Models of decision-making
based on logical counterfactuals
Vladimir Slepnev
Goals
Modeling human decision-making is not really a goal for me.
Neither is building efficient AI algorithms.
What is the “correct” decision, never mind the reasoning cost?
What is the “correct” way of decision-making that we’re trying to approximate?
Overview
A simple decision problem
“Would you like some chocolate?”
A simple decision problem
“Would you like some chocolate?”
Desiderata for a model:
Our proposed model
Our proposed model
A mutually recursive definition of U and A:
“If you say "yes", I will give you chocolate, otherwise I won't.”
“If I can prove that saying “yes” leads to chocolate, then I say “yes”, otherwise “no”.”
All self-references occur within Gödel number quotes, therefore such U and A exist, by the Diagonal Lemma.
Analysis
It’s easy to prove that U and A are both true.
Analysis
It’s easy to prove that U and A are both true.
What if we changed the problem a little? Reward “no” with chocolate:
Now A is false (as long as PA is consistent), and U is again true.
It feels like A is trying to make U true, in order to get some chocolate :-)
But does it generalize?
Newcomb’s problem
Newcomb’s problem
We will define these sentences in PA:
We will use these equations:
Newcomb’s problem
“The predictor predicts that I one-box iff I actually one-box.”
“I get the contents of the first box iff I two-box.”
“I get the contents of the second box iff the predictor predicted that I would one-box.”
“If I can get the contents of both boxes by one-boxing, then I one-box;
otherwise, if I can get both boxes by two-boxing, then I two-box;
otherwise, if I can get only box 2 by one-boxing, then I one-box;
otherwise, if I can get only box 2 by two-boxing, then I two-box;
otherwise, if I can get only box 1 by one-boxing, then I one-box;
otherwise I two-box.”
Newcomb’s problem
The completed equations:
It’s easy to prove that A is true, B1 is false, and B2 is true.
Thus, our approach favors one-boxing.
Absent-minded driver problem
(Slightly modified from Piccione and Rubinstein, 1997)
Absent-minded driver problem
We will define these sentences in PA:
U11 ↔ U22 ↔ ⊥
U12 ↔ A1∧￢A2
U21 ↔ ￢A1∧A2
A1 ↔ ?
A2 ↔ ?
Absent-minded driver problem
A1 ↔ ?
A2 ↔ ?
“If A1∧A2 provably implies that all Uij are true, then make A1 and A2 true;
otherwise, if A1∧￢A2 provably implies that all Uij are true, then make A1 true and A2 false;
{...}
otherwise, if A1∧A2 provably implies that exactly three of Uij are true, then make A1 and A2 true;
{…}”
The equations begin like this:
Other proposed models
Using Gödel-Löb provability logic instead of PA:
Using computer programs that look for proofs, instead of arithmetic formulas:
Further work
From decision theory to game theory
From perfect certainty to uncertainty
Questions?