When the Pool Fights Back (Part II): Approximating Equilibria in Complex MFGs

Trapped between two shores: bounding liquidity games when theory fails.

Introduction: From Theory Breakdown to Smart Workarounds

In [Part I],

When the Pool Fights Back (Part I): Modeling Liquidity with Transaction Costs

we introduced a more realistic mean field game (MFG) model for decentralized trading, where agents interact with a liquidity pool while facing transaction costs. This small twist, modeling the bid/ask spread due to pool fees, added a surprising level of complexity. Most notably, it broke one of the structural assumptions needed to guarantee the existence of an MFG equilibrium.

So where does that leave us?

In this second part, we take a deeper look at how to work around this theoretical obstacle. While the original game is no longer covered by classical existence theorems, we show that by carefully constructing two auxiliary games, we can still extract useful insights. These auxiliary models approximate the original one from below and above, and both do admit a well-defined mean field equilibrium.

This strategy, bounding a hard problem with two easier ones, gives us a practical way to recover approximate equilibria even when the theory fails us. And it raises a broader question: in crypto, where market frictions and design quirks are everywhere, how often is approximation not just a fallback, but the best tool we have?

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Two Similar Mean Field Games: A Game of Bounds

The key challenge in our model is the structure of the utility function. In order to apply classical results from mean field game theory (like Theorem 3.5 from Carmona and Lacker) the utility function needs to be separable: it must decompose into a part that depends on the control variable and a part that depends on the distribution of controls. Formally, this means

\(f(t,x,\mu,q,a) = f_1(t,x,\mu,a) + f_2(t,x,\mu,q)\)

 In our case, this is no longer true. The term Λ

\(\Lambda(t, x, \mu, q, a) := a \left(1-\frac{(1+\phi)^2}{2\phi}\right) \frac{k_0}{(X_0- \int_0^t \int id dq_sds )^2(X_0- \phi\int_0^t \int id dq_sds )^2} \)

in the utility function couples the control a and the distribution q in a multiplicative way, which breaks separability and prevents us from directly applying the standard theory.

Instead of trying to fix the structure of f, we sandwich it between two auxiliary functions that do satisfy the separability assumption.

To do this, we isolate the problematic term Λ and build two bounding functions:

• Lower bound (Λ₁): We apply Young’s inequality to the multiplicative term, decomposing it into a sum of squares. This produces a conservative approximation, possibly underestimating trader losses but restoring separability.

  \(\Lambda_1 := -\frac{1}{2}\left[\left(a \left(\frac{(1+\phi)^2}{2\phi}-1\right)\right)^2 + \left(\frac{k_0}{(X_0- \int_0^t \int id dq_sds )^2(X_0- \phi\int_0^t \int id dq_sds )^2}\right)^2\right]\)

• Upper bound (Λ₂): We use the fact that trader inventories can’t fall below X₀ − T⋅M, and replace the denominator in Λ with its minimum value. This gives a more pessimistic approximation that likely overestimates costs.

  \(\Lambda_2 := -a \left(\frac{(1+\phi)^2}{2\phi}-1\right) \frac{k_0}{(X_0+TM)^4}\)

So, we have

\(\Lambda_1 \leq \Lambda \leq \Lambda_2.\)

Lets call Γ to the term

\(\Gamma(t, x, \mu, q, a) := k_0 \int id dq \frac{(1+\phi)X_0-2\phi\int_0^t\int id dq_sds}{(X_0- \int_0^t \int id dq_sds )^2(X_0- \phi\int_0^t \int id dq_sds )^2}.\)

Then, by plugging Λ₁ and Λ₂ into the original utility function, we define two new functions:

• f₁ = xΓ + Λ₁ − h

• f₂ = xΓ + Λ₂ − h

Both f₁ and f₂ satisfy the full set of assumptions needed to guarantee existence of a solution to the mean field game. In other words, even though the original problem is intractable under classical MFG theory, these two auxiliary problems are not.

This bounding approach allows us to keep analyzing the system, not exactly, but with mathematical rigor and control over the approximation error.

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Back to Our Original Problem: A Sandwich Theorem for Equilibria

Now that we have two well-behaved approximations, the games defined by utility functions f₁ and f₂, we return to the original game, defined by the non-separable utility function f.

While we can’t prove that f admits a mean field equilibrium in the classical sense, we can say something powerful: the value function (the inventory of the representative agent) of the original game is tightly enclosed between the value functions of the auxiliary games. That is,

\(V_{f_1} \leq V_f \leq V_{f_2},\)

where

\(V_{f_i} = \sup_{\alpha \in \mathbb{A}} J_{f_i}(\alpha).\)

with

\(J_{f_i}(\alpha) := \mathbb{E}\left[\int_0^T f_i(t, X_t, \mu_t, q_t, \alpha_t) dt - l(X_T) \right],\)

This is a key insight. It tells us that, although the original game is more complex, we can still bound its optimal value from above and below using two simpler games for which we know how to compute equilibria.

Even more importantly, this bound leads to a practical consequence: if the value functions V_f₁ and V_f₂ are close enough, say, their difference is less than ε, then the strategies that are optimal in either f₁ or f₂ are ε-optimal strategies for the original problem.

This leads to what we call an ε-Nash equilibrium via bounding games. Even though the original utility function f doesn’t allow a clean equilibrium construction, we can still say:

“Here’s a strategy that performs within ε of the true optimum, even under the full complexity of f.”

This is not just a workaround it’s a stability result. It shows that the system’s behavior doesn’t collapse when we step outside idealized assumptions. Instead, we can control and quantify how far we are from the theoretical optimum, and still trust the strategies we compute.

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Conclusion: Bending Theory Without Breaking Insight

Introducing transaction costs into our mean field game framework forced us to step outside the comfort zone of classical MFG theory. The clean separability that guaranteed equilibrium existence is gone, replaced by a more tangled structure where controls and distributions interact directly.

Yet, by constructing two auxiliary games that bound the original one, we recovered a practical notion of equilibrium, an ε-Nash equilibrium, and a way to approximate the true optimal strategies. This “sandwich” approach doesn’t just patch the theory; it offers a systematic method to handle real-world frictions that break idealized assumptions.

From a DeFi perspective, this is crucial. Liquidity pools are never frictionless, and designing robust strategies or protocols means understanding behavior in the presence of fees, slippage, and other market imperfections. Our results show that even when the math gets messy, structure and predictability can be rescued through approximation and bounding techniques.

Looking ahead, our next steps include expanding the model to incorporate liquidity providers, arbitrageurs, and other strategic agents, as well as validating these theoretical insights through numerical experiments. The ultimate goal is a framework that not only describes agent behavior but also guides the design of safer and more efficient DeFi markets.