How quantum circuits can mitigate the tragedy of the commons

Institutions are a central pillar of economics. However, to me it always felt quite unintuitive how they are constructed. In this post, I explore how a quantum protocol can generate the necessary institutions to avoid the "tragedy of the commons". This builds upon the previous post in which the I discussed the possibility to solve the prisoners dilemma through institutions generated by the quantum mechanical extension in the Eisert–Wilkens–Lewenstein game. The new post extends the complexity of the previous example substantially as it has to work with continuous variables and the dependency between the different rounds. The successful construction gives a cooking recipe to find institutions for complex economical games.

The commons and its two benchmarks

A common-pool resource is one that many people draw from and nobody can be excluded from: a fish stock, a groundwater basin, an atmosphere that absorbs emissions. Each user's take reduces what is left for everyone else, and none of them is charged for it. I will run the whole argument on the most visual instance, two islands fishing one ground. The general model — and the textbook fishery John Leach sets up in §8.1 of A Course in Public Economics — is worked through in a companion post. Here is only the notation the rest of this one needs.

Island $j$ sends $b_j$ boats onto the shared ground; the fleet on the water is $B = b_1 + b_2$ and the stock is $s$. Write $u(B,s)$ for what the whole fishery earns in one season, and give each island the share matching how hard it worked,

Over many seasons an island wants its discounted total $U_j = \sum_t \beta^{t} u_{j,t}$, with $\beta$ near 1 a patient islander and $\beta = 0$ one who does not think about next season at all.

With $N$ users drawing independently, the companion post shows the symmetric equilibrium sitting at

The damage $D$ corresponds the whole loss your extra boat inflicts on everyone else. The equilibrium condition then reads: you collect the average $u/B$ for your extra boat, but are charged only $1/N$ of the damage it does. $N \to \infty$ gives $u = 0$ — everyone keeps adding boats until no profit is left. $N = 1$ gives $\partial_B u = 0$ — one owner feeling the whole cost of crowding, which is the planner in the economists' sense. The planner reaches a more economic fleet but has plenty of practical issues that come from the centralization of decision making. You can read more about the problem in this previous post.

A quantum protocol gives a third option. A while ago I wrote about the quantum prisoner's dilemma: take the Eisert–Wilkens–Lewenstein scheme, entangle two qubits, let each player act on their own qubit, disentangle — and mutual cooperation becomes stable. Nobody designed an enforcement mechanism. You write down the standard entangling operation, allow the natural moves, and an institution falls out of the machinery. In the previous post I exemplified it with a crooked lawyer with a filing cabinet.

The circuit

The continuous-variable version of EWL is due to Li, Du and Massar — two field modes instead of two qubits, and instead of a discrete gate each player picks a real number.

Both modes start in vacuum. A referee applies a two-mode squeezer $\hat J(\gamma)$, where the squeezing parameter $\gamma \ge 0$ is fixed once, in advance. Each island displaces its own mode by the number of boats it wants to send. The referee undoes the squeezing with $\hat J(\gamma)^\dagger$, and homodyne detection reads out each mode. Grau-Climent et al. showed that what comes out of the detectors is the licensed fleet $b_j^{c}$ that island $j$ is actually credited with:

We can analyze the two interesting limits, starting with $\gamma \rightarrow 0$ first. Then we have:

So we can already see that we really start to care about the number of boats that the other neighbor has sent out. Let us see what happens in the extreme case $\gamma \rightarrow \infty$. Then we have:

This is exactly the situation of the planner as we care about both boats equally. So we can directly expect that the squeezing leads to cooperation and solve the tragedy in a similiar fashion as a planner did. However, not by some external institution but by simply sharing the ressources in a controlled way.

Solving the tragedy of the commons

We can now look how each individual island would optimize its discounted payoffs for a given squeezing parameter and we get:

So the payoff is distributed. Note that the licensed fleets sum to the declared total, $b_1^{c} + b_2^{c} = b_1 + b_2 = B$, so the fleet on the water is unchanged and only its attribution moves. Write $\alpha = \frac{1+e^{-2\gamma}}{2}$ for the weight each island keeps on its own declaration, so that $\partial_{b_1} b_1^{c} = \alpha$ and $\partial_{b_1} B = 1$. Each island now optimizes its own $u_{j}^{c}$ over its declaration $b_j$:

with the damage $D = \frac{u}{B} - \partial_B u$ as in the companion post. Setting this to zero, and using that the two islands are symmetric so that $b_1^{c} = B/2$:

So the squeezing heals the tragedy as it effectively reduces the number of independent islands through the squeezing parameter. The tragedy disappears at $N_{\text{eff}} = 1$, where only one island is effectively left and the condition becomes $\partial_B u = 0$, so we optimize total profits.

The equivalent classical game

This mechanism is well-known in economics as cross-ownership. Each firm holds some equity in other one and Reynolds and Snapp wrote down how this solves the tragedy as described above. So we are really back to the situation we had in the Prisoners' dilemma. The quantum circuit created a mechanism that is well-known in economy and proven to enhance cooperation. It works in both cases, and in both cases economics had already found the generated institution before.

Outlook

This finishes the second test of the possibility to generate economic institution through quantum circuits. In both, writing down a standard entangling operation produced a working institution, and in both that institution had an entirely classical description — a filing cabinet, a clerk with a ledger. Here it reaches the planner's outcome knowing only $\gamma$, and nothing about the resource.

What it cannot change is patience because $\beta$ appears nowhere in $N_{\text{eff}}$. The protocol removes the externality between the users and leaves the single-owner problem that remains at whatever patience they had to start with — it reaches the patient planner if they are patient, and the myopic one if they are not. So in the future, I am likely to look for further possibilities to use quantum circuits as a cooking recipe for games which require coordination. Maybe I will come across situations which were solved quite as nicely within economics.

If you have thought about that I would like to hear from you.