Test your wits

Yes. Game Theory is mathematics. Mathematics is not Game Theory. Thus one can be a mathematician and have not studied Game Theory. I listed several mathematicians who were also Game Theorists who invented the dilemma and how they pick 'defect'. Can you name any mathematicians who pick 'cooperate'? How many of them have studied Game Theory?As I said, I've never seen this economics vs mathematics divide on this issue. Mathematicians in my experience simply apply Game Theory and betray. In your opinion however, "The mathematicians are seeking a solution that reduces the entire amount of time spent in jail." - which would tell me that these particular mathematicians haven't studied game theory...unless they are using super-rationalism as justification. No, it's called a dilemma because "The unique equilibrium for this game is a Pareto-suboptimal solution”that is, rational choice leads the two players to both play defectly even though each player's individual reward would be greater if they both played cooperately.": wiki.I mean, if we assume that the object of the game is to go to prison for maximum amount of time, everything changes. If we assume the object of the game is to kill the guards and create a prisoner uprising to overthrow the shackles of oppression, then the optimum solution changes. Really there is only one game: What is the best rational strategy for any given prisoner. The answer: betray.Every mathematician agrees that betrayal is the best rational strategy. Some mathematicians argue that rational strategies aren't the way forward and we should use super-rational strategies. In which case the solution is a controversial cooperate. But you weren't talking about super-rational strategies. I have no idea what you are talking about here.The optimal solution might well be to BOTH cooperate, but you cannot communicate with the other prisoner. Since you cannot cooperate, mathematically speaking your best strategy is to betray. If they choose to betray you, the last thing you want to do is cooperate. If they choose to cooperate you can skip spending 6 months in prison by betraying them. In each and every case, betrayal gets you the best outcome. The one where we have to assume that both prisoner's are rational agents. The original and classic Prisoner's Dilemma as put forward by Game Theorists. In the post which you categorically stated was 'incorrect' I even clarified in an attempt to undercut soft complications by saying:

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Assuming that I'm socially short-sighted or just self-serving

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To remind us what you said

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Incorrect. If you keep silent and they also do, you only get six months. If you betray and they also do, you get five years.

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It is better to keep silent. This is an old puzzle. If you want, I can go through the math....

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Now you seem to be suggesting that it isn't that I was incorrect at all, but that it depends on what game we are playing. Yes, it is an old puzzle but I haven't had the pleasure of seeing your maths that demonstrates it is better to keep silent. If you and the other prisoner are working as a team, or if you have set things up so that there are additional rewards for cooperating and penalties for betrayal, then it may be best to cooperate.But we don't have these things set up. You don't even know what the probabilities are of your opponent picking cooperate. If you choose cooperate, you are taking a huge risk of going to prison for a long time -> especially given the fact that no matter what you pick the other prisoner would be best served by betraying you, and the other prisoner knows that this is true of your position too.Can you find any websites that support your position? I can only find ones that don't and what you write intrigues me. Can you find this economists vs mathematicians study?

I have studied Game Theory for my mathematics degree, and Modulous is right. Whatever option your opponent chooses you are better off if you choose to defect. It is only in the iterated case, when the rewards of future cooperation outweigh the short-term gain of defection that cooperation becomes the better option in mathematical (or economic) terms.Perhaps you are thinking of psychologists or sociologists ? Mathematics assumes no link between your choice and your opponents, while your argument assumes that your opponent will choose the same as you.

Modulous responds to me:

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Can you name any mathematicians who pick 'cooperate'?

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Well, there's me. And then there's all the professors in my various probability and statistics classes.

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How many of them have studied Game Theory?

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All of them. That's why they were the professors of the classes. They were teaching it.

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As I said, I've never seen this economics vs mathematics divide on this issue.

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And I have. So now what?

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In your opinion however, "The mathematicians are seeking a solution that reduces the entire amount of time spent in jail." - which would tell me that these particular mathematicians haven't studied game theory...unless they are using super-rationalism as justification.

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There's another option, you know....

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No, it's called a dilemma because "The unique equilibrium for this game is a Pareto-suboptimal solution”that is, rational choice leads the two players to both play defectly even though each player's individual reward would be greater if they both played cooperately."

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That's not a dilemma. And you do know that Wikipedia isn't really a source, but since you seem to want it, I notice you didn't pay attention to your own article: Hmmm..."important concept in economics"...I wonder why there might be a distinction between the economic interpretation and the mathematical one....But, let's continue with your own source: Hmmm...wasn't that what I was saying? That there is the game of self-interest and the game of group-interest, that they are not the same, and players that go for group-interest will do better than those that go for self-interest?

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I mean, if we assume that the object of the game is to go to prison for maximum amount of time, everything changes.

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Indeed, which is why the economic interpretation is different from the mathematical one. The economic interpretation is to minimize the individual one. The way it is usually presented in all my classes on the subject, the idea was to reduce the total amount of time spent in jail.But even so, let's look at the sentence before the one you quoted: That's not true. The only concern is not always maximixing his/her own payoff. While it is a common one and most game scenarios are presented as such, sometimes the goal is different.

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If we assume the object of the game is to kill the guards and create a prisoner uprising to overthrow the shackles of oppression, then the optimum solution changes.

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(*sigh*)Let's not play dumb. The only options are to cooperate or defect. The only payouts are various lengths of time.

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Every mathematician agrees that betrayal is the best rational strategy.

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(*sigh*)As an individual strategy, yes. Who said that was the game?

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Even when shown that the optimal solution is to cooperate, they betray. They think that somehow the rules don't apply to them and they always end up with both getting five years.

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I have no idea what you are talking about here.

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That even when they know that they can optimize for all, they will choose to betray in order to optimize for the individual. This even goes to the iterative game. Even when they know that the winning solution for the iterative game is to play tit-for-tat/forgive/don't get nasty first, they will be likely to betray. It's why we keep running into the tragedy of the commons.

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The optimal solution might well be to BOTH cooperate, but you cannot communicate with the other prisoner.

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Who said you were? You both know the outcome. If you're playing the game of getting you both out, then you know what to do.

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Since you cannot cooperate, mathematically speaking your best strategy is to betray.

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"Cannot"? Why not?

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What game are you trying to play?

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The one where we have to assume that both prisoner's are rational agents.

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That's not an answer. That's an assumption of the game. A rational player playing a game of minimizing jail time for all will cooperate.

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Now you seem to be suggesting that it isn't that I was incorrect at all, but that it depends on what game we are playing.

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As one who does support for a living, this is always a problem: Unstated assumptions. I was assuming that the game was as it was presented to me. Instead, we're now talking about something else.When I do a training, I steal something that I saw in Foxtrot:A train leaves Station A where the clock reads 10 am, arriving at Station B, 180 miles away, at where the clock reads 2 pm.Now, the common question at this point is, "What is the average speed of the train?" and the expected answer is "45 mph." But instead, the question I put is:What do we need to assume in order to determine the average speed of the train?There's quite a list:The clocks are working.
The stations are in the same time zone.
The train arrives on the same day.
The track goes directly to the station and doesn't waver.
The track is taking the short way instead of going around the other side of the globe first.
The track is following the curvature of the earth and isn't tunneling through as a chord.
Ignoring relativistic effects of moving bodies in a gravity field.I was assuming a different game than you were.

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You don't even know what the probabilities are of your opponent picking cooperate.

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There aren't any. You assume a rational player. That's the point.

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Can you find any websites that support your position? I can only find ones that don't and what you write intrigues me.

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(*blink!*)You've never heard of "sub-optimization"? OK.The problem of suboptimization

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Can you find this economists vs mathematicians study?

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If you can wait, I'll ask my colleague as he is the one who presented it to me. He, too, is a mathematician who crossed to the dark side and went into economics.He got better.By the way, a good textbook game theory, involving games with multiple choices ("Gladyn and Don inherit a car worth $800. The evils of communism being well known to them, they agree to settle the ownership by means of sealed bids. The high bidder gets the car by paying his brother the amount of the high bid. If the bids are equal”which they may well be, because they agree to bid in hundred-dollar quantities”the ownership is determined by the toss of a coin, there being no exchange of funds. Gladyn has $500 on hand, whereas Don has $800. How should they bid?"...this is a 5x8 game with multiple saddle points):The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy by J. D. Williams.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

PaulK responds to me:

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I have studied Game Theory for my mathematics degree,

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As did I for mine.

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and Modulous is right.

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Assuming an individual game. There is another possible game to play.

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Mathematics assumes no link between your choice and your opponents, while your argument assumes that your opponent will choose the same as you.

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Not quite. In introductory game theory, you usually assume that you are going for an individual game. There are other games that have you seek to find a group outcome.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Assuming an individual game. There is another possible game to play.

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Exactly as I stated: Edited by PaulK, : No reason given.

PaulK responds to me:

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Assuming an individual game. There is another possible game to play.

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Exactly as I stated

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Indeed, but I am pointing out that there is even a third, non-iterative game to play. I don't think I was thinking of the same game.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Feel free to provide a better one of your own. So when you said earlier that this solution was flat 'Incorrect' for selfish agents and that you could show me the maths to demonstrate that this was so, you were actually wrong? What you meant to say was 'As you say, this is the correct solution for selfish agents. However, I can use lots of words to explain why I think 'cooperate' can seen as the best option.' Sounds familiar. It's like when I said the best option is to betray "Assuming that I'm socially short-sighted or just self-serving", and then you said "Incorrect." Also, reading where assumptions are stated is very important. Yes, of course. That's why I said that the best solution for the prisoner's was to betray despite it being sub-optimal. Just like the bit you quoted said. The prisoners are placed into competition with one another resulting in a sub-optimal solution.Edited by Modulous, : No reason given.

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Indeed, but I am pointing out that there is even a third, non-iterative game to play. I don't think I was thinking of the same game.

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Likely you were thinking of what amounts to a different game (in mathematical terms) from the Prisoner's Dilemma. But what's the point of bringing it up ?The iterative version is interesting because it keeps the same payoff matrix. A game using a different matrix would just be a different game. It may or may not be interesting in its own right but any relationship to the Prisoner's Dilemma is likely to be in terms other than the strictly mathematical.

Modulous responds to me:

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So when you said earlier that this solution was flat 'Incorrect'

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So this is the third time I'm saying it. Will you pay attention this time?I was assuming a different game.

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and that you could show me the maths to demonstrate that this was so

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Which I did.

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you were actually wrong?

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No. For the fourth time...let's see if you remember:I was assuming a different game.

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What you meant to say was 'As you say, this is the correct solution for selfish agents. However, I can use lots of words to explain why I think 'cooperate' can seen as the best option.'

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Incorrect. What I meant to say was actually what I said:There are other games that can be played with this scenario. The specific game you are playing will determine the best strategy. The way this scenario has always been presented to me, the game was to reduce time across the board.Why is this so hard for you?

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It's like when I said the best option is to betray "Assuming that I'm socially short-sighted or just self-serving", and then you said "Incorrect."

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(*sigh*)Do you truly not understand the difference between the assumptions involved in game theory in and of itself and the assumptions involved in the specific game you are trying to play?Game theory assumes rational players. But you will need to decide upon what type of game you are going to play in order to determine what a "rational player" will do. Even in this game, which you freely admit to accepting, a single-shot instance of the game is different from an iterative version of the game.Since you seem to be able to accept it in that other instance, why are you having such a hard time with the idea that there is yet a third type of game to play?

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Also, reading where assumptions are stated is very important.

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That's why they're called "unstated assumptions." I've pointed out that I was dealing with unstated assumptions four times now. Will you pay attention now that I've said it a fifth time?

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Yes, of course. That's why I said that the best solution for the prisoner's was to betray despite it being sub-optimal.

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No! For a single-shot game where you're trying to maximize your own outcome, betrayal is optimal! That's the entire point! Didn't you read the quotation I gave you? Sub-optimalization recognizes that when you try to maximize a single part, you rarely manage to optimize the whole.But what do you mean by "the whole"? Until you make that decision, you cannot know if sub-optimalization will achieve grand optimization.So if you're only going for maximizing your own return, sub-optimalization and grand optimization are the same: Betray.But is that the game you're playing? Are you sure? It's an unstated assumption.Hmmm...that's six times. Will you remember it now?

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Just like the bit you quoted said. The prisoners are placed into competition with one another resulting in a sub-optimal solution.

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Yes, but only if the game being played is to reduce time overall. Sound familiar? That's the game I was talking about. It apparently wasn't the game you were referring to (seventh time...might you consider that fact before responding?)If the game is to reduce your own time, then the sub-optimal solution is also the grand optimal solution.Is that the game you're playing? Are you usre? It's an unstated assumption.Eighth time. Will that be the charm?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

PaulK responds to me:

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Likely you were thinking of what amounts to a different game (in mathematical terms) from the Prisoner's Dilemma.

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No, I was thinking of the Prisoner's Dilemma as it has always been presented to me: As an exercise in showing how individual strategies do not lead to grand strategies.

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But what's the point of bringing it up ?

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Because it's always been the way the game was presented to me.

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The iterative version is interesting because it keeps the same payoff matrix. A game using a different matrix would just be a different game.

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Indeed. Where did I say I changed the payoff matrix?

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It may or may not be interesting in its own right but any relationship to the Prisoner's Dilemma is likely to be in terms other than the strictly mathematical.

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Huh? Can you give any indication in anything I have said that would signify that I have changed the matrix?Be specific.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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1. Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal. If one testifies for the prosecution against the other and the other remains silent, the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?

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You know the one problem I have with the prisoner's dilemma is that ignores reality to much. Some of the underlying assumptions present in the philosophical problem like an assumed same background, underlying beliefs and desires of the criminals just doesn't nuance the situation enough. I mean in any given situation one criminal versus another criminal is going to have different motivations and or reasons for committing a given crime. And with that each criminal will experience a different set of loss or gains depending on the length of the sentence. I.e. if one has a family and the other doesn't the intrinsic motivation for each one based on family circumstances alone would argue for a more nuanced approach.

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No, I was thinking of the Prisoner's Dilemma as it has always been presented to me: As an exercise in showing how individual strategies do not lead to grand strategies.

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No, that doesn't seem to be the issue.

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Indeed. Where did I say I changed the payoff matrix?

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Quite explicitly in Message 39Reducing time "across the board" would produce a quite different payoff matrix since the time served by your opponent would be a loss to you, as well as him.

You can play whatever game you like Rrhain. You can keep repeating you were playing a different game all you like. Whenever you are ready you may return to Message 11...the message you replied to. In Message 11 you can see how I explicitly stated the assumptions inherent in the standard Prisoner's Dilemma. They weren't exhaustive, obviously, but I think you'll find they cover the important parts and my subsequent replies should have clarified them considerably. You replied to Message 11, where I stated my assumptions, by saying "Incorrect". You have since indicated that my solution is correct given my stated assumptions, you don't need to repeat yourself eight times - a simple concession that you were actually wrong in Message 19 would suffice. Betrayal is the optimal solution for the 'subsystem', which is why it is the correct answer to the standard dilemma. It is, as you have stated, optimal for the total system (or global optimization) for them both to cooperate. Unfortunately for the poor prisoners the warden/sheriff has put them in a position where they cannot conspire to shoot for global optimization but have to shoot for the optimal solution for their immediate subsystem which turns out not to be globally optimal. Optimizing a problem's subsystems do not lead to global optimization (The Principle of Suboptimization).You insist that you were playing a game of 'find the globally optimal solution', which is rather pointless since it is stated pretty much in the game parameters but that's fine. The Prisoner's Dilemma as it is presented to everybody else in the world is all about how given the circumstances 'Defect' is what rational agents would do, despite them both knowing it is not the globally optimal solution. I find this variety more interesting than your version.You are the one who offered F. Heylighen as a suitable authority. Here's what he has to say about the Prisoner's Dilemma:

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The problem with the prisoner's dilemma is that if both decision-makers were purely rational, they would never cooperate. Indeed, rational decision-making means that you make the decision which is best for you whatever the other actor chooses. Suppose the other one would defect, then it is rational to defect yourself: you won't gain anything, but if you do not defect you will be stuck with a -10 loss. Suppose the other one would cooperate, then you will gain anyway, but you will gain more if you do not cooperate, so here too the rational choice is to defect. The problem is that if both actors are rational, both will decide to defect, and none of them will gain anything. However, if both would "irrationally" decide to cooperate, both would gain 5 points. This seeming paradox can be formulated more explicitly through the principle of suboptimization.

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I believe that is exactly what I was saying.Edited by Modulous, : No reason given.

The answer to the first question is dependant. In repeat scenarios, the best decision is to be the good guy, or in this case, be silent and not rat out your partner. After enough repeats, you will eventually end up at the top of the pile. This follows reality far more closely than your presented game theory scenario--in which they only meet once. If they only meet once, they should both rat the other out, because you cannot trust the other guy to stay silent. This doesn't lead to the best or the worst outcome possible, but its still inefficient (I'm not sure how important that is to game theory, given that I don't really study it). It is however, the best possible outcome for both contestants.The second question I'm thinking is also game theory. Personally, I wouldn't trust it. If I don't drink, I either end up with money or no money and my life. If I do drink, I will have money or no money, pain, no pain, life, or death. I don't think I'll drink, nor would I have the intention of doing so. I'll earn my money honestly.If I was a game theoretician I could answer the second one correctly, but alas I am not.

A little of topic, but that's interesting. Usually European universities do not offer game theory to undergraduates, so it's pretty good of your universities to offer it. Makes a good break from the definitions => theorem => corollary stuff.