If we hope to understand this apparent paradox, we must examine each of its components very carefully. While the end result appears to be paradoxical, each component, under careful consideration, is quite straightforward.
It helps a great deal to understand that the components of the cost-benefit equation are usually independent. In fact, the VP situation that is discussed here is principally due to this independence. On the other hand, if benefits are dependent on the individual's costs, then the situation is not likely to be a VP. Since this essay is about the VP, near independence will be assumed.
To keep things simple, I will ignore for the moment the fact that some of the terms in the following equations are known only with some probability. When probabilities are introduced, the equation is often referred to as the expected utility function. Experimental evidence indicates that humans make serious errors in their calculations when probabilities are involved. For a discussion on this issue, see John Harvey's essay, "HEURISTIC JUDGMENT THEORY". Also, this easy to understand essay on Decision Theory touches this and other points.
Again, it must be emphasized that this model is based on what a rational person would do. That is, a rational person would surely examine the costs and benefits of any potential action before she acts. That people sometimes act irrationally and sometimes do not have the time to make rational judgments is acknowledged but is not within the scope of the model presented here. Further, there is some question as to whether a rational person's mental computational ability is up to the task of evaluating the more complex utility functions, especially when it involves probabilities. For the moment, we will assume that we do have that capability.
Let us define a few symbols to make the reasoning more concise and precise.
C = The direct personal cost of contribution
BG = Benefit to the individual derived from being a member of the group
BI = A direct benefit to the individual
R = Net return to the individual
Then for any action taken,
R = BG + BI - C
Again, I must emphasize that the most important fact to recognize in understanding the VP is that the components of R can be, and usually are, quite independent. I believe that a misunderstanding of this fact is the reason that many people have a hard time accepting and understanding the VP.
Further, since a person must act now on the basis of a future return, these variables represent perceived, not actual values. Obviously, the individual acts on what he or she perceives the costs and benefits to be, not what they actually are. This is very significant and will be discussed further in the following pages.
BG is the marginal benefit to the individual derived from being a member of the group and the result of this particular action. Specifically, BG is the individual's portion of the reward received by the group as a result of this particular contribution. BG could be a function of C but does not have to be. Societal benefits generally accrue to the individual whether the individual makes a contribution or not (unless no one or an insufficient number contributes).
If the group reward is not in joint supply (there is a finite amount that has to be divided by the participants), BG is determined by taking the marginal reward to the group, as a result of the individuals particular action, and dividing it by the number of members of the group. Of course, this assumes that everyone shares equally -- which is often not the case but is an adequate assumption for this analysis. It should be noted that some group rewards are in joint supply (one person's consumption does not reduce the amount available to everyone else) and BG would not be a function of the number of members. For example, let us assume that a reward for voting is freedom of the individual. The fact that others may enjoy that freedom does not subtract in anyway from my enjoyment.
More examples will be presented in detail later but for now a couple will be provided to illustrate the independence.
Our community wants to build a Community Center and to do it from contributions. I can contribute or not but in either case I still get to use the Center. Unless, of course, no one contributes (more precisely, the contributions are below some minimum value), in which case the community center will not be built.
I may volunteer or not but in either case, if the levee holds, my home will be saved just like everyone else's.
BI is the marginal benefit that the individual receives directly from her action without regard to the group benefit. An example follows.
A public spirited individual contributes $25 to Public Television and receives a Viewer's Guide. The guide is an immediate and significant benefit above and independent of the group benefits received from being able to watch the station.
BI can be very complex to evaluate since it includes intangible "feel good" rewards. BI can vary enormously between individuals all doing the same thing. One person may receive great personal satisfaction for making a community contribution while another may not.
As will be discussed further in this essay, BI is greatly impacted by anonymity. In fact, anonymity appears to be at the very heart of the freerider part of the VP. If you make a public contribution and your effort is completely anonymous, many individuals would receive or feel very little reward. But if everyone knows about your contribution, BI in the form of personal notice and satisfaction can be substantial. In fact, the personal reward resulting from contributions to a small group in which each individual is known by the others can be of such value as to completely eliminate the paradox!
C is the cost to the individual for performing a particular action. For example, C would include the cost of driving to the polling booth for the voter. C can be quite small or even zero. Again, I must emphasize that BG may have little or no dependence on C.
Like BG and BI, C should always be evaluated in a marginal sense. That is, what additional return will I get for this additional contribution? For example, a minimum contribution of $25 may get me coverage from the local volunteer fire department - an excellent investment. An additional $25 contribution may provide for only a very slight improvement to service and the return on this marginal investment to the contributor is very poor. Yet to the community, the second $25 is just as good as the first $25!
Like BI, C is also greatly dependent on anonymity. If you live in a small community and you are observed chunking your trash out on the public highways, your personal cost -- in guilt feelings -- can be considerable whereas in a large anonymous community, for many, the cost would be zilch.
An aspect of the extended VP, more common than not, is the situation in which the return to the group exceeds the contribution of the sum of the individuals (the PD effect). Of course, this is the basis for the overwhelming desire of most responsible citizens to have individuals contribute to the common good. The return we get from everyone or nearly everyone voting far exceeds the cost of the sum of the individual efforts.
Another example is the act of building a house. There are many activities in which a group of individual's efforts on a task is far more productive than an individual effort. Let us say you need to make a measurement. One person holding one end of the tape and another person holding the other is more than twice as fast as one person doing it alone. Or maybe there is a need to raise a wall. Several people working together can quickly raise the wall where one person would take considerable time to do it. We might reasonably conclude then that 5 people working together can build 5 houses faster than each person building their own house.
So, while group efforts can and often do result in a return less than the investment, most reasonable group efforts are characterized by the synergistic effect, which is a characteristic shared by the so-called Prisoner's Dilemma situation.
Cooperative efforts can be classified into two distinct types: those that are not in joint supply in which the return to an individual is diminished by the return given to other individuals and those in joint supply in which the return to the individual is the same regardless of the benefits it provides to other individuals. An example of the second type of reward would be the repair of the levee that saves the town. That my neighbor's house is saved does not impact my benefit of having my house saved. In this case, the group reward is not divided by the number of members to obtain the individual reward, BG.
The freedom that we enjoy in this democracy is a common good characterized by being in joint supply. Some people work very hard in preserving that freedom and without their efforts, it might just become extinct. Yet once that freedom is obtained, many others, including freeriders, can share it without any cost to anyone else.
Public Television is another example of a product in joint supply. I can view the products of the Public Television effort without any cost to anyone else -- including the producers!
Another characteristic of common goods is the characteristic of "Impossibility of Exclusion". Such common goods, once provided, are available to everyone and no one can be excluded. Far more discussion on this and the previous subject, you are encouraged to read Russell Hardin's book listed in the reference section.
When individuals contribute to a group effort, the result is often non-linear (i.e., not proportional) with respect to the input. An example of linear results would be something like an investment group where the return would be in proportion to every unit of input. Another example of a linear relation would be the classic "Tragedy of the Commons" situation where the addition fertilize to the pasture would result in a return to all proportional to the input. An example of non-linearity would be the case of a petition effort to get a bill passed in Congress. In this case the result is binary, all or nothing, with no direct relation to an incremental input.
Many phenomena such as elections have a result that is binary in nature. The result is either true or false depending on the input reaching a minimum value. A politician is elected only if he receives a majority of the votes. This has particular impact on the phenomena of the VP in that it is highly unlikely that one vote will have any effect on the outcome. In fact, the number of votes can vary over a wide range without changing the outcome.
This means, for example, if you live in California and the votes collected back East already exceed what is needed to win the election, your efforts in voting have no impact on the result.
This situation is best illustrated by a simple experiment. Suppose that you had a balance scale with the balance pans filled with marbles with the scale having a sensitivity such that a one marble difference caused the scale indicator to tilt over against one of its stops. If an equal number of marbles is in each pan, then the scale indicator is at center. Otherwise, the pointer is either at the left or right stop.
Suppose there are a few more marbles in one pan than the other (few being more than 2). I can remove a marble from either pan and nothing happens. Or I can transfer a marble from one pan to the other and still nothing happens. This example perfectly illustrates the VP for the situation where the results are binary.
The PD game itself has both a binary input and a binary output. The input is either Cooperate or Defect and the output to each individual is one of 4 values.
Actual voting can be analyzed from two points of view: "voting to save democracy" or "voting to elect a candidate". Most citizens seem to feel that the main purpose of voting is to save democracy ("you have a moral duty to vote") but most academic studies seem to concentrate on the purpose of group choice, i.e., selecting a candidate or option. Let us look a the second view first.
It is difficult for people to understand what an incredibly small chance there is of a major election ending in a tie.
When there are probabilities involved, we must modify the cost/benefit equation to represent the expected value. This is done by multiplying the expected return by the probability of that return.
A realistic assessment of major elections in the USA will result in the conclusion that the probability of a tie is infinitesimally small. A national election is complicated by the "Electoral College" and, if the election ends in a near tie, a recount will be called for anyway! While such elections just do not end in ties, even if close, a recount may be called!
Some scholars have made some calculations on the probability of a tie using various assumptions. Mathematical analysis of a simple model can show that the probability is not infinitely small but in fact about 1 in 10,000 or so if you assume that the probability for each candidate winning is exactly equal. For further study, see the book, Democracy and decision, by Brennan and Lomasky listed in the Reference section.
One way to better understand the insignificance of one vote is to imagine that you are an active supporter of a certain candidate and you are out on the trail soliciting votes. How much would you pay for just one additional vote?
A better understanding of the VP might "put to bed" the specious argument heard so often in the last election that goes something like this, "I really would like to vote for Perot, but I realize that my vote would be wasted (since he is not likely to get enough votes to win). So I will vote for Clinton". This bit of choice reasoning, apparently used by millions of voters, supposedly made a major impact on the vote count in the last election. Note the fallacies: (1) Since no particular individual's vote will impact the election results, that individual would receive greater satisfaction by voting his or her "conscience". (2) The fact that many people considered a vote for Perot as being wasted and therefore switched their vote to another candidate significantly impacted the vote count for Perot and conceivably caused him to lose. We will never know.
On further thought, we see that many other SD's have this same characteristic. People will act if they think the possibility of success is about a toss up. For example, many people claim that our government is becoming more and more oppressive and is violating our constitution. When I say, "Why don't you do something about it?", they respond with, "Because any effort will fail at this time. When I think that there are enough people willing to help and there is some chance of success, I will join the fray". For another example, consider the "impending break of the levee" situation. If the levee is a long ways from overflowing, they don't need my help in sandbagging. If the levee is already overflowing, my effort would be a waste. But if it is at the point of almost overflowing, then I feel that the few bags I throw just might save the levee. The conclusion is that in many SD's a person would be rational to contribute if the situation is that one more contribution just might be what is needed for success. Unfortunately, this is the exceptional and rare case.
In the real world, randomness is the rule rather than the exception. When the number of things in a collection is very large, the addition or removal of one of these things may be less than the normal random variation of the total quantity. This would make the addition or removal of one or two objects undetectable.
There are situations in which the impact of one event is just insignificant compared to the normal random variations. The amount of water I use to take a shower is far less than the normal variations of the volume of water in the reservoir due to other causes. Therefore, my taking of a shower has no measurable impact on the water situation. That is, the water utility would be unable to detect -- at the source -- that I have taken a shower.
Another example would be a misguided effort to reduce the national debt by a personal contribution. What if everyone in the USA sent in a thousand dollars to be applied to the debt? A few years ago, it would have paid it off and even now it would make a major reduction in it. But let us look as just one person making a thousand dollar contribution. It would be undetectable in the total debt. There are other fluctuations that would totally obliterate its affect.
While random variations can make detection impossible for one event, another factor is involved in the detection: the sensitivity of the detector. Even if there was no random variation of the water volume in the reservoir, no means of measuring the volume is sensitive enough to detect the usage of one shower by one individual.
But we are not in general talking about some device that does detection, but we are talking about human beings. If the event is not detectable by humans, then it is likely of no practical significance. The rock star on the stage cannot detect whether I applaud or not. Most humans cannot detect if I say "aye" or not in a voice vote of 50 or more people.
The VP seems to occur mostly where there are large numbers of anonymous members in a group. Those two factors -- group size and anonymity -- need to be examined more carefully.
A thoughtful person upon first examining the VP might speculate that the paradox results from the sheer size of the group. "My vote doesn't count because there are so many voters, the situation makes my vote insignificant".
So, how many votes does it take to make your vote insignificant?. The answer is simple: your vote only counts when there is a tie regardless of the total number.
More precisely, regardless of how small the number, your vote only counts when there is a tie, plus or minus one vote. Consider that there are 4 voters and you are one of them. If you did not vote and A got 2 votes and B got 1 then your vote could have caused a tie if you voted for B or done no good if you voted for A. If you didn't vote at all, then A wins. Regardless of the number of votes, this situation obviously prevails.
However, the "benefit" terms of the cost/benefit equations include such things as "good feelings", as discussed above which is a function of group size. That is, if I live in a small community, the "good feelings" rewards I receive are much more significant than when I am part of a large anonymous group. This negative impact on cooperation by anonymity has been discussed extensively in the literature.
BI, the direct benefit to the individual and C, the cost to the individual, contain a component that we will call psychological rewards or punishment. For most individuals, psychological rewards and/or punishment are very powerful components in the cost/benefit equation. In fact, the factors account for most of the "irrational" but good behavior that civilization depends upon to exist!
Let's look at an example. Suppose your church wants to add a new audio/video room that will provide free access to educational materials. They wish to do this by means of contributions. How do you think the results would compare between allowing the members to contribute anonymously or to contribute to a basket being passed while all are sitting in their pews? I'm afraid anonymous contributions would not do very well at all.
Given that there are enormous social pressures to "do the right thing", what is the effect of anonymity in the group? It practically nullifies the rewards for such contributions. If I contribute to a cause and the contribution is anonymous, then these psychological factors are not at play. Other factors, particularly guilt, must account for this cooperative behavior.
In fact, this is why governments find it necessary to tax rather than rely on contributions for all causes, regardless of how worthy they may be. Once there is real anonymity, most people do not cooperate.
The understanding that anonymity nullifies the psychological pressures to "do the right thing", then explains why people in small towns act in socially desirable ways and people in big cities typically do not. As long as most people in your group are fully aware of your actions, you will most likely act responsibly with regard to both personal and group activities.
If the logic presented so far in this essay is sound -- especially given the fact that a single individual's actions are of no consequence to the outcome and there is anonymity -- then we must conclude that society will have a problem with "freeriders". And of course it does, with enormous costs in money, time and security.
While some actions are more sinister than others, we all freeload to some extent. Some of us might cheat Sears by taking back a product for exchange or refund when we actually did the damage. Why not? Sears is a big corporation and one returned item will not make any difference. Besides, they don't know me from Adam. Of course, I wouldn't even think of doing this to someone that knows me personally.
We cheat the insurance companies and the health plans that our dollars collectively support.
We take advantage of every benefit from the government we can whether we are justified or not.
In the view of some, more sinister examples are the cheating on welfare and the wasting of public funds and the goofing off by government employees.
Freeloading is a rational action when the "benefit-cost" value is positive. Public programs provided by the government, insurance companies and health plans provide great benefits compared to the cost to the freeloader. Society can increase the cost to the potential freeloader by changing the mental make-up of the individuals or by increased controls and punishments. It is most important to realize that these increased costs to the freeloader usually also greatly increase the cost of the benefits to everyone else.
An interesting aspect of the freeloader phenomena is that the freeloader can not exist without the contributions of those who do not freeload. The hippie living on welfare and using the public medical facilities depends on the existence of the straight people that they hold in contempt. That is, without the host, the parasite dies.
There are many asymmetries between the individuals of a group or between the group itself and the individual. For example, let us say that I can barely make ends meet and I contribute $100 to Public Television. My $100 is a great personal loss to me but an insignificant gain to the Public Television organization.
How individuals value a cost or reward often varies greatly. Laws passed that makes some presently legal activity illegal, to help fight crime, often do more harm to the general population than to the criminal, for the criminal was already in violation of existing laws while the lawful citizen now has to restrict certain activities. Further the criminals are in relatively small numbers compared to the rest of society. The total group loss, therefore, is often much greater than the reward.
The VP can raise its ugly head in time as well as space. For example, in a long project in which the end date is subject to significant variability, what difference would taking a day off make? That is, could the fact that an individual took a day off be detected at the end of the project? Not likely. (But what about "n + 1" days, and so on?).
Why should I make significant sacrifices for the benefit of those yet to come? Even if I consume a great part of the Earth's resources and just leave garbage and contamination, I will likely not live to see the consequences. It is difficult for a rational person to give up very much for the generations that come after his or her death.
There is the possibility that our actions today may spell the end of humanity. What if our selfish actions today results in the destruction of the survival resources of the earth? What if the war machines we build create a very high probability that the Earth will be destroyed? Should I sacrifice my safety and immediate financial rewards?
Here we have a double whammy of the VP! First, will anything I do as an individual affect what the mass of humanity receives in rewards? No. Second, will anything I do affect future generations to come? Possibly, but I will not be here to find out.
When most people hear the argument for the VP the first time, the most common reaction is, "But what if everyone did that?". Obviously, if everyone declined to vote, democracy would fail. Still the argument is specious. The impact of "everyone doing it" would radically change the analysis of any logical discussion. What if everyone decided to withdraw their money from the bank? What if everyone decided to quit buying new cars? What if everyone decided to not go to work tomorrow? What if everyone decided to read this article?
Thousands of examples can be given in which a certain action is harmless when committed by you and me but becomes a disaster if "everyone does it". Like I said, a specious argument.
If you owned millions of shares of IBM stock and you decided to put them up for sale, you would surely realize that your action is likely to result in lowering of the value of that stock. Many owners of IBM stock, including aging widows barely getting by, would suffer as the result of your sale offer. By the way, you would also feel the negative impact of this large sale if you retained some of the stock.
But what if you only sold a couple of shares? Most of us would agree, that offering of a couple of shares to the market is not likely to impact the value of the stock. But what if thousands of others followed your lead and did the same thing? If that happened, as with your large sale, the price is likely to fall.
Consider another case: let us say the Red Cross broadcasts a mass appeal for more blood as a result of needs coming from some disaster. What if you didn't feel quite up to giving blood at this time? Would their appeal fail? Of course not. But what if everyone followed your example?
While most people clearly understand the above arguments for the cases presented, they seem to have difficulty understanding the ramifications of other problems that are characterized by the same phenomena -- the classic example being voting in a national election. Your vote in a national election has even less impact on the results of that election than the sale of one share of IBM stock would have on the price of IBM stock! And far less impact than your withholding of a pint of blood from the Red Cross.
Suppose that there are a thousand needy orphan children and a thousand contributors willing to give $100 each. Would it make a difference if this money were first pooled and then divided evenly among the orphans vs. the direct contribution of one contributor to an orphan? Yes, for with the "pool" method, it appears that my $100 will make little difference to any orphan and I might not contribute, whereas, I can see the great value coming from a direct contribution. 
1. See Parfit's book, Reasons and Persons, pages 76-82.
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