I wouldn’t say that is a clear exception. There are perfectly normal, subjective probability ways to make sense of mixed strategies in game theory. For example, this paper by Aumann and Brandenburger provides epistemic conditions for Nash equilibria, that don’t require objective probabilities to randomize. From their paper:
“Mixed strategies are treated not as conscious randomizations, but as conjectures, on the part of other players, as to what a player will do.” (p. 1161)
In slightly more detail:
“According to [our] view, players do not randomize; each player chooses some definite action. But other players need not know which one, and the mixture represents their uncertainty, their conjecture about his choice. This is the context of our main results, which provide sufficient conditions for a probability of conjectures to constitute a Nash equilibrium.” (p. 1162)
Interestingly, this paper is very motivated by embedded agency type concerns. For example, on page 1174 they write:
“Though entirely apt, use of the term “state of the world” to include the actions of the players has perhaps caused confusion. In Savage (1954), the decision maker cannot affect the state; he can only react to it. While convenient in Savage’s one person context, this is not appropriate in the interactive, many-person world under study here. Since each player must take into account the actions of the others, the actions should be included in the description of the state. Also the plain, everyday meaning of the term “state of the world” includes one’s actions: Our world is shaped by what we do. It has been objected that prescribing what a player must do at a state takes away his freedom. This is nonsensical; the player may do what he wants. It is simply that whatever he does is part of the description of the state. If he wishes to do something else, he is heartily welcome to do it, but he thereby changes the state.”
In general, getting back to reflective oracles, indeed I think that is one way that one might try to provide a formalism underlying some application of game theory! And I think it is a very interesting one. But, as the Aumann and Brandenburger paper shows, there are totally normal ways to do this without fundamental chance. They have some references in their paper to other papers with this perspective, and it forms one of many motivations for the approach of epistemic game theory.
And, in general, I would resist the inference from “this kind of reasoning requires the world to be a certain way” to “the world must be a certain way”.
I wouldn’t say that is a clear exception. There are perfectly normal, subjective probability ways to make sense of mixed strategies in game theory. For example, this paper by Aumann and Brandenburger provides epistemic conditions for Nash equilibria, that don’t require objective probabilities to randomize. From their paper:
“Mixed strategies are treated not as conscious randomizations, but as conjectures, on the part of other players, as to what a player will do.” (p. 1161)
In slightly more detail:
“According to [our] view, players do not randomize; each player chooses some definite action. But other players need not know which one, and the mixture represents their uncertainty, their conjecture about his choice. This is the context of our main results, which provide sufficient conditions for a probability of conjectures to constitute a Nash equilibrium.” (p. 1162)
Interestingly, this paper is very motivated by embedded agency type concerns. For example, on page 1174 they write:
“Though entirely apt, use of the term “state of the world” to include the actions of the players has perhaps caused confusion. In Savage (1954), the decision maker cannot affect the state; he can only react to it. While convenient in Savage’s one person context, this is not appropriate in the interactive, many-person world under study here. Since each player must take into account the actions of the others, the actions should be included in the description of the state. Also the plain, everyday meaning of the term “state of the world” includes one’s actions: Our world is shaped by what we do. It has been objected that prescribing what a player must do at a state takes away his freedom. This is nonsensical; the player may do what he wants. It is simply that whatever he does is part of the description of the state. If he wishes to do something else, he is heartily welcome to do it, but he thereby changes the state.”
In general, getting back to reflective oracles, indeed I think that is one way that one might try to provide a formalism underlying some application of game theory! And I think it is a very interesting one. But, as the Aumann and Brandenburger paper shows, there are totally normal ways to do this without fundamental chance. They have some references in their paper to other papers with this perspective, and it forms one of many motivations for the approach of epistemic game theory.
And, in general, I would resist the inference from “this kind of reasoning requires the world to be a certain way” to “the world must be a certain way”.
Edit: Lightly edited for typos.