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The price of anarchy is a ratio between the efficiency of a centralized solution and a solution with decentralized decision makers. The implicit assumption seems to be that the centralized solution is always at least as efficient as the decentralized solution, due to phenomena such as Braess's Paradox, and thus the ratio must be at least 1. However, it is often said that command economies are less efficient than free market economies. Central planners cannot seem to get the parameters of an economy pinned down effectively, relative to markets under first welfare theorem assumptions about elasticity, market impact, etc. How does this fit into the price of anarchy model? Shouldn't this be a case in which the price of anarchy is less than 1?

A related question, and perhaps even a special case of the above, a lot of macroeconomic models seem to be aimed at getting market outcomes to hug centrally planned outcomes as closely as possible (i.e., the motivation for first welfare theorem conditions), but if centrally planned outcomes empirically appear to be worse, why is the focus not instead on evaluating how centralized systems stack up using market systems as the gold standard? This seems to be closer to how lay persons conceive of the relation between centralized and decentralized economic systems. Is this type of analysis also performed in macroeconomics?

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Any real-life command economy will likely do worse than a free market economy because of the massive information asymmetries and incentive problems arising between the government and individual firms, workers, and consumers.

However, PoA models don't compare actual decentralized systems with actual centralized systems. They compare the theoretical solution under decentralized decision making (typically Nash equilibrium) with the theoretical solution under centralized decision making with complete information (typically just the welfare optimum). Since the central decision maker under the latter system could just implement the Nash equilibrium of the former system, the welfare achieved in the centralized system is at least as large as in the decentralized system, i.e. the PoA is at least 1.

As I understand it, PoA models are interesting for small systems where the complete information assumption holds at least approximately in real life, but it doesn't make a lot of sense to apply them to a whole economy.

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You seem to be falling for a fallacy of equivocation here. Centralized solution to the PoA model is not the same as having command economy. For example, some classic PoA models are about managing transportation, there a central authority could very well be a firm like Amazon that manages its delivery trucks as opposed to some politburo commissar. You can't necessarily generalize the result to the whole society. This is because in a single firm people might have the right incentives and little information asymmetry, while one more macro levels there is very high informational asymmetry between people and government and government officials will not have the right incentives as individuals within a price system. Analogously, you can't even generalize first welfare theorem to whole society, since an economy without any government and anarchy would likely completely disintegrate as well due to presence of market failures and need for at least some basic public goods like defense.

Regarding the second question, I think, at least when it comes to macro models they do not typically try to make economy hug centrally planned outcomes. First welfare theorem is not about centrally planed economy. First welfare theorem says that if there are no market failures free market equilibrium will be pareto optimal and is allocatively efficient and maximizes (Marshallian) welfare. Second welfare theorem states that if social planner has access to individualized lump sum tax then resources can be arbitrary distribution of resources, but even this talks about distribution not necessarily command economy.

Models typically use stylized economy under welfare theorem as a goal as that is the most efficient way economy can theoretically work. Hence any deviations from that indicates some allocative inefficiencies.

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  • $\begingroup$ Regarding the first paragraph, you seem to be giving reasons why centralization is more efficient in some systems and decentralization is more efficient in others, which I think is right. Is your point then that economists apply PoA models only to those systems in which centralization is more efficient, merely as a methodological constraint? If not, then I still don't see why the lower bound of a PoA should be 1. $\endgroup$
    – user10478
    Dec 7, 2022 at 5:47
  • $\begingroup$ As for the second question, I did get this impression from a single lecture series (youtu.be/5ZoX77T6gs8?t=4683), in which the professor kept comparing free market outcomes against a social planning benchmark. It may have been little more than his own narrative baked into the course. $\endgroup$
    – user10478
    Dec 7, 2022 at 5:50
  • $\begingroup$ @user10478 my point is rather that even the definition of centralization in PoA and centralization as in central planning is not the same. But yes, an economist would not apply PoA model to whole economy. Also PoA model with informational asymmetries and incentive problems explicitly modeled would likely not give you the same result. My point is that the 1 lower bound is something that holds in this model given its assumptions. For example, statement that interior angles of triangle add up to 180 only holds within a stylized model of triangle on flat space you can't extrapolate it to triangles $\endgroup$
    – 1muflon1
    Dec 7, 2022 at 10:16
  • $\begingroup$ on earth surface as earth is curved. I will have to check the video to get a context, but note when we talk about social planners in economics we do not necessarily mean centralized planners like in USSR for example, the social planner from 2nd welfare theorem still uses markets and then just redistributes the outcome using lump sum tax. Also social planner in economics models is assumed to be all knowing, all loving/benevolent and all powerful (unless stated otherwise by introducing information asymmetries etc). Social planner in such models is in essence secular version of god $\endgroup$
    – 1muflon1
    Dec 7, 2022 at 10:20
  • $\begingroup$ even in 2nd welfare theorem IRL people would likely not obey the social planner and would resist the lump sum taxes, but 2nd welfare theorem assumes that the social planner just states what taxes are and people will immediately obey without question. Thats not how the real world works $\endgroup$
    – 1muflon1
    Dec 7, 2022 at 10:24

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