Harsanyi’s Impartial Observer Theorem and Income Redistribution
Utilitarian Impartiality and Its Uncertain Distributive Implications
David Friedman has an interesting post on libertarian arguments for income redistribution. Friedman offers a partial defense of utilitarianism where he notes that utilitarianism comes closer than Rawls’s theory of justice of being derived compellingly. In other words, the implications of utilitarianism regarding income distribution are more plausible than those of Rawls’s theory as they build on firmer foundations. Friedman’s main argument for this claim is that a version of utilitarianism, not Rawls’s difference principle, is correctly deduced from the conditions of the original position where people are put behind a veil of ignorance that makes them ignore their social situation as well as their personal characteristics. Friedman refers here to an important theorem demonstrated by John Harsanyi that Friedman discusses in more detail in another post.
John Harsanyi
Harsanyi’s theorem is sometimes called in the literature the “impartial observer theorem.” Friedman is not alone in claiming that Harsanyi, not Rawls, had the conclusions right regarding what people would choose in the original position. Ken Binmore for instance also considers that decision makers behind a veil of ignorance would be utilitarians, at least if their choices can then be enforced by a political authority.[1] As it happens, things are not so straightforward. Rawls and Harsanyi are actually making quite different assumptions regarding the rationality of the individuals in the original position. Also, they are not choosing among the same sets of objects. Rawls’s decision-makers are choosing among a restricted set of justice principles – among which figure the difference and the utilitarian principles. As we shall see below, Harsanyi’s agents are choosing between social states and, it is claimed, their choices are made according to the (average) utilitarian principle.
I don’t want to return to this controversy. Rawls and Harsanyi had a short and mostly unproductive exchange in the 1970s and since then economists (who tend to side with Haransyi) and philosophers (many of them are at least prone to offer a qualified defense of Rawls) have essentially talked past to each other. What interest me here is that Friedman’s call to Harsanyi’s theorem is presumed to support his view that there is no good philosophical argument in favor of a large income redistribution of the kind Rawls’s difference principle would justify. I think actually that the implications of Harsanyi’s theorem for income redistribution are far from clear and not necessarily in accordance with Friedman’s normative views.
Let me start with a brief and broad statement of Harsanyi’s impartial observer theorem.[2] Suppose that we have individuals put behind a “thin veil of ignorance”. Individuals placed in this situation ignore both their social position and their personal identity. The former means that individuals don’t know, inter alia, their income level, their health state, their social capital, and the value of any other variable over which their preferences are defined. The latter means that they also ignore their personal preferences, especially how they trade-off between the different variables. The veil of ignorance is however said to be “thin” (while Rawls’s is said to be “thick”) because individuals nonetheless know the distribution of social positions in any given social state. In other words, depending on the social state that is actually chosen, they know what are the probabilities that they end up with a certain level of income, a certain health state, and so on. What people are asked to choose behind this thin veil of ignorance is a social state among the (possibly infinite) range of states that are feasible, based on the limited information at their disposal.
Now comes a bunch of core assumptions that are crucial in the derivation of the utilitarian result. First, individuals are assumed to be Bayesian rational, meaning that their preferences over social states satisfy a certain number of axioms such that they can be represented by the expectation of a (class of) utility functions. In simpler terms, individuals are able to order social states according to the expected utility that is derived from the social positions that one can occupy in this state. The computation of this expected utility is based on the probabilities I refer to above. Second, because individuals ignore their personal identities and thus their personal preferences, they have to order social states based on what Harsanyi calls their “extended preferences”. Basically, an extended preference is of the kind “I prefer to be individual A in social position/state X rather than individual B in social position/state Y”. Hence, extended preferences guarantee that individuals can make the kind of interpersonal comparisons of utility without which utilitarianism makes no sense. And here comes a key claim, based on what has been proved to be mistaken reasoning:[3] all individuals behind the veil of ignorance have the same extended preferences!
The third and last assumption is that individuals, ignoring who they actually are, assume that they have an equiprobable chance of being anyone in the population. In other words, Harsanyi is assuming Laplace’s indifference principle according to which when you have no specific information about the probability of mutually exclusive events, you should ascribe to them the same probability – an assumption which fits uneasily with Harsanyi’s general commitment to Bayesianism. Based on the characterization of the original position and on those three assumptions, it is relatively easily seen that (i) all individuals behind the veil of ignorance would order social states the same way and thus pick the same one as the best; (ii) the representative individual will rank social states according to the average expected utility of persons in those states, computed by making the sum of each’s expected utility divided by the number of persons. Note that because extended preferences are assumed to be uniform across the population, all individuals make the same interpersonal comparisons of utility, and so the computation of the average expected utility can be based indifferently on extended or personal preferences.
The conclusion is therefore that behind Harsanyi’s veil of ignorance, individuals would choose as if they were (average) utilitarians. I shall not discuss here whether this conclusion is compelling given all the above assumptions, nor whether the theorem is normatively relevant in light of all the assumptions it rests on. Let’s focus on its implications with respect to income distribution. I shall make two remarks.
First, the theorem implies utilitarianism about utility, not about income. That means that the social choice should be insensitive to the distribution of utility, not of income. This is a standard feature of utilitarianism actually. If you assume that money has decreasing marginal utility, then utilitarianism pushes for a partial (or even full if everyone has the same utility function) equalization of income. A redistribution of a unit of money from a wealthy person toward a poorer person (who remains poorer) will often be utility-improving everything else equals. The same applies to Harsanyi’s impartial observer theorem. Income level is one of the variables that constitute utility. Under plausible assumptions about personal preferences, the social state that maximizes average expected utility would be one where income distribution is relatively equal.
Second, even if we grant all the assumptions made in the theorem, it is actually not true that every “impartial observer” will make the same utilitarian choice. As noted in an old article by Prasanta Pattanaik,[4] impartial observers may differ with respect to their risk attitudes. Relatively risk-averse impartial observers will prefer social states where the utility distribution is less dispersed in the population. Because utility is correlated with income, those same impartial observers will also tend to prefer social states where the income distribution is less dispersed. Interestingly, Harsanyi’s other utilitarian theorems may come to the rescue to at least solve the ensuing problem of aggregating different extended preferences over (extended) lotteries of social positions (or states) and persons. But it remains that there is absolutely no guarantee that the resulting social choice will not be one with a high level of income redistribution.
[1] Ken Binmore, Natural Justice (Oxford University Press, 2005).
[2] Harsanyi has another utilitarian theorem, generally called the aggregation theorem. Both have developed in parallel, and they are sometimes confused. They are however two distinct ways to defend utilitarianism starting the assumptions of Bayesian rationality. I’ll have nothing to say here about the aggregation theorem.
[3] John Broome, “A Cause of Preference Is Not an Object of Preference,” Social Choice and Welfare 10, no. 1 (1993): 57–68. Philippe Mongin, “The Impartial Observer Theorem of Social Ethics,” Economics and Philosophy 17, no. 02 (2001): 147–79.
[4] Prasanta K. Pattanaik, “Risk, Impersonality, and the Social Welfare Function,” Journal of Political Economy 76, no. 6 (1968): 1152–69.