5 ms·

I still don't understand, you seem to be responding with jargon. The point made was that your 'equilibrium' is not valid because a pirate voting that way decre

by sago 11y ago

I still don't understand, you seem to be responding with jargon. The point made was that your 'equilibrium' is not valid because a pirate voting that way decreases their expected return. I don't see how you need a different 'equilibrium concept' for your proposed solution, when the basic assumption of rationality excludes it. Not trying to be mean, but your response seems obfuscatory.

cousin_it 11y ago
If everyone votes "yes", one pirate deciding to vote "no" doesn't affect their expected return. Therefore everyone voting "yes" is a subgame-perfect equilibrium. Ideas like Nash equilibria and subgame-perfect equilibria are the only known formalizations of rational behavior in multiplayer games. I'm not obfuscating, I just don't know any other kind of math that would work...
- 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...

by sago 11y ago

cousin_it 11y ago

If everyone votes "yes", one pirate deciding to vote "no" doesn't affect their expected return. Therefore everyone voting "yes" is a subgame-perfect equilibrium. Ideas like Nash equilibria and subgame-perfect equilibria are the only known formalizations of rational behavior in multiplayer games. I'm not obfuscating, I just don't know any other kind of math that would work...

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...