Consequentialism and Formalism in Rational and Social Choice Theory

Rational and social choice theory (RCT and SCT respectively) in economics are broadly consequentialists. Consequentialism can be characterized as the view that all choice alternatives should be evaluated in terms of their consequences and that the best alternatives are those which have the best consequences. This is a very general view which allows for many different approaches and frameworks. In SCT, welfarism is for example a particular form of consequentialism largely dominant in economics and utilitarianism is a specific instance of welfarism. In RCT, expected utility theory and revealed preference theory are two accounts of rational decision-making that assume that choices are made on the basis of their consequences.

Consequentialism is also characterized by a variety of principles or axioms that take different and more or less strong forms depending on the specific domain of application. The most important are the following:

Complete ordering (CO): The element of any set A of alternatives can be completely ordered on the basis of a reflexive and transitive binary relation ≥.

Independence (I): The ranking of any pair of alternatives is unaffected by a change in the likelihood of consequences which are identical across the two alternatives.

Normal/sequential form equivalence (NSE): The ordering of alternatives is the same whether the decision problem is represented in normal form (the alternative is directly associated to a consequence or a probability distribution of consequences) or in sequential form (the alternative is a sequence of actions leading to a terminal node associated to a consequence or a probability distribution of consequences).

Sequential separability (SS): For any decision tree T and any subtree Tn starting at node n of T, the ordering of the subset of consequences accessible in Tn is the same in T than in Tn.

Pareto (P): If two alternatives have the same or equivalent consequences across some set of locations (events, persons), then there must be indifference between the two alternatives.

Independence of irrelevant alternatives (IIA): The ordering of any pair of alternatives is independent of the set of available alternatives.

All these axioms are used either in RCT or in SCT, sometimes in both. CO, I, NSE, SS and IIA are almost always imposed on individual choice as criteria of rationality. CO and IIA, together with P, are generally regarded as conditions that Arrowian social welfare functions must satisfy. I is also sometimes considered as a requirement for social welfare functionals, especially in the context of discussions over utilitarianism and prioritarianism.

It should be noted that they are not completely independent: for instance, CO will generally require the satisfaction of IIA or of NSE. Regarding the former for instance, define a choice function C(.) such that, for any set S of alternatives, C(S) = {x|x ≥ y for all y  S}, i.e. the alternatives that can be chosen are those and only those which are not ranked below any other alternative in terms of their consequences. Consider a set of three alternatives x, y, z and suppose that C(x, y) = {x} but C(x, y, z) = {y, z}. This is a violation of IIA since while x y and (not y x) when S = (x, y), we have y x and (not x y) when S = (x, y, z). Now suppose that C(x, z) = {z}. We have a violation of the transitivity of the negation of binary relation ≥ since while we have (not z y) and (not y x), we nevertheless have z x. However, this is not possible if CO is satisfied.

All these axioms have traditionally been given a normative interpretation. By this, I mean that they are seen as normative criteria of individual and collective rationality: a rational agent should or must have completely ordered preferences over the set of all available alternatives, he cannot on pain of inconsistency violate I or NSE, and so on. Similarly, collective rationality entails that any aggregation of the individuals’ evaluations of the available alternatives generates a complete ordering satisfying P and IIA and possibly I. Understood this way, these axioms characterize consequentialism as a normative doctrine setting constraints on rational and social choices. For instance, in the moral realm, consequentialism rules out various forms of egalitarian accounts which violate I and sometimes P. In the domain of individual choice, it will regard criteria such as minimization of maximum regret or maximin as irrational. Consequentialists have to face however several problems. The first and most evident one is that reasonable individuals regularly fail to meet the criteria of rationality imposed by consequentialism. This has been well-documented in economics, starting with axiom I in Allais’ paradox and Ellsberg’s paradox. A second problem is that the axioms of consequentialism sometimes lead to counterintuitive and disturbing moral implications. It has been suggested that criterion of individual rationality should not apply to collective rationality, especially CO and I (but also P and IIA).

These difficulties have led consequentialists to develop defensive strategies to preserve most of the axioms. Most of these strategies refer to what I will call formalism: in a nutshell, they consist as regarding the axioms as structural or formal constraints for representing, rather than assessing, individual and collective choices. In other words, rather than a normative doctrine, consequentialism is instead best viewed as a methodological and theoretical framework to account for the underlying values that ground individual and collective choices. As this may sound quite abstract, I will discuss two examples, one related to individual rational choice the other to social choice, both concerned with axiom I. The first example is simply the well-known Ellsberg’s paradox. Assume you are presented with two consecutive decision-problems, each time between a pair of alternatives. In the first one, we suppose that an urn contains 30 red balls and 60 other balls which can be either black or yellow. You are presented with two alternatives: alternative A gives you 100$ in case a red ball is drawn and alternative B gives you 100$ in case a black ball is drawn. In the second decision-problem, the content of the urn is assumed to be the same, but this time alternative C gives you 100$ in case you draw either a red or yellow ball and alternative D gives you 100$ in case you draw either a black or yellow ball.

Alternative/event E1: Red ball is drawn E2: Black ball is drawn E3: Yellow ball is drawn
A 100$ 0$ 0$
B 0$ 100$ 0$
Alternative/event E1: Red ball is drawn E2: Black ball is drawn E3: Yellow ball is drawn
C 100$ 0$ 100$
D 0$ 100$ 100$

Axiom I entails that if the decision-maker prefers A to B, then he should prefer C to D. The intuition is that if one prefers A to B, that must mean that the decision-maker ascribes a higher probability to event E1 than to event E2. Since the content of the urn is assumed to be the same in both decision-problems, this should imply that the expected gain of C (measured either in money or in utility) should be higher than D’s. The decision-maker’s ranking of alternatives should be independent of what happen in case event E3 holds, since in each decision-problem the alternatives have the same outcome. However, as Ellsberg’s experiment shows, while most persons prefer A to B, they prefer D to C which is sometimes interpreted as the result of some ambiguity-aversion.

The second example has been suggested by Peter Diamond in a discussion of John Harsanyi’s utilitarian aggregation theorem. Suppose a doctor has two patients waiting for kidney transplantation. Unfortunately, only one kidney is available and it is not expected that another one will be before quite some time. We assume that the doctor, endorsing the social preference of the society, is indifferent between giving the kidney to one or the other patient. The doctor is considering choosing between three allocation mechanisms: mechanism S1 gives the kidney to patient 1 for sure, mechanism S2 gives the kidney to patient 2 for sure, while in mechanism R he tosses a fair coin and gives the kidney to patient 1 if tails but to patient 2 if heads.

Alternative/event E1: Coin toss falls Tails E2: Coin toss falls Heads
S1 Kidney is given to patient 1 Kidney is given to patient 1
S2 Kidney is given to patient 2 Kidney is given to patient 2
R Kidney is given to patient 1 Kidney is given to patient 2

Given that it is assumed that the society (and the doctor) is indifferent between giving the kidney to patient 1 or 2, axiom I implies that the three alternatives should be ranked as indifferent. Most people have the strong intuition however that allocation mechanism R is better because it is fairer.

Instead of giving up axiom I, several consequentialists have suggested instead to reconcile our intuitions with consequentialism through a refinement of the description of outcomes. The basic idea is that, following consequentialism, everything in the individual or collective choice should be featured in the description of outcomes. Consider Ellsberg’s paradox first. If we assume that the violation of I is due to the decision-makers’ aversion to probabilistic ambiguity, then we modify the tables in the following way:

Alternative/event E1: Red ball is drawn E2: Black ball is drawn E3: Yellow ball is drawn
A 100$ + sure to have a 1/3 probability of winning 0$ + sure to have a 1/3 probability of winning 0$ + sure to have a 1/3 probability of winning
B 0$ + unsure of the probability of winning 100$ + unsure of the probability of winning 0$ + unsure of the probability of winning
Alternative/event E1: Red ball is drawn E2: Black ball is drawn E3: Yellow ball is drawn
C 100$ + unsure of the probability of winning 0$ + unsure of the probability of winning 100$ + unsure of the probability of winning
D 0$ + sure to have a 2/3 probability of winning 100$ + sure to have a 2/3 probability of winning 100$ + sure to have a 2/3 probability of winning

The point is simple. If we consider that being unsure of one’s probability of winning the 100$ is something that makes an alternative less desirable everything else equals, then this has to be reflected in the description and valuation of outcomes. It is then easy to see that ranking A over B but D over C no longer entails a violation of I because the outcomes associated to event E3 are no longer the same in each pair of alternatives. A similar logic can be applied to the second example. If it is collectively considered that the fairness of the allocation mechanism is something valuable, then this must be reflected in the description of outcomes. Then, we have

Alternative/event E1: Coin toss falls Tails E2: Coin toss falls Heads
S1 Kidney is given to patient 1 Kidney is given to patient 1
S2 Kidney is given to patient 2 Kidney is given to patient 2
R Kidney is given to patient 1 + both patients are fairly treated Kidney is given to patient 2 + both patients are fairly treated

Once again, this new description allows to rank R strictly above S1 and S2 without violating I. Hence, the consequentialist’s motto in all the cases where one axiom seems to be problematic is simply “get the outcome descriptions right!”.

A natural objection to this strategy is of course that it seems to make things too easy for the consequentialist. On the one hand, it makes the axioms virtually unfalsifiable as any choice behavior can be trivially accounted for by a sufficiently fine grain partition of the outcome space. On the other hand, all moral intuitions and principles can be made compatible with a consequentialist perspective, once again provided that we have the right partition of the outcome space. However, one can argue that this is precisely the point of the formalist strategy. The consequentialist will argue that this is unproblematic as long as consequentialism is not seen as a normative doctrine about rationality and morality, but rather as a methodological and theoretical framework to account for the implications of various values and principles on rational and social choices. More precisely, what can be called formal consequentialism can be seen as a framework to uncover the principles and values underlying our moral and rational behavior and judgments.

Of course, this defense is not completely satisfactory. Indeed, most consequentialists will not be comfortable with the removal of all the normative content from their approach. As a consequentialist, one wants to be able to argue what it is rational to do and to say what morality commends in specific circumstances. If one wants to preserve some normative content, then the only solution is to impose normative constraints on the permissible partitions of the outcome space. This is indeed what John Broome has suggested in several of his writings with the notion of “individuation of outcomes by justifiers”: the partition of the outcome space should distinguish outcomes if and only if they differ in a way that makes it rational to not be indifferent between them. It follows then that theories of rational choice and social choice are in need of a substantive account of rational preferences and goodness. Such an account is notoriously difficult to conceive. A second difficulty is that the formalist strategy will sometimes be implausible or may even lead to some form of inconsistency. For instance, in the context of expected utility theory, Broome’s individuation of outcomes depends on the crucial and implausible assumption that all “constant acts” are available. This leads to a “richness” axiom (made by Savage for instance) according to which all probabilistic distribution of outcomes should figure in the set of available alternatives, including logically or materially impossible alternatives (e.g. being dead and in a good health). In sequential decision-problems, the formalist strategy is bounded to fail as soon as the path taken to reach a given outcome is relevant for the decision-maker. In this case, to include the path taken in the description of outcomes will not be always possible without leading to inconsistent descriptions of what is supposed to be the same outcome.

These difficulties indicate that formalism cannot fully vindicate consequentialism. Still, it remains an interesting perspective both in rational and social choice theory.

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Isaac Levi on Rationality, Deliberation and Prediction (2/3)

This is the second of a three-part post on the philosopher Isaac Levi’s account of the relationship between deliberation and prediction in decision theory and which is an essential part of Levi’s more general theory of rationality. Levi’s views potentially have tremendous implications for economists especially regarding the current use of game theory. These views are more particularly developed in several essays collected in his book The Covenant of Reason, especially “Rationality, prediction and autonomous choice”, “Consequentialism and sequential choice” and “Prediction, deliberation and correlated equilibrium”. The first post presented and discussed Levi’s main thesis that “deliberation crowds out prediction”. This post discusses some implications of this thesis for decision theory and game theory, specifically the equivalence between games in dynamic form and in normal form. On the same basis, the third post will evaluate the relevance of the correlated equilibrium concept for Bayesianism in the context of strategic interactions. The three posts are collected under a single pdf file here.

 

In his article “Consequentialism and sequential choice”, Isaac Levi builds on his “deliberation crowds prediction” thesis to discuss Peter Hammond’s account of consequentialism in decision theory presented in the paper “Consequentialist Foundations for Expected Utility”. Hammonds contends that consequentialism (to be defined below) implies several properties for decision problems, especially (i) the formal equivalence between decision problems in sequential (or extensive) form and strategic (or normal) form and (ii) ordinality of preferences over options (i.e. acts and consequences). Though Levi and Hammonds are essentially concerned with one-person decision problems, the discussion is also relevant from a game-theoretic perspective as both properties are generally assumed in the latter. This post will focus on point (i).

First, what is consequentialism? Levi distinguishes between three forms: weak consequentialism (WC), strong consequentialism (SC) and Hammond’s consequentialism (HC). According to Levi, while only HC entails point (i), both SC and HC entail point (ii). Levi contends however that none of them is defensible once we take into account the “deliberation crowds out prediction” thesis. We may define these various forms of consequentialism on the basis of the notation introduced in the preceding post. Recall that any decision problem D corresponds then to a triple < A, S, C > with A the set of acts (defined as functions from states to consequences), S the set of states of nature and C the set of consequences. A probability distribution over S is defined by the function p(.) and represents the decision-maker DM’s subjective beliefs while a cardinal utility function u(.) defined over C represents DM’s preferences. Now the definitions of WC and SC are the following:

Weakly consequentialist representation – A representation of D is weakly consequentialist if, for each a 󠄉 ∈ A, an unconditional utility value u(c) is ascribed to any element c of the subset Ca  C and where we allow for a  ∈ Ca. If a is not the sole element of Ca, then the representation is nontrivially weakly consequentialist.

(WC)   Any decision problem D has a weakly consequentialist representation.

Strongly consequentialist representation – A representation of D is strongly consequentialist if, (i) it is nontrivially weakly consequentialist and (ii) given the set of consequence-propositions C, if ca and cb are two identical propositions, then the conjuncts aca and bcb are such that u(aca) = u(bcb).

            (SC)     Any decision problem D has a strongly consequentialist representation.

WC thus holds that it is always possible to represent a decision problem as a set of acts to which we can ascribe unconditional utility value to all consequences each act leads to, and where an act itself can be analyzed as a consequence. As Levi notes, WC formulated this way is undisputable.* SC has been endorsed by Savage and most contemporary decision theorists. The difference with WC lies in the fact that SC holds a strict separation between acts and consequences. Specifically, the utility value of any consequence c is independent of the act a that brought it. SC thus seems to exclude various forms of “procedural” account of decision problems. Actually, I am not sure that the contrast between WC and SC is as important as Levi suggests for all is required for SC is to have a sufficiently rich set of consequences C to guarantee the required independence.

According to Levi, HC is stronger than SC. This is due to the fact that while SC does not entail that sequential form and strategic form decision problems are equivalent, HC makes this equivalence its constitutive characteristic. To see this, we have to refine our definition of a decision problem to account for the specificity of the sequential form. A sequential decision problem SD is constituted by a set N of nodes n with a subset N(D) of decision nodes (where DM makes choice), a subset N(C) of chance nodes (representing uncertainty) and a subset N(T) of terminal nodes. All elements of N(T) are consequence-propositions and therefore we may simply assume that N(T) = C. N(D) is itself partitioned into information sets I where two nodes n and n’ in the same I are indistinguishable for DM. For each n ∈ N(D), DM has subjective beliefs measured by the probability function p(.|I) that indicates DM’s belief of being at node n given that he knows I. The conditional probabilities p(.|I) are of course generated on the basis of the unconditional probabilities p(.) that DM holds at each node n ∈ N(C). The triple < N(D), N(C), N(T) > defines a tree T. Following Levi, I will however simplify the discussion by assuming perfect information and thus N(C) = ∅. Now, we define a behavior norm B(T, n) for any tree T and any decision node n in T the set of admissible options (choices) from the set of available options at that node. Denote T(n) the subtree starting from any decision node n. A strategy (or act) specifies at least one admissible option for all decision nodes reachable, i.e. B(T(n), n) must be non-empty for each n ∈ N(D). Given that N(T) = C, we write C(T(n)) the subset of consequences (terminal nodes) that are reachable in the subtree T(n) and B[C(T(n)), n] the set of consequences DM would regard as admissible if all elements in C(T(n)) were directly available at decision node n. Therefore, B[C(T(n)), n] is the set of admissible consequences in the strategic form equivalent of SD as defined by (sub)tree T(n). Finally, write φ(T(n)) the set of admissible consequences in the sequential form decision problem SD. HC is then defined as follows:

(HC)   φ(T(n)) = B[C(T(n)), n]

In words, HC states the kind of representation (sequential or strategic) of a decision problem SD used is irrelevant in the determination of the set of admissible consequences. Moreover, since we have assumed perfect information, its straightforward that this is also true for admissible acts which, in sequential form decision problems, correspond to exhaustive plans of actions.

Levi argues that assuming this equivalence is too strong and cannot be an implication of consequentialism. This objection is of course grounded on the “deliberation crowds out prediction” thesis. Consider a DM faced with a decision problem SD with two decision nodes. At node 1, DM has the choice between consuming drug (a1) or abstaining (b1). If he abstains, the decision problem ends but if the adduces, he then has the choice at node 2 between continuing taking drugs and becoming addict (a2) or stopping and avoiding addiction (c2). Suppose that DM’s preferences are such that u(c2) > u(b1) > u(a2). DM’s available acts (or strategies) are therefore (a1, a2), (a1, c2), (b1, a2) and (b1, c2).** Consider the strategic form representation of this decision problem where DM has to make a choice once for all regarding the whole decision path. Arguably, the only admissible consequence is c2 and therefore the only admissible act is (a1, c2). Assume however that if DM were to choose a1, he would fall prey to temptation at node 2 and would not be able to refrain from continuing consuming drugs. In other words, at node 2, only option a2 would actually be available. Suppose that DM knows this at node 1.*** Now, a sophisticated DM will anticipate his inability to resist temptation and will choose to abstain (b1) at node 1. It follows that a sophisticated DM will choose   (b1, a2) in the extensive form of SD, thus violating HC (but not SC).

What is implicit behind Levi’s claim is that, while it makes perfect sense for DM to ascribe probability (including probability 1 or 0) to his future choices at subsequent decision nodes, this cannot be the case for his choice over acts, i.e. exhaustive plans of actions in the decision path. For if it was the case, then as (a1, c2) is the only admissible act, he would have to ascribe probability 0 to all acts but (a1, c2) (recall Levi’s claim 2 in the previous post). But then, that would also imply that only (a1, c2) is feasible (Levi’s claim 1) while this is actually not the case.**** Levi’s point is thus that, at node 1, the choices at node 1 and at node 2 are qualitatively different: at node 1, DM has to deliberate over the implications of choosing the various options at his disposal given his beliefs over what he would do at node 2. In other words, DM’s choice at node 1 requires him to deliberate on the basis of a prediction about his future behavior. In turn, at node 2, DM’s choice will involve a similar kind of deliberation. The reduction of extensive form into strategic form is only possible if one conflates these two choices and thus ignores the asymmetry between deliberation and prediction.

Levi’s argument is also relevant from a game-theoretic perspective as the current standard view is that a formal equivalence between strategic form and extensive form games holds. This issue is particularly significant for the study of the rationality and epistemic conditions sustaining various solution concepts. A standard assumption in game theory is that the players have knowledge (or full belief) of their strategy choices. The evaluation of the rationality of their choices both in strategic and extensive form games however requires to determine what the players believe (or would have believed) in counterfactual situations arising from different strategy choices. For instance, it is now well established that common belief in rationality does not entail the backward induction solution in perfect information games or rationalizability in strategic form games. Dealing with these issues necessitates a heavy conceptual apparatus. However, as recently argued by economist Giacomo Bonanno, not viewing one’s strategy choices as objects of belief or knowledge allows an easier study of extensive-form games that avoids dealing with counterfactuals. Beyond the technical considerations, if one subscribes to the “deliberation crowds out prediction”, this is an alternative path worth exploring.

Notes

* Note that this has far reaching implications for moral philosophy and ethics as moral decision problems are a strict subset of decision problems. All moral decision problems can be represented along a weakly consequentialist frame.

** Acts (b1, a2) and (b1, c2) are of course equivalent in terms of consequences as DM will never actually have to make a choice at node 2. Still, in some cases it is essential to determine what DM would do in counterfactual scenarios to evaluate his rationality.

*** Alternatively, we may suppose that DM has at node 1 a probabilistic belief over his ability to resist temptation at node 2. This can be simply implemented by adding a chance node before node 1 that determines the utility value of the augmented set of consequences and/or the available options at node 2 and by assuming that DM ignores the result of the chance move.

**** I think that Levi’s example is not fully convincing however. Arguably, one may argue that since action c2 is assumed to be unavailable at node 2, acts (a1, c2) and (b1, c2) should also be regarded as unavailable. The resulting reduced version of the strategic form decision problem would then lead to the same result than the sequential form. This is not different even if we assume that DM is uncertain regarding his ability to resist temptation (see the preceding note). Indeed, the resulting expected utilities of acts would trivially lead to the same result in the strategic and in the sequential forms.  Contrary to what Levi argues, it is not clear that that would violate HC.