Quantitative Counterfactual Reasoning Research Paper

1. Counterfactuals

Counterfactuals are a species of conditionals. They are propositions or sentences, expressed by or equivalent to subjunctive conditionals of the form ‘if it were the case that A, then it would be the case that B,’ or ‘if it had been the case that A, then it would have been the case that B’; A is called the antecedent, and B the consequent. Counterfactual reasoning typically involves the entertaining of hypothetical states of affairs: the antecedent is believed or presumed to be false, or contrary-to-the-fact, but its truth is imagined or supposed. Counterfactual reasoning is thus a form of modal reasoning, kindred to reasoning about necessity or possibility, and in constrast to reasoning about the way things actually are.

The philosophical study of conditionals goes back at least as far as the Stoics of ancient Greece, although their systems of logic apparently did not accord the counterfactual any emphasis. The rise in interest in counterfactuals has been a rather recent phenomenon, as it started to become clear to philosophers that counterfactuals are implicated in a host of other important concepts—laws of nature, confirmation, causation, scientific explanation, knowledge, perception, dispositions, free action, etc. The significance of counterfactuals has also become increasingly appreciated in the social sciences—by psychologists (e.g., in understanding emotions such as regret), by historians (e.g., in studying the incremental contribution by some commodity to economic growth), in the law (e.g., in apportioning responsibility), etc. However, a widely agreed upon analysis of counterfactuals has proved elusive. The so-called problem of counterfactuals, raised notably by Chisholm and by Goodman in the 1940s, is that of providing adequate (in particular, noncircular) truth conditions for the counterfactual.

Let ‘A□→B’ denote a counterfactual with antecedent A and consequent B. Fixing the truth value of A and B clearly does not fix the truth value of A□→ B. For example, ‘if I were to drop this fragile glass, it would break’ is true, ‘if I were to drop this fragile glass, it would turn into a hamster’ is false, yet the antecedents (‘I drop this fragile glass’) and consequents (‘the glass breaks,’ ‘the glass turns into a hamster’) are all false. This rules out as an adequate analysis the material conditional AↄB (defined to be equivalent to ‘not-A or B’), which is automatically true when A is false. According to metalinguistic approaches, A□→B is true if and only if A, together with suitable further premises (on one version, stating appropriate laws of nature and initial conditions), implies B. Mackie takes counterfactuals to be themselves elliptical presentations of such arguments, and thus lacking truth values altogether. The advent of possible worlds semantics for modal logics in the 1960s gave new direction to the analysis of counterfactuals. Truth conditions for them are proposed in classic works by Stalnaker (1968) and Lewis (1973).

Stalnaker motivates his account by Ramsey’s recipe for counterfactual reasoning (‘Ramsey’s test’): First, add the antecedent (hypothetically) to your stock of beliefs; second, make the minimal adjustments required in the rest of your beliefs to maintain consistency; finally consider whether or not you then believe the consequent. Stalnaker’s truth conditions are intended to apply generally to conditionals of the form ‘if A, then B,’ which Stalnaker denotes ‘A B,’ pragmatics distinguishing between indicative conditionals and counterfactuals. He assumes that for each world w there is an ordering of worlds according to their similarity, or ‘closeness,’ to w. Call a world at which A is true an A-world; the proposition A is regarded as the set of A-worlds. His guiding idea is that A>B is true at a world w if and only if B is true at the closest A-world to w.

Lewis rejects Stalnaker’s assumption that, for any given w and A, there is exactly one closest A-world to w. His truth conditions take as primitive the notion of comparative closeness of worlds: A□→B is true at w if and only if some AB-world is closer to w than any AB-world. For both Stalnaker and Lewis, the counter-factual is vacuously true if the antecedent is impossible.

2. Probabilities Of Counterfactuals

One important line of research considers counterfactuals within the quantitative framework of probability theory. In particular, there has been much interest in the connection between probabilities of conditionals (of which counterfactuals are an important class) and conditional probabilities. Let P be your subjective probability function, so that P(A□→B) is your degree of belief in A□→B. There are various reasons (linguistic intuitions, considerations stemming from Ramsey’s test, etc.) for thinking that counterfactual reasoning is, or should be, subject to the following constraint: (*) P(A□→B)=P(B|A) for all A, B in the domain of P such that P(A)>0.

Stalnaker proposed a version of this identity, which can be shown to vindicate his truth conditions above, and to vitiate Lewis’s. In response, Lewis (most famously in 1976) proved a number of ‘triviality results’: under certain assumptions, the only classes of probability functions that can sustain (*) consist solely of members that take at most four values, and that are thus trivial. Since then, a number of authors have strengthened these results (see Eells and Skyrms 1994 for a collection of relevant articles).

Nonetheless, an identity somewhat in the spirit of (*) does hold for the Stalnaker conditional. Start with probability function P. Among other things, P assigns probabilities to individual worlds. Define a new probability function PA as follows: for each world w, PA shifts the probability that P assigns to w to the closest A-world to w. PA is said to be derived from P by imaging on A. Some authors model counterfactual reasoning by imaging (eschewing conditional probabilities as in (*)). Lewis (1976) shows that there is an intimate connection between the Stalnaker conditional and imaging: P(A>B) =P_A(B) (for all P, A and B). The exploration of tenable variants of (*) is likely to continue to be a lively area of research.

3. Counterfactual Reasoning In Decision Theory

The distinction between probabilities of conditionals and conditional probabilities is important for decision theory, a normative theory of how one’s beliefs and desires in tandem determine what one should do. Such a theory typically combines an agent’s utility function u and probability function P to give a figure of merit for each possible action, called the expectation or desirability of that action. Let S₁,S₂,…,S_n be a partition of possible states of the world. Evidential decision theory, as presented by Jeffrey (1966), measures the choiceworthiness of action A by a conditional probability-weighted average of the utilities of its possible outcomes (the conjunctions A and S_i):

It thus prescribes that an agent acts so as to maximize V. Various authors believe that evidential decision theory gives incorrect verdicts in Newcomb problem cases, in which an action is thought to be evidence for the obtaining of a given state, without in any way causing that state (see Newcomb’s problem). This has led to the development of rival versions of decision theory, sharing the name causal decision theory, that replace the ‘evidential’ conditional probability weights with weights that are supposed to capture the agent’s degrees of belief about the causal efficacy of the action in bringing about each possible state of affairs. Gibbard and Harper (1981), following Stalnaker, use probabilities of counterfactuals as the weights

They then advocate the maximization of U. Proponents of causal decision theory believe that it agrees with evidential decision theory in unproblematic cases, and that it delivers intuitively correct answers in Newcomb problem cases.

4. Counterfactual Reasoning In Game Theory

Counterfactual reasoning can play a crucial role in the rational deliberation of two or more agents strategically playing against each other, their pay-offs de-pending on what they all do—the province of game theory. Paralleling the distinction we found in decision theory, Stalnaker (1996) distinguishes two sorts of counterfactual reasoning, causal and epistemic (the latter could also be called evidential). In the former, agents reason about what would happen if they or others were to act in ways that they know or believe to be nonactual. In the latter, they consider how their own or others’ beliefs would change if they were to learn things that they expect not to learn, for example, contrary to what is standardly taken to be common knowledge (and thus regarded as certain) by all the players, an irrational choice is hypothetically made. Indeed, such (epistemic) counterfactual reasoning leads naturally to some important refinements of Nash equilibrium, such as Selten’s subgame perfect and (‘trembling hand’) perfect equilibrium.

Various models have been proposed of how an agent’s beliefs are disposed to change in the face of unexpected actions by others, and the consequences of such dispositions for rational action—see, for ex-ample, Bicchieri (1988) (for extensive form games) and Stalnaker (1996) (for normal form games). Closely related is Shin’s (1989) theory of equilibrium in normal-form games. He gives a counterfactual-based account of rationality—a rational agent never acts in such a way that he would have done better had he acted differently—and proves that his account is equivalent, under certain assumptions, to those that employ game theory’s standard solution concepts of Nash equilibrium and correlated equilibrium. Skyrms (1998) integrates the themes that we have discussed in this and the previous two sections. He gives a probabilistic theory of subjunctive conditionals that finds its inspiration in (*), with ‘assertability value’ replacing the ‘P’ on the left-hand side, and expectation of conditional chance replacing the right-hand side. He then uses this theory to give a unified treatment of counterfactual reasoning in individual decision theory, and in normal form and extensive form games.

Finally, in what might be taken as an indicator of a future trend in this field, more and more sophisticated probabilistic models are being developed for the representation of conditional beliefs in games. The economists Battigalli and Siniscalchi (1998), for ex-ample, construct a space of infinite hierarchies of conditional probability systems in which one can represent any statement about the players’ dispositions to hold (arbitrarily high-order) beliefs regarding the other players.

Bibliography:

1. Battigalli P, Siniscalchi M 1998 Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games. EUI Working Paper ECO No. 98 29, European University Institute, France, Italy
2. Bicchieri C 1988 Strategic behavior and counterfactuals. Synthese 76: 135–69
3. Eells E, Skyrms B (eds.) 1994) Probability and Conditionals. Cambridge University Press, Cambridge, UK
4. Gibbard A, Harper W L 1981 Counterfactuals and two kinds of expected utility. In: Harper W L, Stalnaker R, Pearce G (eds.) Ifs. D. Reidel, Boston
5. Jeffrey R C 1966 The Logic of Decision. Chicago University Press, Chicago
6. Lewis D 1976 Probabilities of conditionals and conditional probabilities. Philosophical Review 85: 297–315
7. Lewis D K 1973 Counterfactuals. Basil Blackwell, Oxford, UK
8. Shin H S 1989 Two notions of ratifiability and equilibrium in games. In: Bacharach M, Hurley S (eds.) Foundations of Decision Theory. Blackwell, Oxford, UK
9. Skyrms B 1998 Subjunctive conditionals and revealed preference. Philosophy of Science 65: 545–74
10. Stalnaker R 1968 A theory of conditionals. In: Cornman J (ed.) Studies in Logical Theory. Basil Blackwell, Oxford, UK
11. Stalnaker R 1996 Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy 12: 133–63