After someone chooses 30 once, they still get to choose something different in future rounds. In the strategy profile I claim is a Nash equilibrium, they’ll set it to 100 next round like everyone else. If anyone individually deviates from setting it to 100, then the equilibrium temperature in the next round will also be 100. That simply isn’t worth it, if you expect to be the only person setting it less than 100. Since in the strategy profile I am constructing everyone does set it to 100, that’s the condition we need to check to check whether it’s a Nash equilibrium.
I guess the unstated assumption is that the prisoners can only see the temperatures of others from the previous round and/or can only change their temperature at the start of a round (though one tried to do otherwise in the story). Even with that it seems like an awfully precarious equilibrium since if I unilaterally start choosing 30 repeatedly, you’d have to be stupid to not also start choosing 30, and the cost to me is really quite tiny even while no one else ever ‘defects’ alongside me. It seems to be too weak a definition of ‘equilibrium’ if it’s that easy to break—maybe there’s a more realistic definition that excludes this case?
The other thing that could happen is silent deviations, where some players aren’t doing “punish any defection from 99”—they are just doing “play 99″ to avoid punishments. The one brave soul doesn’t know how many of each there are, but can find out when they suddenly go for 30.
After someone chooses 30 once, they still get to choose something different in future rounds. In the strategy profile I claim is a Nash equilibrium, they’ll set it to 100 next round like everyone else. If anyone individually deviates from setting it to 100, then the equilibrium temperature in the next round will also be 100. That simply isn’t worth it, if you expect to be the only person setting it less than 100. Since in the strategy profile I am constructing everyone does set it to 100, that’s the condition we need to check to check whether it’s a Nash equilibrium.
I guess the unstated assumption is that the prisoners can only see the temperatures of others from the previous round and/or can only change their temperature at the start of a round (though one tried to do otherwise in the story). Even with that it seems like an awfully precarious equilibrium since if I unilaterally start choosing 30 repeatedly, you’d have to be stupid to not also start choosing 30, and the cost to me is really quite tiny even while no one else ever ‘defects’ alongside me. It seems to be too weak a definition of ‘equilibrium’ if it’s that easy to break—maybe there’s a more realistic definition that excludes this case?
The other thing that could happen is silent deviations, where some players aren’t doing “punish any defection from 99”—they are just doing “play 99″ to avoid punishments. The one brave soul doesn’t know how many of each there are, but can find out when they suddenly go for 30.