Posts Tagged ‘expected utility’

Strong Independence in Decision Theory

Thursday, 21 July 2016

In the course of some remarks on Subjective Probability by Richard C. Jeffrey, and later in defending a claim by Gary Stanley Becker, I have previously given some explanation of the model of expected-utility maximization and of axiomata of independence.

Models of expected-utility maximization are so intuïtively appealing to some people that they take one of these models to be peculiarly rational, and deviations from any such model thus to be irrational. I note that the author of a popular 'blog seems to have done just that, yester-day.[0]

My own work shows that quantities cannot be fitted to preferences, which pulls the rug from under expected-utility maximization, but there are other problems as well. The paradox that the 'blogger explores represents a violation of the strong independence axiom. What I want to do here is first to explain again expected-utility maximization, and then to show that the strong independence axiom violates rationality.

A mathematical expectation is what people often mean when they say average — a probability-weighted sum of measures of possible outcomes. For example, when a meteorologist gives an expected rainfall or an expected temperature for to-morrow, she isn't actually telling you to anticipate exactly that rainfall or exactly that temperature; she's telling you that, given observed conditions to-day, the probability distribution for to-morrow has a particular mean quantity of rain or a particular mean temperature.

The actual mathematics of expectation is easiest to explain in simple cases of gambling (which is just whence the modern, main-stream theories of probability itself arose). For example, let's say that we have a fair coin (with a 50% chance of heads and a 50% chance of tails); and that if it comes-up heads then you get $100, while if it comes-up tails then you get $1. The expected pay-out is .5 × $100 + .5 × $1 = $50.50 Now, let's say that another coin has a 25% chance of coming-up heads and a 75% chance of coming-up tails, and you'd get $150 for heads and $10 for tails. Its expected pay-out is .25 × $150 + .75 × $10 = $45 More complicated cases arise when there are more than two possible outcomes, but the basic formula is just prob(x1m(x1) + prob(x2m(x2) + … + prob(xnm(xn) where xi is the i-th possible outcome, prob(xi) is the probability of that i-th possible outcome, and m(xi) is some measure (mass, temperature, dollar-value, or whatever) of that outcome. In our coin-flipping examples, each expectation is of form prob(headspayout(heads) + prob(tailspayout(tails)

One of the numerical examples of coin-flips offered both a higher maximum pay-out ($150 v $100) and a higher minimum pay-out ($10 v $1) yet a lower expected pay-out ($45 v $50.50). Most people will look at this, and decide that the expected pay-out should be the determining factor, though it's harder than many people reälize to make the case.

With monetary pay-outs, there is a temptation to use the monetary unit as the measure in computing the expectation by which we choose. But the actual usefulness of money isn't constant. We have various priorities; and, when possible, we take care of the things of greatest priority before we take care of things of lower priority. So, typically, if we get more money, it goes to things of lower priority than did the money that we already had. The next dollar isn't usually as valuable to us as any one of the dollars that we already had. Thus, a pay-out of $1 million shouldn't be a thousand times as valuable as a pay-out of $1000, especially if we keep in-mind a context in which a pay-out will be on top of whatever we already have in life. So, if we're making our decisions based upon some sort of mathematical expectation then, instead of computing an expected monetary value, we really want an expected usefulness value, prob(x1u(x1) + prob(x2u(x2) + … + prob(xnu(xn) where u() is a function giving a measure of usefulness. This u is the main-stream notion of utility, though sadly it should be noted that most main-stream economists have quite lost sight of the point that utility as they imagine it is just a special case of usefulness.

A model of expected-utility maximization is one that takes each possible action aj, associates it with a set of probabilities {prob(x1|aj),prob(x2|aj),…,prob(xn|aj)} (the probabilities now explicitly noted as conditioned upon the choice of action) and asserts that we should chose an action ak which gives us an expected utility prob(x1|aku(x1) + prob(x2|aku(x2) + … + prob(xn|aku(xn) as high or higher than that of any other action.

If there is a non-monetary measure of usefulness in the case of monetary pay-outs, then there is no evident reason that there should not be such a measure in the case of non-monetary pay-outs. (And, likewise, if there is no such measure in the case of non-monetary pay-outs, there is no reason to suppose one in the case of monetary pay-outs, where we have seen that the monetary pay-out isn't really a proper measure.) The main-stream of economic theory runs with that; its model of decision-making is expected-utility maximization.

The model does not require that people have a conscious measure of usefulness, and certainly does not require that they have a conscious process for arriving at such a measure; it can be taken as a model of the gut. And employment of the model doesn't mean that the economist believes that it is literally true; economists across many schools-of-thought regard idealizations of various sorts as approximations sufficient for their purposes. It is only lesser economists who do so incautiously and without regard to problems of scale.

But, while expected-utility maximization may certainly be regarded as an idealization, it should not be mistaken for an idealization of peculiar rationality nor even for an idealization of rationality of just one variety amongst many. Expected-utility maximization is not rational even if we grant — as I would not — that there is some quantification that can be fitted to our priorities.

Expected-utility maximization entails a proposition that the relevant expectation is of potential outcomes which are taken themselves to be no better or worse for being more or less probable. That is to say that what would be the reälized value of an outcome is the measure of the outcome to be used in the computation of the expectation; the expectation is simply lineär in the probabilities. This feature of the model follows from what is known as the strong independence axiom (underscore mine) because Paul Anthony Samuelson, having noticed it, conceptualized it as an axiom. It and propositions suggested to serve in its stead as an axiom (thus rendering it a theorem) have been challenged in various ways. I will not here survey the challenges.

However, the first problem that I saw with expected-utility maximization was with that lineärity, in-so-far as it implies that people do not benefit from the experience of selecting amongst discernible non-trivial lotteries as such.[1]

Good comes from engaging in some gambles as such, exactly because gambling more generally is unavoidable. We need practice to gamble properly, and practice to stay in cognitive shape for gambling. Even if we get that practice without seeking it, in the course of engaging in our everyday gambles, there is still value to that practice as such. A gamble may become more valuable as a result of the probability of the best outcome being made less probable, and less valuable as a result of the best outcome becoming more certain. The value of lotteries is not lineär in their probabilities!

It might be objected that this value is only associated with our cognitive limitations, which limitations it might be argued represented a sort of irrationality. But we only compound the irrationality if we avoid remedial activity because under other circumstance it would not have done us good. Nor do I see that we should any more accept that a person who needs cognitive exercise to stay in cognitive shape is thus out of cognitive shape than we would say that someone who needs physical exercise to stay in physical shape is thus out of physical shape.

[0 (2016:07/22)] Very quickly, in a brief exchange, he saw the error, and he's corrected his entry; so I've removed the link and identification here.

[1] When I speak or write of lotteries or of gambling, I'm not confining myself to those cases for which lay-people normally use those terms, but applying to situations in which one is confronted by a choice of actions, and various outcomes (albeït some perhaps quite impossible) may be imagined; things to which the term lottery or gamble are more usually applied are simply special cases of this general idea. A trivial lottery is one that most people would especially not think to be a lottery or gamble at all, because the only probabilities are either 0 or 1; a non-trivial lottery involves outcomes with probabilities in between those two. Of course, in real life there are few if any perfectly trivial lotteries, but a lot of things are close enough that people imagine them as having no risk or uncertainty; that's why I refer to discernible non-trivial lotteries, which people see as involving risk or uncertainty.

Crime and Punishment

Thursday, 31 December 2015

My attention was drawn this morning to What Was Gary Becker's Biggest Mistake? by Alex Tabarrok, an article published at Marginal Revolution back in mid-September.

Anyone who's read my paper on indecision should understand that I reject the proposition that a quantification may be fit to the structure of preferences. I'm currently doing work that explores the idea (previously investigated by Keynes and by Koopman) of plausibility orderings to which quantifications cannot be fit. I'm not a supporter of the theory that human behavior is well-modelled as subjective expected-utility maximization, which is a guiding theory of mainstream economics. None-the-less, I am appalled by the ham-handed attacks on this theory by people who don't understand this very simple model. Tabarrok is amongst these attackers.

Let me try to explain the model. Each choice that a person might make is not really of an outcome; it is of an action, with multiple possible outcomes. We want these outcomes understood as states of the world, because the value of things is determined by their contexts. Perhaps more than one action might share possible outcomes, but typically the probability of a given outcome varies based upon which action we choose. So far, this should be quite uncontroversial. (Comment if you want to controvert.) A model of expected-utility maximization assumes that we can quantify the probability, and that there is a utility function u() that takes outcomes as its argument, and returns a quantified valuation (under the preferences of the person modelled) of that outcome. Subjective expected-utility maximization takes the probabilities in question to be judgments by the person modelled, rather than something purely objective. The expected utility of a given action a is the probability-weighted sum of the utility values of its possible outcomes; that is p1(au(o1) + p2(au(o2) + … + pn(au(on) where there are n possible outcomes (across all actions), oi is the i-th possible outcome (from any action) and pi(a) is the probability of that outcome given action a.[1] (When oj is impossible under a, pj(a) = 0. Were there really some action whose outcome was fully determinate, then all of the probabilites for other outcomes would be 0.) For some alternative action b the expected utility would be p1(bu(o1) + p2(bu(o2) + … + pn(bu(on) and so forth. Expected-utility maximization is choosing that action with the highest expected utility.

Becker applied this model to dealing with crime. Becker argued that punishments could be escalated to reduce crime, until potential criminals implicitly regarded the expected utility of criminal action to be inferior to that of non-criminal action. If this is true, then when two otherwise similar crimes have different perceived rates of apprehension and conviction, the commission rate of the crime with the lower rate of apprehension and conviction can be lowered to that of the other crime by making its punishment worse. In other words, graver punishments can be substituted for higher perceived rates of apprehension and conviction, and for things that affect (or effect) the way in which people value successful commission of crime.

The simplest model of a utility function is one in which utility itself increases linearly with a quantitative description of the outcome. So, for example, a person with $2 million dollars might be said to experience twice the utility of a person with $1 million dollars. Possession of such a utility function is known as risk-neutrality. For purposes of exposition, Becker explains his theory with reference to risk-neutral people. That doesn't mean that he believed that people truly are risk neutral. Tabarrok quotes a passage in which Becker explains himself by explicit reference to risk-neutrality, but Tabarrok misses the significance — because Tabarrok does not really understand the model, and confuses risk-neutrality with rationality — and proceeds as if Becker's claim hangs on a proposition that people are risk-neutral. It doesn't.

Becker's real thought doesn't even depend upon all those mathematical assumptions that allow the application of arithmetic to the issue. The real thought is simply that, for any contemplated rates of crime, we can escalate punishments to some point at which, even with very low rates of apprehension and conviction, commission will be driven below the contemplated rate. The model of people as maximizers of expected utility is here essentially a heuristic, to help us understand the active absurdity of the once fashionable claim that potential criminals are indifferent to incentives.

However, as a community shifts to relying upon punishment from relying upon other things (better policing, aid to children in developing enlightened self-interest, efforts at rehabilitation of criminals), the punishments must become increasingly … awful. And that is the moral reason that we are damned if we simply proceed as Becker said that we hypothetically could. A society of monsters licenses itself to do horrific things to people by lowering its commitment to other means of reducing crime.

[1] Another way of writing pi(a) would be prob(oi|a). We could write ui for u(oi) to and express the expected utility as p1(au1 + p2(au2 + … + pn(aun but it's important here to be aware of the utility function as such.