Distributive Justice, Bargaining Theory and Morality (Index)

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This is an index for my recent posts on bargaining theory, distributive justice and morality. I list them here in order in which I wrote them. Since I didn't originally intend for this to be a series, they may have a somewhat disjointed feel.


Index
1. Principles of Distributive Justice

2. The Nash Bargaining Solution

3. Egalitarian and Utilitarian Social Contracts

4. Axiomatic Bargaining, Moral Constructivism and Infant Mortality

The End of Skeptical Theism? (Part 9) - Warranted Beliefs and Epistemic Defeaters

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This post is part of my series The End of Skeptical Theism? For an index, see here.

Previous entries in this series have afforded us the opportunity to consider the unwelcome implications of the skeptical theist response to Rowe's evidential problem of evil. In particular, we have seen how the skeptical theist's appeals to cognitive limitation and unrepresentative sampling may force a regular theist to give up their epistemic rights to a whole series of beliefs that are essential to their theism.

These unwelcome implications are, I believe, enough to put to rest the entire ST-enterprise. Nevertheless, there is a further problem with ST that I want to explore in the final stage of this series. The problem has a somewhat more specialist flavour as it focuses specifically on the difficulties that ST poses for Alvin Plantinga's religious epistemology.

My primary source for this material is the following article:

• Paul Jude Naquin "Theism's Pyrrhic Victory" (2002) 40 The Southern Journal of Philosophy 557

I have come across similar thoughts in some of Evan Fales's writings. But since Fales leaves those thoughts relatively undeveloped, Naquin's article is the place to go.

Naquin's argument is essentially that ST provides an undercutting defeater for Alvin Plantinga's model of warranted Christian belief. To understand this argument, we first need to understand the nature of Plantinga's religious epistemology. Then we need to understand the nature of an epistemic defeater. Finally, we need to consider Plantinga's alleged defeater to the naturalistic worldview.

1. Warranted Christian Belief: The Extended A/C Model

I already have two posts on commonsenseatheism.com which provide, in my opinion, a reasonable summary of Alvin Plantinga's religious epistemology. The first part covers the motivations behind Plantinga's work, and the second part covers Plantinga's specific models of warranted belief. I would encourage you to read those and come back. There is also, for the undaunted, this epic interview with Tyler Wunder covering the same material in much more depth.

For present purposes, I will attempt a more bite-sized summary of Plantinga's work. Put simply, Plantinga's goal is to show how it is possible to have religious knowledge without the assistance of evidence.

"Knowledge", for Plantinga, is taken to denote warranted true belief. But classically, knowledge was said to be justified true belief, where justification was understood to arise from some subjective (internal) deduction or induction from available evidence and/or foundational beliefs. The legacy of 20th century epistemology has been to dismiss as insufficient (for knowledge) the process of internal justification; warrant is simply whatever replaces justification in a completed theory of knowledge.

Plantinga's account of warrant is based on the notion of proper function. He says that one has warranted beliefs whenever the following four conditions are met:

1. The beliefs are produced by a properly functioning cognitive faculty.
2. The cognitive environment in which the belief is formed is sufficiently similar to the one for which the cognitive faculties were designed.
3. The cognitive faculties were the product of a design plan that was aimed at truth.
4. The design plan was one with a high objective probability of producing truth-apt cognitive faculties.

The thing to note about these conditions is that they are all external to the agent. This allows Plantinga to say that one can have knowledge of something without knowing exactly why or how one has that knowledge. To put it another way: one can have knowledge without evidence. To be sure, there is an internal component to Plantinga's account that focuses on the coherence of one's noetic structure, but it is relatively less important.

With the proper function account tucked under his arm, Plantinga proceeds to develop a model of Christian belief that meets the four conditions of warrant. He argues that if God created the world, we can expect him to have created us with reliable, truth-apt cognitive faculties. We can also expect him to create cognitive faculties (e.g the sensus divinitatus) that would produce religious beliefs, including the belief in God and in the major doctrines of Christianity, in appropriate environments.

This gives rise to the Extended Aquinas/Calvin Model of warranted Christian belief. This is illustrated in the diagram below. It grants the Christian the possibility that their beliefs are true even if they have no evidence for their truth.

2. Defeaters

Although Plantinga is adamant that a Christian can have warranted properly basic beliefs about the major doctrines of their religion, he does not go so far as to say that these beliefs are unchallengeable. He accepts that one could be presented with defeaters that would, rightly, undercut one's confidence in certain beliefs.

Three types of defeater are identified by Plantinga:

• A Proper Function Defeater: This arises when someone must doubt or deny the truth of a proposition in order to maintain normal functioning. An example might be a person with a debilitating illness who must believe their chances of survival are higher than they actually are in order to live a normal life.

• The Purely-Alethic Defeater: This works from the third-person perspective. It gives an objective observer a reason to doubt the reliability of some cognitive faculty. For example, suppose your friend Crystal is convinced that she is being persecuted by the CIA. You know that Crystal has been diagnosed as a paranoid schizophrenic and is prone to such delusions. Thus, you have reason to doubt her beliefs. Note, however, that she does not (yet) have reason to doubt her own beliefs.

• The Humean Defeater: This is a purely alethic defeater that reflexively applies to your own belief-forming faculties. For example, suppose you currently believe that the furniture around you is singing and dancing. Ordinarily, these kinds of perceptions might count as warranted. However, a few moments before this you took some LSD, which you know to be a powerful hallucinogenic. Thus, you have reason to doubt that your current perceptual beliefs are true.

Plantinga goes on to note that all three of these defeaters could arise in a single case. The case is a purely hypothetical one involving a drug called XX. The drug damages one's cognitive faculties and induces hallucinations. Your friend takes the drug but cannot function properly if he believes his faculties are impaired. He tells you that his doctor told him he was immune to the effects. But you realise that this encounter with the doctor is likely to be a hallucination. Thereby, you acquire a purely alethic defeater for that proposition. You share this with your friend and he acquires a Humean defeater for his belief.

The importance of all this is that if one wishes to challenge Plantinga's A/C model, one must supply some sort of defeater to the believer. This defeater must cause them to doubt that their beliefs satisfy the four conditions of warrant. This is exactly what Naquin tries to do.

3. Naturalism Defeated

Plantinga uses his approach to warrant and defeat to develop a defeater to the naturalistic worldview. Plantinga argues that the naturalist must think that their cognitive faculties were designed by the process of evolution. 

In order for the beliefs formed by these cognitive faculties to be warranted, the evolutionary design process must be one with a high objective probability of producing truth-oriented cognitive faculties. The problem, as Plantinga sees it, is that the probability that evolution would give rise to such faculties is either low or inscrutable. This is because evolution aims a survival value not truth.

There are reasons to doubt Plantinga's argument (to defeat his defeater). For example, it may be that there is considerable overlap or coincidence between what is good for survival and what is true. This coincidence would raise the probability that we have reliable cognitive faculties. For present purposes, these counter-arguments are irrelevant. We can grant Plantinga the success of his argument.

What we need to focus on is the distinction Plantinga makes between theism and naturalism. On the one hand, he thinks he has provided a defeater for all aspects of naturalistic cognition by showing how there is no reason to think evolved cognitive faculties satisfy the four conditions of warrant. On the other hand, he thinks he can vindicate Christian theism because there is every reason to think that the Christian God created us with cognitive faculties that satisfy the four conditions of warrant.

This is where Naquin thinks Plantinga goes wrong. He thinks that by using the basic precepts of ST, it is possible to construct a defeater for Christian theism that directly mirrors the defeater that Plantinga constructed for naturalism. We'll begin looking at that argument in the next part.

Axiomatic Bargaining, Moral Constructivism and Infant Mortality

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In some recent posts, I have looked at the role that bargaining theory can play in our understanding of distributive justice. In this post, I want to summarise the earlier discussion and connect it with some broader points in moral philosophy. I also want to consider an example of a moral use of bargaining theory.


1. The Bargaining Problem Redux
To start things off, let's consider once more the basic bargaining game I outlined earlier. It was called "Divide the Money". In it, two players, A and B, are given a sum of money (say, $100) to divide between them. The rules of the game stipulate that both must demand a fraction of the money simultaneously. If the sum of those fractions is 1 or less than 1, the money is divided accordingly; if it is greater than 1, neither gets anything.

This is an example of a game in which the players can achieve some surplus by coordinating their behaviour on a set of strategies (in this case a pair of fractional demands) that allows them to share the money. These kinds of games are commonplace in social settings. Indeed, one could argue that society originated from our need to solve these distributive games.

Following the standard methods of game theory, when trying to solve this game we should look for the pair of strategies that is in equilibrium. The problem is that there are multiple possible equilibria in this game. Indeed, any pair of fractional demands that adds up to one is a potential equilibrium.

This presents us with a problem, despite the importance of these kinds of distributive games to society, we cannot find a unique solution to them.


2. The Axiomatic Approach
This is where bargaining theory steps in. It tries to use the basic analytical tools from game theory and decision theory to find unique solutions to bargaining problems. In an earlier post, we considered the axiomatic and cooperative approach to bargaining pioneered by John Nash. This cooperative approach assumes that once a bargain has been struck it is possible for the parties to implement it through joint action. It also works from a set of axioms, which are nothing more than assumptions about the behaviour of rational actors.

(Note: the non-cooperative approach has not been considered. I may do this at a later date, but don't hold your breath.)

We considered Nash's own bargaining solution in some detail. While it clearly succeeds in finding a unique solution to the bargaining problem, it does so with several questionable axioms in tow. In particular, Nash's assumption that removing irrelevant options would not change the behaviour of the players seems to be false. At best, this assumption could be adopted as a conventional constraint on the agreement by the parties. But if this is allowed, other conventional constraints could be adopted (in the form of axioms) which could lead to different solutions.

3. Moral Constructivism and Bargaining Theory
One reason that I have for being interested in bargaining theory is its potential for operationalising a type of moral constructivism, at least when it comes to distributive questions. I have an independent series of posts looking at moral constructivism. To briefly recap, moral constructivism is the view that moral truths are constructed, not discovered.

Two type of moral constructivism were discussed in that other series: (i) restricted constructivism and (ii) thoroughgoing (or metaethical) constructivism.

Restricted constructivism begins with a set of moral principles or intuitions that it then inputs into some mechanism. This mechanism, in turn, produces a set of moral judgments about a given area (e.g. distributive justice). This account is "restricted" in the sense that it does not provide a complete theory of the origin or nature of moral truths. It only provides a restricted account of the origin of moral truths in a given subject area. The moral inputs into the constructive mechanism could come from anywhere.

This is obviously in contrast to metaethical constructivism. This tries to provide a complete answer to all (or most) metaethical questions concerning the origin and nature of moral truths. It does so by trying to provide a formal account of practical reason and showing how moral judgments follow from this formal account. I have recently begun a series looking at one form of metaethical constructivism: Alan Gewirth's argument to the principle of generic consistency.

My feeling is that the axiomatic approach to bargaining provides a good way of operationalising restricted constructivism: the axioms are equivalent to the moral inputs and the formalisms of bargaining theory are equivalent to the constructive mechanism. The combination of the two produces a solution which is equivalent to a moral judgment.

I actually considered how this could be done in the previous post on utilitarian and egalitarian bargaining solutions. That post showed how John Rawls' device of the "Original Position" could be incorporated into a Nash-style bargaining solution.

If the axiomatic approach is to constitute a form of restricted constructivism, it is important that the axioms do reflect or cash-out certain moral intuitions. I want to consider one more (quick) example of this from the work of John Roemer.

4. Solving the Problem of Infant Mortality
Roemer asks us to imagine a charitable agency who deal specifically with the problem of child mortality in the developing world. The agency receives donations and it must decide how to spend those donations. Thus it is charged with solving a distributive problem.

Suppose it can spend its money on two types of intervention (x, y) that help reduce infant mortality. And suppose also that there are five countries (A - E) in which these interventions can be distributed. Country A has an infant mortality rate of 10%, Country B 8%, Country C 6%, Country D 4% and Country E 2%.

Roemer suggests that the following axioms apply to the agency's decision:

• (1) The ultimate distribution should be efficient in the technical sense that there is no way of reallocating resources so as to raise infant survival rates in one country without lowering them in another.

• (2) The distribution should be fair in two senses: (a) it is monotonic (any increase in the agency's resources should not lead to lower survival rates in any one country) and (b) it is symmetric (if countries have equal resources and technologies for deploying the intervention techniques, then the agency's distribution to these countries should be in proportion to their populations).

• (3) The distribution should be neutral in that it should not take into account any information that is irrelevant to infant survival.

• (4) The distribution should be consistent in that it should allocate the interventions (x, y) between the countries according to some consistent rule.

• (5) The distribution should be sensitive to the budget, technology-levels and resource base of the countries and the agency.

Roemer argues that these axioms embody an (eclectic) set of moral intuitions that we have about procedural and distributive justice. Furthermore he argues that only one distributive solution can satisfy all five conditions at the same time. This is the "leximin solution" which allocates resources in the following way:

First, reduce the infant mortality rate of Country A to that of Country B. Then, if there are resources left, allocate resources between A and B so as to reduce their infant mortality rates to that of C. And so on, until the budget has been exhausted.

5. Conclusion
As mentioned above, I think the axiomatic approach is one plausible way to operationalise restricted constructivism. More may need to be said about the types of moral issues or problems to which it can be properly applied, however, I will not be saying this for the time being.

The question I want to consider next is whether there is any plausible way of using the axiomatic approach to operationalise a metaethical form of constructivism. One person who has attempted to do just this is David Gauthier. I propose to look at his most famous work, Morals by Agreement, in a later series.

Egalitarian and Utilitarian Social Contracts

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What originally began as one post on the various principles of distributive justice has quickly morphed into a series on bargaining theory and the social contract. In a previous post, I tried to explain Nash's bargaining solution.

In this post, the Nash bargaining solution will be used as the basis for understanding the difference between egalitarian and utilitarian social contracts. The discussion is based on Chapter 2 of Ken Binmore's Natural Justice.


1. The Social Contract and The Meeting Game
Societies are frequently called upon to solve coordination problems. These are bargaining problems where people need to coordinate their behaviour in order to achieve some mutually advantageous result.

I always refer back to Thomas Hobbes to explain this concept. Hobbes argued that in a "state of nature" (i.e. a state without a social contract) humanity would be trapped in an unending cycle of violence ("the war of all against all"). This could only be avoided if each individual gave up certain rights and agreed to create a strong central authority with a monopoly on violence (the Leviathan). Hobbes was thus arguing that there is a mutually advantageous result (avoiding the unending cycle of violence) that can only be achieved through coordinated behaviour. Achieving the necessary coordination is the purpose of a social contract.

The key moral question arising from this is the following: if there are gains to be made from coordinated behaviour, how are those gains to be distributed among the parties to the agreement?

We can use a very simple example of a coordination problem to explore the possible answers to that distributive question. The example is the Meeting Game:

Alice and Bob, who met through an online dating agency, have agreed to meet each other for the first time. They have ten possible locations at which they could meet. Although they are both eager for the meet-up, they have differing views on the most desirable location (this could be due to transportation, desire for publicity, ease of "bail out" etc.). Indeed, two locations are so undesirable to at least one of the players that they would rather not meet at all than meet there. Alice and Bob need to agree upon some meeting point.

The table below represents their respective payoffs for the different locations as well as the status quo payoffs (i.e. what the parties get if they don't agree to some meeting point).

It should be obvious from this that Alice would prefer not to meet than to meet at location 1; ditto for Bob at location 10. All other locations are on the negotiation table as they would lead to at least some gain over the status quo. To fit this in with our discussion of the social contract, we can think of the status quo as being equivalent to the state of nature.

The meeting game can also be represented geometrically, as in the following diagram (see this post for more on what such a representation is intended to show).

The blue curve with the ten marked points represent the ten solutions to the game (corresponding with those in the table). The point D represents the status quo or state of nature. If we assume the parties have equal bargaining power, we can easily locate the Nash bargaining solution. It occurs at the point N, which is where the product of gains over the status quo is highest,

• (34 - 4) x (22 - 2) = 600

This is higher than any of the nearest rivals. For example, at points 6 and 8, the product of gains over the status quo is,

• (28 - 4) x (26 - 2) = 576

• (40 - 4) x (18 - 2) = 576

If you are wondering how equality of bargaining power can be incorporated into the Nash bargaining solution, I invite you to consider the equations that were formulated in the previous post. Look at the variables h and k in those equations. These can be thought to represent the bargaining power of the parties. If we set h = 1/2 and k = 1/2, we can model the equality of bargaining power.

2. Adding some moral content: The Original Position

The Nash solution outlined above has no moral content. It says that a certain solution to the coordination will be reached under a particular set of assumptions. But it does not say whether that solution is fair or morally good. 

The amorality of the solution is apparent once you realise (according to Nash's assumptions) that if the scales used to measure each player's utility are recalibrated, the solution remains the same. So for example, if we change Alice's payoff scale so that each unit is now worth twice what it used to be worth, the solution is still the same even though she now gets double the satisfaction. This would strike many as being deeply unfair since it fails to take into consideration how the benefits of coordination are distributed. But under Nash's theory it can't be otherwise, as Binmore puts it:

Players can't alter their bargaining power by changing the scale they choose to measure their utility, any more than a physicist can change how warm a room is by switching from degrees Celsius to degrees Fahrenheit.

There are ways to add some moral content to the Nash solution. Binmore considers one such way: using the device of the original position. The original position, as many know, comes from the work of John Rawls. Rawls argued that, in order to assess the justice of a given distribution of social goods, we have to ask whether we would agree to that distribution if we were bargaining from behind a veil of ignorance. This is the "original position".

The veil of ignorance prevents us from knowing where we will end up in society. In the example of the meeting game, this would mean that Alice and Bob have to negotiate their agreement without knowing which one is Alice and which one is Bob. In other words, with the possibility that their identities are randomly reassigned after the agreement is reached. As a result, whatever agreement each proposes will need to keep in mind both sets of payoffs.

The original position allows for complete impartiality when negotiating over possible schemes of distribution. If one thinks that impartiality is morally desirable, it constitutes the addition of a moral component to our understanding of the Nash bargaining solution (and the social contract).


3. The Need for Social Indices
In order for the original position to play a meaningful role in how we think about coordination problems such as the Meeting Game, there must be some way to compare the utility functions of different individuals (i.e. to allow for interpersonal comparison). This is what makes it legitimate to think about the reassignment of identities after the agreement.

Unfortunately, the standard method for deriving utility functions does not allow for such interpersonal comparison. This is because, as mentioned previously, the scales chosen for personal utility functions are arbitrary. We cannot say they are the same from person-to-person. The temperature analogy works again here: it would not be possible to compare temperature measurements across different locations if one did not know which temperature scale was being used. The same holds for utility functions.

Binmore argues that what we need is a "social index". This would allow us to say how much one of Alice's utils is worth in terms of Bob's utils, i.e. it would allow us to determine the ratio of Alice's utils to Bob's. The social index would represent society's total judgment on the relative worth of the two individuals. There could be a huge number of factors influencing such a judgment, but from a game theoretical perspective all that matters is the ratio.

(Note: John Harsanyi has an argument -- or a bit of "dogma" if you prefer -- which suggests that if the parties have access to the same information we can legitimately assume they have the same utility functions. Binmore thinks our capacity for empathy may allow for this. Make of it what you will.)


4. Utilitarian and Egalitarian Solution
Once we have a social index, we can proceed to consider the solution to the Meeting Game that would be reached in the original position. Even here things are uncertain, with two plausible candidates. To examine these, let's assume that the social index makes one of Bob's utils worth two of Alice's.


I. The Utilitarian Solution
A utilitarian would maintain that the best solution (given the veil of ignorance) is the one in which utility is maximised. In other words, the point at which the sum of the index-weighted payoffs is greatest. This occurs at location 3 in the Meeting Game because,

• (16/2) + (34/1) = 42

is greater than any of the alternatives e.g. location 4,

• (19/2) + (32/1) = 41 and 1/2

This solution is represented geometrically below.

The dotted line of slope - 1/2 moves down one unit for every two units it moves across (I didn't draw this precisely). This depicts the fact that one of Bob's utils is worth two of Alice's. The utilitarian solution occurs at the point where this line touches the boundary of the set of possible solutions.

II. The Egalitarian Solution

An egalitarian (like Rawls) would maintain that the best solution (given the veil of ignorance) is the one that gives each player an equal gain over the status quo. 

Obviously, this gain over the status quo need to be weighted for each player in accordance with the ratio of utils. If the ratio was 1:1, the egalitarian solution would occur at location 6, because,

• (28 - 4)/1 = (26 - 2)/1

However, in the case where one of Bob's utils is worth two of Alice's, there is no single solution. This can be seen in the geometric representation given below.

The dotted-line of slope +1/2 through the point D represents the social index. This line only intersects the efficient boundary of the set of possible solutions at the point E, somewhere between location 7 and location 8. The best that Alice and Bob can do is to randomise between the two locations (maybe meeting at location 7 on weekdays and location 8 at weekends).

5. Some Conclusions

This post has shown three things. First, it has shown how the Nash bargaining solution can be used to understand the coordination problems that lie at the heart of the social contract. Second, it has shown how Rawls's device of the original position can be used to inject some moral content into the Nash solution. Third, it has outlined the differences between a utilitarian solution to a coordination problem and an egalitarian solution.

Before finishing up, it's worth noting two important differences between the utilitarian and egalitarian solutions:

• Utilitarian solutions do not care about the status quo (or "state of nature"). They think that how we got to the position we are now in is irrelevant to deciding where we should go next. Egalitarian solutions care greatly about the status quo, and evaluate all potential social changes by comparison with it.

• Utilitarians and egalitarians look at the social index in different ways. This is obvious from the two different geometrical representations. For a utilitarian, a decrease in someone's social index (i.e. an increasing of the worth of one of their utils in comparison with someone else's) increases their contribution to the weighted utilitarian sum. Thus a utilitarian favors giving more help to those with lower social indices. Conversely, for an egalitarian, a decrease in social index decreases the utility awarded by the egalitarian solution. Thus an egalitarian favours helping those with higher social indices.

I'll look at more issues arising from bargaining theory in later posts.

The Nash Bargaining Solution

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This post tries to explain the Nash bargaining solution. This is part of some work I am doing on distributive justice, see here for more. It is worth noting at the outset that the Nash bargaining solution is distinct from the Nash equilibrium. Since both concepts are mentioned in this post, it will be important to keep the distinction in mind (I'll give a fairly simple explanation of a Nash equilibrium in a moment).


1. Why the need for a Bargaining Solution?
Consider the following simple bargaining game:

Two players, A and B, are given 100 dollars to divide between them. According to the rules of the game, they must each make a demand for some fraction of the 100 dollars simultaneously. If the sum of their fractional demands is less than or equal to 1, the division will go ahead; if it is greater than one, the money is lost to both.

If you were a player in this game, how much should you demand? 50 dollars? 75 dollars? 25 dollars? Or something else? Game theory tries to help us to answer those questions. It is the formal analysis of strategic interactions.

When trying to decide what to do, game theory offers some simple guiding principles. First off, if possible, you should try to play a strictly dominant strategy, i.e. a strategy that yields the best result for you no matter what the other player does. As it happens, the divide-the-money game does not have a strictly dominant strategy: if my opponent demands $50, I am best off if I demand the same. But if my opponent demands $75, I am not best off demanding $50, instead I should demand $25.

Alternatively, you should try to play the strategy that is the best response to some rationalisable strategy of your opponent. In other words, using some plausible assumptions about what I think my opponent will demand, I should play the strategy that would be the best response to that demand. So if I think my opponent will demand $50, I should demand $50; if I think my opponent will demand $25, I should demand $75; and so on.

A game has a Nash Equilibrium whenever players can play best responses to each others' best responses. Under assumptions of rational choice theory, rational players are most likely to play a Nash equilibrium when it is possible to do so. The divide-the-money game has a large (potentially infinite) number of Nash equilibria. Indeed, every division of the money that adds up to exactly $100 would constitute a Nash equilibrium.

Think about it: my best response to my opponent's demanding $75 is to demand $25. At the same time, his best response to my demanding $25 is to demand $75. The same reasoning applies to every division in which the sum of our demands equals $100.

The image below depicts the potential solutions to the divide-the-money game. The blue shaded region represents the set of potential distributions (less than or equal to 100). The solid black curve represents the efficient frontier of the set of possible distributions. Every point along this efficient frontier is a potential Nash equilibrium.

The analysis so far is disappointing. The basic principles of game theory show that there is no unique solution to the bargaining game. This is where the Nash bargaining solution comes into play. It tries to show how, given certain assumptions, there can be a unique solution.


2. Nash's Bargaining Solution
To understand the Nash bargaining solution, we will look at a general model of a bargaining game as opposed to a specific numeric example. The math is fairly elementary. I take this version from here (without doubt the best serious introduction to game theory).

Again, there are two players, A and B, who seek to split some total value v, through some sort of negotiated agreement. If no agreement is reached, A will get a and B will get b. These figures represent the payoff that the players can achieve on their own, independent of a negotiation. Sometimes these are referred to as BATNAs (Best Alternatives to Negotiated Agreement).

In order for the negotiated agreement to be worthwhile, it must be the case that v is greater than the sum of the BATNAs. In other words: a+b < v. This means that there is a positive surplus (v - a - b) to be achieved from the negotiation.

In the final agreement, A will get fraction h of v and B will get fraction k of v. The sum of these fractions will equal one (h + k = 1). If x represents the total sum that A ends up with, and y represents the total sum that B ends up with, we can formulate the following equations:

• x = a + h(v - a - b), or x - a = h(v - a - b)

• y = b + k(v - a - b), or y - b = k(v - a - b)

These expressions are known as the Nash formulas. They just say that the positive surplus (v - a - b) is divided between the players in the proportions h:k:

• (y - b) / (x - a) = k / h

or in slope-intercept form:

• y = b + k/h(x - a) = (b - (ak/h)) + k/h(x)

This can be represented geometrically as follows.

In this diagram, P represents the disagreement point. It is what can be achieved independently of a negotiated agreement. All divisions of v in the proportions h:k lie along the straight line joining P and Q. The efficient frontier is represented by the thick green line joining (v, 0) and (0, v).

The Nash Bargaining solution is represented by the point Q (the intersection of the two lines). As can be seen, it is a unique point and hence constitutes a unique solution to the bargaining game. The coordinates of this point (r, s) are the parties' payoffs after the agreement.



3. Why this Solution?
To this point, nothing has been said about the credibility of the Nash bargaining solution. Indeed, the formulas cited above are agnostic about the mechanism through which the agreement is actually reached.

The credibility, such as it is, of the solution comes from certain assumptions (axioms) that Nash formulated about the bargaining process. First, Nash assumed that the bargaining game was cooperative as opposed to non-cooperative. These terms are used in a technical sense in game theory and, as is the case with most technical terms, they bear little resemblance to the mundane everyday use of those terms.

In particular, it is not the case that players in a non-cooperative game are unable to cooperate with one another. Nor is the case that players in a cooperative game are unable to be non-cooperative. Instead, a non-cooperative game is said to arise whenever players make their decisions and play their strategies without prior communication or agreement. A cooperative game arises whenever players can consult with another and plan joint action.

Game theorists generally think that non-cooperative games are more significant because the conditions necessary for cooperative games are rarely met. However, they may be met if there is some sort of enforcement mechanism that would ensure that the joint agreement was implemented.

Nash modeled bargaining as a cooperative, as opposed to a non-cooperative, game and this allowed him to reach his solution. (Note: there is an argument for thinking the same solution would apply in certain non-cooperative games. However, I'm not going to explain it here.)

In addition to this, Nash made several key assumptions about how the players would act. Before listing them, it should first be noted that the standard general assumption of decision theory is that actors have utility functions for payoffs, and that they seek to maximise their utility (I hope to cover the different forms of utility function at a later date). Nash's additional assumptions were as follows:

• Efficiency: The players will exploit the full value v. They will leave no portion of v undistributed.

• Independence of Irrelevant Alternatives (IIA): This means that if options that none of the players would have chosen are removed from the game, there is no change in outcome. Suppose, for example, player A has three offers they could make x, y and z. Suppose, further, that A thinks z would be best. Now, along comes a third party who removes offer x from A's pool of possible offers. According to IIA, nothing will change since A was never going to choose x anyway.

• Symmetry: If the players' utility functions for v are the same, they should each receive the same outcome.

• Independence of (Linear) Utility Rescaling: The utility functions for players can be calibrated or scaled in somewhat arbitrary ways. With this assumption, Nash is saying that if we recalibrate or rescale the utility function on a linear basis, we do not change the solution to the game. This is a good thing since utility functions are arbitrarily scaled. Note that this would not apply if the rescaling were non-linear. A non-linear scaling would imply that the player has changed their attitude towards risk and so would actually change their behaviour. To give an example, imagine a game with two possible payoffs: (i) $25 for sure; and (ii) $0 with 75% certainty and $100 with 25% certainty. The expected payoff from (i) and (ii) is the same (i.e. $25). However, a risk averse player would prefer (i) over (ii) and would, as a result, have a concave utility scale. By way of contrast, a risk-loving player would prefer (ii) over (i) and would have a convex utility scale.

Through these assumptions, Nash guarantees a unique solution to the bargaining game. The solution will lie along the efficient frontier of the set of feasible solutions, and will be the point at which the product of the the players' gains over the disagreement point is highest. In the example given above, this would imply that (r - a)(s - b) is higher than any other value along the line (v, 0) to (0, v). 

To put it another way, it is the point at which the aggregate utility of the players is maximised. This solution is unsurprising given the economic basis of Nash's theory.

4. Do the Assumptions Hold Up?

Nash's assumptions are the key to his solution. Without them, he would not be able to find a unique solution to the bargaining game. So are they reasonable assumptions?

One of the more interesting findings from experimental economics is that IIA is almost certainly false. One might think that if A is preferred to B and B is preferred to C, the removal of C from the set of possible choices would make no difference. As it turns out, there are all sorts of weird effects that can arise when choices are added and removed. Dan Ariely's discussion of the Economist's subscription rates is instructive in this regard.

Although IIA is false, it might be possible for players to adopt it as a conventional constraint on their bargaining. But if we are going to do that, it might be possible for other conventions that yield different, but still unique, solutions to be adopted. Indeed, Kalai and Smorodinsky have done just that with their "monotonicity" axiom. This was used by David Gauthier in his book Morals by Agreement (well worth reading, if you get a chance). I'll be covering Gauthier's basic argument soon enough.

Nash's other assumptions seem more plausible. The other potentially problematic one is the symmetry assumption. For it to be true, different players would have to have identical utility functions. But one may wonder whether it is possible for such symmetry to really exist.


5. Conclusion
Despite these limitations, the Nash bargaining solution is still a noteworthy achievement. It shows that unique solutions to bargaining games such as those discussed above are not implausible.

One may ask: why am I interested in this? Well, the answer is that bargaining theories such as this can make important contributions to our understanding of social contracts. I'll look at those contributions in later entries.

The Dialectical Necessity of Morality: Methodology and Terminology

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This post is part of my series on Deryck Beyleveld's book The Dialectical Necessity of Morality. In the book, Beyleveld tries to argue in favour of Alan Gewirth's Principle of Generic Consistency (PGC). For an index, see here.

As this is the first truly substantive entry in the series, I will present a brief sketch of the argument to the PGC. I will follow this with a discussion of the methodology underlying the argument. I will close by running through some of the terminology that will be essential for understanding the argument.


1. The Argument in Brief
As stated in the introduction, the PGC is (allegedly) the supreme moral principle that must be accepted by all potentially purposive agents (PPAs). The following is a quick sketch of the argument to the PGC:

• (1) To be a PPA is to claim: I do X voluntarily for some purpose E (premise).

• (2) E is good (from 1).

• (3) Freedom and well-being are generically necessary conditions of agency, i.e. they are necessary no matter what purposes I may wish to pursue (premise).

• (4) My freedom and well-being are necessary goods (from 2 and 3).

• (5) I (even if no one else) have a claim right to my freedom and well-being (from 4).

• (9) Other PPAs (PPAOs) have the same claim right (argument for this is omitted).

• (13) I must accept that every PPA has a claim right to its freedom and well-being.

As should be apparent from the numbering, several critical portions of the argument are left out of this sketch. Still, enough is presented to give you a feel for what is trying to be proved. Beyleveld (and Gewirth) are trying to show that anyone who thinks that they are a PPA must, on pain of inconsistency, accept that all PPAs have claim rights to their freedom and well-being. This is a very strong claim.

2. Methodology
Beyleveld adopts a particular style of argument which he terms dialectic. He contrasts this with an assertoric method. A dialectic argument is something which appeals directly to the subjectivity of the agent and deduces conclusions from premises to which an agent would (subjectively) be inclined to agree. An assertoric argument would deduce its conclusions from statements about the (objective) properties of agents.

Furthermore, the argument is dialectically necessary, not merely contingent. This is because the premises are ones to which any PPA, in virtue of being a PPA, must agree.


There is an important point to be made about the methodology underlying the argument, particularly if one wishes to assess its metaethical pretensions. Since this is exactly what I wish to assess, I am going to make it, however, it should be noted that, as far as I am aware, neither Beyleveld nor Gewirth see fit to do the same.


The key questions in metaethics are, roughly, do moral terms like "good", "bad", "right", and "wrong" have truth values? Do they refer to actual states of affairs? Are those states of affairs mind-dependent or mind-independent? How can they be known?

If one rejects non-cognitivist positions (i.e. positions maintaining that moral terms do not have truth values) then one must accept that terms such as good, bad, right and wrong can be correctly applied to certain states of affairs. This is to embrace the existence of some identity-relationship between a term such as "good" and an actual state of affairs.

The methodological question is: how can this identity-relationship be established? There are two ways in which this can be done.

First, if the term has some universally agreed-upon referent, then one can investigate the referent and establish the relationship between it and the term. The classic example of this method in action comes from the identity-relationship between "water" and "H2O". Because the term water had a universally agreed-upon referent (the colourless, odourless, liquid that everyone drinks) scientists could work out its chemical structure and then establish the equivalency between H2O and water.

Second, if the term has some widely agreed-upon meaning, then one can derive a set of platitudes about the term and see which state of affairs best instantiates those platitudes. This method is very popular in metaethics where there is considerable disagreement about the referents of moral terms, but some reasonable agreement about the meaning of those terms (or what would need to be the case for them to successfully refer). Indeed, the method is used by Michael Smith in his book The Moral Problem, by Robert M. Adams in his attempt to develop a modified divine command theory, and, in a negative way, by Richard Joyce in his argument in favour of error theory.

It should come as no surprise then to learn that the second method is (implicitly) adopted in Beyleveld's work. He begins with the following four platitudes about moral "oughts" and criteria of practical reasonableness:

• Universality: "X ought to be done" applies, at least in part, to every agent.

• Other-regardingness: A moral criterion of practical reasonableness will take favourable account of the interests of agents other than, or in addition to, myself.

• Conative Independence: The prescription "X ought to be done" holds irrespective of the agent's occurrent wishes and desires.

• Over-ridingness: A moral criterion of practical reasonableness will take precedence over all other criteria of practical reasonableness. 

Having singled-out these four moral platitudes, the success or failure of the argument to the PGC is easily measured: it will succeed if the PGC satisfies those four platitudes; it will fail if it does not. Consequently, it is difficult to understate the importance of these four platitudes to Beyleveld's (and Gewirth's) enterprise.

One final point about methodology and we will be done. In addition to the four platitudes, there are three central questions in moral philosophy that it is hoped the argument to the PGC can answer. They are:

• The Authoritative Question: Why should one be moral?

• The Distributive Question: Whose interests, other than one's own, should be favourably considered when acting?

• The Substantive Question: Which interests should be taken into consideration?

3. Terminology

One of the more annoying things about the book is the terminology (particularly the acronyms) with which it is overloaded. Because of this, I thought it might be useful to provide some simple explanations of the terminology used throughout the book. I'll add to this as I go along and I will repeat the explanations whenever I introduce a term in future entries.

• Potentially Purposive Agents (PPAs): These are the agents to whom the argument is directed. A PPA is someone who voluntarily acts so as to achieve purposes or goals. Also "Other PPAs" (PPAOs)

• Subjective Viewpoint on Practical Reasonableness (SPR): A PPA's personal view, theory or criterion by which it assesses which purposes are permissible/impermissible for it to pursue. Three SPRs are highlighted: (i) the Adeonticist-SPR, according to which there are no impermissible purposes; (ii) the Deontic Amoralist-SPR, according to which there are some restrictions on purposes but none that involve duties to others; and (iii) the Deontic Moralist-SPR, according to which there are other-regarding restrictions on purposes.

• Claim Rights: The object of the argument is to prove that a PPA must accept that all PPAOs have claim rights to freedom and well-being. A claim right is something that comes with a correlative duty which is supposed to apply to others. A negative claim right comes with a correlative duty to not interfere with the person making the claim. A positive claim right comes with a correlative duty to provide assistance to the person making the claim. A claim right is to be contrasted with a weak right or a "mere liberty" which is a permission to do something with no correlative duty.

• Generically Necessary Requirements of Agency: These are things that all PPAs, irrespective of purposes they try to fulfil, will require. 

• Freedom: One of the two (allegedly) necessary requirements of agency. The specific focus is on dispositional freedom, which refers to the general capacity or ability to control one's behaviour by unforced choice. This is to be contrasted with occurrent freedom which refers to the unforced operation of that capacity at a particular moment. It is argued that dispositional freedom is necessary to have any purposivity at all, but that occurrent freedom can be waived in order to achieve specific purposes.

• Well-being: The second of the (allegedly) necessary requirements of agency. Well-being is said to have three levels. Basic well-being covers the proximate necessary preconditions of performing an action, such as physical fitness and integrity, and mental equilibrium and confidence. Non-subtractive well-being covers whatever the agent needs to maintain what it already has that is good. Additive well-being covers whatever the PPA needs to increase its existing level of purpose-fulfillment. Basic well-being is an absolute necessity for a PPA, nonsubtractive and additive well-being are important for general success as a PPA. One can imagine a hierarchy of well-being with basic at the top and additive at the bottom.

That list barely scratches the surface but it does cover the more important stuff. 

In the next post we will begin to work our way through the stages in the argument to the PGC.

The Dialectical Necessity of Morality: Introduction

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And so it begins.

Today marks the launch what will either be the longest or shortest series in the history of this blog: my attempt to go through Derek Beyleveld's book The Dialectical Necessity of Morality. It will be long-lived if I have the patience and perseverance to get to through all of it; it will be short-lived if, as is too often the case, I run out of interest after a few entries. Still, something will be learned no matter what the duration.


What's this About?
Beyleveld's book is a meticulous and pain-staking effort to defend the legitimacy of Alan Gewirth's Principle of Generic Consistency (PGC). The PGC is, supposedly, the supreme moral principle that must guide the actions of all potentially purposive agents (PPAs -- incidentally, there are a lot of acronyms to get to grips with).

The derivation of the PGC is perhaps the exemplar of the Kantian constructivist position in metaethics (a topic I have covered before). It is constructivist because it tries to construct moral values out of the attitudes of practically rational agents. It is metaethical because it does not presuppose the existence of any moral values but instead tries to show how moral values can come into being and how they can be known. Finally, it is Kantian (as opposed to Humean) because it argues that, when properly understood, practical rationality is consistent with the existence of one, absolute and universal, set of moral values.

If successful, the derivation of the PGC would be a major achievement. Indeed, it would be the Holy Grail of moral philosophy. This should make us suspicious: the PGC has been around since the 1970s and Beyleveld's book itself dates from 1991, surely it would be more widely known if it was actually successful? Interestingly, Beyleveld acknowledges this suspicion in his introduction. Nevertheless, he thinks the derivation of the PGC is ultimately successful and he wants to show us why.

I'm certainly willing to admire Beyleveld's efforts. Although I have only read about 100 pages so far, I can say with some surety that his book is nothing if not an extremely impressive exercise in analytical philosophy. It consists of two parts. In the first part, he presents what he thinks is the strongest argument to the PGC. In the second part, he proceeds to identify and rebut 66 objections to this argument. The objections having been collected from various scholarly articles written in the period 1971-1990.

I begin this series with no real preconceptions. I am neither committed nor closed to the possible success of the argument to the PGC. I am somewhat sceptical, and in what I have read so far I think I have managed to identify one major lacunae in the argument, but I am conscious of the fact that somewhere within the 66 objections and rebuttals there may be one directed at my concerns. I can only wait, read and see.

This post will serve as an index to the series.

Index:

• Part 1: Methodology and Terminology