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by 258 122 days ago | Alex Appel and Abram Demski like this | link | parent

Since Briggs [1] shows that EDT+SSA and CDT+SIA are both ex-ante-optimal policies in some class of cases, one might wonder whether the result of this post transfers to EDT+SSA. I.e., in memoryless POMDPs, is every (ex ante) optimal policy also consistent with EDT+SSA in a similar sense. I think it is, as I will try to show below.

Given some existing policy \(\pi\), EDT+SSA recommends that upon receiving observation \(o\) we should choose an action from \[\arg\max_a \sum_{s_1,...,s_n} \sum_{i=1}^n SSA(s_i\text{ in }s_1,...,s_n\mid o, \pi_{o\rightarrow a})U(s_n).\] (For notational simplicity, I’ll assume that policies are deterministic, but, of course, actions may encode probability distributions.) Here, \(\pi_{o\rightarrow a}(o')=a\) if \(o=o'\) and \(\pi_{o\rightarrow a}(o')=\pi(o')\) otherwise. \(SSA(s_i\text{ in }s_1,...,s_n\mid o, \pi_{o\rightarrow a})\) is the SSA probability of being in state \(s_i\) of the environment trajectory \(s_1,...,s_n\) given the observation \(o\) and the fact that one uses the policy \(\pi_{o\rightarrow a}\).

The SSA probability \(SSA(s_i\text{ in }s_1,...,s_n\mid o, \pi_{o\rightarrow a})\) is zero if \(m(s_i)\neq o\) and \[SSA(s_i\text{ in }s_1,...,s_n\mid o, \pi_{o\rightarrow a}) = P(s_1,...,s_n\mid \pi_{o\rightarrow a}) \frac{1}{\#(o,s_1,...,s_n)}\] otherwise. Here, \(\#(o,s_1,...,s_n)=\sum_{i=1}^n \left[ m(s_i)=o \right]\) is the number of times \(o\) occurs in \(\#(o,s_1,...,s_n)\). Note that this is the minimal reference class version of SSA, also known as the double-halfer rule (because it assigns 1/2 probability to tails in the Sleeping Beauty problem and sticks with 1/2 if it’s told that it’s Monday).

Inserting this into the above, we get \[\arg\max_a \sum_{s_1,...,s_n} \sum_{i=1}^n SSA(s_i\text{ in }s_1,...,s_n\mid o, \pi_{o\rightarrow a})U(s_n)=\arg\max_a \sum_{s_1,...,s_n\text{ with }o} \sum_{i=1...n, m(s_i)=o} \left( P(s_1,...,s_n\mid \pi_{o\rightarrow a}) \frac{1}{\#(o,s_1,...,s_n)} \right) U(s_n),\] where the first sum on the right-hand side is over all histories that give rise to observation \(o\) at some point. Dividing by the number of agents with observation \(o\) in a history and setting the policy for all agents at the same time cancel each other out, such that this equals \[\arg\max_a \sum_{s_1,...,s_n\text{ with }o} P(s_1,...,s_n\mid \pi_{o\rightarrow a}) U(s_n)=\arg\max_a \sum_{s_1,...,s_n} P(s_1,...,s_n\mid \pi_{o\rightarrow a}) U(s_n).\] Obviously, any optimal policy chooses in agreement with this. But the same disclaimers apply; multiple policies satisfy the right-hand side of this equation and not all of these are optimal.

[1] Rachael Briggs (2010): Putting a value on Beauty. In Tamar Szabo Gendler and John Hawthorne, editors, Oxford Studies in Epistemology: Volume 3, pages 3–34. Oxford University Press, 2010.

by 258 105 days ago | Abram Demski and Jessica Taylor like this | link

Caveat: The version of EDT provided above only takes dependences between instances of EDT making the same observation into account. Other dependences are possible because different decision situations may be completely “isomorphic”/symmetric even if the observations are different. It turns out that the result is not valid once one takes such dependences into account, as shown by Conitzer [2]. I propose a possible solution in . Roughly speaking, my solution is to identify with all objects in the world that are perfectly correlated with you. However, the underlying motivation is unrelated to Conitzer’s example.

[2] Vincent Conitzer: A Dutch Book against Sleeping Beauties Who Are Evidential Decision Theorists. Synthese, Volume 192, Issue 9, pp. 2887-2899, October 2015.






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