Humane Studies Review

Volume 7, Number 1 Winter 1991/92

Indivisibility, Probability and Game Theory

by Anthony de Jasay


A rejoinder to Robert Sugden' s Review
The central project of my book is to prove that, contrary to conventional theory, the public goods or "collective action" problem is not a dilemma. Thus, if providing a good publicly is superior to falling back on market substitutes for it, it would not be unreasonable to expect the good to be voluntarily provided (if the state were not there to provide it anyway). In his review, Dr. Sugden judges this project to be a failure, partly because he disagrees with my assumptions, partly because he finds my argument defective in the techniques of economics and game theory. I have some sympathy with his general verdict, for I do not think that mine is the definitive statement; most radical first attempts fail in that sense. It is not hard to find serious weaknesses in my admittedly (and regrettably) complex construction. The ones Dr. Sugden found, however, are not among them.

With regard to our assumptions, we have what seems to me a legitimate difference of view that cannot be resolved by recourse to logic. Some goods are physically indivisible, or "lumpy." A bridge that is too short to reach the other shore, a fire brigade lacking a fire engine, a workers' cafeteria where last- comers get no main course, a school without a teacher just won' t do, and I imagine that Dr. Sugden, too, would concede that such goods yield a drastically reduced marginal utility short of a critical threshold quantity. Others, however, are physically divisible, their supply is continuously variable both ways and so is their utility. Yet if their relevant "public" is given, and no member of this public must be excluded if he wishes to use the good, even such goods must be supplied in a minimum size, flow or standard; to be truly public, the television or radio transmitter must afford reception to every home, the national health service must keep waiting lists short enough not to drive people to private health insurance, and so forth. My contention is that indivisibility in this latter sense is implicit in publicness. Dr. Sugden, invoking parks, denies this.

I would be content to agree to differ, not so much about the physical world, but about the relevant customs and social ethics. But there is a rub: the reviewer is out for more. For he declares that my argument misses the point, the point being that public goods are divisible, each marginal contribution to their provision does yield a marginal benefit according to some continuous function relating the two, and since each marginal benefit is dissipated among many beneficiaries, no non-altruist in his senses can possibly choose to contribute unless forced; q.e.d.

Now this, of course, is merely asserting once more, in different words, that public goods present a prisoners' dilemma, which is the very assertion I am contesting and in whose stead I propose to put a competing and, I believe, richer theory. Whatever my argument does, and whatever other points it misses, it cannot be said to miss this one. In claiming that it does, Dr. Sugden is stubbornly trying to steer public goods theory back into its old prisoners' dilemma channel, where the free rider conduct of homo oeconomicus is pre-ordained, and voluntary provision can only be explained by ascribing to people psychological properties transcending economic man. He is clearly entitled to do this, as I am entitled to my less heroic psychological assumptions. But since an argument that misses the point is a defective one, we look to Dr. Sugden to show the defect. Changing assumptions, and doing it above board, is not one.

The review is no more successful in proving defects and "crucial mistakes" when it invokes probability and game theory. Its verdict of failure, if just, is just for the wrong reasons, and is very much the kettle calling the pot black. Dr. Sugden believes that he is vindicated by an "inescapable law of probability." But there is nothing inescapable about the law on which he rests his case. The law, as he phrases it, applies in the special case where the probability distribution is not skewed in favor of any subset of the set of alternatives. It is the classroom world of the unbiased die, the perfect roulette wheel and the randomizer. In general theory, as in the real world, there is room for any shape of (statistical or subjective) distribution. If there are reasons for it to be skewed in a particular way, no "inescapable law" will prevent it.

Consider, however, the example of the random distribution chosen by Dr. Sugden, and the question of symmetry. There is symmetry between pledging and not pledging in the following sense: for all alternatives where some number of other people from 0 to 49 make the pledge, making it is weakly superior to not making it. At worst, the pledge is not called up; I get no benefit and pay no cost. At best, I cause the critical number of pledges to be reached, get the benefit and pay its cost. For all alternatives where between 50 and 100 other people make the pledge, not making it is strongly superior, for I get the benefit and do not pay the cost. It is obviously possible, however, for a rational person to hold that the probability distribution is markedly skewed towards fewer than 50 people making the pledge, in which case his superior strategy is not to bank on the chance of free riding, but to play it safe and pledge.

Back-of-the-envelope calculations can be perfectly suited to unmask a plainly defective argument. But they must not start off with a gross error of logic in formulating the problem. If, in a cost-benefit calculation, an uncertain benefit is to be deflated by its probability (in the unskewed example, 0.16), so must the uncertain cost be, since it may or may not be incurred. Inexplicably, however, Dr. Sugden' s numbers show it as certain (probability of 1) that the pledge will be called up. Hence, it would take an uncertain benefit of 13.5 (for 0.16 times 13.5 makes 2.16) to offset the certain cost of 2. The whole "point," however, is that the pledge is not certain to be called up, and, to use his parlance, this point has been missed. If this is the result of "mastering the techniques" of probability theory, the effort hardly seems worthwhile.

In any event, even without its misunderstood laws and mistaken calculations, Dr. Sugden' s analysis does not encompass the really interesting question, which is the relation of the expected utilities of the dovish (to pledge) and the hawkish (not to pledge) strategy. The former aims at the inside, moderate pair of possible payoffs, the latter at the more extreme pair which "straddles" them. The former is the "sucker," the latter the "free rider" strategy. Their expected utilities vary as the probability distribution of alternative behaviors by others varies. As argued in the book, the free-rider utility function intersects the sucker utility function from below. This is the crux of the matter. The review ignores and sidesteps it. Dr. Sugden merely notes the Cretan Liar-type paradox involved, as he does the Sorites-problem involved in indivisibility, and opines that I get bogged down in both. This is as it may be, but it is not the main part of the story.

Now, a word on game theory. "Straddle" indeed has the same payoff structure as "Chicken" or "Hawk-and-Dove." An appendix in the book is mostly devoted to discussing their differences. What we need to note here is that in Straddle, the interactions of the participants are parametric, not strategic. Consequently, to borrow Dr. Sugden' s reproving language again, if he had "mastered the technicalities," he would not be asking me for a strategy equilibrium in the Nash sense. In a parametric interaction, I have the best strategy given what I expect others to do. I do not expect them to choose the strategy that would seem best for them if they expected me to choose my best one. Whether parametric or strategic is the right way to go is perhaps arguable, but -- as economists and game theorists could tell Dr. Sugden -- they should not be confused.

Lastly, dreary technicalities aside, I cannot disprove that in the absence of a state, voluntary provision of public goods might fail unless the benefit were much greater than the cost. But unless it were much greater, compulsory provision by the state would be hard to justify, too, unless coercion were regarded as harmless and costless in terms of our conceptions of morals, justice, and welfare. Against public goods that fail to meet the test of voluntariness, the school of Murray Rothbard and David Friedman, and lesser pro-marketeers, may well have a nearly cast-iron case.

Anthony de Jasay is also the author of The State (New York: Basil Blackwell, 1985) and Choice, Contract, Consent: A Restatement of Liberalism (London: Institute of Economic Affairs, 1991).

Copyright 1992 by the Institute for Humane Studies.

Return to Winter 1991/92 Humane Studies Review menu IHS Publications or IHS homepage