Thirteen Ways to Split a Cake*


Axiomatic bargaining is presented in MWG in the context of welfare economics (Ch. 22), the aim being the formulation of reasonable criteria for dividing gains resulting from cooperative endeavors (the “joint surplus”). It is further presented as an application of cooperative game theory, in which an arbitrator distributes the joint surplus in a manner that reflects “fairly” the bargaining strength of the different agents (although we could conceive of a situation where the parties are bargaining without an external party). If bargaining fails, the outcome is the parties’ own fallback positions (the threat point, or status quo).

What is a bargaining solution (according to MWG)? It is a unique payoff representing a particular outcome (technically, an element of the set U of all possible allocations of utility resulting from all possible distributional outcomes). The implicit assumption (in itself debatable, as we will discuss) is that the bargaining solution “depends on the set of feasible alternatives only through the resulting utility values” (MWG, p838). We note here further that the utility payoffs resulting from the distribution of the surplus are defined as utility gains, i.e. utility increases over and above the utility of the status quo (the threat point).

The discussion of axiomatic bargaining in MWG (pp838-846) consists of a survey of various axioms (desirable properties that the bargaining solution must possess) and solutions that satisfy these criteria. The Nash bargaining solution is then presented as the only bargaining solution that simultaneously satisfies all such desirable properties.

Unfortunately, the MWG discussion of axiomatic bargaining in general, and the Nash solution in particular, is not as enlightening as it could be. It is a good example of the somewhat limited perspective adopted in the teaching of microeconomics at the graduate level, with a focus on developing a technical apparatus, but little in the way of critical reflection on the considerations which determine the validity of the formulation (or set-up) of a problem (in this instance, a bargaining problem), as well as the meaning and significance of solutions (in this instance, the Nash solution).

Let us develop these points further and reflect upon what may or may not be missing from the MWG presentation. Section 22.E begins with an overview of the various bargaining axioms, which we outline below at the risk of being dry:

  • Independence of utility origin (IUO) [Def. 22.E.2]: A change in the origin of measurement of utility will not affect the bargaining solution.
  • Independence of utility units (IUU) [Def. 22. E.3]: A change in the measurements of utility units will not affect the bargaining solution.
  • Pareto property [Def. 22.E.4]: No further Pareto improvements can be made.
  • Symmetry property [Def. 22.E.5]: If agents are identical (the set U is symmetric), the gains from cooperation are split equally (the payoffs are equal).
  • Individual rationality [Def. 22.E.6]: The bargaining solution does not give any agent less than that agent would receive in the threat point.
  • Independence of irrelevant alternatives (IA) [Def. 22.E.7]: If the set U of possible utility allocations is shrunk (such that some trading possibilities are eliminated, but none are added), and retains the feasibility of the bargaining solution as a trading possibility in the smaller utility set, this bargaining solution remains unchanged. The two bargaining games defined over the two utility sets have the same bargaining solution.

Together, the IUO and IUU properties demand that if two versions of the same bargaining problem differ only in the units and origin of utility measurement, the bargaining solution should stay the same. As a result, there must be invariance of the bargaining solution to linear transformations of the utility functions of each party (invariance here means that the bargained solutions will differ in terms of utility payoffs, but those payoffs will be related via the same utility transformation, and will require the same distribution of underlying resources). This might not be thought surprising since linear transformations of their utility functions represent the same underlying ordinal preferences for each. On the other hand, not every transformation that preserves those preferences leaves the solution unchanged (specifically non-linear ones, on which more below).

To the extent changes of units or origins need not be common (i.e. the demand is for invariance to independent changes in units or origins), this framework is usually also said to preclude interpersonal comparisons of utilities. MWG highlights this point in an important comment found in Def. 22.E.3. The IUO and IUU properties tell us that “although the bargaining solution uses cardinal information on preferences, it does not in any way involve interpersonal comparisons of utilities.” (MWG, p.840). Is it in fact so straightforward? This matters because if even bargaining problems can be addressed without interpersonal comparisons then this tells us that an economics without them (i.e. the version of economics presented in MWG) can get quite far.

In what sense does the bargaining solution use cardinal information on preferences and what does that mean exactly? In what sense can it be said that the bargaining solution does not involve interpersonal utility comparisons? More fundamentally, what are the normative merits of allowing or disallowing interpersonal comparisons of utilities? These points are neither trivial nor obvious.

Cardinal vs. Ordinal Information on Preferences

One might think that the bargaining solution should be required to be preserved for all utility functions representing the same underlying ordinal preferences of the parties (i.e. including utility representations generated from arbitrary, non-linear, monotonic transformations) but in that case, it would be preferences over alternatives and not utility scores which would matter. If a cake is to be divided between two persons and both persons prefer more cake to less then the two individuals would have identical ordinal preferences and they could have different bargaining outcomes only as a result of faring differently in the event of no agreement (the threat point). The Nash bargaining solution goes beyond this to depend on the utility representation generating the utility possibility set. The Nash bargaining solution can lead to different bargaining solutions for different utility functions adopted to represent unchanged underlying preferences (even though the solution does not change when the utility functions are changed in certain specified ways). This might at first be thought highly troubling.

The recognition that the bargaining solution uses cardinal information on preferences helps us to understand better this issue. What is the exact meaning of this cardinality requirement? In their discussion of the Nash bargaining solution, Gaertner and Klemisch (1992) write that “It is postulated that each individual expresses his (her) preferences over alternatives in X in terms of numerical utilities, more precisely, in terms of a von Neumann-Morgenstern utility function which is determined up to positive affine transformations of the utility scale. The von Neumann-Morgenstern type of utility function is cardinal so that the consideration of utility differences becomes meaningful.” (Gaertner and Klemisch, 1992: 62). Cardinality here involves, first, the notion that the numerical values generated by the utility representation are consequential to the bargaining outcome, and second, the idea that if these numerical values are changed simply by varying the ‘zero’ or the unit of measurement, then such variations are not consequential to the bargaining outcome!

If the consumption of a commodity (say milk) secures a utility level of 4 utils, and the consumption of some other commodity (say coffee) secures a utility level of 8 utils, this does not mean that drinking coffee procures twice as much pleasure as drinking milk (that would require a ratio scale, in which the zero is fixed, which is not being done here). Rather, cardinality in the current interpretation means that the relative distance between utility measurements can be meaningfully interpreted. If drinking tea scores 10 utils, the increase in happiness procured by tea over coffee (2 utils) is only half of the increase in happiness produced by coffee over milk (4 utils). Changes in the zero or the units of measurement (i.e. affine transformations) will not affect the validity of such a statement. In effect, cardinality as interpreted here means that intrapersonal comparisons of utility differences are possible: we can interpret “how much happier” one outcome makes a given person in relation to a status quo than does another outcome. Moreover, the Nash bargaining solution relies only on intrapersonal information of this type. Information concerning whether one person’s utility level is higher or lower than another’s (or by how much) is irrelevant to determining that outcome, as if such variations result from changes in the zero or the units of the scale then they do not affect the relevant intrapersonal information.

We can now understand better the sense in which the Nash bargaining solution is said not to depend on interpersonal comparison. It is not influenced by the relative levels of utility experienced by different persons but only by the relative differences in utility experienced by the same person in alternate comparisons of outcomes.

Does the Nash Solution Really Avoid Engaging in Interpersonal Utility Comparisons?

The Nash solution considers that the only information relevant to the bargaining allocation consists in ratios of utility gains (more specifically marginal utility as a percentage of total utility gain). Let us state the obvious. If we consider a situation of two parties with different utility functions U1 and U2, and let x represent the share of the joint surplus going to the first party, the Nash solution maximizes the product U1(x)U2(1-x), with the FOC U1’(x)/U1(x)=U2’(1-x)/U2(1-x), where the utility levels are measured as gains in utility relative to the threat point. The cardinal utility requirement means that an affine transformation of either party’s utility function cannot change the measurement of marginal utility as a percentage of total gain for that party. This implies that the value of x which satisfies the condition will not change.

This example makes clear that the FOC does not involve any direct marginal utility tradeoff between the parties. Yet upon further consideration, it is not obvious that the solution completely obviates the need for some kind of interpersonal utility comparison. The FOC does compare across persons the ratio of marginal utility to total gain. A non-linear utility transformation that preserves the preference ordering but changes this ratio will indeed change the solution. The Nash solution is therefore hardly indifferent between all representations of utility for the parties. One way to understand what aspects of the representation of utilities it does and does not register as salient, and how this is related to interpersonal comparison, is to recognize that in effect it divides the set of all possible utility functions into equivalence classes. Each equivalence class consists of a set of utility functions that can be generated from one another through affine transformations. Substituting a utility function of one of the agents by another utility function from the equivalence class to which it belongs has no effect on the solution but substituting it by a member of another equivalence class does. If we think of each of these equivalence classes as represented by a person (Tracy, Dick or Ahmad) then it certainly does matter whether Tracy or Dick are bargaining or Tracy and Ahmad are doing so. The specific features which distinguish Dick from Ahmad (realized in the form of different relative differences in utility) prove salient in determining how much Tracy should get. Why is this not an instance of interpersonal comparison? Interpersonal comparison here does not take the form of saying that Jack has greater utility than Jill, or that Jack will gain more utility than Jill if given an extra piece of cake. Rather, it takes the form of saying that Jack’s gain from being given two slices of cake relative to his gain from being given one is greater (or lesser) than Jill’s. Although the focus of the comparison is different, comparison it surely is.

What Are The Normative Merits of the Nash Solution?

The FOC shows that if the parties have identical utility functions, they will split the gains resulting from cooperative behavior evenly. If they have different utility functions, the bargainer with the more concave utility function (often interpreted in terms of greater risk aversion) will, other things being equal, receive less. This result is not immediately obvious, but can be seen through inspection of the FOC. A simple thought experiment involving a risk-neutral party (with no diminishing marginal utility) facing a risk-averse one (with diminishing marginal utility) will suffice to see the point. In this sense, it could be said that the Nash solution awards implicit “relative bargaining power” to the agent with the lowest risk aversion. This is, for example, the interpretation in Hargreaves and Varoufakis (1995: 12). However, other than the recognition given to the concavity of the utility function, bargaining power is assumed to be equal for all parties.[1] Why should the Nash solution reward the less risk-averse bargainer more generously? It is not straightforward to see why attitudes toward risk should provide an explanation (and certainly not a justification) for the Nash allocation. It is not obvious as to why this carries normative merits as a fair solution, and why it is more desirable than more explicit interpersonal utility comparisons based on other criteria (although to be fair while this is a consequence of the axioms assumed by Nash, the normative merits of those axioms should in the first instance be considered on independent grounds).

I have made a case that the Nash bargaining solution does in fact involve interpersonal comparison (contrary to the standard presentation). Indeed, insofar as any bargaining framework allocates resources according to a first order condition that equates across persons expressions which depend in some way upon the specific utility representations chosen, it could be said subtly to involve such comparison. Any such framework will generate equivalence classes such that substituting one utility function representing the same underlying preferences for another will not change the allocation generated by the bargaining rule as long as the substitute is chosen from the same equivalence class but will matter otherwise.

It is difficult to imagine that any sensible judgment regarding distributive justice can be made without engaging in some kind of comparison amongst persons (see a comment to that effect in Gaertner and Klemisch, 1992: 65). Luce and Raiffa (1957) may be a bit more cautious, but do acknowledge that although interpersonal comparisons have not always been given a ‘rigorous’ meaning, they may be warranted in some situations: “we feel that an abstraction completely omitting such considerations is perhaps departing too far from reality in certain contexts.” (Luce and Raiffa, 1957: 131, italics mine). Attempting to do without interpersonal utility comparisons leads to reintroducing them (as we have seen) through a back door, without taking more directly relevant normative considerations, such as the relative ‘needs’ of the two parties, into account. In the Nash solution utility payoffs are strictly defined as utility increases over and above the status quo, with no consideration of the fairness of this status quo. To borrow Sen’s example (Sen, 1970: 121), in a labor market with unemployment, workers may accept very poor terms of employment and subhuman wages because starvation might be their only alternative. This does not mean that the Nash solution would necessarily lead to a socially desirable outcome, if all that is being considered is the equalization of marginal utility as a percentage increase of total utility gain.

For that matter, more procedural considerations which are independent of utility information can play a role, and these can go in an entirely different direction. Consider the following example from Hargreaves and Varoufakis (1995: 20). Two parties A and B have identical utility functions and are vacationing in a Greek island. They are walking together when A finds $100 on the ground. A passport is needed to exchange the $100 into Euros. A does not have a passport with her, but B does. Though the Nash solution predicts that the parties should split the $100 bill equally, it is quite possible that they would both be of the opinion that A deserves more than 50% of the proceeds, since A actually found the $100 bill. Such considerations of “fairness” external to the utility informational landscape can easily ben seen as relevant to determining a desirable outcome. To take an even more accustomed example, inheritance customs and laws often have a predisposition to divide an estate equally. The claim that one inheritor has greater needs (or can use the estate to better effect) may play at best a role in allocation and very often plays none, as a result of an equal claim to resources arising from procedural considerations.

That economic theory can provide some insight into empirical situations in which bargaining occurs seems plausible. It has been used to considerable benefit in understanding intra-household dynamics, labor relations and other areas, where understanding the relevance of the threat point in determining bargained outcomes or otherwise fruitfully employing some concepts of bargaining theory, has proved informative (see e.g. Agarwal, 1997). It seems less evident that it can provide much guidance in determining how resources should be allocated. Unavoidably, one must ask what is lost as well as gained when a problem is approached with undue narrowness.

References

Agarwal, B. (1997), “Bargaining” and Gender Relations: Within and Beyond the Household, Feminist Economics, Vol. 3(1): 1-51.

Bowles, S. (2004), Microeconomics: Behavior, Institutions and Evolution, [Ch. 5].

Gaertner, W. and Klemisch-Ahlert, M. (1992). Social Choice and Bargaining Perspectives on Distributive Justice. Springer-Verlag.

Hargreaves, S.P. and Varoufakis, Y. (1995). Game Theory: A Critical Introduction. Routledge, 1995. [Ch. 4].

Luce, R.D. and Raiffa, H. (1957) Games and Decisions: Introduction and Critical Survey. Dover (Reprint). [Ch. 13].

Sen, A. (1970). Collective Choice and Social Welfare. Holden-Day, San Francisco.

Svejnar, J. (1986). Bargaining Power, Fear of Disagreement, and Wage Settlements: Theory and Evidence from U.S. Industry. Econometrica, Vol. 54(5): 1055-1078.

[1] We note that there are some attempts in the literature (which require dropping the symmetry axiom) to incorporate exogenous differences in bargaining powers by developing a more generalized Nash product where each party has attached a factor which reflects the strength of their bargaining power (see for example Svejnar, 1986).

* Thirteen Ways of Looking at a Blackbird, Wallace Stevens, http://www.writing.upenn.edu/~afilreis/88/stevens-13ways.html

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