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Right, so you need a more restrictive equilibrium concept than subgame-perfect equilibrium. It's a good math exercise to try fleshing out exactly which concept

Right, so you need a more restrictive equilibrium concept than subgame-perfect equilibrium. It's a good math exercise to try fleshing out exactly which concept is implied by your common sense reasoning (which I'm not arguing against!) Some folks have suggested trembling hand equilibria and other ideas, none of which are obvious from a naive reading of the problem. Maybe it's more helpful to think about such problems in terms of "how would I write a program to solve all problems in this class?", rather than "how would I act using common sense in this particular problem?" That often makes things clearer.

darkmighty 11y ago
> "...no pirate has an incentive to deviate unilaterally (one vote can't change the voting outcome)..." I would strongly challenge that assumption as having any realism (even for perfectly rational agents and no communication): if an agent decided to vote, he could assume all other perfectly rational agents (another usual assumption) would also take the same decision (by symmetry), so he can safely vote yes. This symmetry could only be broken if the agents have access to randomness.
- cousin_it 11y ago
  Right, so we need to figure out what kinds of reasoning by symmetry are allowed under perfect rationality (in all possible games, not just this one). For example, should a rational person cooperate in the Prisoner's Dilemma because the opponent will be forced to cooperate by symmetry? In that case, won't irrational people eat rational people for breakfast, thus devaluing the idea that rational behavior maximizes utility? If you take the time to think it though, you might well reinvent superrationality, updateless decision theory and other fascinating things. In fact, you'll quickly get to questions that I have no idea how to solve!
  - darkmighty 11y ago
    Yes I don't think this assumption makes a lot of sense in general if you don't assume your perfectly rational players have access to randomness (I do think it's the best assumption if the players can't communicate or have any entropy). In the case of Prisoner's Dilemma, if the cooperate/defect outcome had sufficiently large cooperation bonus for fixed other payoffs (large enough T>>R in wikipedia's notation), then with my assumptions it's clear that each prisoner should flip a coin to decide, and each one gets ~T/4 expected payoff. But the most glaring problem is of course assuming every player is perfectly rational. Personally I'd only assume that if your players are all game theorists with enough of time and paper :) I probably wouldn't even assume myself as rational. In conclusion, I believe equating maximin with optimal play/perfect rationality is misguided, but maximin is a good safe bet.
  - the_af 11y ago
    What do you mean, "in all possible games"? The assumptions that hold for this game don't hold for the Prisoner's Dilemma. Like someone else said, you seem to be speaking in jargon. "Superrationality", "updateless decision theory", none of this matters in order to understand the Pirate Game. The rules of the game are very simple, maybe the pirates thing is throwing you off? That's just a theme for the puzzle (people aren't rational like this, and of course pirates aren't, and the whole situation is extremely artificial and unrealistic). The non-intuitive solution is derived from the rules as stated without any need for additional theory.
- tamana 11y ago
  This is Hofstadter's SuperRationality. It's main contribution isnto highlight the arbitrary simplifying assumptions of traditional game theory
  - darkmighty 11y ago
    Ah thanks it's exactly what I was thinking. It clearly isn't a good model for games in real life, but yes it does show the assumptions of traditional game theory can be misleading. Optimal play can only be defined with relation to your knowledge of the precise inner workings of your opponents.
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.

darkmighty 11y ago

> "...no pirate has an incentive to deviate unilaterally (one vote can't change the voting outcome)..." I would strongly challenge that assumption as having any realism (even for perfectly rational agents and no communication): if an agent decided to vote, he could assume all other perfectly rational agents (another usual assumption) would also take the same decision (by symmetry), so he can safely vote yes. This symmetry could only be broken if the agents have access to randomness.

cousin_it 11y ago
Right, so we need to figure out what kinds of reasoning by symmetry are allowed under perfect rationality (in all possible games, not just this one). For example, should a rational person cooperate in the Prisoner's Dilemma because the opponent will be forced to cooperate by symmetry? In that case, won't irrational people eat rational people for breakfast, thus devaluing the idea that rational behavior maximizes utility? If you take the time to think it though, you might well reinvent superrationality, updateless decision theory and other fascinating things. In fact, you'll quickly get to questions that I have no idea how to solve!
- darkmighty 11y ago
  Yes I don't think this assumption makes a lot of sense in general if you don't assume your perfectly rational players have access to randomness (I do think it's the best assumption if the players can't communicate or have any entropy). In the case of Prisoner's Dilemma, if the cooperate/defect outcome had sufficiently large cooperation bonus for fixed other payoffs (large enough T>>R in wikipedia's notation), then with my assumptions it's clear that each prisoner should flip a coin to decide, and each one gets ~T/4 expected payoff. But the most glaring problem is of course assuming every player is perfectly rational. Personally I'd only assume that if your players are all game theorists with enough of time and paper :) I probably wouldn't even assume myself as rational. In conclusion, I believe equating maximin with optimal play/perfect rationality is misguided, but maximin is a good safe bet.
- the_af 11y ago
  What do you mean, "in all possible games"? The assumptions that hold for this game don't hold for the Prisoner's Dilemma. Like someone else said, you seem to be speaking in jargon. "Superrationality", "updateless decision theory", none of this matters in order to understand the Pirate Game. The rules of the game are very simple, maybe the pirates thing is throwing you off? That's just a theme for the puzzle (people aren't rational like this, and of course pirates aren't, and the whole situation is extremely artificial and unrealistic). The non-intuitive solution is derived from the rules as stated without any need for additional theory.
tamana 11y ago
This is Hofstadter's SuperRationality. It's main contribution isnto highlight the arbitrary simplifying assumptions of traditional game theory
- darkmighty 11y ago
  Ah thanks it's exactly what I was thinking. It clearly isn't a good model for games in real life, but yes it does show the assumptions of traditional game theory can be misleading. Optimal play can only be defined with relation to your knowledge of the precise inner workings of your opponents.

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.