Abstract
Many will find the answer easy—though they may disagree with each other on which the answer is. A standard line on the prisoner’s dilemma rests on dominance: What you do won’t affect what Twin does. Twin may rat or keep mum, but in either case, you yourself will do better to rat. Whichever Twin is doing, you would spend less time in jail if you were to rat than if you were to keep mum. Therefore the rational way to minimize your own time in jail is to rat.
In the nearly 20 years since this article was written there has been a revolution in the understanding of causal and counterfactual reasoning. This revolution had its roots in early work by Rubin (1974), Holland (1986) and Robbins (1986), which gave rise to the so-called “potential outcomes” framework. At roughly the same time the closely related “structural equations/causal graphs” approach was being developed and used to great effect by Spirtes et al. (1993), and Pearl (2000). In both treatments counterfactual reasoning plays a leading role in causal inference, just as in causal decision theory. While the core claims of this article remain true, and the basic structure of causal decision theory remains intact, these new models of provide us with far more sophisticated ways of representing and identifying causal relationships than were available and widely known when we wrote. As a result, some of our remarks about “the need for new advances in understanding of localization in relation to rational belief” have been rendered moot. Readers are encouraged to investigate these new developments, which we see as great advances.
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Notes
- 1.
An interesting model of Prisoner’s Dilemma with a twin can be found in Howard (1988). Howard, who endorses a version of the auspiciousness argument, shows how to write a Basic program for playing the game which is capable of recognizing and cooperating with programs that are copies of itself.
- 2.
Nozick (1969) introduced PDT and other cases of this kind, focusing his discussion on Newcomb’s problem, which he credits to physicist William Newcomb. He makes many of the points that causal theorists have come to accept, but recognizes only one kind of expected utility, the one we are calling auspiciousness. Stalnaker originated causal expected utility in a 1972 letter published only much later (Stalnaker 1981). Gibbard and Harper (1978) proselytize Stalnaker’s proposal, and Lewis (1981) gives an alternative formulation which we discuss below. Gibbard and Harper (1978) and Lewis (1979b) also discuss PDT along with Newcomb’s problem.
- 3.
More precisely, A i is the proposition that one performs a particular one of the alternative acts open to one in one’s circumstances. We reserve the notation ρ(S|A j ) for a more general use later in this chapter.
- 4.
- 5.
Adams (1975) examines pairs like this.
- 6.
- 7.
Fine (1975) gives this example to make roughly this point.
- 8.
Shin (1991a), for instance, devises a metric that seems suitable for simple games such as “Chicken”.
- 9.
Lewis (1973) constructs a system in which worlds may tie for most similar, or it may be that for every A-world, there is an A-world that is more similar. He thus denies Conditional Excluded Middle: It fails, for instance, when two A-worlds tie for most similar to the actual world, one a C-world and the other a ¬C-world.
- 10.
Skyrms (1980) offers another formulation, invoking a distinction between factors that are within the agent’s control and factors that aren’t. Lewis (1981) discusses both Skyrms and unpublished work of Jordan Howard Sobel, and Skyrms (Skyrms 1984), 105–6, compares his formulation with those of Lewis (1981) and Stalnaker (1981).
- 11.
- 12.
Gibbard and Harper (1978), 140–2, construct an example of such a case.
- 13.
See, for example, Horgan (1981).
- 14.
Jeffrey (1983) p. 16.
- 15.
- 16.
Piccione and Rubinstein (1997) present another kind of case in which considerations of ratifiability may be invoked: the case of the “absent-minded driver” who can never remember which of two intersections he is at. One solution concept they consider (but reject) is that of being “modified multi-selves consistent”. In our terms, this amounts to treating oneself on other occasions as a twin, selecting a strategy that is ratifiable on the following assumption: that one’s present strategy is fully predictive of one’s strategy in any other situation that is subjectively just like it. This turns out to coincide with the “optimal” strategy, the strategy one would adopt if one could choose in advance how to handle all such situations.
- 17.
- 18.
Note that such a distribution determines a unique mixed act for each player. Thus, it makes no difference whether one talks about the players’ acts or the players’ beliefs being in equilibrium.
- 19.
- 20.
See Shin (1991b), for instance, for an interesting ratificationist gloss on Selten’s notion of a “perfect” equilibrium.
- 21.
- 22.
Harper (1991), 293.
- 23.
This representation will not be unique (except in the rare case where \( \mathcal{V} \) is unbounded), for, as a simple calculation shows, if the function \( \mathcal{V}(A)=\sum\rho({S_i}/A)u(A,S_i) \) represents a preference ordering, and if k is such that 1 + k \( \mathcal{V} \)(X) > 0 for all propositions X in the algebra over which \( {\mathcal{V}_k}(A)=\sum\rho k({S_i}/A){u_k}(A,S_i) \) is defined, then \( {V}_k(A)={\displaystyle \sum \rho k}\left({S}_i/A\right){u}_k\left(A,{S}_i\right) \) will also represent the ordering, when pk(X) = ρ(X)(l + k \( \mathcal{V} \)(X)) and \( \mathcal{V} \) k (X) = [\( \mathcal{V} \)(X)(1 + k)]/(l + k \( \mathcal{V} \)(X)).
- 24.
- 25.
Notice that states are being viewed here as functions from acts to outcomes, whereas acts are taken as unanalyzed objects of choice (that is, as propositions the agent can make true or false as she pleases). This contrasts with Savage’s well-known formalization in which acts are protrayed as functions from states to outcomes, and states are left as unanalyzed objects of belief. Less hangs on this distinction than one might think. When one adopts the perspective of Jeffrey (1967, 1983) and interprets both states and actions as propositions, and views outcomes as conjunctions of these propositions, the two analyses become interchangeable.
- 26.
Here we are following Gibbard (1986).
- 27.
An explicit construction of K can be found in Gibbard and Harper (1978).
- 28.
The set C always takes the form C = Ω ∼ I, where the ideal I is a collection of Ω-propositions that contains the contradictory event (X&¬X), is closed under countable disjunctions, and which contains (X&Y) whenever X ∈ I and Y ∈Ω.
- 29.
- 30.
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Joyce, J.M., Gibbard, A. (2016). Causal Decision Theory. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_23
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