Riches and Rationality

ABSTRACT

A one-boxer, Erica, and a two-boxer, Chloe, engage in a familiar debate. The debate begins with Erica asking Chloe, ‘If you’re so smart, then why ain’cha rich?’ As the debate progresses, Chloe is led to endorse a novel causalist theory of rational choice. This new theory allows her to forge a connection between rational choice and long-run riches. In brief, Chloe concludes that it is not long-run wealth but rather long-run wealth creation that is symptomatic of rationality.

KEYWORDS:

• causal decision theory
• ‘Why ain’cha rich?’
• decision instability

Disclosure Statement

No potential conflict of interest was reported by the author.

Notes

1 The probability that Newcomb predicted that they would $X$, given that they $X$, is 90%.

2 Neither Erica nor Chloe is risk-averse.

3 This argument is made by Gibbard and Harper [1978].

4 Erica’s views here agree with Ahmed’s [2014: sec. 7.3].

5 Erica is appealing to the weak law of large numbers. If she were more careful, she would have said ‘For any ϵ > 0, the probability that my average return and my expected return differ by no more than ϵ approaches 100% as we play the game more and more times.’

6 This reply is offered by Lewis [1981b] and Joyce [1999], among others.

7 Erica forgets to note that these $K_i$s are what Lewis [1981a] calls ‘dependency hypotheses’, and they are mutually exclusive and jointly exhaustive. (By the way, Erica doesn’t think that she has to calculate her expected return by using these kinds of states; any other set of mutually exclusive and jointly exhaustive states would do just as well. Even so, this is how she does it.)

8 Skyrms [1980] evaluates options with $\mathbf{C}\mathbf{E}\mathbf{R}$. Other versions of causal decision theory use similar expectations. See Lewis [1981a].

9 See Gallow [ms. forthcoming] for discussion of additional complications in choices with three or more options.

10 Harper [1986: 33] says this.

11 This deliberative vacillation is recommended by Skyrms [1990] and Joyce [2012]. See also Arntzenius [2008].

12 Chloe needn’t worry—there will always be an equilibrium. See [Skyrms 1990] and Arntzenius [2008]. (There could be more than one equilibrium. Chloe should worry about that, but it doesn’t occur to her.)

13 $Pr\left(B\right)=\left[Pr\left(K_B\right)-Pr\left(K_B|W\right)\right]÷\left[Pr\left(K_B|B\right)-\mathrm{P}\mathrm{r}\left(K_B\mid W\right)\right]$

14 Chloe assumes that, for any proposition $\phi$ and any partition $\left\{Z_1,Z_2,\dots ,Z_N\right\}$, $V\left(\phi \right)=\sum _{i}Pr\left(Z_i|\phi \right)\cdot V\left(\phi Z_i\right)$. That’s why she allows herself to replace ‘$V\left(S^{\ast }K\right)$’ in $\mathbf{C}\mathbf{E}\mathbf{R}\left(S^{\ast }\mid S\right)$ with $\sum _{A}Pr\left(A|S^{\ast }K\right)\cdot V\left(A{S}^{\ast }K\right)$. Because Chloe doesn’t intrinsically value strategies, $V\left(A{S}^{\ast }K\right)=V\left(AK\right)$.

15 In a choice between two options, a strategy will be rational according to Chloe’s Third Proposal iff that strategy is an equilibrium, in the sense of Skyrms’s [1990] deliberational causal decision theory. The proposal is similarly endorsed by Arntzenius [2008] and Joyce [2012].

16 Similar decisions are discussed in [Egan 2007].

17 She sets $\mathbf{C}\mathbf{E}\mathbf{R}\left(X\right)$ equal to $\mathbf{C}\mathbf{E}\mathbf{R}\left(Y\right)$ and solves for $Pr\left(X\right)$, getting $Pr\left(X\right)=7/8$.

18 She writes this: $Pr\left(K_X|S_e\right)=Pr\left(K_X|X{S}_e\right)\cdot Pr\left(X|S_e\right)+Pr\left(K_X|Y{S}_e\right)\cdot Pr\left(Y|S_e\right)=10\text{\%}\cdot \left(7/8\right)+90\text{\%}\cdot \left(1/8\right)=1/5$.

19 This is what you would expect because $\mathbf{E}\mathbf{R}\left(S_e\right)=Pr\left(Y|S_e\right)\cdot 20+Pr\left(X|K_X{S}_e\right)\cdot Pr\left(K_X|S_e\right)\cdot 100=\left(1/8\right)\cdot 20+\left(7/16\right)\cdot \left(1/5\right)\cdot 100=11.25$.

20 Given a choice between two options, both Gallow [ms. forthcoming] and Barnett [ms.] say to minimize your Chloe loss. Due to complications that Erica and Chloe don’t have time to discuss, the former manuscript accepts a slightly different theory for choices between three or more options.

21 Thanks to Ray Briggs, Daniel Drucker, Adam Elga, James Shaw, and two anonymous reviewers for helpful conversations and feedback on this material.