of a serious magnitude. Of course, most experimenters can't afford to make such large rewards. In view of that, I would question their results.[2]
In the introduction, it was stated that this essay would attempt to examine the general case of problems that have the "no technical solution" (NTS) characteristic defined by Garrett Hardin in his classic essay, "The Tragedy of the Commons". This section will examine the general case.
For the most general definition of the NTS problems, the only common characteristic seems to be the non-zero-sum of the payoff matrix. That is, in competitive games resulting in social dilemmas, the results are typically characterized by a greater payoff for cooperating players. Just because I win, you don't have to lose. In fact, we could both win -- or both lose.
Non-zero-sum payoffInsignificance of contribution[3] (common, but not necessary)
Anonymity[4] (usual, but not necessary)
Nonlinear (Binary) Payoff[5]
Individual plays against the collective group[6]
The rewards are a public good[7]
Non-zero-sum payoffIncremental Input
Incremental Output
Individual plays against the collective group
Non-jointness of supply
Non-zero-sum payoff
Very nonlinear
Jointness of supply (usually)
Paradox of the infinitesimal change
Paradox of the infinitesimal change
When a thug steals from a bank, the theft obviously impacts all of the community that uses the bank. But does the thug belong to that community? Possibly not. In many cases like this, the perpetrator does not believe that he/she is taking from his own group but only taking from another group (possibly an enemy). In this case, the NTS phenomena does not apply.
However, there are problems in which there is some component of NTS involved.
Many would argue that much of the actions open to individuals of today's society are not really voluntary. For example, do we really have a choice in contributing to the poor when our contribution is actually obtained by taxation?
Closer examination of many "non-voluntary" contributions reveal that there is some, if not a lot, of voluntary aspects involved.
Requirements dictated by law can be avoided, for example. Many aspects of environmental protection are often avoided simply because the police cannot watch everyone at all times.
Laws can be changed. After all this is a democracy. We have to assume that the taxes we have and the environmental restrictions are there because that is what most people want. Therefore, these things must be voluntary.
Lifestyle choices can be made. If you don't like all the restrictive laws of places like California, you can move. If you don't move, there is some reason to believe that you voluntarily support such laws.
Many people that have the means to do so, accepting that an individual investment into the solution to a public problem nets a very minuscule return, take the matter in their own hands and sponsor a private solution. For example, if the community's public water supply is running low, rather than contributing to the public fund, a person may elect to put in his own pump.
Unfortunately, this is not a practical solution to the VP in the general case. It is easily shown that in many - if not most - real world situations, a cooperative effort is the most efficient. It is not practical for everyone to build their own roads and phone system. As we discussed earlier, investments often produce greater return when the input from many individuals are combined.
Even with a heavy load of freeloaders, group effort is often more productive than individual effort. Public television is probably a good example.
The SD seem to belong to a general class of problem that arises whenever an object is actually a composite of sub-objects which are essentially alike but very small compared to the composite. Consider the beach and the sand it is made of. Does removing one grain of sand make any difference to any practical description of the beach? Considering the fluctuations of the wind and water and other random phenomena, it is impossible to detect the removal or addition of one grain. The dilemma is that the event of one grain removed is not detectable but the accumulation of millions of grains removed is detectable. For further emphasis, consider the possibility that you owned the beach. Would you object if one grain were removed? Assume N grains have been removed. Would you object to the Nth +1 grain being removed?
Small increments of time added or subtracted are insignificant in a long term project. There is no way at the end of the project to determine if any particular person just stared at the wall one complete day, for example. But what if this were repeated thousands of times? You see the problem.
Setting arbitrary levels in a continuum creates the same dilemma. In our society, certain benefits accrue to a person if they have earned income less than some precise but arbitrary level. Let us say you are the administrator and you have a case where the potential recipient makes one cent more than that cutoff point. How can you justify not increasing the level by one more cent? It was arbitrary, right? Here is a person in need that would receive help if you raised it one cent. Why not?
A public good, in contrast to a private good (such as an apple), has two characteristics. First, it exhibits jointness of supply. In other words, when one person consumes the good, the same amount of the good still exists. Second, a public good is nonexcludable. That means in this case that no member of a community can be excluded from consuming the good. Security and environmental protection are two examples of public goods which all can (or should be able to) consume equally.
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