Donald Hobson comments on Confusions re: Higher-Level Game Theory

Donald Hobson 3 Jul 2021 14:58 UTC
LW: 4 AF: 2
AF
In a game with any finite number of players, and any finite number of actions per player.
Let $O = A_{1} \times A_{2} \times . . .$ the set of possible outcomes.
Player $i$ implements policy $P_{i} : P (O) \to A_{i}$ . For each outcome in $o \in O$ , each player searches for proofs (in PA) that the outcome is impossible. It then takes the set of outcomes it has proved impossible, and maps that set to an action.
There is always a unique action that is chosen. Whatsmore, given oracles for
$Q_{i} (U) = \cup_{U \subseteq V} P_{i} (V) : P (O) \to P (A_{i})$
Ie the set of actions you might take if you can prove at least the impossility results in $U$ and possibly some others.
Given such an oracle $Q_{i}$ for each agent, there is an algorithm for their behaviour that outputs the fixed point in polynomial (in $| O |$ ) time.