Morals by Agreement (Part 4): Bargaining and Impartiality

This post is part of my series on David Gauthier's Morals by Agreement. For an index, see here.

In the two previous entries we have looked at Gauthier's proposed solution to the bargaining problem, namely: minimax relative concession (MRC). According to this solution, when rational players have to reach some agreement on how to distribute a cooperative surplus, they should initially claim as much of the surplus as they could possibly have, and then agree on the distribution that minimises the maximum relative concessions they have to make from this initial claim.

If that summarisation is in any way confusing, I advise you to go back and read parts two and three. If it's not confusing, we can proceed to consider the moral implications of this theory.


1. What is it that Gauthier wants to do again?
As mentioned in part one, the goal of Gauthier's book is to find the deep connection between morality and rationality. Knowing when we have found the deep connection will depend on how we understand the terms "rationality" and "morality".

For Gauthier, "rationality" means what it means to economists, decision theorists, and game theorists: people should do what they most want to do. In slightly more formal terms, this means that rational agents should choose actions that maximise their utility. (Utility is simply the worth of an action or outcome to an agent -- the utility scale is calculated on a strictly individualistic basis). So, if there is such a thing as morality, it will have to be compatible with the utility-maximising conception of rationality.

But what is morality? For Gauthier, the distinctive feature of moral behaviour is its impartial nature. In other words, a moral act or a moral outcome is notable in that it treats agent's in an equal manner (no favour or bias is shown to particular agents).

It follows that if Gauthier's project of finding the deep connection between morality and rationality is to succeed, he must show how impartiality is possible for a utility-maximising agent. The theory of minimax relative concession is part (but only part!) of his attempt to do this.

In the remainder of this entry, three issues will be addressed: (i) why the MRC-solution is something to which utility-maximising agents can agree; (ii) why the MRC-solution allows us to realise impartial outcomes; and (iii) why the MRC-solution is only part of the complete picture.


2. Why is the MRC-solution Rational?
The bargaining process arises when there is some value to be obtained from cooperation that goes over and above what can be obtained from independent action. As such, there is the opportunity for every agent to increase their utility by cooperating. The problems arise when deciding how exactly the cooperative surplus should be distributed.

Gauthier identifies the following four conditions of rational bargaining (remember: a concession is an offer by a prospective cooperator for less than their initial claim; a concession point is the outcome that would result from a given set of concessions, one from each cooperator):

• (1) Rational Claim: every player should claim the cooperative surplus that yields them the maximum utility, with the sole caveat being that they cannot claim the surplus if they would not be party to the cooperative interaction required to create it. In other words, they can't claim or demand something from the other rational players in order to secure cooperation, if they themselves wouldn't agree to that claim or demand.

• (2) Concession Point: Given claims satisfying condition (1), every player must suppose that there is a feasible concession point that every rational player is willing to entertain (since they want the benefit of the cooperative surplus, but they know they can't each get their maximum claim, they must suppose there is some concession point that they can agree upon).

• (3) Willingness to Concede: Each player must be willing to entertain a concession in relation to a feasible concession point, if its relative magnitude is no greater than that of the greatest concession that he supposes some other rational player is willing to entertain.

• (4) Limits of Concession: No person is willing to entertain a concession in relation to a concession point if he is not required to do so by conditions (2) and (3).

I think conditions (1) and (2) are relatively straightforward. (1) is simply the application of the utility maximising model of rationality to the specific context of a bargaining problem; and (2) merely draws out from this the fact that if cooperation is to be possible at all, there must be a concession point that all can agree upon. Otherwise, there would be no point to cooperation.

Condition (3) highlights the equal rationality of the players. Since each player is seeking to maximise their utility (and by correlation minimise their concessions) no player can expect another player to make a concession unless they would be willing to make a similar concession.

Condition (4) is saying that no utility maximising player will be willing to entertain a concession unless: (i) there is a feasible concession point that all could agree upon and (ii) he/she is not being asked to make unnecessary, or unnecessarily large, concessions.

These conditions of rational bargaining -- which are compatible with the utility-maximising conception of rationality -- combine to show that the MRC-solution is one on which rational actors can be expected to agree. How so? Well, conditions (2), (3) and (4) imply that every rational agent should be willing to entertain a concession point up to and including (but no greater than) the minimax relative concession. At the same time, if the proposed outcome is not the MRC-point, it would mean that some player is being asked to concede more than they can be expected to concede. Consequently, no rational player is going to agree to an outcome that is different from the MRC-point.


3. Why is the MRC-solution Impartial?
Gauthier's argument for the impartiality of the MRC-solution is slightly more complicated and, as it's late enough as I'm writing this, I'm going to skimp on the details and give a pretty cursory summary. Basically, Gauthier argues that a solution is impartial if it gives the same relative treatment to people, whenever similar treatment is possible.

There are two cases to consider. The first is where the surplus produced by cooperation is a fully transferable good (i.e. can be easily transferred between the parties). In this case, equal relative shares should be distributed between the parties as they contribute equally to the creation of the surplus.*

The second case is that of the non-fully-transferable good. In this case, it will not be possible to equally distribute the entire surplus, only the transferable portion can be so distributed.

The MRC-solution covers both scenarios: when it is possible to fully transfer the good, an equal relative share will coincide with the MRC; likewise, in the non-transferable case, the MRC-solution will allow for equal relative shares of the transferable portion with the non-transferable portion going to whoever accrues it. It is the best that any rational player can expect to obtain, and it also affords them equal relative treatment.


4. What more needs to be done?
Although the MRC-solution is a significant step along the road to showing the deep connection between rationality and morality, it does not bring us all the way to our desired destination. Two additional things need to be done.

First, the MRC-solution only shows the impartiality of the agreement reached through rational bargaining. It does not show the rationality of complying with that agreement in the long run. Gauthier's theory of constrained maximisation tries to deal with this problem.

Second, the impartiality of the MRC-solution is relative to the initial bargaining position (IBP) of the parties. If this IBP is seriously partial or unequal, it will be reflected in the final outcome. Hence, some restrictions may need to be placed on what counts as an IBP. Gauthier tries to specify those restrictions in a later chapter.

I will be looking into the idea of constrained maximisation in due course. I have no intentions to look at the stuff on the IBP.

* Gauthier has an argument covering cases where the initial contribution seems to be unequal. In those cases he shows how the MRC-solution leads to a distribution which is equal, but proportionate to the contribution. I won't explain that here, it occurs on pp. 140-141 of the book.

Morals by Agreement (Part 3): Some Examples of Gauthier's Bargaining Solution

This post is part of my series on David Gauthier's Morals by Agreement. For an index, see here.

I am currently looking at Gauthier's proposed solution to the bargaining problem, something he calls minimax relative concession. The previous post outlined how to approach the bargaining problem and how to discover the MRC-solution. We can summarise it as follows:

• (1) Define the outcome space, i.e. all the feasible solutions to the bargaining game, and, if possible, draw it on an x-y axis.

• (2) Locate the initial bargaining position (IBP), i.e. the outcomes the parties could achieve without reaching agreement.

• (3) Locate the claim point, i.e. the point representing the maximum that each player can demand. This will usually be a point outside the outcome space, as players will initially demand more than can be distributed between them.

• (4) Let the players make concessions from this claim point and then compare the relative magnitude of those concessions.

• (5) The solution will be the point at which the maximum relative concession is as small as it is possible for it to be, hence the minimax relative concession.

In this post, we will look at two numerical examples of this solution-concept in action. These are discussed by Gauthier in Chapter 5.

1. Jane and Brian go to a Party

Jane has been invited to a party by Anne. She would really like to go but is worried that Brian might be there. She doesn't like him and would prefer not to go if he would be there. Brian has also been invited to the party, but he doesn't want to go unless Jane is going too.

Based on these preferences, we can draw-up the following payoff-table (or outcome matrix) for this game. The figures are interval measures of the utility each player derives from the four possible outcomes. They are measured by asking the players to consider lotteries over the different outcomes. For example, Jane's 2/3 payoff in the bottom left quadrant represents her indifference between that outcome and a lottery with a 2/3 chance of achieving her preferred outcome (top right) and a 1/3 chance of achieving her least favoured outcome (top left).

I won't get into it here, but it turns out that there is no pure strategy equilibrium in this game ("pure strategy" = definitely staying at home, or definitely going). There is, however, a mixed strategy equilibrium ("mixed strategy" = choosing the options with a certain probability).

Consider, if Jane chooses to go to the party with a probability of 1/4 (or 0.25) and to stay at home with a probability of 3/4 (or 0.75), then Bob's expected utilities from his possible choices will be:

• Stay at home:       [(1/4 x 0) + (3/4 x 1/2)] = 3/8

• Go to the party:    [(1/4 x 1) + (3/4 x 1/6)] = 3/8

Since the expected utilities from each option are the same, either response is utility maximising. A similar argument can be made for Jane's expected utilities if Brian chooses to stay at home with a probability of 1/2 and goes to the party with a probability of 1/2.

Consequently, the outcome resulting from the choice of these two mixed strategies is in equilibrium: each is a utility-maximising response to the other.

2. Jane and Bob Negotiate a Party-going Agreement

The analysis to this point has been straightforward game theory. Now we are going to look at the same outcome space through the lens of bargaining theory and try to locate the MRC solution. 

The first thing we need to is draw the outcome space and locate the initial bargaining position. In this case, the IBP will be the outcome that the parties could expect to obtain without an agreement. That will be the pair of outcomes associated with the mixed strategy equilibrium that has just been described, i.e. (1/2, 3/8).

Once we have drawn the outcome space and located the IBP, we can define the range of admissible outcomes that rational players would agree upon. These will occur on the optimal boundary (the line between (0, 1) and (1, 0)) between the points (1/2, 1/2) and (5/8, 3/8). Each party will initially try to claim as much as is possible. This means Brian will demand 1/2 and Jane will demand 5/8. This is illustrated below.

Obviously, the claim point is not an admissible outcome so the parties need to make some concessions. If we draw a straight line connecting the claim point to the IBP, then every point along that line will represent an equal relative concession from the players. This line intersects the optimal boundary at the point (9/16, 7/16). At this point, the relative concession for each player is 1/2 (I leave the math to the reader). This is the MRC-solution, because any outcome which gave more to Jane would force a greater relative concession from Brian and vice versa.

What does this solution mean in practice? Well, according to Gauthier, it means that Jane should be allowed to go to the party, and Brian should be allowed to play a mixed strategy with 7/16 probability of going to the party and 9/16 probability of staying at home.

3. Ernest and Adelaide Make a Deal

A second example works with monetary payoffs instead of utilities and allows us to explore the difference between relative concessions and absolute concessions.

Suppose that Ernest and Adelaide have the opportunity to co-operate in a mutually beneficial way, provided they can agree how to share their potential gains. Adelaide would receive a maximum net benefit of $500 from the joint venture, provided she receives all the gains after covering Ernest's costs. On the other hand, Ernest could only obtain a maximum net benefit of $50, provided he receives all the gains after covering Adelaide's costs. In this case we assume that neither can obtain anything without cooperation and so the IBP is (0, 0). We assume the possible outcomes lie along the curve in the following diagram.

Each party will initially claim as much as is possible for them to claim, i.e. the maximum net benefit. Obviously, this would not be desirable for the other party as they would then receive no gain from the joint enterprise. Concessions will have to be made by both sides.

Again, we follow the familiar method and draw a straight line connecting the claim point to the IBP. This line will intersect the optimal boundary of the outcome space at the point (353, 35). This amounts to an equal relative concession from each part of approximately 0.3. This is illustrated below.

Now the legitimate question arises: what about the absolute magnitudes of the gains and the concessions? Should they change how we think about the solution? After all, wouldn't Ernest be entitled to complain that he is not gaining anywhere near as much as Adelaide?

Here, we run into some interesting possibilities. Although Ernest could indeed make the complaint just outlined, Adelaide could also complain that, in the final agreement, Ernest is conceding far less than she is ($15 compared with $147). So, in some sense, the greater gain is offset by the greater loss. 

Gauthier points out that this kind of absolute comparison is only possible in a few cases (where utilities map directly onto monetary outcomes). And in those cases, if we are tempted by some principle of equal gain, we should always bear in mind the principle of equal loss (as we just did). What makes MRC an acceptable solution to the bargaining problem is its ability to automatically balance relative loss and gain.

That's it for now. In the next post, we will try to relate the MRC-solution to the broader issues in moral and political philosophy that Gauthier is trying to address.

Morals by Agreement (Index)

This post serves as an index to my series on David Gauthier's book Morals by Agreement.

Index
1. Gauthier's Approach

2. Minimax Relative Concession

3. Some Examples of MRC

4. Bargaining and Impartiality

Morals by Agreement (Part 2): Minimax Relative Concession

This post is part of my series on David Gauthier's Morals by Agreement? The first part is available here.

Gauthier's book tries to show the deep connection between rationality and morality. "Rationality", for Gauthier, means what it means to economists, decision theorists and game theorists. But to show the deep connection between it and morality, he is not afraid to reformulate certain key parts of the traditional theory.

In particular, he tries to show (i) how rational cooperation is possible through the concept of constrained maximisation (CM) and (ii) how rational agents would agree to cooperate through the bargaining solution known as minimax relative concession (MRC).

I promised I would address both of those concepts in this series, but I have some difficulty knowing where to begin. My personal feeling is that it makes more sense to discuss CM first, and MRC second. However, Gauthier does things the other way around, and, at the end of the day, who am I to argue. MRC it is.

I reckon it will be best to spread the discussion of MRC out over a few posts. So this post will sketch out the theory in somewhat formal terms; the next post, will look at some worked examples; and another post will consider the moral implications of the theory.

My discussion is based on Chapter 5 of Morals by Agreement. It is quite long, but straightforward and (I hope) informative.


1. The Outcome Space
I'm not going to say anything about the moral and and social importance of cooperation and bargaining since I've discussed it before. Instead, I'm going to cut straight to the chase and describe the formal concepts needed to understand Gauthier's theory.

First, allow me to introduce you to something we are going to call the outcome space. It is depicted in the diagram below. You may recognise it from a previous post where it depicted the payoffs (or utilities) that two players attached to particular outcomes in the Meeting Game. On this occasion, it is meant to stand-in for the outcome space in any bargaining game.

The Outcome Space and the Optimal Boundary

The X-axis represents the payoffs for Player 1 and the Y-axis represents the payoffs for player 2. The area enclosed by the blue line represents the space of possible outcomes. Every point within that space is an outcome on which the players can agree. However, the blue line itself represents the efficient (or optimal -- Gauthier prefers to say "optimal") boundary or frontier of this space. Every point along this line would constitute an optimal agreement.

There are obviously bargaining games involving more than two players (n > 2). The outcome spaces for such games are not easily represented in visual terms. One must rely on the math. Fortunately, we are going to stick with the two-person example.

Defining and representing the outcome space is the first thing to do whenever you are modeling a bargaining problem. Once it has been defined, you can start adding some complications to your model. This is what we are going to do next.


2. The Initial Bargaining Position
The first complication we are going to add to the representation is the inclusion of the initial bargaining position (IBP). This has been referred to in previous posts as either the disagreement point or the Best Alternative to Negotiated Agreement (BATNA).

Actually, I need to qualify that. It's not quite right to say that these two terms are equivalent to the IBP because, in a later chapter, Gauthier defines what counts as an IBP in a slightly different manner. I'm not going to get into that here. If you're interested, read the book, or ask me about it in the comments section.

Anyway, the IBP represents what the parties bring to the negotiation table. It is the outcome they can achieve without reaching any agreement. This might mean different things in different contexts. The important point is that it changes how we think about the bargaining process and the outcome space. No longer are all points in the outcome space possible agreements. Instead, only those points that lead to a gain over the IBP are possible. After all, the players aren't (voluntarily) going to agree to something that makes them worse off.

The diagram below has added an IBP to the outcome space. The dotted lines carve out the segment of the outcome space that is now in play. We can even narrow that down further by saying that only those points on the optimal boundary, between the dotted-lines, are outcomes that rational players would agree upon.

Initial Bargaining Position

3. The Claim Point
Now that we have defined the outcome space and narrowed down the range of possible agreements, we can get into the meat of the bargaining process itself. This begins with each player making a claim to their preferred outcome.

Working with the utility-maximising conception of rationality, we can say that rational bargainers will initially claim the maximum they can. This maximum will be the point along the optimal boundary, between the dotted-lines, that represents the most utility for that player. This is depicted in the diagram below for both X and Y.

Initial Claims

Now there is an obvious problem. If each player demands the maximum for themselves, we will end up with a pair of initial claims that goes over and above the values of the possible outcomes. This pair of claims will be called the claim point and it is illustrated in the following diagram.

The Claim Point

4. Making Concessions
Since the claim point is not a possible outcome in the bargaining game, the players will have to make concessions. Anybody who has haggled with a seller at a market is familiar with this process. You initially offer the seller far less than they are willing to accept; they initially demand far more than you are willing to pay; and you both start making concessions until you arrive at an agreed price.

In terms of our diagram, the concessions will be points below the claim point that one player thinks the other might accept. These will be called concession points.

The question before us is: what kinds of concessions would it be rational for players to make and agree upon? This is where Gauthier's theory of rational bargaining starts to get interesting.

One of the problems with determining the rational concessions is that we have to find some way to make comparisons between the concessions made by the players. This is a difficulty since, as discussed before, the utility scales for each player are somewhat arbitrary and so you don't know whether you are comparing like with like.

The easiest analogy here is to imagine comparing temperatures on two different scales (fahrenheit and centigrade). You can only do this if you have some function that converts a measurement of temperature on one scale into a measurement on the other. This would then allow for like-with-like comparison.

How can this be accomplished in the case of utility scales? Well, we could just assume that the players utilities are being measured in the same units. This is essentially what John Harsanyi does and it might be a reasonable assumption under certain conditions (Harsanyi said it would be when players have been exposed to the same information). An alternative proposal, from Ken Binmore, is to come up with a social index that allows you to say how much the utils on one person's scale are worth in terms of the utils on another person's scale. I looked at this before.

Gauthier's solution is neater. He says that instead of comparing the absolute magnitude of the concessions made by the players, we should compare the relative magnitude of the concessions.

This might require a little explanation. The absolute magnitude is simply the difference between the outcome that would be obtained at the claim point and the outcome that would be obtained at the concession point. The relative magnitude is the ratio of the absolute magnitude (just described) to the difference between the outcome at the claim point and the outcome at the IBP.

Take an abstract example: Suppose Player 1's outcome at the IBP is U*; his outcome at the claim point is U1; and his outcome at the concession point is U2. Then, the relative magnitude of his concession will be:

• [(U1 - U2) / (U1 - U*)]

This will be a number between 0 and 1. It will be 1 if the concession point is, for that player, the same as the IBP; it will be 0 if the concession point is, for that player, the same as the claim point; and it will be a fraction (or decimal) if the concession point is somewhere in between.

The advantage with using ratios like this for comparison is that they are pure numbers, not tied to any particular scale. As a result, you don't need to worry about whether you are comparing like with like.


5. Minimax Relative Concession
Now that we have a method for comparing the concessions of the bargainers, we can proceed to identify the agreement that they would reach. According to Gauthier, the agreement would be one in which the maximum relative concession is minimised. Hence, the theory is called minimax relative concession.

In most cases, the MRC will be equal for each player. In other words, both players end up making the same relative concession (this might be quite different in absolute terms). There is an easy graphical representation of this if we go back to the earlier diagrams.

Consider once more the claim point in these diagrams. This point represents the maximum that each player could get from the deal (at the expense of the other player). Now draw a straight line connecting the claim point to the IBP. Every point along that line will represent an outcome requiring equal relative concessions. If that line intersects the optimal boundary of the outcome space, we have the MRC for this bargaining game.

Optimal Solution with Equal Relative Concessions

That's it; that's the theory of minimax relative concession. In the next post, we will look at some numerical examples.

Morals by Agreement (Part 1): Gauthier's Approach

I have recently been pursuing some questions arising from the intersection between bargaining theory and moral and political philosophy.

In a previous post, I argued that many of the most popular uses of bargaining theory in moral and political philosophy are restricted in form. That is to say: they do not attempt to provide a complete answer to the core metaethical questions concerning the ontological basis of moral truths. Instead, they attempt to show how one set of moral judgments can be derived from another set of rational and moral intuitions and principles. This is certainly a valuable endeavour, but it is obviously incomplete.

In this series, I want to look at David Gauthier's unrestricted use of bargaining theory. This will be based on his book Morals by Agreement. This first post will look at the basic methods and concepts that lie behind Gauthier's thesis.


1. A Metaethical Inquiry?
Gauthier's stated goal is to find out where morality comes from. This is a quintessentially metaethical goal: it does not focus on normative ethical questions such as "Should I eat meat?"; it focuses on questions about the ontological significance of normative statements such as "This state of affairs is good/bad".

To pursue his goal, Gauthier follows the platitude-to-state-of-affairs methodology that I have discussed before. This methodology begins with a set of platitudes about moral truths (derived from the agreed-upon semantics of moral terms) and checks to see whether any actual state of affairs would satisfy those moral platitudes. In Gauthier's case, the relevant moral platitude is the impartial nature of moral oughts: he thinks that the distinctive feature of moral prescriptions is that they are not tethered to the beliefs and desires of any particular agent.

One might think that focusing on this single moral platitude could lead to an impoverished account of morality. I am inclined to agree, but I am willing to take Gauthier's point that impartiality is central to most people's understanding of what successful moral theory would contain.


2. Rationality and Morality
In pursuing his metaethical goal, Gauthier tries to show the "deep" connection between rationality and morality. To be precise, he tries to show how moral prescriptions are simply a proper subset of rational prescriptions. This is what makes Gauthier's account different from the restricted approach I was considering earlier. Those accounts tried to combine sets of moral and rational prescriptions without questioning the relationship between them.

Rationality is as good a place as any to locate the foundations of morality because a theory of rationality has a sort of inbuilt normativity to it. After all, a theory of rationality will specify the kinds of things that motivate and provide an agent with reasons for action. Furthermore, the theory will have some connection with empirical reality as it will attempt to capture the process of practical reason that is embodied in actually existent agents.

Gauthier notes that there are two ways to develop the deep connection between rationality and morality:

• The Kantian Approach: this works from a universal or transcendentalist account of practical reason and shows how this account gives rise to impartial moral prescriptions. My series on Alan Gewirth's Principle of Generic Consistency will be exploring this approach.

• The Social Science Approach: this works from the account of rationality that has been developed in the social sciences (economics, decision theory, and game theory) and shows how impartial moral prescriptions can be derived from it. This is Gauthier's preferred approach.

Although some have argued that Gauthier is seriously confused in his understanding of rationality, there are advantages to his approach over the Kantian one. Chief among them is the fact that the social science account of rationality has been formalised in (often painstaking) mathematical terms, and has some empirical support and tractability (although this is certainly questionable).

3. Rational Choice and Cooperation

Cooperation and coordination are essential to society. Indeed, they are the glue that binds society together. In addressing the possibility of impartial moral prescriptions, Gauthier zones in on a particular type of cooperative problem that befalls society, namely: the Prisoners' Dilemma (PD).

Many will be familiar with this problem from the famous story told about the two prisoners who are in separate holding cells, and who are each told they can avoid jail-time if they rat out the other guy. The problem is that if they both rat each other out they get a lengthy jail sentence, whereas if they both stay silent they get a short jail sentence.

Although this story is memorable, it is important to realise that the PD is a general form of cooperative problem, not something that only applies to the specific set of circumstances in the story. To make this point, the diagram below (click to enlarge) describes a PD that has arisen in professional cycling. I took this from an article by Michael Shermer that appeared some time back in Scientific American.

The important features of the PD, for Gauthier's purposes, are the following:

• There is some gain to be made by opting for mutual cooperation over mutual defection. In other words, there is a cooperative surplus that the agents can obtain if they work together that they would not be able to obtain if they worked independently.

• There is some gain to be made by opting for individual defection over mutual cooperation. In other words, one agent can obtain even more if he defects while the other agent cooperates. Furthermore, the agent who is on the receiving end of this unilateral defection receives even less than they would have received through mutual defection.

One may wonder: why does Gauthier focus on the PD? There are, after all, other types of cooperative problems that do not have these features and that can play an equally important role in social life.

As it happens, I think there is a good reason for Gauthier's focus on the PD. Because it has the two features just described, the PD is the ultimate testing ground for a rationalistic account of impartial moral prescriptions. Why? Because the standard analysis of the PD is that rational players, who seek to maximise their utility, should choose mutual defection over mutual cooperation. This is because mutual defection ensures the maximum payoff, no matter what anybody else does.

4. Gauthier's Solutions

So in order to succeed, Gauthier must show how the standard decision-theoretic analysis of the PD is wrong and how mutual cooperation is, in fact, the rational strategy. He must then go on to show that the actual distribution of the cooperative surplus is impartial in form.

Gauthier tries to do this by presenting two key revisions of rational choice theory and bargaining theory. They are:

• (a) Constrained maximisation: This is what allows rational actors to opt for cooperation over defection, even in PDs.

• (b) Minimax Relative Concession: This is Gauthier's contribution to bargaining theory. It describes the type of distribution that fully rational actors could be expected to agree upon.

I'll cover both of these concepts in later entries.