III. The Name of the Game

 

 Voluntary Provision as a Prisoner's Dilemma.

 

The previous chapter argued that a society of egoists will not be able to provide themselves with public goods voluntarily. The present chapter will reaffirm this conclusion by considering what binary two-person game best models the public goods situation. Egoists in a Prisoner's Dilemma are trapped. Egoists in a game of coordination are not. This chapter defends the analysis of public goods as essentially a Prisoner's Dilemma.

 

The Prisoner's Dilemma is a two-person game having a certain strategic structure. The following matrix is an example of a Prisoner's Dilemma game. The first number in each cell represents the preference rank to the Row chooser, with "1" as the most preferred and "5" as the least preferred.

 

  Prisoner's Dilemma

Column Chooser

 

A

B

Row Chooser

A

3,3

0,5

 

B

5,0

1,1

 

Table 1

 

The payoff structure which defines a Prisoner's Dilemma has the characteristic that we can label rows and columns so that row A strongly dominates row B, column A strongly dominates column B and both players strongly prefer (B,B) to (A,A). The Prisoner's Dilemma gets its name from a story often told to illustrate games of this structure. A prosecutor arrests two suspects and gets them to confess to committing a serious crime by separating them and making the same offer to each suspect. "If you confess, and your partner does not, I will let you go free, while your partner will serve five years in jail. If you do not confess and your partner does, it will be the reverse. On the other hand if you both confess, you will each serve three years, while if you both don't confess, you will each serve only one year (since I have enough evidence to convict each of you of a lesser crime)."

 

In a Prisoner's Dilemma game individual rationality and collective welfare "conflict", that is, individual rationality counsels each person to act in a way that results in an outcome which is socially inefficient. The principle of individual choice which recommends defection in Prisoner Dilemma situations is a very strong one, what we will call the "Dominance Principle". The Dominance Principle, as we shall define it, recommends avoiding strongly dominated strategies. In some instances, this eliminates all but one strategy. In even more special situations, the one remaining strategy is also a dominating strategy, a strategy that dominates all other strategies. This stronger property is true of the Prisoner's Dilemma. In our matrix, no matter how the other player chooses, each player is better off choosing A. B is a strongly dominated strategy and A is a strongly dominating strategy. The Dominance Principle recommends to each player that she choose strategy A.

 

We have on the other hand a principle of social choice, the Pareto Principle. The Pareto principle states that outcome a is better than outcome b if and only if no one prefers b to a and at least one person prefers a to b. A Strong Pareto Principle states that outcome a is better than b if and only if everyone strongly prefers a to b. In a Prisoner's Dilemma, the player could achieve an outcome (B,B) which is, according to the Pareto Principle, better than the outcome recommended by the Dominance Principle (A,A). In fact, there exists an outcome which is better than the outcome recommended by the Dominance Principle, even according to the Strong Pareto Principle .

 

So this is a stark contrast, but only a seeming conflict (hence the scare quotes around "conflict"). In fact there is no real contradiction. The Pareto Principle doesn't tell any individual what is best for her to do. It merely states which outcomes of collective decisions are better than which other outcomes. An individual must choose among strategies and cannot choose among outcomes since she cannot choose for others.

 

It is interesting to note that historically the Prisoner's Dilemma game was invented in an attempt to reveal the limitations, not of the dominance principle, but of the Nash equilibrium concept. Recall that a strategy pair is in Nash equilibrium if each is a best reply to the other (where "best" means no other strategy has an outcome the player would strongly prefer). Any strongly dominating strategy is also a unique Nash equilibrium. So in a Prisoner's Dilemma, the Strong Pareto Principle also "conflicts" with the Nash equilibrium concept. We will see that sometimes a "conflict" exists only between a unique Nash equilibrium and the Pareto optimum, while there is no conflict between the dominance principle and the Pareto optimum.

 

Several writers have noticed that many social situations, especially those which invite free riders, have this characteristic flaw, where the exercise of individual rationality produces Pareto inefficient outcomes. In particular, leaving the provision of public goods to voluntary initiative seems to resemble a prisoner's dilemma.

 

Here is a crude and flawed argument for using the Prisoner's Dilemma to model a public good situation, but one which gives the general flavor of a more qualified analysis. In a public good situation, each person is better off not contributing, regardless of what the other people in the group do. If others contribute, then the public good will be provided, so one's contribution is not needed. If others do not contribute, then the public good will not be provided, so one's contribution is wasted. While each is wise not to contribute, when everyone does this, each is worse off than she would be were everyone to contribute.

 

Viewing the voluntary provision of a public good as a Prisoner's Dilemma is overly simplified in at least three respects. First, there are frequently more than two players, while the Prisoner's Dilemma is, by definition, a two-person game. A perfectly exact representation would include as many dimensions as there are players. Second, contributions are not necessarily all or none, but may range nearly continuously over, for example, the amount of money donated. So each dimension of an exact representation ought to include as many choices as there are possible contribution levels. Third, if at least a certain amount is required to provide the public good, one person's contribution might make the difference between provision and no provision. There are often significant thresholds in the production of a public good.

 

While it is not difficult to describe a public good situation involving n people using an n-dimensional space to represent all the possible choices and outcomes, little is lost by summing the choices of all others together and presenting the different possible summations along one dimension of a two dimensional array.

 

In response to the second criticism, it is worth noting that not all choices affecting the provision of a public good are close to continuous. Voting, which provides the good of superior leadership, is a discrete, all or nothing choice. In situations that involve continuous or nearly continuous contributions we must resort to the Nash-Cournot analysis of the previous chapter. An important feature of the Prisoner's Dilemma game--dominance--is lost. It is no longer true that no matter how others behave, one is better offer contributing the Nash amount. However, we may fairly describe the game as a Prisoner's Dilemma because there is still only one Nash equilibrium and this point is not Pareto optimal, but inferior to at least one other available outcome. So while the conflict between the dominance principle and the Pareto principle is lost, the conflict between the unique Nash equilibrium and the Pareto principle remains.

 

If there is a threshold, below which there is insufficient cooperation to provide a public good, then one can reduce the decisions of "everyone else" to three columns: what others do is either enough alone to provide the good, not enough alone, or just enough to provide the good with the added participation of the Row Chooser. Of course collecting the choices of everyone else into the role of one player is an oversimplification which strains the notion of a Nash equilibrium. But assuming that it is better for "everyone else" to provide just enough if Row Chooser is also providing, but not provide if Row Chooser is not providing, then there are two Nash equilibria to this game: no one provides or Row provides while everyone else provides just enough. The latter Nash equilibrium is Pareto efficient, hence, this 2x3 game is not a Prisoner's Dilemma.

 

But realistically, the chance of everyone else providing just enough to put the Row Chooser on the edge of a threshold is highly unlikely. Eliminating this unlikely third alternative returns the game to a Prisoner's Dilemma. Not contributing dominates contributing, for Row Chooser and for everyone else.

 

Suppose, for example, that if enough people sign a petition, a certain issue will appear on the ballot. With a large population, the chances are extremely small that just the right number of people will sign so that one more signature will make a difference. Unless the issue affects the egoist very directly and severely, she will be justified in disregarding this remote possibility. She may reasonably reduce what everyone else does to two alternatives: enough others will sign or not enough will sign. Either way, she is better off not signing the petition, if there is a significant personal cost to signing.

 

So while the initial argument for describing the voluntary provision of public goods as a Prisoner's Dilemma is overly simplified, it can be defended as a reasonable approximation to the truth.

 

 Voluntary Provision as Coordination.

 

Voluntary provision has been compared to most of the prominent two-person binary games: Prisoner's Dilemma, Assurance, Chicken, and Battle of the Sexes. All of these games except the Prisoner's Dilemma are games of coordination, that is, games in which there are two or more Nash equilibria. If the voluntary provision of a public good were merely a matter of coordination, rational egoists would be able to easily solve the problem of provision. I will defend the view that while games of coordination may well describe a small group of rational egoists providing a public good, they misrepresent the predicament of egoists interacting in large numbers.

 

Of course, not all games requiring coordination are without conflict. In some, the conflict of interests is quite pronounced. However, even where there is conflict, conventions, threats, assurances and other measures may assist in achieving the cooperation necessary for adequate provision. For our purposes, there is no point in further dividing games or situations into categories according to whether conflict or coincidence of interests predominate, or even whether there is conflict or no conflict. The point is that rational egoists will more easily achieve Pareto efficient outcomes when the only problem is one of coordination.

 

We will consider each game model in turn.

 

 Assurance

 

The game of Assurance or Stag Hunt has payoffs similar to those in Table 2 (payoffs here and the matrices to follow are in utiles, with higher numbers being more preferred):

 

 

Assurance

Column Chooser

 

A

B

Row Chooser

A

10,10

0,3

 

B

3,0

3,3

 

Table 2

 

Rousseau tells a story in his Discourse on Inequality about a group of hunters who realize that everyone's participation is required to take a deer. However, if a rabbit passes near one of them, he doesn't hesitate to pursue it, thereby forfeiting everyone's chance of bagging the deer. While each prefers a deer to a rabbit, none is confident that everyone else will remain at their posts.

 

Jack Hirschleifer tells a different story about a circular island named, "Anarchia", which is protected around its perimeter by a ring of dikes. Each inhabitant owns a wedge shaped slice of land which will be flooded if any one person does not build a sufficiently high dike. Unlike the participants in Rousseau's stag hunt, each inhabitant of Anarchia expects the others to contribute adequately to the island's sea defense by building and maintaining a dike sufficiently high to protect her own land. We might further specify that the higher everyone's dikes, the greater the protection against flooding. So the amount of protection afforded all of the inhabitants is limited by the height of the lowest dike.

 

Both of these public good situations are Assurance games. They differ primarily in the expectations of the participants. The hunters believe that if a rabbit crosses anyone's path, the hunter will pursue it. The inhabitants of Anarchia trust that each landowner will build and maintain a high dike to protect his property. The hunters expect defection and the landowners expect cooperation.

 

In both examples, the production of the public goods depends on universal cooperation. All of Rousseau's hunters must remain at their post to catch the stag. The unanimity requirement in Hirshleifer's Anarchia is less direct, but still present. Everyone must build a dike of at least a certain height to receive a certain level of protection. One might say that in Anarchia, there are many thresholds, each requiring unanimous participation, each associated with a certain amount of public good provision and each requiring unanimous participation.

 

Unfortunately, many public good situations are not like this. They either do not have thresholds or the thresholds are such as to allow some beneficiaries not to contribute to the public good at all. If a group can exceed a threshold without universal cooperation, then we have conditions more appropriately modelled by the game of Chicken.

 

 Chicken

 

The conventional story associated with Chicken is that of two male youths who drive their cars at high speeds directly towards each other. The first to swerve looses face for being a coward or "chicken". The payoff matrix of which this story is an interpretation is given in Table 3 below. The distinctive characteristic of Chicken is that mutual cooperation is an attractive, but unstable arrangement. The prospect of mutual cooperation invites a race between the players to see who can first commit herself to defection, thereby forcing the other player to remain as the sole cooperator.

 

 

Chicken

Column Chooser

 

A

B

Row Chooser

A

3,3

2,4

 

B

4,2

1,1

 

Table 3

 

The standard interpretation of Chicken does not involve a public good, but this is easily remedied. Take Hume's example of two neighbors who would like to drain a meadow. Suppose, contrary to Hume's description, that each person can rely on the other person to do all of the work if necessary. If one of the two sees that her neighbor cannot attend to the matter, she will not abandon the whole project, as Hume suggests, but supply all of the required labor. This makes mutual cooperation (both digging a ditch to drain the meadow) and mutual defection (neither doing any digging) both unstable. We have not just one Nash equilibrium, but two (if we limit ourselves to pure strategies). So each neighbor has an incentive to be the first to commit herself to inaction, thereby forcing the other person to carry the full burden.

 

One may observe demonstrations of commitment typical of Chicken games in public good situations involving more than two players, if the number of players remains small. For example, the history of whaling exhibits both of these properties.

 

The International Whaling Commission was formed in 1947, but was unable to prevent the commercial extinction of the Blue Whale. Major whaling nations exceeded their quotas, while other nations persisted in spite of these defections, as one would expect from the instability of mutual cooperation. Japan and the USSR, used their veto power and threats of withdrawal from the Commission to block more protective conservation measures. In effect, they were exhibiting their commitment to non-cooperation. Although insufficient restraint eliminated the commercial viability of the Blue Whale, Japan and the USSR did succeeded in driving the other nations out of the remaining market for Sperm Whale and Fin, Sei and Brides Whales, forcing them into de facto cooperation.

 

Closely related to Chicken is another game theoretic model, the War of Attrition. In Wars of Attrition players are rewarded for outlasting all other players. Hence, each player chooses among a continuous range of options regarding how long they are willing to wait before "cooperating". The standard interpretation of the Chicken game matrix is really not a Chicken game, but a War of Attrition. As the game is supposedly played, each driver may choose to swerve at any time, so the range of choices is not merely to swerve or not to swerve, but a continuous range of options. The driver that swerves last wins the game.

 

Public goods with threshold levels of production are sometimes Wars of Attrition. The temperature in a room is a public good for occupants of the room: one cannot easily make the room comfortable for some occupants, but not others. If a room is too cold because of an open window, it may require only one person to close the window. Typically, occupants will wait some length of time before someone takes the initiative to close the window. This possibility is what makes the situation a War of Attrition. As in all Wars of Attrition, while the range of timing strategies is continuous, the rewards for persistence are discrete: we suppose that the window is closed all the way, or however much is appropriate for the circumstances. There is no extra effort involved in completing the job, once someone has begun the task. The volunteer does not close the window some partial amount or close it in a way that will invite others to attempt an adjustment (were such a conflict possible, it would constitute a different kind of situation).

 

In Wars of Attrition with public goods, the person who can supply the good with the least cost will be the first to accept this responsibility. Waiting serves as a tacit signal of each person's true costs. Overt signaling would be pointless either because communication costs are so high or because rational egoists would have an incentive to misrepresent their true costs of providing the good. If one assumes Nash conjectures, then there will exist an equilibrium in which each person waits an amount of time such that if she waited any longer the cost of waiting would exceed the benefit of the public good, and if she waited any less, she would excessively reduce the chance that another person with lower costs would supply the good.

 

Clearly, Chicken and the related War of Attrition model provide insight into the voluntary provision of some public goods. Why then do I not think that Chicken is a good way to model voluntary provision in general and especially for our central examples?

 

First, successful commitment in Chicken (not a War of Attrition) requires affordable communication. Communication within large groups is not cheap. So it is unlikely that one will see people attempting to demonstrate their commitment to defection in large groups. However, this objection is not conclusive.

 

One might infer from the cost of communicating in large groups not that Chicken fails to apply, but that the appropriate model is a game of Chicken in which the players must choose their strategies in the absence of prior communication. With that restriction the most salient Nash equilibria may be a symmetric mixed strategy. Mixed strategy or not, without prior communication, there may be frequent disasters of mutual defection. Chicken games do not necessarily have rosy outcomes.

 

Further, sometimes a kind of communication is achievable even in large groups. When public goods require renewed participation, the availability of the good itself provides information about the level of cooperation and whether there is enough to meet the threshold. The outcome of one period provides relevant information on which people may act in the next period. This minimal exchange of information may be enough to create stability when sufficient numbers cooperate and instability when the numbers are too low. In a War of Attrition, as discussed, one may easily signal one's commitment tacitly by waiting. So communication is not always an insurmountable problem even if the size of the group is large. However, the difficulty of signaling commitment in large groups suggests that one will not observe in large groups the kind of strategic pre-play maneuvering one finds in two-person Chicken games.

 

Second, Chicken and War of Attrition presuppose lumps (thresholds) in the production of a public good). There are several reasons why the provision of large scale public goods will tend to be more smooth than lumpy. When there is a tradition of many contributors giving a large total, there is often little doubt that donated funds will be sufficient to provide the good in one form or another. The real question is whether the organization providing the good will raise enough money to supply a certain amount of the good or enough to create a good of a certain quality. The amount and quality can vary by relatively small units. A nature preserve can be an acre larger or an acre smaller; a non-commercial broadcast can be just a bit better or just a bit worse.

 

Realistically, some lumps are likely to remain. However, even if a public good increases in discrete steps, the effect of a marginal increase in donated money may still be continuous. The expected benefit of increased donations may be continuous even if public goods themselves come in discrete sizes or degrees of quality. This is because the effect of making a slightly larger contribution cannot be known with certainty, but is rather a matter of subjective chance. Slightly larger donations will slightly increase the probability of producing an extra unit of the good, even if that unit has a discrete price.

 

The effect of a donation is a probabilistic relation for two reasons. First, whatever the decision making process of the supplier, donors are in ignorance of this process. So even if each donor has perfect information about the other donors' actions, she still does not know exactly how income to the supplier translates into output of public goods. She can only surmise that the greater the income, the greater the output. She can only conjecture that the more she gives the more likely the supplier is to produce one more lumpy unit of the good.

 

Second, the decision to purchase an extra unit is never rigidly determined by the exact level of income (which fluctuates from day to day). Expenditure decisions are partially a matter of judgement. The provider of a public good always has the discretion to incur a slightly larger or smaller deficit or a larger or smaller surplus. Whether the supplier believes she can afford to provide an extra unit is not a perfectly deterministic function of income. The greater the income, the more likely it is that the supplier will increase production to a certain amount. Production is a probabilistic function of income. The probability of adding an extra unit of a good varies continuously with donor generosity. Hence, even with lumpy goods, expected output is a continuous function of donor input.

 

Where production functions are continuous and convex, the reaction curve models of the previous chapter will apply. Chicken games do not well model situations with continuous production functions. We have already seen how the Prisoner's Dilemma does a good job of modelling such situations.

 

Many charitable organizations and not-for-profit institutions set fund raising goals. Suppose that a public good provider initiates a drive to raise a fixed amount of money. Does this create a threshold that converts a Prisoner's Dilemma game into one of Chicken? It might if the threat not to provide the service if the goal is not met were credible. Unfortunately, in most instances, the threat is not credible. If the organization receives one dollar less than its goal, it will not return the money and shut down. Further, if it receives more than the announced goal, the money will have some probable effect, as argued above.

 

A more credible threat would involve a delay, rather than a full cancellation of services or production. Suppose a radio station threatens not to resume broadcasting until pledges of support reach a certain level. This introduces a distinct threshold. Furthermore, if the station broadcasts frequent updates announcing how close they are to reaching their goal, this would allow for common knowledge of everyone's current commitments. So the station is in a good position to reduce the barriers to communication. By strategically delaying their programming, can a station expect to raise voluntary donations equal to the value of the service they provide? They can if the listeners must call in their pledges in a known order, rather than all at once or haphazardly. The situation is similar to Rubinstein's model of the bargaining process, discussed in previous chapter. Note, however, that having a rigid structure which fixes the order in which a determinate number of people may make pledges is highly artificial.

 

 Battle of the Sexes

 

Finally, consider Battle of the Sexes, as represented by the matrix in Table 4. The usual story associated with this game is one of a couple that would like to spend their evening together. The husband prefers to see a boxing match, while the wife prefers to see the ballet. But each prefers going to either event with the other than going to either alone.

 

  Battle of the Sexes

Column Chooser

 

A

B

Row Chooser

A

4,3

2,2

 

B

2,2

3,4

 

Table 4

 

If a public good has a stepwise production function, then the situation resembles a Battle of the Sexes. The argument in favor of Battle of the Sexes is essentially the same as for Chicken. With a lumpy good excessive support would be a waste, as would slightly too little support, so there is the opportunity for some beneficiaries to refrain from support, thereby forcing other beneficiaries to contribute. Just who gets to refrain and who must contribute is a source of conflict between beneficiaries, just as the choice of which performance to see on a particular evening is a source of conflict between the husband and wife.

 

While Battle of the Sexes and Chicken both fail to apply to non-lumpy goods, I believe that Chicken has fewer inadequacies and better represents realistic public good situations. I shall argue for this by justifying a series of transformations of the Battle of the Sexes matrix. I will suggest that each alteration is either an improvement or a trivial change. Since I have already provided reasons for preferring the Prisoner's Dilemma to Chicken, this comparison of Chicken and Battle of the Sexes will serve to demonstrate the superiority of the Prisoner's Dilemma model to the Battle of the Sexes model.

 

Compare Table 4 with Table 5.

 

 

Column Chooser

 

A

B

Row Chooser

A

2,2

4,3

 

B

3,4

2,2

 

Table 5

 

This first transformation is trivial. Table 5 is Table 4 with the Column labels switched. Now we will depress the payoffs in the northwest corner and elevate those in the southeast corner.

 

 

 

Column Chooser

 

A

B

Row Chooser

A

1,1

4,3

 

B

3,4

3,3

 

Table 6

 

Table 6 better fits a typical public goods situation because presumably both players prefer mutual cooperation (B,B) to mutual defection (A,A). Furthermore, Column does not prefer that Row defect (A) if Column cooperates (B), nor does Row prefer that Column defect (A) if Row cooperates (B), as Table 5 suggests.

 

One last change produces an even more realistic approximation of the public goods situation, if we admit some degree of continuity in the production function. Not only does Column not prefer that Row defect (A) if Column cooperates (B), but she slightly prefers that both cooperate (B,B) than that Column cooperate and Row defect (A,B). This is because Row's contributing produces slightly more of the good.

 

 

Column Chooser

 

A

B

 

Row Chooser

A

1,1

4,2

 

B

2,4

3,3

 

Table 7

 

Table 7 is, in fact, the game of Chicken. I conclude that Chicken and Battle of the Sexes are not very different from each other and that in so far as they do differ, Chicken is a better representation of public goods provision than Battle of the Sexes. Since I have already argued that the Prisoner's Dilemma is a closer fit to our paradigm instances of the public goods problem, I have shown that it is a closer fit than Battle of the Sexes.

 

Hence, the Prisoner's Dilemma offers the best general model for the voluntary provision of public goods, especially when the size of the group is large.

 


 

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