Consider two players as two concurrent processes: each can make any of three decisions. If you consider their decisions separately, it’s total of 9 options, and the state space that you construct to analyze them will contain 9 elements. Reasoning with uncertainty can then consider events on this state space, and preference is free to define prior+utility for the 9 elements in any way.
But consider another way of treating this situation: instead of 9 elements in the state space, let’s introduce only 6: 3 for the first player’s decision and 3 for the second player’s. Now, the joint decision of our players is represented not by one element of the state space as in the first case, but by a pair of elements, one from each triple. The options for choosing prior+utility, and hence preference, are more limited for this state space.
In the first case, it’s unclear what could the probability of being one of the players mean: each element of the state space corresponds to both players. In the second case, it’s easy: just take the total measure of each triple.
When the decisions are dependent, the second way of treating this situation can fail, and the expressive power of expected utility become insufficient to express resulting preference.
There is an interesting extension to the question of whether indexical probability is always meaningful: is the probability of ordinary observations, even in a deterministic world, meaningful? I’m not sure it is. When you solve the decision problem, you consider preference over strategies, and a strategy includes the instructions for what to do given either observation. In the space of all possible strategies, each point considers all branches at each potential observation, just like in the example with triples of decisions above, where all 9 elements of the state space describe the decisions of both players. There doesn’t seem to be a natural way to define probability of each of the possible observations at a given observation point, starting from a distribution representing preference over possible strategies.
In the case of probability of ordinary observations, I think you can assign probabilities if your preferences over possible strategies satisfy some conditions, the major one being what you prefer to happen in one branch has to be independent of what you prefer to happen in another branch, i.e., the Axiom of Independence. If we ignore counterfactual-mugging type considerations, do you see any problems with this? If so can you give an example?
This is exactly the difference that allows to have the 6-element state space, as in the example with indexical uncertainty above, instead of more general 9-element state space. You place the possibilities in one branch by the side with the possibilities in the other branch, instead of considering all possible combinations of possibilities. It’s easy to represent various situations for which you assign probability as alternatives, lying side by side in the state space: the alternatives in different possible worlds, or counterfactuals, as they never “interact”, seem to be right to model by just considering as options, independently. The same for two physical systems that don’t interact with each other: what’s the difference between that and being in different possible worlds? - And a special case of this situation is indexical uncertainty. One condition for doing it without problem is independence. But independence isn’t really true, it’s approximation.
It’s trivial to set up the situations equivalent to counterfactual mugging, if the participants are computer programs that don’t run very far. It’s possible to prove things about where a program can go, and perform actions depending on the conclusion. What do you do then? I don’t know yet, your comment brought the idea of meaninglessness of probability of ordinary observations just yesterday, before that I didn’t notice this issue. Maybe I’ll finally find a situation where prior+utility isn’t an adequate way of representing preference, or maybe there is a good way of lifting probability of observations to probability of strategies.
Consider two players as two concurrent processes: each can make any of three decisions. If you consider their decisions separately, it’s total of 9 options, and the state space that you construct to analyze them will contain 9 elements. Reasoning with uncertainty can then consider events on this state space, and preference is free to define prior+utility for the 9 elements in any way.
But consider another way of treating this situation: instead of 9 elements in the state space, let’s introduce only 6: 3 for the first player’s decision and 3 for the second player’s. Now, the joint decision of our players is represented not by one element of the state space as in the first case, but by a pair of elements, one from each triple. The options for choosing prior+utility, and hence preference, are more limited for this state space.
In the first case, it’s unclear what could the probability of being one of the players mean: each element of the state space corresponds to both players. In the second case, it’s easy: just take the total measure of each triple.
When the decisions are dependent, the second way of treating this situation can fail, and the expressive power of expected utility become insufficient to express resulting preference.
There is an interesting extension to the question of whether indexical probability is always meaningful: is the probability of ordinary observations, even in a deterministic world, meaningful? I’m not sure it is. When you solve the decision problem, you consider preference over strategies, and a strategy includes the instructions for what to do given either observation. In the space of all possible strategies, each point considers all branches at each potential observation, just like in the example with triples of decisions above, where all 9 elements of the state space describe the decisions of both players. There doesn’t seem to be a natural way to define probability of each of the possible observations at a given observation point, starting from a distribution representing preference over possible strategies.
In the case of probability of ordinary observations, I think you can assign probabilities if your preferences over possible strategies satisfy some conditions, the major one being what you prefer to happen in one branch has to be independent of what you prefer to happen in another branch, i.e., the Axiom of Independence. If we ignore counterfactual-mugging type considerations, do you see any problems with this? If so can you give an example?
This is exactly the difference that allows to have the 6-element state space, as in the example with indexical uncertainty above, instead of more general 9-element state space. You place the possibilities in one branch by the side with the possibilities in the other branch, instead of considering all possible combinations of possibilities. It’s easy to represent various situations for which you assign probability as alternatives, lying side by side in the state space: the alternatives in different possible worlds, or counterfactuals, as they never “interact”, seem to be right to model by just considering as options, independently. The same for two physical systems that don’t interact with each other: what’s the difference between that and being in different possible worlds? - And a special case of this situation is indexical uncertainty. One condition for doing it without problem is independence. But independence isn’t really true, it’s approximation.
It’s trivial to set up the situations equivalent to counterfactual mugging, if the participants are computer programs that don’t run very far. It’s possible to prove things about where a program can go, and perform actions depending on the conclusion. What do you do then? I don’t know yet, your comment brought the idea of meaninglessness of probability of ordinary observations just yesterday, before that I didn’t notice this issue. Maybe I’ll finally find a situation where prior+utility isn’t an adequate way of representing preference, or maybe there is a good way of lifting probability of observations to probability of strategies.