The history of coordination spans billions of years, and we've been finding new ways to help each other out for as long as there have been more than one of us. From multicellularity to the evolution of brains, from the development of social and moral instincts to their codification in laws and contracts, from the emergence of currency to the invention of stocks and bonds and options and every other modern financial instrument, we have accumulated countless ways to work together.

Every time a group gets a little better at coordinating, moving closer to the Pareto frontier, they get a little more economically efficient. The field of thermodynamics has a model for the most perfectly efficient engine theoretically possible, given the limitations of entropy. Does economics have an equivalent result for the most perfectly efficient coordination mechanism, given the limitations of voluntary participation by agents with different interests?

Thermal efficiency is limited by the fact that we can't control individual particles directly. At least not without increasing entropy more somewhere else than we were able to reduce it among those particles. If we could, we could use that ability to produce limitless free energy forever. Literally forever, the "inevitable heat death of the universe" would be a cute little thing our distant ancestors worried about.

Economic efficiency is limited by the fact that we can't control individual agents directly. And much more so by our intelligence; our ability to find good solutions among a large space of candidates. If we could identify the socially optimal joint policy, and if we could direct all agents to implement it, this would be an instant win condition, bringing about utopia for as long as there is organized matter in the universe. World peace is easy when you can just direct people to get along.

Economists call the ratio between "the highest social welfare a central planner could achieve" and "the actual social welfare that results from people following their incentives" the Price of Anarchy. The better an economic mechanism is, the lower the price of anarchy (inefficiency caused by the lack of a central planner), and the better job that mechanism is doing at reshaping individual incentives to coordinate on socially-better joint policies. If a mechanism can incentivize agents to voluntarily coordinate on a solution that's as good as what a central planner would have come up with, that is a perfectly economically efficient mechanism.

Does economics have any of those? Under some assumptions, yes! One of them is the second-price auction, and its generalization the VCG mechanism. One assumption these mechanisms make is that participants all act independently, rather than coordinating to advance their interests at the expense of others. And of course, they also assume the enforcement of property and civil rights that limit the scope of interactions to peaceful, voluntary transactions.

Can we always introduce a voluntary mechanism that leads to perfect economic efficiency? Unfortunately we can't. Consider Alice deciding how to split $100 between herself and Bob, where Bob has nothing Alice wants and no power to influence the outcome whatsoever. Our hypothetical central planner might want Bob to get some of that money, but unless Alice wants that too there's no voluntary mechanism we can use to align her incentives with the good of the group. At least when considering this situation as a one-shot interaction. There might be hope if Alice also expects to interact with Carol, who is willing to spend some of her resources incentivizing Alice to treat Bob well.

Asymmetric power dynamics can lead to socially-suboptimal equilibria, which no voluntary mechanism can shift all the way to optimality. There is a maximum efficiency for voluntary mechanisms, or any other system we can't control perfectly, and sometimes that maximum is less than 100%.

So what is the Carnot engine of economics? The theoretically optimally efficient voluntary mechanism? I claim that it's bargaining, and specifically bargaining over the joint policy space $\Pi$.

The Space of Joint Policies

I'm going to go into a lot of detail about why we can't actually do this in practice. But supposing we could directly optimize the joint policy, we could bargain over which Pareto optimal one to implement, and implement it. This accounts for every conceivable way we could try to achieve socially-better outcomes.

The joint policy space $\Pi$ contains everything that all agents can collectively do. It contains every possible reaction to every possible sequence of observations by every agent that might possibly exist. It contains every threat of retaliation, every promise of reward. Every contract they might sign, and every way they might enforce it. Every coalition they might form, every social norm they might adopt. Every mechanism they might implement; voluntary and involuntary, actual and counterfactual. 

Every system of property rights and every system of economics built atop them. Every use of natural resources, every currency and financial instrument they might ever invent. Every government of every form, staffed by every combination of every mind and form of life that can exist. Every law, every scientific establishment, every organization and sport and religion and culture and form of entertainment. Every technology they might develop, every change they can make to their environment. 

Every book in every possible language, every work of art in every possible medium. Every possible conversation between every possible being, of every possible philosophy. Every theorem they might prove and every mathematical insight they might ever have. Every persuasive argument and every inspiring speech and children's TV show and style of education. Every way of resolving every type of disagreement, every peace and war and cooperation and competition. 

Every game they might play amongst themselves for all possible stakes, every long moral reflection and every way they might extrapolate their volition. Every computation they might perform, every self-modification they might make, every successor-agent and autonomous system they might deploy. Every mind control dictatorship, every beautiful transcendence into their most eudaimonic selves. Everything that can be done is in $\Pi$ somewhere.

The space of deterministic policies for a single agent is double-exponentially large. It looks like $|A{|}^{|O{|}^T}$: the number of possible actions per timestep $|A|$, raised to the power of $|O|$ (the number of possible observations per timestep), raised to the power of $T$ (the number of timesteps in your future horizon). 

This is ridiculously huge, and even with the amazing power of calculus we're nowhere close to being able to optimize policies of agents with anything close to human input-output bandwidths and non-trivial goals. We haven't found the optimal policy for chess, a fun little game with only ${10}^{43}$ board positions we use to teach children about strategic thinking. And we know what the rules for chess are! I love Updateless Decision Theory, and also any implementation will need something like a "best policy I've found so far" that agents can use quickly after being booted up, which can be refined over time.

The space of all deterministic joint policies for $n$ agents is $n$ times bigger than that. One policy for each agent; $n|A{|}^{|O{|}^T}$. And of course "the number of actions" a person can take and "the number of observations" a person can make are each also exponentially enormous no matter how we define a "timestep" for an analog system like the human brain. And those are just the deterministic policies: the actual joint policy space includes all the convex combinations of all of those policies we can reach by randomizing our actions.

Implementing the optimal joint policy is the best we can theoretically do, but in practice there is going to be a vast gulf between "the best that an infinitely intelligent rational agent could do in this situation" and "the best course of action I could think of at the time." Intelligence is the bottleneck to economic efficiency; becoming the unchallenged dictator of Earth is trivial by comparison.

Coordination is extremely computationally difficult. We've been developing new ways to coordinate for billions of years, with each advancement making us better at pursuing our goals including finding better ways to coordinate. Each advancement building on others, compounding into a hyper-exponentially growing population with access to a hyper-exponentially growing collection of technological, cultural, and economic resources. And we've barely scratched the surface of what's theoretically possible.

Bargaining Over Joint Policies

If a joint policy $\pi \in \Pi$ isn't considered feasible by a group of agents, it's because at least one member thinks they can do better for themselves in the absence of coordination than they'd get from implementing $\pi$. Only joint policies that all agents prefer to the one they'll implement in the absence of a negotiated agreement are feasible.

This might have already ruled out all of the socially optimal joint policies. If so, there is no voluntary mechanism we can use to reach them; at least one agent will simply refuse to participate and go with their preferred alternative. But if even one socially optimal joint utility is left in the feasible set $F$, it's conceivable that the actual group of negotiating agents, with all of their notions of fairness, will implement a joint policy which our hypothetical central planner agrees is optimal.

If so, great! Bargaining has achieved perfect economic efficiency. If not, it's because the actual group of negotiating agents decided to do something they like even more. And they made that decision after considering all possible voluntary and involuntary mechanisms they could implement. There is no voluntary mechanism our hypothetical central planner could suggest, which the group as a whole prefers to adopt. The agents actually making the decision simply disagree with the hypothetical central planner about what way of life is best.

Alas, the price of anarchy.