Mykhailo Odintsov

Software engineer and small time DS/ML practitioner.

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Open problem: how can we quantify player alignment in 2x2 normal-form games?

So, something like "fraction of preferred states shared" ? Describe preferred states for P1 as cells in the payoff matrix that are best for P1 for each P2 action (and preferred stated for P2 in a similar manner) Fraction of P1 preferred states that are also preferred for P2 is measurement of alignment P1 to P2. Fraction of shared states between players to total number of preferred states is measure of total alignment of the game.

For 2x2 game each player will have 2 preferred states (corresponding to the 2 possible action of the opponent). If 1 of them will be the same cell that will mean that each player is 50% aligned to other (1 of 2 shared) and the game in total is 33% aligned (1 of 3), This also generalize easily to NxN case and for >2 players.

And if there are K multiple cells with the same payoff to choose from for some opponent action we can give 1/K to them instead of 1.

(it would be much easier to explain with a picture and/or table, but I'm pretty new here and wasn't able to find how to do them here yet)