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 a ∧ ca and b ∧ cb are such that u(a ∧ ca) = u(b ∧ cb).
(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.
Bargaining Game 