Good catch. To tell the truth, I didn’t even think about mixing strategies in G_M and G_S, only playing deterministically and purely “on top of” mixed strategies in G. When we add mixing, G_S does turn out to be stronger than G_M due to correlated play; your construction is very nice.
Your final result is correct, here’s a proof:
1) Any Nash equilibrium of G_S or “new” G_M plays a correlated strategy profile of G (by definition of correlated strategy profile, it’s broad enough) that on average gives each player no less than their security value (otherwise they’d switch).
2) Any such “good” profile can be implemented as a Nash equilibrium of G_S in mixed strategies, using your construction above and the usual method of punishment by security level. If all of the enemy’s strategies are quine-compatible with yours, the profile gets played exactly thanks to the niceties of your construction. If any small part of his strategies are incompatible with yours, that’s enough to bring him down on average.
3) For “new” G_M you just submit the profile and the punishment fallback. So “new” G_M doesn’t actually need mixed strategies, our latest deus ex machina is too strong.
Thanks, Benja and cousin_it, for working out the math. The “new” G_M looks correct to me (and let’s just call it G_M from now, since I see no reason why the joint machine can’t be programmed with a correlated strategy profile). To make it a little more realistic, we can add a step after the joint machine is constructed, where each player has a choice to transfer his resources or not. It’s pretty obvious that adding this step makes no difference in the outcome, since the players can program the joint machine to execute their fallback strategies if one of the player fails to transfer.
Benja’s method for G_S to enforce correlated play in G seems to require simultaneous source-swapping. Otherwise, if I choose my random number first, then the second player can pick his number to favor him, right? It seems that the common quined program has to use some cryptographic method to generate a common random number in a way that’s not vulnerable to manipulation by the players.
Good catch. To tell the truth, I didn’t even think about mixing strategies in G_M and G_S, only playing deterministically and purely “on top of” mixed strategies in G. When we add mixing, G_S does turn out to be stronger than G_M due to correlated play; your construction is very nice.
Your final result is correct, here’s a proof:
1) Any Nash equilibrium of G_S or “new” G_M plays a correlated strategy profile of G (by definition of correlated strategy profile, it’s broad enough) that on average gives each player no less than their security value (otherwise they’d switch).
2) Any such “good” profile can be implemented as a Nash equilibrium of G_S in mixed strategies, using your construction above and the usual method of punishment by security level. If all of the enemy’s strategies are quine-compatible with yours, the profile gets played exactly thanks to the niceties of your construction. If any small part of his strategies are incompatible with yours, that’s enough to bring him down on average.
3) For “new” G_M you just submit the profile and the punishment fallback. So “new” G_M doesn’t actually need mixed strategies, our latest deus ex machina is too strong.
Thanks, Benja and cousin_it, for working out the math. The “new” G_M looks correct to me (and let’s just call it G_M from now, since I see no reason why the joint machine can’t be programmed with a correlated strategy profile). To make it a little more realistic, we can add a step after the joint machine is constructed, where each player has a choice to transfer his resources or not. It’s pretty obvious that adding this step makes no difference in the outcome, since the players can program the joint machine to execute their fallback strategies if one of the player fails to transfer.
Benja’s method for G_S to enforce correlated play in G seems to require simultaneous source-swapping. Otherwise, if I choose my random number first, then the second player can pick his number to favor him, right? It seems that the common quined program has to use some cryptographic method to generate a common random number in a way that’s not vulnerable to manipulation by the players.