8 ms·

Oh don't get me wrong, my objection is not because I don't understand Nash Equilibria (my PhD was in evolutionary dynamics, so it's been a while, and I was neve

by sago 11y ago

Oh don't get me wrong, my objection is not because I don't understand Nash Equilibria (my PhD was in evolutionary dynamics, so it's been a while, and I was never a game theoretician, but I'm not totally ignorant of the field). My problem is that the mathematical tools you are name checking don't apply in this case. Even your quote itself says 'changing ... strategy', and goes on: "If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged..." which is not the situation in this case. Strategies for infinite games are not necessarily optimal if the game is finite, and vice versa.

cousin_it 11y ago
I'm sorry, now I don't follow. If not subgame-perfect equilibrium, what do you think is the right solution concept for the pirate game? Maybe it doesn't have a name, but can you describe it (at least informally)?
- kmonad 11y ago
  Can you explain a little more how this subgame-perfect equilibrium should be reached? From the game's description it appears that, all other things equal, I as a pirate reject a proposed coin distribution if I could kill the proposer. Thus, if offered 0 coins, I would always choose to reject that plan.
  - cousin_it 11y ago
    Well, one pirate changing their vote to "no" wouldn't actually kill anyone. So it sounds like you want a utility function over outcomes (prefer outcomes where someone is killed) and another, separate utility function over actions (prefer voting to kill someone even if that won't actually kill them). That's a pretty unusual setup, because game theory usually just gives you a utility function over outcomes and lets you derive strategic behavior from that, without fiddling with individual actions. Though I guess most people solving the problem have tried to merge the two utility functions somehow? That's probably the root of the disagreement in this thread...
    - kmonad 11y ago
      You keep saying "change their vote to no". Perhaps it would be educational why you think they should default to "yes" in the first place when offered 0 coins. It really doesn't seem that the problem you see necessarily arises from this specific set of rules for this game. Offered 0 --> vote no; no would at worst be equal and at best be better than yes.
      - lmm 11y ago
        The point is that everyone voting yes is a possible equilibrium solution (along with the "normal" solution), so a priori there's no rule that tells you which one is the equilibrium the pirates would reach.
        sago 11y ago
        No, the point being made is there is no 'equilibrium' because the game is not repeated. Equilibrium meaning 'if everyone votes this way, there is no benefit for any individual to vote differently' - that doesn't apply when the pirates are trying to determine what to vote for the one and only time they'll ever vote on this question, not knowing what anyone else will be voting.
        cousin_it 11y ago
        The idea of Nash equilibrium isn't just for repeated games, in fact it was invented for one-shot games. (Now I understand why you keep mentioning "infinite games"...)
        sago 11y ago
        It isn't, but it requires knowledge of the other strategies: there is at least a 'starting' strategy. That isn't the case here. Which is why you've been unable to express your disagreement with the puzzle without resorting to language of 'changing' votes, or assuming that you know what other people are going to vote. None of which are reasonable assumptions here.
- sago 11y ago
  The solution is contained in the post. I can't find a problem with it. It was you who were suggesting this is a naive analysis. Equilibria just seem to me to be the wrong tool for the job, you have to (as you have) invent a related infinite game with additional axioms and expectations before that kind of analysis kicks in. Check your responses. If you can't describe what you're trying to say without mentioning anyone 'changing' their strategy, or assuming that a strategy is default, then you've got the wrong framework, I'd suggest (modulo my admitted lack of specific expertise).

by sago 11y ago

cousin_it 11y ago

I'm sorry, now I don't follow. If not subgame-perfect equilibrium, what do you think is the right solution concept for the pirate game? Maybe it doesn't have a name, but can you describe it (at least informally)?

kmonad 11y ago
Can you explain a little more how this subgame-perfect equilibrium should be reached? From the game's description it appears that, all other things equal, I as a pirate reject a proposed coin distribution if I could kill the proposer. Thus, if offered 0 coins, I would always choose to reject that plan.
- cousin_it 11y ago
  Well, one pirate changing their vote to "no" wouldn't actually kill anyone. So it sounds like you want a utility function over outcomes (prefer outcomes where someone is killed) and another, separate utility function over actions (prefer voting to kill someone even if that won't actually kill them). That's a pretty unusual setup, because game theory usually just gives you a utility function over outcomes and lets you derive strategic behavior from that, without fiddling with individual actions. Though I guess most people solving the problem have tried to merge the two utility functions somehow? That's probably the root of the disagreement in this thread...
  - kmonad 11y ago
    You keep saying "change their vote to no". Perhaps it would be educational why you think they should default to "yes" in the first place when offered 0 coins. It really doesn't seem that the problem you see necessarily arises from this specific set of rules for this game. Offered 0 --> vote no; no would at worst be equal and at best be better than yes.
    - lmm 11y ago
      The point is that everyone voting yes is a possible equilibrium solution (along with the "normal" solution), so a priori there's no rule that tells you which one is the equilibrium the pirates would reach.
      - sago 11y ago
        No, the point being made is there is no 'equilibrium' because the game is not repeated. Equilibrium meaning 'if everyone votes this way, there is no benefit for any individual to vote differently' - that doesn't apply when the pirates are trying to determine what to vote for the one and only time they'll ever vote on this question, not knowing what anyone else will be voting.
        cousin_it 11y ago
        The idea of Nash equilibrium isn't just for repeated games, in fact it was invented for one-shot games. (Now I understand why you keep mentioning "infinite games"...)
        sago 11y ago
        It isn't, but it requires knowledge of the other strategies: there is at least a 'starting' strategy. That isn't the case here. Which is why you've been unable to express your disagreement with the puzzle without resorting to language of 'changing' votes, or assuming that you know what other people are going to vote. None of which are reasonable assumptions here.
sago 11y ago
The solution is contained in the post. I can't find a problem with it. It was you who were suggesting this is a naive analysis. Equilibria just seem to me to be the wrong tool for the job, you have to (as you have) invent a related infinite game with additional axioms and expectations before that kind of analysis kicks in. Check your responses. If you can't describe what you're trying to say without mentioning anyone 'changing' their strategy, or assuming that a strategy is default, then you've got the wrong framework, I'd suggest (modulo my admitted lack of specific expertise).