Free Market Failures

The invisible hand is the idea that free markets will, without any external ruling power, provide wealth and prosperity. This idea is often criticised by presenting some cases of market failures, showing that free markets do not provide the global optimum solution.

Examples are natural monopolies, where the cost to produce the infrastructure is so large that competition is not possible over one human life-span. Examples are railroads, and to a lesser degree, water and electricity. Once a railroad track is built, no same person would try to build a competing track next to it, as that would only ruin both railroad companies.
On the contrary, big players sometimes build their infrastructure next to new competitors to push them out of a market. In the end, free market solutions remain expensive and resources are wasted.

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Another example are the locations of selling points, for example lemonade stands. If there are two competing stands for a street, the customers would like to see them located at 1/3 and 2/3 of the street so that everyone has easy access. But this situation is not a stable nash equilibrium. If one stand moves to 1/2 while to other remains at 2/3, then the first will make more profit at the expense of the other. So the nash equilibrium will be both stands sitting at 1/2 next to each other. Now both make less profits as the people on the far end of the road might not get lemonade anymore and rather stay at home.
This effect can often be seen by the location of fast food chains that tend to stick close to each other.

These cases are often used to attack free market economy. And while they are game theoretically valid, this attack is not well thought out. We have proven that free markets do not provide the optimum solution. So the alternative that might lead to the optimum is introducing central power (free markets are the only case without central power).

But in order to justify central power we have to show that this will actually provide the optimum, or at least a better situation than the free markets. This step is never done by the critics of a free economy. The proudly present where free markets fail, but then turn a blind eye to the many failings of controlled economies.

If you want to advocate for controlled markets, you need to show how you avoid failures. And simply stating that the central authority will always make optimum decisions is not enough. Humans are flawed, humans can be corrupted, and humans are not always very smart.

Mathematically, controlled markets are much harder to describe as there is an uncontrollable element in them, but in the real world we can observe where they fail. Mismanagement by the central power is the least problem. In game theory, every participant will try to interact with the central authority to gain an advantage. This introduces a creeping corruption into the system. It may manifest in entry barriers that create situations similar to natural monopolies where a free market would work fine. It may also lead to privatising profits and distributing losses in bailouts.

If you want to ague in game theory against free markets, then you should do the same for regulated markets. And doing that quickly shows us that we are not even close to stating a centralised alternative that has a nash equilibrium close to the global optimum.