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Hello,

I hope this question is appropriate for Mathoverflow. Gandhi said, "Be the change that you wish to see in the world". I don't understand anything in Game/optimization theory (I don't know exactly what theory), but I am curious whether there are mathematical models, in which if the individual makes his local contribution the best, without thinking about the interaction with the decisions of the other, this makes the whole system act in the best possible way.

Thank you, Sasha

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    $\begingroup$ Sigmund's "The calculus of selfishness" (see cscs.umich.edu/~crshalizi/reviews/calculus-of-selfishness.html for a review) might be a good read. $\endgroup$
    – Guntram
    Commented Sep 4, 2012 at 18:03
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    $\begingroup$ Yes... if you change your notion of "best possible" accordingly. Gerhard "Still It Makes Little Sense" Paseman, 2012.09.04 $\endgroup$
    – Gerhard Paseman
    Commented Sep 4, 2012 at 18:31
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    $\begingroup$ Did you make any effort to read something on game theory (if only a couple of pages on wikipedia) before asking this question? It seems not. Vote to close. $\endgroup$
    – user9072
    Commented Sep 4, 2012 at 18:51
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    $\begingroup$ @quid: +1. the question could be interesting, but it seems to me that the OP did not make the effort that would make it an interesting question. $\endgroup$
    – DamienC
    Commented Sep 4, 2012 at 19:05
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    $\begingroup$ A general remark: given the context of the quote, it seems highly unlikely that answers about "selfish economics" are what the OP has in mind. It seems more appropriate to consider a model in which all agents have exactly the same optimization function, and must determine how to achieve it without coordination (and possibly with each agent having limited, "local" information). Cooperative hat puzzles seem like a possible candidate here; see en.wikipedia.org/wiki/Hat_puzzle . Unfortunately, for a hat puzzle, the "local algorithm" tends to look quite different from the global problem. $\endgroup$
    – Charles Staats
    Commented Sep 4, 2012 at 20:14

2 Answers 2

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The simplest example is an Edgeworth Box economy, or more generally a Walrasian equilibrium.

The fundamental insight is that when multiple individuals are faced with the same (linear) prices, they choose consumption bundles (or production bundles or something else, depending on the model) where certain level hypersurfaces are tangent to a particular hyperplane. Moreover, equilibrium requires these tangencies to occur at a common point. On the other hand, for one interpretation of "best possible" (i.e. Pareto optimality), the condition for a best possible outcome is that these level hypersurfaces should be tangent to each other.

Because hypersurfaces tangent to the same hyperplane at the same point are tangent to each other, the result follows. This simple insight has been vastly generalized and underlies all of modern welfare economics.

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  • $\begingroup$ +1. I suspect that "welfare economics" is precisely the search phrase needed to find the pre-requisite reading that a lot of the commenters desired. $\endgroup$
    – R Hahn
    Commented Sep 5, 2012 at 0:15
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I think in several cases "best self interest" can lead to overall poorer solutions than solutions with interaction.

More formally, the self-interest maximizing version falls into the domain of traditional "Non-cooperative game theory" while the one with interactions falls into "Cooperative game theory".

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