Above, I compare different decision theories to FDT. At the same time, I claim that in a deeper sense, FDT is ill-defined. One may doubt whether that is a coherent line of reasoning. Therefore, instead of a comparison to FDT, I propose to frame these observations as being about stability to precommitments. Details follow.
Definition: A precommitment game is a decision problem in which 1. We are given some (the precommitment policies). We will denote . 2. We are given : for each precommitment , it says to which policy this is precomitting. 3. For any , if then : if a precommitting decision is made, then it is the only decision. 4. Denote , . (Notice that .) We are given a relation s.t. (i) implies and (ii) and implies . This tells us which precommitted outcomes correspond to which unprecomitted outcomes.
The restriction of to is called the underlying decision problem of .
Definition: An EDT precommitment game is a precommitment game which is also an EDT problem (i.e. extensive form and equipped with ) s.t. the following property holds. Denote (the “external” i.e unprecomittable policies) and . We require that there is some and s.t. 1. is a convex combination of and .[1] 2. is in the convex hull of where ranges over .
The underlying decision problem is then an EDT decision problem with the belief .
as above with is called formally causal when for any and
(Is there a natural generalization without the assumption ? I don’t know.)
Proposition: EDT is precommitment-stable in formally causal precommitment games. That is, in any such game there is which is EDT-optimal.
For example, XOR blackmail can be formalized as an EDT precommitment game which is not formally causal and EDT is not precommitment-stable there (the only optimal policy is precommitting to reject).
Definition: [EDIT: The treatment of CDT here is problematic, see child post.] A CDT precommitment game is a precommitment game which is also a CDT problem (in the sense that we are given and formally causal in the second argument) s.t. the following property holds. For any and , there is some s.t. and .
The underlying decision problem is then a CDT decision problem with and .
as above is called policy-bottlenecked when for any , . (Because this condition holds when the decision nodes form a bottleneck in the causal network s.t. the outcome depends only on nodes on the downstream side.)
Proposition: CDT is precommitment-stable in policy-bottlenecked precommitment games. That is, in any such game there is which is CDT-optimal.
For example, Newcomb’s paradox can be formalized as a CDT precommitment game is which is not policy-bottlenecked and CDT is not precommitment-stable there (the only optimal policy is precommitting to one-box).
Definition: A DDT precommitment game is a precommitment game which is also a DDT problem (in the sense that we are given ) s.t. the following property holds. For any and , if is supported on then there exists s.t. and . is called pseudocausal when we can also guarantee that .
The underlying decision problem is then a DDT decision problem with .
Proposition: IDDT is precommitment-stable in pseudocausal precommitment games. That is, in any such game there is which is IDDT-optimal.
It should be straightforward to also formulate an analogous claim with plain DDT and iterated pseudocausal precommitment games.
To make the claim that DDT/IDDT is precommitment-stable more often than EDT and CDT, we need to somehow compare different decision theories on the same game. For this purpose, we have the following translations.
Definition: Given an EDT precommitment game with , its DDT-translation is defined by setting
Proposition: If is formally causal then its DDT-translation is pseudocausal. Moreover, plain DDT is then precommitment-stable even without iteration.
Definition: Given a CDT precommitment game, its DDT-translation is defined by setting
Proposition: If is policy-bottlenecked then its DDT-translation is pseudocausal. Moreover, plain DDT is then precommitment-stable even without iteration.
The above treatment of “CDT precommitment games” is problematic: the concept made sense in the context of “FDT to CDT translation” but it’s not clear what it’s doing here (i.e. what is the first argument?) Here is a better treatment.
Definition: A CDT decision problem is the following data. We have a set of variables and for each we have its range , its set of parents and its kernel
The parent relation must induce an acyclic directed graph. We also have a selected subset of decision variables and a selected subset of outcome variables s.t. . For each there is a special element (denoting that the decision wasn’t made) and we denote . We are a given a loss function
This is connected to our overall formalism by setting and . We also define
The CDT counterfactuals and decision-rule are defined via a do-operator that forces to take the value unless the value was in which case it remains .
Definition: A CDT precommitment game is a CDT decision problem in which there is some special (the precommitment decision) s.t. , denoting , 1. 2. For some we have (where denotes the decision to not precommit). 3. For every it holds that and is s.t. if has any value other than then takes the value . 4. For every , there is a special s.t. (i) (ii) (iii) is the only variable with parent (iv) is defined s.t. if has value or then has the same value as , whereas if has value then has the value .
This is connected to our abstract notion of precommitment game by setting , and for . We define to hold whenever (i) for some (ii) (iii) for every , .
The underlying decision problem of the precommitment game is constructed by deleting and identifying and in the obvious way.
The game is said to be trivial when all variables with parent are of the form or .
Proposition: CDT is precommitment-stable in trivial precommitment games.
Definition: Given a CDT precommitment game with , its DDT-translation is defined by setting and
Proposition: If is a trivial CDT precommitment game then its DDT-translation is pseudocausal. Moreover, plain DDT is precommitment stable on the translation even without iteration.
Above, I compare different decision theories to FDT. At the same time, I claim that in a deeper sense, FDT is ill-defined. One may doubt whether that is a coherent line of reasoning. Therefore, instead of a comparison to FDT, I propose to frame these observations as being about stability to precommitments. Details follow.
Definition: A precommitment game
1. We are given some
2. We are given
3. For any
4. Denote
The restriction of
Definition: An EDT precommitment game
1.
2.
The underlying decision problem is then an EDT decision problem with the belief
(Is there a natural generalization without the assumption
Proposition: EDT is precommitment-stable in formally causal precommitment games. That is, in any such game there is
For example, XOR blackmail can be formalized as an EDT precommitment game which is not formally causal and EDT is not precommitment-stable there (the only optimal policy is precommitting to reject).
Definition: [EDIT: The treatment of CDT here is problematic, see child post.] A CDT precommitment game
The underlying decision problem is then a CDT decision problem with
Proposition: CDT is precommitment-stable in policy-bottlenecked precommitment games. That is, in any such game there is
For example, Newcomb’s paradox can be formalized as a CDT precommitment game is which is not policy-bottlenecked and CDT is not precommitment-stable there (the only optimal policy is precommitting to one-box).
Definition: A DDT precommitment game
The underlying decision problem is then a DDT decision problem with
Proposition: IDDT is precommitment-stable in pseudocausal precommitment games. That is, in any such game there is
It should be straightforward to also formulate an analogous claim with plain DDT and iterated pseudocausal precommitment games.
To make the claim that DDT/IDDT is precommitment-stable more often than EDT and CDT, we need to somehow compare different decision theories on the same game. For this purpose, we have the following translations.
Definition: Given an EDT precommitment game with
Proposition: If
Definition: Given a CDT precommitment game, its DDT-translation is defined by setting
Proposition: If
Below we only use the case
The above treatment of “CDT precommitment games” is problematic: the concept
Definition: A CDT decision problem is the following data. We have a set of variables
The parent relation must induce an acyclic directed graph. We also have a selected subset of decision variables
This is connected to our overall formalism by setting
The CDT counterfactuals and decision-rule are defined via a do-operator that forces
Definition: A CDT precommitment game is a CDT decision problem in which there is some special
1.
2. For some
3. For every
4. For every
This is connected to our abstract notion of precommitment game by setting
The underlying decision problem of the precommitment game is constructed by deleting
The game is said to be trivial when all variables with parent
Proposition: CDT is precommitment-stable in trivial precommitment games.
Definition: Given a CDT precommitment game with
Proposition: If