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#31
05-09-2007, 09:40 PM
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Re: Is zero-sum the default economic position?
[ QUOTE ]
[ QUOTE ] [ QUOTE ] [ QUOTE ] [ QUOTE ] Why wouldn't the Nash equilibrium it depend at least a little bit on the relative penalties? If the penalties range from going to bed without supper to life imprisonment, can't a scenario be constructed where the best choice for both is silence? I have never understood why the optimal strategy is stated without such things being taken into account. Haven't read it in a while though. Example, 2 prisoners each 25 years old: Punishment 1: both prisoners remain silent = sent to bed without supper tonight Punishment 2: both prisoners confess = 40 years in prison Punishment 3: one prisoner confesses and testifies against the other = no penalty for himself, but life imprisonment for the silent prisoner In the above scenario, the optimum strategy is clearly to risk the negligible punishment of being sent bed without supper tonight along with the chance of the somewhat worse alternative of life in prison compared to 40 years in prison (release at age 65). There isn't a great deal of difference between spending one's years from 25-65 in prison, versus life in prison; but there is a huge difference between being sent to bed without supper and either of the other alternatives. If the 40 years in prison versus life in prison isn't a convincing comparison then change the example to 80 lashes versus 85 lashes...versus going to bed without supper for one night (or versus one lash). Can anyone explain why the oft-touted "answer" to the non-iterative PD neglects the consideration of the relative severity of outcomes? [/ QUOTE ] In the game you described your optimum strategy is still to defect. If you defect and the other guy doesn't, you get off scott free. If you defect and he also defects, you get 40 years (which is better than if you had cooperated and he defected). The point is that whatever the other guy decides, you will be better off by defecting. This doesn't change by making the difference in punishments more severe. [/ QUOTE ] The bolded part isn't always true, depending upon the punishments specified (unless I am missing something) because you have to weigh the outcomes and chances the other will defect, and consider that he can think too: For prisoners X and Y: X not defect, Y not defect---> 1 lash each X defect, Y defect---> 100 lashes for X, 100 lashes for Y X defect, Y not defect---> 0 lashes for X, 101 lashes for Y X not defect, Y defect---> 101 lashes for X, 0 lashes for Y Considering each lash as of equal negative value: The optimal strategy would be to realize that the difference between 0 and 1 is slight, and the difference between 100 and 101 is slight; but the difference between 1 and 100 is great; next, consider that the other prisoner is a thinking player also, capable of realizing all of this; then realize that each player is laying 100-1 if he chooses to defect (because if you choose to defect you stand to gain at most only one lash differential benefit regardless of the other prisoner's choice, whereas if you choose to not defect you stand to gain 100 lashes differential benefit if he cooperates with you by also not defecting). Therefore, ideal strategy must weigh the relative punishments for each scenario and take into account that the other player can think also: if the benefits of cooperation far outweigh the slight differential benefit of defection, and if the punishment if you both defect is enormous to both of you, and the punishment if you both do not defect is minor, then the the optimal strategy would be to opt for the minor punishment and hope your counterpart is able to also figure this out, that it is worth the risk of getting at most one extra lash in order to not be laying 100-1 odds that the other player won't defect. If this is wrong, could you please explain how it is wrong using the specific number of lashes for each case specified in the example above? Thanks for taking the time to discuss this. [/ QUOTE ] This reminds me of this thread. Hopefully that will answer all your questions. [img]/images/graemlins/grin.gif[/img] [/ QUOTE ] Thanks for your help in linking, but that thread is 25 pages long, and even the OP in it is long. Cliff notes as it relates to my example above, perhaps? [/ QUOTE ] First off, let me just say, if you have a few hours, you really should read that thread. Its amazing. Its really a 2p2 classic. Cliff notes: A bunch of guys use intuition to try to win their way out of a PD-esque problem, fail, get yelled at by game theorists, NEs are explained dozens of times, normal people laugh at stupid game theorists, everyone realizes the real world doesn't work like that, no one is happy. 
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#32
05-09-2007, 09:43 PM
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Re: Is zero-sum the default economic position?
[ QUOTE ]
[ QUOTE ] [ QUOTE ] [ QUOTE ] [ QUOTE ] [ QUOTE ] Why wouldn't the Nash equilibrium it depend at least a little bit on the relative penalties? If the penalties range from going to bed without supper to life imprisonment, can't a scenario be constructed where the best choice for both is silence? I have never understood why the optimal strategy is stated without such things being taken into account. Haven't read it in a while though. Example, 2 prisoners each 25 years old: Punishment 1: both prisoners remain silent = sent to bed without supper tonight Punishment 2: both prisoners confess = 40 years in prison Punishment 3: one prisoner confesses and testifies against the other = no penalty for himself, but life imprisonment for the silent prisoner In the above scenario, the optimum strategy is clearly to risk the negligible punishment of being sent bed without supper tonight along with the chance of the somewhat worse alternative of life in prison compared to 40 years in prison (release at age 65). There isn't a great deal of difference between spending one's years from 25-65 in prison, versus life in prison; but there is a huge difference between being sent to bed without supper and either of the other alternatives. If the 40 years in prison versus life in prison isn't a convincing comparison then change the example to 80 lashes versus 85 lashes...versus going to bed without supper for one night (or versus one lash). Can anyone explain why the oft-touted "answer" to the non-iterative PD neglects the consideration of the relative severity of outcomes? [/ QUOTE ] In the game you described your optimum strategy is still to defect. If you defect and the other guy doesn't, you get off scott free. If you defect and he also defects, you get 40 years (which is better than if you had cooperated and he defected). The point is that whatever the other guy decides, you will be better off by defecting. This doesn't change by making the difference in punishments more severe. [/ QUOTE ] The bolded part isn't always true, depending upon the punishments specified (unless I am missing something) because you have to weigh the outcomes and chances the other will defect, and consider that he can think too: For prisoners X and Y: X not defect, Y not defect---> 1 lash each X defect, Y defect---> 100 lashes for X, 100 lashes for Y X defect, Y not defect---> 0 lashes for X, 101 lashes for Y X not defect, Y defect---> 101 lashes for X, 0 lashes for Y Considering each lash as of equal negative value: The optimal strategy would be to realize that the difference between 0 and 1 is slight, and the difference between 100 and 101 is slight; but the difference between 1 and 100 is great; next, consider that the other prisoner is a thinking player also, capable of realizing all of this; then realize that each player is laying 100-1 if he chooses to defect (because if you choose to defect you stand to gain at most only one lash differential benefit regardless of the other prisoner's choice, whereas if you choose to not defect you stand to gain 100 lashes differential benefit if he cooperates with you by also not defecting). Therefore, ideal strategy must weigh the relative punishments for each scenario and take into account that the other player can think also: if the benefits of cooperation far outweigh the slight differential benefit of defection, and if the punishment if you both defect is enormous to both of you, and the punishment if you both do not defect is minor, then the the optimal strategy would be to opt for the minor punishment and hope your counterpart is able to also figure this out, that it is worth the risk of getting at most one extra lash in order to not be laying 100-1 odds that the other player won't defect. If this is wrong, could you please explain how it is wrong using the specific number of lashes for each case specified in the example above? Thanks for taking the time to discuss this. [/ QUOTE ] This reminds me of this thread. Hopefully that will answer all your questions. [img]/images/graemlins/grin.gif[/img] [/ QUOTE ] Thanks for your help in linking, but that thread is 25 pages long, and even the OP in it is long. Cliff notes as it relates to my example above, perhaps? [/ QUOTE ] First off, let me just say, if you have a few hours, you really should read that thread. Its amazing. Its really a 2p2 classic. Cliff notes: A bunch of guys use intuition to try to win their way out of a PD-esque problem, fail, get yelled at by game theorists, NEs are explained dozens of times, normal people laugh at stupid game theorists, everyone realizes the real world doesn't work like that, no one is happy. [/ QUOTE ] It sounds like it should be a good read some day. Meanwhile, I am still in the dark as to why the optimal solution in the example I posed above would (supposedly) be to lay 100-1 odds.
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#33
05-09-2007, 09:43 PM
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Re: Is zero-sum the default economic position?
Probable cliff notes: the forumulation of equilbrium and optimal situations wrt to game theory ignore relative consequences. +1 last is the same as +100 lashes.
A formulation that takes into account relative punishments is no longer game theory, its....something less...OR maybe?
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#34
05-09-2007, 09:54 PM
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Re: Is zero-sum the default economic position?
A lot of people are used to zero sum situations in thier regular lives.
Growing up, parents buying their brother or sister a gift meant less money to spend on their gift. This is reinforced on the schoolyard a lot. At your job you may have a bonus pool that gets divided amongst employees. Another employee getting bonus means yours is smaller. Same could be said of a corporate hierarchy were only one person out of many can be promoted.
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#35
05-09-2007, 09:54 PM
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Re: Is zero-sum the default economic position?
[ QUOTE ]
Probable cliff notes: the forumulation of equilbrium and optimal situations wrt to game theory ignore relative consequences. +1 last is the same as +100 lashes. [/ QUOTE ] +1 dollar is the same as +100 dollars according to game theory? [ QUOTE ] A formulation that takes into account relative punishments is no longer game theory, its....something less...OR maybe? [/ QUOTE ] Change it to relative dollar rewards in a game show setting, then: Parameters: Cooperation in not defecting wins a million dollars apiece. Successful sole defection wins a million dollars plus a hundred dollar bonus, with the other player getting nothing. Dual defection wins a hundred dollars apiece. Wouldn't you not defect and hope your counterpart would see the wisdom in that too? A player who defects is risking a million dollars to win a hundred dollars, is laying 10,000-1 odds. Isn't optimal strategy in the above scenario (given a thinking other player) to not defect? Heck even if you were against another player who would just choose randomly the optimal strategy would be to not defect.
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#36
05-09-2007, 10:00 PM
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Re: Is zero-sum the default economic position?
[ QUOTE ]
[ QUOTE ] Probable cliff notes: the forumulation of equilbrium and optimal situations wrt to game theory ignore relative consequences. +1 last is the same as +100 lashes. [/ QUOTE ] +1 dollar is the same as +100 dollars according to game theory? [ QUOTE ] A formulation that takes into account relative punishments is no longer game theory, its....something less...OR maybe? [/ QUOTE ] Change it to relative dollar rewards in a game show setting, then: Parameters: Cooperation in not defecting wins a million dollars apiece. Successful sole defection wins a million dollars plus a hundred dollar bonus, with the other player getting nothing. Dual defection wins a hundred dollars apiece. Wouldn't you not defect and hope your counterpart would see the wisdom in that too? A player who defects is risking a million dollars to win a hundred dollars, is laying 10,000-1 odds. Isn't optimal strategy in the above scenario (given a thinking other player) to not defect? Heck even if you were against another player who would just choose randomly the optimal strategy would be to not defect. [/ QUOTE ] Using words like 'hope' demonstrates where you are going wrong. Nothing you do can impact his decision, not even hoping. So, knowing you can have absolutely no impact on his decision, you get to eliminate some of the choices. Now, its a matter of 100,000,100 or 100,000,000 and you are telling me I should take the latter.
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#37
05-09-2007, 10:04 PM
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Re: Is zero-sum the default economic position?
John,
That's a common natural reaction (indeed, if most people think this way, most people will be generally better off in PD situations, even if it is irrational). But the key game theory point to realize is that in a single-shot game, your strategy does not affect the strategy of the other player. Not defecting doesn't affect the other player's chance of not defecting. Defecting doesn't make him more likely to screw you. Whatever he does, in the PD you will be strictly better off by defecting. When you talk about laying odds, it doesn't make sense. You're not risking one thing for another by defecting. You ALWAYS are better off.
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#38
05-09-2007, 10:06 PM
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Re: Is zero-sum the default economic position?
[ QUOTE ]
John, That's a common natural reaction (indeed, if most people think this way, most people will be generally better off in PD situations, even if it is irrational). But the key game theory point to realize is that in a single-shot game, your strategy does not affect the strategy of the other player. Not defecting doesn't affect the other player's chance of not defecting. Defecting doesn't make him more likely to screw you. Whatever he does, in the PD you will be strictly better off by defecting. When you talk about laying odds, it doesn't make sense. You're not risking one thing for another by defecting. You ALWAYS are better off. [/ QUOTE ] An entire lifetime of social conditioning (and many, many lifetimes of evolution) have combined to make this extremely counter-intuitive to most (all?) people.
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#39
05-09-2007, 10:09 PM
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Re: Is zero-sum the default economic position?
[ QUOTE ]
Well, I think the question is while the examples given by OP (increased number of billionares, Walmart), while they do not necessarily have to have a "loser", they don't necessarily have to be "win-win", and if you look at people that know what they are talking about criticize these things they don't do it from a zero sum perspective (though I agree you do hear stuff like what the OP referred to a lot) [/ QUOTE ] Actually economics is basically about win-win. Even your "losers" are winners for the most part in the U.S. anyway.
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#40
05-09-2007, 10:12 PM
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Re: Is zero-sum the default economic position?
[ QUOTE ]
[ QUOTE ] Well, I think the question is while the examples given by OP (increased number of billionares, Walmart), while they do not necessarily have to have a "loser", they don't necessarily have to be "win-win", and if you look at people that know what they are talking about criticize these things they don't do it from a zero sum perspective (though I agree you do hear stuff like what the OP referred to a lot) [/ QUOTE ] Actually economics is basically about win-win. Even your "losers" are winners for the most part in the U.S. anyway. [/ QUOTE ] Quiet, you. Big corporations are stripping the middle class of their right to own BMW's as we speak. It's practically Haiti.
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