Wednesday, October 13, 2010

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.

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