Is it Rational to be Bayesian Rational?

Most economists and many decision theorists equate the notion of rationality with Bayesian rationality. While the assumption that individuals actually are Bayesian rational has been largely disputed and is now virtually rejected, the conviction that Bayesianism defines the normative standard of rational behavior remains fairly entrenched among economists. However, even the normative relevance of Bayesianism has been questioned. In this post, I briefly survey one particular and interesting kind of argument that has been particularly developed by the decision theorist Itzhak Gilboa with different co-authors in several papers.

First, it is useful to start with a definition of Bayesianism in the context of economic theory: the doctrine according to which it is always rational to behave according to the axioms of Bayesian decision theory. Bayesianism is a broad church with many competing views (e.g. radical subjectivism, objective Bayesianism, imprecise Bayesianism…) but it will be sufficient to retain a generic characterization through the two following principles:

Probabilism: Bayesian rational agents have beliefs that can be characterized through a probability function whose domain is some state space.

Expected Utility Maximization: The choices of Bayesian rational agents can be represented by the maximization of the expectation of a utility function according to some probability function.

Gilboa’s critique of Bayesianism is uniquely concerned with probabilism though some of its aspects could be easily extended to the expected utility maximization principle. Probabilism can itself be characterized as the conjunction of three tenets:

(i) Grand State Space: each atom (“state of nature”) in the state space is assumed to resolve all uncertainty, i.e. everything that is relevant for the modeler is specified, included all causal relationships. Though in Savage’s version of Bayesian decision theory, states of nature where understood as “small worlds” corresponding to some coarse partition of the state space, in practice most economists implicitly interpret states of nature as “large worlds”, i.e. as resulting from the finest partition of the state space.

(ii) Prior Probability: Rational agents have probabilistic beliefs over the state space which are captured by a single probability measure.

(iii) Bayesian updating: In light of new information, rational agents update their prior to a posterior belief according to Bayes’s rule.

While the third tenet may be disputed, included within the realm of Bayesianism (see for instance Jeffrey’s probability kinematics or views entertained by some objective Bayesians), it is the first two that are targeted by Gilboa. More exactly, while each tenet taken separately seems pretty reasonable normatively speaking, problems arise as soon as one decides to combine them.

Consider an arbitrary decision problem where it is assumed (as economists routinely do) that all uncertainty is captured through a Grand State Space. Say, you have to decide between choosing to bet on what is presented to you as a fair coin falling on heads and betting on the fact that the next winner of the US presidential will be a Republican. There seem to be only four obvious states of nature: [Heads, Republican], [Heads, Democrat], [Tail, Republican], [Tail, Democrat]. Depending on your prior beliefs that the coin toss will fall on Heads (maybe a 1:1 odd) and that the next US president will be a Republican (and assuming monotonic preferences in money), your choice will reveal your preference for one of the two bets. Even if ascribing probabilities to some of the events may be difficult, it seems that the requirements of Bayesian rationality cannot be said to be unreasonable here. But matters are actually more complicated because there are many things that may causally affect the likelihood of each event. For instance, while you have been said that the coin is fair, maybe you have reason to doubt this affirmation. This will depend for instance on who has made the statement. Obviously, the result of the next US presidential elections will depend on the many factual and counterfactual events that may happen. To form a belief about the result of the US elections not only you have to form a belief over these events but also over the nature of the causal relationships between them and the result of the US election. Computationally, the task quickly becomes tremendous as the number of states of nature to consider is quite huge. Assuming that a rational agent should be able to assign a prior over all of them is normatively unreasonable.

An obvious answer (at least for economists and behaviorists-minded philosophers) is to remark that prior beliefs need not be “in the mind” of the decision-maker. What matters is that the betting behavior of the decision-maker reveals preferences over prospects that can be represented by a unique probability measure over as larger a state space as needed to make sense of it. There are many things to be said against this standard defense but for the sake of the argument we may momentarily accept it. What happens however of the behavior of the agents fail to reveal the adequate preferences? Must we conclude then that the decision-maker is irrational? A well-known case leading to such questions is Ellsberg’s paradox. Under a plausible interpretation, the latter indicates that most actual agents reveal through their choices an aversion for probabilistic ambiguity which directly led to the violation of the independence axiom of Bayesian decision theory. In this case, the choice behavior of agents cannot be consistently represented by a unique probability measure. Rather than arguing that such a choice behavior is irrational, a solution (which I have already discussed here) is to adopt the Grand State Space approach. It is then possible to show that with an augmented state space there is nothing “paradoxical” in Ellsber’s paradox. The problem however with this strategy is twofold. On the one hand, many choices are “unobservable” by definition, which fits uneasily in the behaviorist interpretation of Bayesian axioms. On the other hand,  it downplays the reasons that explain the choices that actual agents are actually making.

To understand this last point, it must be acknowledged that Bayesianism defines rationality merely in terms of consistency with respect to a set of axioms. As a result, such an approach completely disregards the way agents form their beliefs (as well as their preferences) and – more importantly – abstains from making any normative statement regarding the content of beliefs. “Irrational” beliefs are merely beliefs that fail to qualify for a representation through a unique probability measure. Now, consider whether it is irrational to fail or to refuse to have such beliefs in cases where some alternatives but not others suffer from probabilistic ambiguity. Also, consider whether it is irrational to firmly believe (eventually to degree 1) that smoking presents no risk for health. Standard Bayesianism will answer positively in the first case but negatively in the second. Not only this is unintuitive but it also seems to be pretty unreasonable. Consider the following alternative definition of rationality proposed by Gilboa:

A mode of behavior is irrational for a decision maker, if, when the latter is exposed to the analysis of her choices, she would have liked to change her decision, or to make different choices in similar future circumstances.

This definition of rationality appeals to the reflexive abilities of human agents and, crucially, to our capacity to motivate our choices through reasons. This suggests first that the axioms of Bayesian decision theory can be submitted both as reasons to make specific choices but also has the subject of the normative evaluation. This also indicates that whatever may be thought of these axioms, Bayesianism lacks an adequate account of beliefs formation. In other words, Bayesianism cannot pretend to constitute a normative theory of rationality because it does not offer any justification neither for the way an agent should partition the state space nor for deciding which prior to adopt. The larger the state space is made to capture all the relevant features explaining an agent’s prior, the lesser it seems reasonable to expect rational agents to be able or to be willing to entertain such a prior.

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Bayesian Rationality and Utilitarianism

In a recent blog, Bryan Caplan gives his critical views about the “rationality community”, i.e. a group of people and organizations who are actively developing ideas related to cognitive bias, signaling and rationality. Basically, members of the rationality community are applying the rationality norms of Bayesianism to a large range of issues related to individual and social choices. Among Caplan’s complaints figures the alleged propensity of the community’s members to endorse consequentialist ethics and more specifically utilitarianism, essentially for “aesthetic” reasons. In a related Twitter exchange, Caplan states that by utilitarianism he refers to the doctrine that one’s duty is to act as to maximize the sum of happiness in the society. This corresponds to what his generally called hedonic utilitarianism.

Hedonic utilitarianism faces many problems well-known to moral philosophers. I do not know if the members of the rationality community are hedonic utilitarians, but there is another route for Bayesians to be utilitarians. This route is logical rather than aesthetic and is grounded on a theorem exposed by the economist John Harsanyi in the 1950s and since largely discussed by philosophically-minded economists and mathematically-minded philosophers. Harsanyi’s initial demonstration was grounded on the von Neumann and Morgenstern’s axioms (actually Marshak’s version of them) of decision under risk but has since been extended to other versions of decision theory, especially Savage’s axioms for decision under uncertainty. The theorem can be briefly stated in the following way. Denote S the set of states of nature, i.e. morally-relevant features that are outside the control of the decision-makers and O the set of outcomes. Intuitively, an outcome is a possible world specifying everything that is morally relevant for the individuals: their wealth, their health, their history, and so on. Finally, denote X the set of “prospects”, i.e. social alternatives or public policies mapping any state s onto an outcome o. We assume that the n members of the population have preferences over the set of prospects and that these preferences satisfy Savage’s axioms. Therefore, the preferences of any individual i can be represented by an expectational utility function: each prospect x is ascribed a utility number ui(x) that cardinally represent i’s preferences. ui(x) corresponds to the probability weighted-sum of utility of all possible outcomes (which correspond to “sure” prospects). Hence, each individual also has beliefs regarding the likelihood of the states of nature that are captured by a probability function pi(.).

Given the individuals’ preferences, each prospect x is assigned a vector of utility numbers (u1(x), …, un(x)). Now, we assume that there is a “benevolent dictator” k (possibly one of the member of the population) whose preferences over X also satisfy Savage’s axioms. It follows that the dictator’s preferences can also be represented by an expectational utility function with each prospect x mapped into a number uk(x). Last assumption: the individuals’ and dictator’s preferences over X are related by a Pareto principle: if every individual prefers (resp. is indifferent) prospect x to prospect y, then the dictator prefers (resp. is indifferent) x to y. Harsanyi’s theorem states that the dictator’s preferences can then be represented by a utility function corresponding to the weighted-sum of the individuals’ utilities for any prospect x. Suppose moreover than utilities are interpersonally comparable and that the dictator’s preferences are impartial (they do not arbitrarily weight more a person’s utility than another’s one), then for any x

uk(x) = u1(x) + … + un(x).

Of course, this is the utilitarian formulae but stated in utility rather than hedonic terms. Note that here utility does not correspond to happiness or pleasure but rather to preference-satisfaction. Harsanyi’s utilitarianism is preference-based. The point of the theorem is to show that consistent Bayesians should be utilitarians in this sense.

It should be acknowledged that what the theorem demonstrates is actually far weaker. A first reason (discussed by Sen among others) is that the cardinal representation of the individuals’ preferences is not imposed by Savage’s theorem. Obviously, the use of other representations of individuals’ preferences will have the effect of making the additive structure unable to represent the dictator’s preferences. Some authors like John Broome have argued however that the expectational representational is the most natural one and fits well with some notion of goodness. There is another, different kind of difficulty related to the Pareto principle. It can be shown that the assumption that the dictator’s preferences are transitive (which is imposed by Savage’s axioms) combined with the Pareto principle imply “probabilistic agreement”, i.e. that all individuals agree regarding their probabilistic assessment over the likelihood of the states of nature. Otherwise, probabilistic disagreement and the Pareto principle would lead to cases where the dictator’s preferences are inconsistent and thus unamenable to a utility representation. Probabilistic agreement is of course a very strong assumption, an assumption that Harsanyi would have been ready to defend without doubt (see the “Harsanyi doctrine” in game theory). Objective Bayesians may indeed argue that rationality entails a unique correct probabilistic assessment. But subjective Bayesians will of course disagree.

What happen if we give up the Pareto principle for prospects (not for outcomes however)? Then, the dictator’s preferences are amenable to being represented by an ex post prioritarian social welfare function such that

uk(x) = ∑spk(s)∑iv(ui(x(s)=o))

where v(.) is a strictly increasing and concave function. This corresponds to what Derek Parfit called the “priority view” and leads to giving priority to the satisfaction of preferences of the less well-off in the population.

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

This is the last 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”. The second post discussed 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, this post evaluates 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 important article “Correlated equilibrium as an Expression of Bayesian Rationality”, Robert Aumann argues that Bayesian rationality in strategic interactions makes correlated equilibrium the natural solution concept for game theory. Aumann’s claim is a direct and explicit answer to a short paper by Kadane and Larkey which argues that the extension of Bayesian decision theory to strategic interactions leads to a fundamental indetermination at the theoretical level regarding how Bayesian rational players will/should play. To these authors, the way a game will be played depends on contextual and empirical features on which the theorist has few things to say. Aumann’s aim is clearly to show that the game theorist endorsing Bayesianism is not committed to such nihilism. Aumann’s paper was one of the first contributions to what is nowadays sometimes called the “epistemic program” in game theory. The epistemic program can be characterized as an attempt to characterize various solution concepts for normal- and extensive-form games (Nash equilibrium, correlated equilibrium, rationalizability, subgame perfection, …) in terms of sufficient epistemic conditions regarding the players’ rationality and their beliefs and knowledge over others’ choices, rationality and beliefs. While classical game theory in the tradition inspired by Nash has followed a “top-down” approach consisting in determining which strategy profiles in a game correspond to a given solution concept, the epistemic approach rather follows a “bottom-up” perspective and asks what are the conditions for a given solution concept to be implemented by the players. While Levi’s essay “Prediction, deliberation and correlated equilibrium” focuses on Aumann’s defense of the correlated equilibrium solution concept, its main points are essentially relevant for the epistemic program as a whole, as I will try to show below.

Before delving into the details of Levi’s argument, it might be useful to first provide a semi-formal definition of the correlated equilibrium solution concept. Denote A = (A1, …, An) the joint action-space corresponding to the Cartesian product of the set of pure strategies of the n players in a game. Assume that each player i = 1, …, n has a cardinal utility function ui(.) representing his preferences over the set of outcomes (i.e. strategy profiles) determined by A. Finally, denote Γ some probability space. A function f: Γ –> A defines a correlated equilibrium if for any signal γ, f(γ) = a is a strategy profile such that each player maximizes his expected utility conditional on the strategy he is playing:

For all i and all strategy ai’ ≠ ai, Eui(ai|ai) ≥ Eui(ai|ai)

Correspondingly, the numbers Prob{f-1(a)} define the correlated distribution over A that is implemented in the correlated equilibrium. The set of correlated equilibria in any given game is always at least as large than the set of Nash equilibria. Indeed, a Nash equilibrium is also a correlated equilibrium while correlated equilibria correspond to the convex hull of Nash equilibria. As an illustration, consider for instance the famous hawk-dove game:

Column

C

D

Row

C

5 ; 5

3 ; 7

D

7 ; 3

2 ; 2

This game has two Nash equilibria in pure strategy (i.e. [D, C] and [C, D]) and one Nash equilibrium in mixed strategy where each player plays C with probability 1/3. There are many more correlated equilibria in this game however. One of them is trivially given for instance by the following correlated distribution:

Column

C

D

Row

C

0

1/2

D

1/2

0

In his paper, Aumann establishes the following important theorem:

Aumann’s theorem – For any game, if

(i)        Each player i has a probability measure pi(.) over the joint-action space A;

(ii)       The probability measure is unique, i.e. p1(.) = … = pn(.);

(iii)      The players are Bayesian rational (i.e. maximizes expected utility) and this is common   knowledge;

then, the players implement a correlated equilibrium corresponding to a function f with the correlated distribution defined by Prob{f-1(a)} = p(a).

The theorem thus shows that Bayesian rational players endowed with common knowledge of their rationality and a common prior belief over the joint-action space must implement a correlated equilibrium. Therefore, it seems that Kadane and Larkey were indeed too pessimistic in claiming that nothing can be said regarding what will happen in a game with Bayesian decision makers.

Levi attacks Aumann’s conclusion by rejecting all of its premises. Once again, this rejection is grounded on the “deliberation crowds out prediction” thesis. Actually, Levi’s makes two distinct and relatively independent critics against Aumann’s assumptions. The first concerns an assumption that I have left implicit while the second targets premises (i)-(iii) together. I will consider them in turn.

Implicit in Aumann’s theorem is an assumption that Levi calls “ratifiability”. To understand what that means, it is useful to recall that a Bayesian decision-maker maximizes expected utility using conditional probabilities over states given acts. In other words, a Bayesian decision maker has to account for the possibility that his choice may reveal and/or influence the likelihood that a given state is the actual state. Evidential decision theorists like Richard Jeffrey claim in particular that it is right to see one’s choice as an evidence for the truth-value of various state-propositions even in cases where no obvious causal relationship seems to hold between one’s choice and states of nature. This point is particularly significant in a game-theoretic context where while the players make choice independently (in a normal-form game), some kind of correlation over choices and beliefs may be seen as plausible. The most extreme case is provided by the prisoner’s dilemma which Levi discusses at length in his essay:

Column

C

D

Row

C

5 ; 5

1 ; 6

D

6 ; 1

2 ; 2

The prisoner’s dilemma has a unique Nash equilibrium: [D, D]. Clearly, given the definition given above, this strategy profile is also the sole correlated equilibrium. However, from a Bayesian perspective, Levi argues that it is perfectly fine for Row to reason along the following lines:

“Given what I know and believe about the situation and the other player, I believe almost for sure that if I play D, Column will also play D. However, I also believe that if I play C, there is a significant chance that Column will play C”.

Suppose that Row’s conditional probabilities are p(Column plays D|I play D) = 1 and p(Column plays C|I play C) = ½. Then, Row’s expected utilities are respectively Eu(D) = 2 and Eu(C) = 3. As a consequence, being Bayesian rational, Row should play C, i.e. should choose to play a dominated strategy. Is there any wrong for Row to reason this way? The definition of the correlated equilibrium solution concept excludes this kind of reasoning because, for any action ai, the computation of expected utilities for each alternative actions ai’ should be made using the conditional probabilities u(.|ai). This corresponds indeed to the standard definition of ratifiability in decision theory as put by Jeffrey: “A ratifiable decision is a decision to perform an act of maximum estimated desirability relative to the probability matrix the agent thinks he would have if he finally decided to perform that act.” In the prisoner’s dilemma, it is easy to see that only D is ratifiable because considering to play C with the conditional probabilities given above, Row would do better by playing D; indeed, Eu(D|Columns plays C with probability ½) > Eu(C|Columns plays C with probability ½).

As Levi recognizes, the addition of ratifiability as a criterion of rational choice leads de facto to exclude the possibility that the players in a game may rationally believe that some form of causal dependence holds between their choices. Indeed, as formally shown by Oliver Board, Aumann’s framework tacitly builds upon an assumption of causal independence but also of common belief in causal independence. For some philosophers and game theorists, this is unproblematic and indeed required since it is constitutive of game-theoretic reasoning (see for instance this paper of Robert Stalnaker). Quite the contrary, Levi regards this exclusion as illegitimate at least on a Bayesian ground.

Levi’s rejection of premises (i)-(iii) are more directly related to his “deliberation crowds out prediction thesis”. Actually, we may even focus on premises (i) and (iii) as premise (ii) depends on (i). Consider the assumption that the players have a probability measure over the joint-action space first. Contrary to a standard Bayesian decision problem where the probability measure is defined over a set of states that is distinct from the set of acts, in a game-theoretic context the domain of the probability measures encompasses each player’s own strategy choice. In other words, this leads the game theorist to assumes that each player ascribes an unconditional probability to his own choice. I have already explained why Levi regards this assumption as unacceptable if one wants to account for the way decision makers reason and deliberate.* The common prior assumption (ii) is of course even less commendable in this perspective, especially if we consider that such an assumption pushes us outside the realm of strict Bayesianism. Regarding assumption (iii), Levi’s complaint is similar: common knowledge of Bayesian rationality implies that each player knows that he is rational. However, if a player knows that he is rational before making his choice, then he already regards as feasible only admissible acts (recall Levi’s claims 1 and 2). Hence, no deliberation has to take place.

Levi’s critique of premises (i)-(iii) seem to extend to the epistemic program as a whole. What is at stake here is the epistemological and methodological status of the theoretical models build by game theorists. The question is the following: what is the modeler trying to establish regarding the behavior of players in strategic interactions? There are two obvious possibilities. A first one is that, as an outside observer, the modeler is trying to make sense (i.e. to describe and to explain) of players’ choices after having observed them. Relatedly, still as an outside observer, he may try to predict players’ choices before they are made. A second possibility is to game-theoretic models as tools to account for the players’ reasoning process prospectively, i.e. how players’ deliberate to make choice. Levi’s “deliberation crowds out prediction” thesis could grant some relevance to the first possibility may not for the second. However, he contends that Aumann’s argument for correlated equilibrium cannot be only retrospective but must also be prospective.** If Levi is right, the epistemic program as a whole is affected by this argument, though fortunately there is room for alternative approaches, as illustrated by Bonanno’s paper mentioned in the preceding post.

Notes

* The joint-action space assumption results from a technical constraint: if we want to exclude the player’s own choice from the action space, we then have to account for the fact that each player has a different action space over which he forms beliefs. In principle, this can be dealt with even though this would lead to more cumbersome formalizations.

** Interestingly, in a recent paper with Jacques Dreze which makes use of the correlated equilibrium solution concept, Aumann indeed argues that the use of Bayesian decision theory in a game-theoretic context has such prospective relevance.

Isaac Levi on Rationality, Deliberation and Prediction (1/3)

This is the first 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 presents and discusses Levi’s main thesis that “deliberation crowds out prediction”. The next two posts will discuss some implications of this thesis for decision theory and game theory, specifically (i) the equivalence between games in dynamic form and in normal form and (ii) 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.

The determination of principles of rational choice is the main subject of decision theory since its early development at the beginning of the 20th century. Since its beginnings, decision theory has pursued two different and somehow conflicting goals: on the one hand, to describe and explain how people actually make choices and, on the other hand, to determine how people should make choices and what choices they should make. While the former goal corresponds to what can be called “positive decision theory”, the latter is constitutive of “normative decision theory”. Most decision theorists, especially the proponents of “Bayesian” decision theory, have agreed that decision theory cannot but be partially normative. Indeed, while today Bayesian decision theory is generally not regarded as an accurate account of how individuals are actually making choices, most decision theorists remain convinced that it is still relevant as a normative theory of rational decision-making. This is in this context that Isaac Levi’s claim that “deliberation crowds out prediction” should be discussed.

In this post, I will confine the discussion to the restrictive framework of Bayesian decision theory though Levi’s account more generally applies to any form of decision theory that adheres to consequentialism. Consequentialism will be more fully discussed in the second post of this series. Consider any decision problem D in which an agent DM has to make a choice over a set of options whose consequences are not necessarily fully known for sure. Bayesians will generally model D as a triple < A, S, C > where A is the set of acts a, S the set of states of nature s and C the set of consequences c. In the most general form of Bayesian decision theory, any a, s and c may be regarded as a proposition to which truth-values might be assigned. In Savage’s specific version of Bayesian decision theory, acts are conceived as functions from states to consequences, i.e. a: S à C or c = a(s). In this framework, it is useful to see acts as DM’s objects of choice, i.e. the elements over which he has a direct control, while states may be interpreted as every features in D over which DM has no direct control. Consequences are simply the result of the combination of an act (chosen by DM) and a state (not chosen by DM). Still following Savage, it is standard to assume that DM has (subjective) beliefs over which state s actually holds. These beliefs are captured by a probability function p(.) with ∑sp(s) = 1 for a finite state space. Moreover, each consequence c is assigned a utility value u representing DM’s preferences over the consequences. A Bayesian DM will then choose the act that maximizes his expected utility given his subjective beliefs and his preferences, i.e.

Maxa Eu(a) = ∑sp(s|a)u(a(s)) =  ∑sp(s|a)u(c).

Two things are worth noting. First, note that the probabilities that enter into the expected utility computation are conditional probabilities of states given acts. We should indeed account for the possibility that the probabilities of states depend on the act performed. The nature of the relationship between states and acts represented by these conditional probabilities is the main subject of conflict between causal and evidential decision theorists. Second, as it is well-known, in Savage’s version of Bayesian decision theory, we start with a full ordering representing DM’s preferences over acts and given a set of axioms, it is shown that we can derive a unique probability function p(.) and a cardinal utility function u(.) unique up to any positive affine transformation. It is indeed important to recognize that Savage’s account is essentially behaviorist because it merely shows that given the fact that DM’s preferences and beliefs satisfy some properties, then his choice can be represented as the maximization of some function with some uniqueness property. Not all Bayesian decision theorists necessarily share Savage’s behaviorist commitment.

I have just stated that in Savage’s account, DM ascribes probabilities to states, utilities to consequences and hence expected utilities to acts. However, if acts, states and consequences are all understood as propositions (as argued by Richard Jeffrey and Levi among others), then there is nothing in principle prohibiting to ascribe utilities to states and probabilities to both consequences and acts. This is this last possibility (ascribing probabilities to acts) that is the focus of Levi’s claim that deliberation crowds out prediction. In particular, does it make sense for DM to have unconditional probabilities over the set A? How having such probabilities could be interpreted from the perspective of DM’s deliberation in D? If we take a third person perspective, ascribing probabilities to DM’s objects of choice seems not particularly contentious. It makes perfect sense for me to say for instance “I believe that you will start again to smoke before the end of the month with probability p”. Ascribing probabilities to others’ choices is an essential part of our daily activity consisting in predicting others’ choices. Moreover, probability ascription may be a way to explain and rationalize others’ behavior. The point of course is that these are my probabilities, not yours. The issue here is whether a deliberating agent has to, or even can ascribe such probabilities to his own actions, acknowledging that such probabilities are in any case not relevant in the expected utility computation.

Levi has been (with Wolfgang Spohn) the most forceful opponent to such a possibility. He basically claims that the principles of rationality that underlie any theory of decision-making (including Bayesian ones) cannot at the same time serve as explanatory and predictive tools and as normative principles guiding rational behavior. In other words, as far as the deliberating agent is using rationality principles to make the best choice, he cannot at the same time use these principles to predict his own behavior at the very moment he is making his choice.* This is the essence of the “deliberation crowds out prediction” slogan. To understand Levi’s position, it is necessary to delve into some technical details underlying the general argument. A paper of philosopher Wlodek Rabinowicz makes a great job in reconstructing this argument (see also this paper by James Joyce). A crucial premise is that, following De Finetti, Levi considers belief ascription as fully constituted by the elicitation of betting rates, i.e. DM’s belief over some event E is determined and corresponds to what DM would consider as the fair price of a gamble where event E pays x$ and event non-E pays y$.** Consider this example: I propose you to pay y$ (the cost or the price of the bet) to participate to the following bet: if Spain win the Olympic gold medal of basketball at Rio this month, I will pay you x$, otherwise I pay you nothing. Therefore, x is the net gain of the bet and x+y is called the stake of the bet. Now, the fair price y*$ of the bet corresponds to the amount for which you are indifferent between taking and not taking the bet. Suppose that x = 100 and that y* = 5. Your betting rate for this gamble is then y*/(x+y*) = 5/105 = 0,048, i.e. you believe that Spain will win with probability less than 0,05. This is the traditional way beliefs are determined in Bayesian decision theory. Now, Levi’s argument is that such a procedure cannot be applied in the case of beliefs over acts on pain of inconsistency. The argument relies on two claims:

(1)       If DM is certain that he will not perform some action a, then a is not regarded as part of the feasible acts by DM.

(2)       If DM assigns probabilities to acts, then he must assign probability 0 to acts he regards as inadmissible, i.e. which do not maximize expected utility.

Clearly, (1) and (2) entail together that only feasible acts (figuring in the set A) are admissible (maximize expected utility), in which case deliberation is unnecessary for DM. If it is the case however, that means that principles of rationality cannot be used as normative principles in the deliberation process. While claim (1) is relatively transparent (even if it is disputable), claim (2) is less straightforward. Consider therefore the following illustration.

DM has a choice between two feasible acts a and b with Eu(a) > Eu(b), i.e. only a is admissible. Suppose that DM assigns probabilities p(a) and p(b) according to the procedure presented above. We present DM with a fair bet B on a where the price is y* and the stake is x+y*. As the bet is fair, y* is the fair price and y*/(x+y*) = p(a) is the betting rate measuring DM’s belief. Now, DM has four feasible options:

Take the bet and choose a (B&a)

Do not take the bet and choose a (notB&a)

Take the bet and choose b (B&b)

Do not take the bet and choose b (notB&b)

As taking the bet and choosing a guarantee a sure gain of x to DM, it is easy to see that B&a strictly dominates notB&a. Similarly, as taking the bet and choosing b guarantee a sure loss of y*, notB&b strictly dominates B&b. The choice is therefore between B&a and notB&b and clearly Eu(a) + x > Eu(b). It follows that the fair price for B is  y* = x + y* and hence p(a) = 1 and p(b) = 1 – p(a) = 0. The inadmissible option b has probability 0 and is thus regarded as unfeasible by DM (claim 1). No deliberation is needed for DM if he predicts his choice since only a is regarded as feasible.

Levi’s argument is by no means undisputable and the papers of Rabinowicz and Joyce referred above make a great job at showing its weaknesses. In the next two posts, I will however take it as granted and discuss some of its implications for decision theory and game theory.

 Notes

* As I will discuss in the second post, Levi considers that there is nothing contradictory or problematic in the assumption that one may be able to predict his future choices.

** A gamble’s fair price is the price at which DM is indifferent accepting to buy the bet and accepting to sell the bet.

Working paper: “Bayesianism and the Common Prior Assumption in Game Theory”

I have a new working paper on a slightly esoteric subject, at least for those unfamiliar with decision theory and game theory: “Bayesianism and the Common Prior Assumption in Game Theory: Toward a Theory of Social Interactions“. If everything goes well, I should present this paper at the 3rd Economic Philosophy Conference which will be held in June at Aix-en-Provence. It is organized by Aix-Marseille University and the GREQAM, but one of the co-organizer is my colleague (and office neighbor) from the University of Reims Champagne-Ardenne Jean-Sébastien Gharbi.

This gives me a good excuse for a slight digression: Jean-Sébastien and myself are trying hard to develop economic philosophy at Reims, and though we are currently small in number (hopefully, not for too long!), there has been some momentum shift recently. We have welcome well-known economic methodologist John Davis last year as a visiting professor and we have a significant  numbers  of recent publications in top-journals in economic philosophy and the history of economic thought (Economics and Philosophy, Journal of Economic Methodology, Revue de philosophie économique, European Journal of the History of Economic Thought, Journal of the History of Economic Thought). This is only beginning, or I hope so!