5 ms·

> If everyone votes "yes", one pirate deciding to vote "no" doesn't affect their expected return. If everyone knows what everyone else is going to do, then nei

by sago 11y ago

> If everyone votes "yes", one pirate deciding to vote "no" doesn't affect their expected return. If everyone knows what everyone else is going to do, then neither option is better or worse, sure. But where did you get this perfect knowledge from? Certainly you couldn't deduce your situation. And why is 'no' your default choice: this isn't a repeated game. You're trying to decide what to vote, you don't have a mind to change, at this stage. And neither does anyone else. Therefore, if everyone knows I can choose to vote either way, your situation is irrational. You're objecting to a problem that isn't the one being considered. It may well be that there is a compelling mathematical argument you're trying to evoke, but at the moment you're not actually putting it forward. You're name-dropping, but not explaining. Well, not in a way I can follow, anyway.

cousin_it 11y ago
Man, I agree with you. This kind of reasoning is bizarre, how could you possibly know the strategies of the other players? And yet it's right. Read the first sentence of https://en.wikipedia.org/wiki/Nash_equilibrium https://en.wikipedia.org/wiki/Nash_equilibrium, which is brilliant in its simplicity: > In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy. Many people have tried to define mathematically what "rational behavior" means when there are multiple players. Nash equilibrium is the one idea that stayed. It's the first chapter of every game theory textbook. Since the 1950s, literally all publications about non-cooperative games have been based on Nash equilibrium. To understand why, you'd have to look at the history of the field, or you could take my word for it. If it's any consolation, it was very unintuitive for me at first, too.
- 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.

by sago 11y ago

cousin_it 11y ago

Man, I agree with you. This kind of reasoning is bizarre, how could you possibly know the strategies of the other players? And yet it's right. Read the first sentence of https://en.wikipedia.org/wiki/Nash_equilibrium https://en.wikipedia.org/wiki/Nash_equilibrium, which is brilliant in its simplicity: > In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy. Many people have tried to define mathematically what "rational behavior" means when there are multiple players. Nash equilibrium is the one idea that stayed. It's the first chapter of every game theory textbook. Since the 1950s, literally all publications about non-cooperative games have been based on Nash equilibrium. To understand why, you'd have to look at the history of the field, or you could take my word for it. If it's any consolation, it was very unintuitive for me at first, too.

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.