Is zero-sum the default economic position?

[Copernicus]

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Technically speaking, some versions of the IPD have multiple equilibria. Both players always defecting is always an equilibrium. In cases where the game is infinitely or indefinitely repeated, cooperation becomes an equilibrium.

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When repetition is introduced, defection ceases to be pareto-optimal and cooperation becomes pareto-optimal; it just requires a little more planning than the the usual "one-step," single iteration moves that game theory likes to concern itself with.

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I think youre misstating something. Without iteration, PD defection is the NE but it is pareto-suboptimal. Cooperation is pareto-optimal without iteration because at least one participant is as well off as he can be (in fact both are) and no one can improve his position (or is the condition weaker than that?)

[Copernicus]

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Even if they save it, that isnt the same as investing it.

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It is. Increased savings shifts the supply of money in the loanable funds market. Money which will then be borrowed by companies to buy new factories, etc...

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Not quite, because there is a reserve requirement for a portion of the savings, how big depends on the nature of the savings.

However, I realized that there is a bigger problem than whether or not its saved or invested, which is allocation. If 100 million people do have an extra $30 to invest they throw it into their IRA or 401(k) and it flows through a broadly diversified low risk portfolio.

On the other hand, if the billionaire has that money, he will invest it more aggressively, either starting up his own businesses or investing in a VC fund that capitalizes more risky ventures. That kind of investing is what fuels the growth in the economy, not swapping a share of GM and a bottle of wine.

[AWoodside]

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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?

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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.

[hmkpoker]

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Technically speaking, some versions of the IPD have multiple equilibria. Both players always defecting is always an equilibrium. In cases where the game is infinitely or indefinitely repeated, cooperation becomes an equilibrium.

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When repetition is introduced, defection ceases to be pareto-optimal and cooperation becomes pareto-optimal; it just requires a little more planning than the the usual "one-step," single iteration moves that game theory likes to concern itself with.

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I think youre misstating something. Without iteration, PD defection is the NE but it is pareto-suboptimal. Cooperation is pareto-optimal without iteration because at least one participant is as well off as he can be (in fact both are) and no one can improve his position (or is the condition weaker than that?)

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You're right, my mistake.

[Copernicus]

I prefer pareto-optimal solutions to equilibrium solutions anyway. Nash is a schizo. [img]/images/graemlins/grin.gif[/img]

He was also an advisor of mine briefly, lol.

[WillMagic]

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I explained to him that this is not the case, as most of a billionaire's "billions" are not locked away in a savings account, but are invested in productive capital; the actual money is still out there in circulation, and the "money" that the billionaire possesses is producing goods and services for others.

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Economics question from an admitted econ noob: If that billionaire's "billions" were instead in the hands of a larger middle class, that money would still be out there in circulation producing goods and services for others, right? I mean everyone I know in the "middle class" has at least a 401k or a Roth IRA.

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It would be out there, but it wouldn't be as productive. The way billionaires become billionaires is providing goods and services that have an excellent combination of quality and price, (Gates, Walton, Mittal) or by allocating funds to managers who do the same. (Buffett, others)

Taking money away from these people makes them less able to satisfy consumer wants at the level they are able to. Not to mention that it thoroughly disincentivizes them from making the effort, knowing that at the end of the day the fruit of their labor will be wasted.

[John Kilduff]

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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?

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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.

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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.

[vhawk01]

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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?

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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.

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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.

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This reminds me of this thread. Hopefully that will answer all your questions. [img]/images/graemlins/grin.gif[/img]

[hmkpoker]

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I prefer pareto-optimal solutions to equilibrium solutions anyway.

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Don't we all [img]/images/graemlins/smile.gif[/img] Pareto-optimal solutions are almost by definition preferable. Equilibrium solutions are going to happen, preferable or not. The trick is to design the structure such that equilibrium is pareto-optimal.

[John Kilduff]

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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?

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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.

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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.

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This reminds me of this thread. Hopefully that will answer all your questions. [img]/images/graemlins/grin.gif[/img]

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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?