Liquidity Pools as Mean Field Games with Transaction Costs

Liquidit y P o ols as Mean Field Games with T ransaction Costs Agustín Muñoz González 1 1 Departamen to de Matematicas, F acultad de Ciencias Exactas y Naturales, Univ ersidad de Buenos Aires, Buenos Aires, Argentina Marc h 18, 2026 Abstract This pap er extends the theoretical framew ork in tro duced in Liquidity Po ols as Me an Field Games: A New F r amework , where the in teractions among traders in a constant pro duct market-making proto col were mo deled using mean field games (MFG). In this extension, transaction costs are incorp orated in to the traders’ in ven tory dynamics, mo d- eling the impact of po ol fees on trading decisions. T raders op erate at a mid-price ad- justed for transaction costs, in tro ducing a new dynamic for the D AI in ven tory . The existence of MF G solutions for this new trader game is established, taking these addi- tional costs into account. The traders’ optimization strategies and the conditions for equilibrium existence are discussed, providing a solid foundation for future researc h. Keywor ds— Mean field games, transaction costs, liquidity p o ols, automated market mak- ers, approximate Nash equilibrium, mean field equilibrium, decentralized finance 1 Con ten ts 1 In tro duction 3 2 Mo del F orm ulation 3 3 Do es a Mean Field Game Solution Still Exist? 7 4 T w o Auxiliary Mean Field Games 10 5 Bac k to the Original Problem 13 2 1 In tro duction In this w ork, we extend the model in tro duced in Liquidity Po ols as Me an Field Games: A New F r amework , incorp orating transaction costs in to the dynamics of traders op erating within a constant pro duct liquidit y p o ol. In the baseline mo del, traders seek to maximize their in ven tory by trading against the reserves of the ETH-DAI p o ol. Ho w ev er, in this new approac h, we consider that traders face additional costs when executing transactions, which affect b oth the price and their optimal strategies. T ransaction costs are modeled as a spread b et ween the bid pric e and the ask pric e , meaning that traders buy at a higher price and sell at a low er price than the p o ol’s equilibrium price. This introduces a new dynamic into the traders’ inv en tory , requiring them to optimize their strategies while considering the impact of fees on their buy and sell decisions. The trader’s new utilit y function incorp orates these costs, adjusting their DAI inv en- tory based on a mid-price adjustment. Despite the inclusion of transaction costs, the mo del preserv es the relationship b et ween price and p o ol reserves, main taining the traditional equi- librium structure where X t Y t = k t , but with an in v arian t k t that no w dep ends on the accum ulated fees in the p o ol. The primary goal of this work is to establish the existence of a mean field equilibrium for this trader game with transaction costs, using techniques similar to those dev elop ed in the original mo del. The traders’ optimization strategies and equilibrium conditions are rigorously presented, laying a strong foundation for future extensions of the mo del. 2 Mo del F orm ulation W e w ork on the filtered probability space (Ω , F T , F , P ) supp orting N + 1 mutually indep en- den t Wiener pro cesses W 0 , . . . , W N . As in the baseline mo del of [ 2 ], we consider N traders who op erate in a liquid it y p o ol holding tw o tokens, ETH and USDT. Let X i t and Y i t denote the ETH and USDT inv entories, resp ectiv ely , of the i -th trader at time t . The ETH inv en tory dynamics for trader i are given b y dX i t = α i t dt + σ i dW i t , (1) where α i t : [0 , T ] → R is the trading rate (the control); σ i denotes the individual v olatility , assumed indep endent of i for simplicity; and σ i dW i t captures random fluctuations in the trader’s holdings due to exogenous mark et even ts or wallet-lev el noise. In the baseline mo del, the USDT in ven tory dynamics w ere given b y d Y i t = − ( α i t P t + c p ( α i t )) dt, with P t the p o ol’s equilibrium price of ETH in USDT at time t . Here, we explicitly incorp o- rate the transaction cost incurred b y traders when using the p o ol’s sw ap service. Sp ecifically , when selling, a trader receives the bid price p b < P , and when buying, pa ys the ask price p a > P . Since the mo del do es not inheren tly distinguish buy from sell orders, we follow [ 3 ] (Eq. (15)) and assume trades o ccur at the mid-price ˜ P := p a + p b 2 . W e therefore prop ose the follo wing USDT inv en tory dynamics: d Y i t = − α i t ˜ P t dt. 3 F rom [ 3 ] we hav e the iden tities p a = 1 ϕ P and p b = ϕP , where ϕ = 1 − τ and 0 < τ < 1 is the p o ol fee. Hence ˜ P = 1 + ϕ 2 2 ϕ P , and the USDT inv en tory dynamics for trader i b ecome d Y i t = − α i t 1 + ϕ 2 2 ϕ P t dt. (2) Although transaction costs are now present, once a sw ap is executed and fees are added to the p o ol, the equilibrium price con tin ues to satisfy P t = Y t X t , (3) where X t and Y t are the ETH and USDT reserv es in the po ol, resp ectively . Note that, b ecause of the fee mec han ism, the “constan t” k t is no longer constant: fees paid b y traders accum ulate in the p o ol, causing k t to increase o ver time. The constant-product relation, ho wev er, still holds: X t Y t = k t . (4) In a p o ol with transaction costs, trades pro ceed in tw o stages. Supp ose that at time t > 0 a trade of ∆ X t ETH units is submitted. In the first stage, the p o ol computes the corresp onding USDT output ∆ Y t via k 0 = ( X 0 + ϕ ∆ X t )( Y 0 − ∆ Y t ) ⇒ ∆ Y t = − k 0 X 0 + ϕ ∆ X t + Y 0 . In the second stage, the fees are added to the p o ol and the in v ariant is up dated: k t = X t Y t = ( X 0 + ∆ X t )( Y 0 − ∆ Y t ) = ( X 0 + ∆ X t ) k 0 X 0 + ϕ ∆ X t . Com bining (3) and (4), the price equation b ecomes P t = k 0 ( X 0 + ϕ ∆ X t )( X 0 + ∆ X t ) . (5) F ollo wing the metho dology of the simpler mo del, we mo del the ETH balance inside the p o ol via the av erage of the agents’ controls: X t := X 0 − 1 N N X i =1 Z t 0 α i s ds, (6) where X 0 is the initial tok en supply provided by the liquidit y pro vider. F or mo del consistency it is essen tial that X t  = 0 for all t ∈ [0 , T ] , as the p o ol must alw a ys retain a minimum reserv e. 4 Remark. In Se ction 3.1, after verifying hyp othesis (S.4) , we wil l have X t  = 0 for al l t ∈ [0 , T ] . Henc e the p o ol maintains at le ast a minimum token fr action at every moment: X t ≥ ϵ 0 , ∀ t ∈ [0 , T ] , (7) for some ϵ 0 > 0 . Setting ∆ X t = − 1 N P N i =1 R t 0 α i s ds , the price dynamics b ecome dP t = d  k 0 ( X 0 + ϕ ∆ X t )( X 0 + ∆ X t )  dt + σ 0 dW 0 t = k 0 1 N N X i =1 α i t ! (1 + ϕ ) X 0 + 2 ϕ ∆ X t ( X 0 + ∆ X t ) 2 ( X 0 + ϕ ∆ X t ) 2 dt + σ 0 dW 0 t , (8) where W 0 t is a Wiener pro cess, σ 0 is a constant volatilit y , and σ 0 dW 0 t represen ts exogenous price sho c ks. Remark. By writing ∆ X t = − 1 N P N i =1 R t 0 α i s ds inside the pric e dynamics, we ar e tr e ating al l tr ades in [0 , t ] as a single aggr e gate tr ansaction ∆ X t . Although the trader executes orders at the mid-price ˜ P , p ortfolio v aluation is p erformed at the p o ol’s arbitraged equilibrium price P . The total inv en tory of trader i at time t is therefore defined, as in the baseline mo del, by V i t = Y i t + X i t P t . Applying Itô’s lemma gives dV i t = d Y i t + X i t dP t + dX i t P t = " X i t k 0 1 N N X i =1 α i t ! (1 + ϕ ) X 0 + 2 ϕ ∆ X t ( X 0 + ∆ X t ) 2 ( X 0 + ϕ ∆ X t ) 2 + α i t  1 − 1 + ϕ 2 2 ϕ  k 0 ( X 0 + ∆ X t )( X 0 + ϕ ∆ X t )  dt + X i t σ 0 dW 0 t + P t σ i dW i t . (9) W e assume that agents are risk-neutral and seek to maximize their exp ected profit from trading in the decentralized mark et: J i ( α 1 , . . . , α N ) = E  V i T − Z T 0 h ( t, X i t ) dt − l ( X i T )  , (10) where h : [0 , T ] × R → R represents the cost of holding inv entory x at time t , and l : R → R is a terminal inv en tory cost. 5 W orking spaces Before formulating the mean field game for infinitely many traders, we in tro duce the relev ant functional spaces. • Let (Ω , F , F = ( F t ) 0 ≤ t ≤ T , P ) b e a complete filtered probability space supp orting a one- dimensional Wiener process W = ( W t ) 0 ≤ t ≤ T and an initial condition ξ ∈ L 2 (Ω , F 0 , P ; R ) . • Let C := C ([0 , T ]; R ) b e the space of con tinuous real-v alued functions on [0 , T ] , equipp ed with the supremum norm ∥ x ∥ := sup s ∈ [0 ,T ] | x ( s ) | . • Given P ( R ) , the space of probability measures on R , and a measurable function ψ : C → R , define P ψ ( R ) =  µ ∈ P ( R ) : Z ψ dµ < ∞  , B ψ ( R ) =  f : Ω → R : sup ω | f ( ω ) | /ψ ( ω ) < ∞  . W e equip P ψ ( R ) with the weak est top ology τ ψ ( R ) making the map µ 7→ R f dµ contin- uous for every f ∈ B ψ ( R ) . • Let the control space A ⊂ R b e a b ounded subset, and let A = { α : [0 , T ] × Ω → A : progressively measurable } b e the set of admissible controls. As in [ 2 ], all control functions α are considered in closed-lo op form. • Finally , let P ( A ) b e the space of probability measures ov er A , equipp ed with the weak top ology τ ( A ) that mak es q 7→ R A f dq con tinuous for each f ∈ B ( A ) . Throughout the pap er we omit the dep endence on ω ∈ Ω to ligh ten notation. Using the deterministic part of the in ven tory dynamics (9) in the infinite-pla y er limit, and defining f : [0 , T ] × R × P ψ ( R ) × P ( A ) × A → R b y f ( t, x, µ, q , a ) = x k 0 Z A ˜ a dq t (˜ a ) · (1 + ϕ ) X 0 − 2 ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 + a  1 − 1 + ϕ 2 2 ϕ  k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 − h ( t, x ) , the functional J to b e maximized can b e rewritten as J i ( α 1 , . . . , α N ) = E  Z T 0 f ( t, X i t , µ, ˆ q N t , α i t ) dt − l ( X i T )  , 6 where µ is a probabilit y measure ov er the state space and ˆ q N t denotes the empirical distri- bution of α 1 t , . . . , α N t . By the symmetry of the mo del, trader i ’s con tribution to ˆ q N is negligible for large N , and we may treat ˆ q N as fixed. W e therefore work with the functional J ( α ) := E  Z T 0 f ( t, X t , µ t , q t , α t ) dt − l ( X T )  , where µ = ( µ t ) 0 ≤ t ≤ T is a flow of probability measures o ver the state space and q = ( q t ) 0 ≤ t ≤ T is a flow of probability measures o v er the control space. T o av oid rep eating notation already established in [ 2 ], w e only recall the MF G definition here. Giv en flo ws µ = ( µ t ) 0 ≤ t ≤ T and q = ( q t ) 0 ≤ t ≤ T of probability measures o ver R and A , resp ectiv ely , and a control α ∈ A , the asso ciated exp ected reward is J µ , q ( α ) := E µ ,α  Z T 0 f ( t, X t , µ t , q t , α t ) dt + g ( X T , µ T )  , where E µ ,α denotes exp ectation under P µ ,α , defined via the Doléans-Dade exp onential E b y d P µ ,α d P = E  Z · 0 σ − 1 b ( s, X s , µ s , α s ) dW s  T . This is a standard sto chastic optimal control problem whose optimal v alue is V µ , q = sup α ∈ A J µ , q ( α ) . (11) Recall that P µ ,α ◦ X − 1 and P µ ,α ◦ α − 1 denote the push-forward measures of P µ ,α under X and α , resp ectively . A pair of flows ( µ , q ) constitutes a me an field game (MF G) solution if there exists a con trol α ∈ A such that V µ , q = J µ , q ( α ) , P µ ,α ◦ X − 1 = µ , and P µ ,α ◦ α − 1 = q for almost every t . F or a more detailed presen tation we refer the reader to [ 2 ]. 3 Do es a Mean Field Game Solution Still Exist? In this section we examine whether the existence hypotheses needed to in v oke Theorem 3.5 of [ 1 ] remain satisfied after the introduction of transaction costs. W e will see that h yp othesis (S.5) , which requires the utilit y function f to decomp ose additively into a term that do es not dep end sim ultaneously on the control distribution q and on the con trol a , fails to hold. Ho wev er, this obstacle can b e circum v ented b y b ounding the original f ab o ve and b elow by auxiliary functions that do satisfy all the required hypotheses. W e retain the same standing assumptions as in [ 2 ]: • The initial ETH in ven tories X i 0 are i.i.d. with common distribution λ 0 ∈ P ( R ) satisfying Z R e p | x | λ 0 ( dx ) < + ∞ , for all p > 0; 7 • The control space A ⊂ R is a compact interv al containing the origin; • The volatilit y of the state pro cess X t is constant and p ositive, σ > 0 ; • The costs h and l are measurable, and there exists a constant c 1 > 0 such that | h ( t, x ) | + | l ( x ) | ≤ c 1 e c 1 | x | , for all ( t, x ) ∈ [0 , T ] × R . (12) Using the notation of the previous section, for ( t, x, µ, q , a ) ∈ [0 , T ] × R × P ψ ( R ) × P ( A ) × A w e set b ( t, x, µ, a ) := a, g ( µ, x ) := l ( x ) , ψ ( x ) := e c 1 | x | , f ( t, x, µ, q , a ) = x k 0 Z A ˜ a dq t (˜ a ) · (1 + ϕ ) X 0 − 2 ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 + a  1 − 1 + ϕ 2 2 ϕ  k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 − h ( t, x ) . T o lighten the notation, we introduce the auxiliary functions Γ( t, x, µ, q , a ) := k 0 Z A ˜ a dq t (˜ a ) · (1 + ϕ ) X 0 − 2 ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 , Λ( t, x, µ, q , a ) := a  1 − 1 + ϕ 2 2 ϕ  k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 , so that f ( t, x, µ, q , a ) = x Γ( t, x, µ, q , a ) + Λ( t, x, µ, q , a ) − h ( t, x ) . W e no w recall the hypotheses of Theorem 3.5 in [ 1 ] and v erify which ones remain v alid in the new mo del. Corollary 3.1. The fol lowing c onditions hold: (S.1) The c ontr ol sp ac e A is a c omp act c onvex subset of R , and the drift b : [0 , T ] × R × P ψ ( R ) × A → R is c ontinuous. (S.2) Ther e exists a str ong solution X to the driftless state e quation dX t = σ dW t , X 0 = ξ , (13) such that E [ ψ 2 ( X )] < + ∞ , σ > 0 , and σ − 1 b ( a ) is uniformly b ounde d. 8 (S.3) The c ost/b enefit function f : [0 , T ] × R × P ψ ( R ) × P ( A ) × A → R is such that ( t, x ) 7→ f ( t, x, µ, q , a ) is pr o gr essively me asur able for e ach ( µ, q , a ) , and a 7→ f ( t, x, q , a ) is c ontinuous for e ach ( t, x, µ, q ) . The terminal c ost function g : R × P ψ ( R ) → R is Bor el me asur able for e ach µ . (S.4) Ther e exists c > 0 such that | g ( x, µ ) | + | f ( t, x, µ, q , a ) | ≤ c ψ ( x ) ∀ ( t, x, µ, q ) ∈ [0 , T ] × R × P ψ ( R ) × P ( A ) . Pr o of. W e v erify each statement in turn. Conditions (S.1) and (S.2) are immediate. F or (S.3) and (S.4) we rep eat the argument of [ 2 ]. Since A is b ounded, there exists M suc h that | α ( t, ω ) | ≤ M for all ( t, ω ) ∈ [0 , T ] × Ω , and hence X t = X 0 − Z t 0 Z A r dq s ( r ) ds ≥ X 0 − T M , ∀ 0 ≤ t ≤ T . (14) Restricting the admissible controls so that M < X 0 T , there exists ϵ 0 > 0 such that X t = X 0 − Z t 0 Z id dq s ds > ϵ 0 , ∀ t ∈ [0 , T ] . (15) This immediately implies X 0 − ϕ Z t 0 Z id dq s ds > ϵ 0 . These low er b ounds guaran tee that f is progressively measurable in ( t, x ) (it is w ell-defined) and contin uous as a function of a . Since g ( x, µ ) is con tin uous and hence Borel measurable, condition (S.3) follows. The lo w est fee observ ed in practice is 0 . 01% (Uniswap v3), and the zero-fee case ϕ = 0 corresp onds to the mo del without transaction costs, so w e may assume c 0 ≤ ϕ ≤ 1 for some c 0 > 0 . W e now b ound f from ab ov e by b ounding Γ and Λ separately: | Γ | =          k 0 Z A ˜ a dq t (˜ a ) · (1 + ϕ ) X 0 − 2 ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2          ≤ 2 k 0 ϵ 4 0 M ( X 0 + T M ) , | Λ | =          a  1 − 1 + ϕ 2 2 ϕ  k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2          ≤ k 0 ϵ 4 0 M  1 + ϕ 2 2 ϕ − 1  , 9 where in the last b ound for | Λ | we used that 1+ ϕ 2 2 ϕ > 1 . Com bining these estimates with (12), | g ( x ) | + | f ( x ) | ≤ | l ( x ) | + | h ( t, x ) | + | x || Γ | + | Λ | ≤ c 1 e c 1 | x | + | x | 2 k 0 ϵ 4 0 M ( X 0 + T M ) + k 0 ϵ 4 0 M  1 + ϕ 2 2 ϕ − 1  ≤ C e C | x | , for the constan t C = max n c 1 , 2 k 0 ϵ 4 0 M ( X 0 + T M ) , k 0 ϵ 4 0 M  1+ ϕ 2 2 ϕ − 1 o . This completes the v erification of (S.4) . The remaining condition required b y the theorem is: (S.5) The function f is of the form f ( t, x, µ, q , a ) = f 1 ( t, x, µ, a ) + f 2 ( t, x, µ, q ) . As noted in the in tro duction to this section, this condition fails: the term Λ couples the con trol a with the con trol distribution q in a w a y that cannot b e separated additiv ely . 4 T w o Auxiliary Mean Field Games In this section we show that, b y replacing the utility function f with suitable auxiliary functions f 1 and f 2 that b ound f from b elow and ab o v e, we can prov e the existence of MF G solutions for t w o related problems. The solutions to these auxiliary problems will then allo w us to draw conclusions ab out the original problem induced b y f . The strategy is to b ound the problematic term Λ = a  1 − 1 + ϕ 2 2 ϕ  k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 b y functions that do satisfy condition (S.5) . It is conv enien t to rewrite this as Λ = − a  1 + ϕ 2 2 ϕ − 1  k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 . F rom the low er b ound (14) we obtain Λ ≤ − a  1 + ϕ 2 2 ϕ − 1  k 0 ( X 0 − T M ) 4 . On the other hand, Y oung’s inequality states that for an y a, b ≥ 0 and conjugate exp o- nen ts p, q > 1 with 1 p + 1 q = 1 , ab ≤ a p p + b q q , 10 or equiv alently , − ab ≥ −  a p p + b q q  . T aking p = q = 2 , and noting that 1 + ϕ 2 2 ϕ − 1 > 0 and k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 > 0 , w e get −  a  1 + ϕ 2 2 ϕ − 1  · k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 ≥ − 1 2       a  1 + ϕ 2 2 ϕ − 1  2 +      k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2      2      . Defining the tw o b ounding terms Λ 1 := − 1 2       a  1 + ϕ 2 2 ϕ − 1  2 +      k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2      2      , Λ 2 := − a  1 + ϕ 2 2 ϕ − 1  k 0 ( X 0 + T M ) 4 , w e hav e Λ 1 ≤ Λ ≤ Λ 2 . Remark 4.1. By r eplacing Λ with Λ 1 we ar e b ounding the pr oblematic factors by the smal lest value they c an attain, which amounts to an upp er b ound on the p otential loss for tr aders. By c ontr ast, r eplacing Λ with Λ 2 substitutes a p otential ly lar ger c ost, thus p enalising tr aders mor e than the original mo del do es. 11 Corollary 4.2. L et f 1 ( t, x, µ, q , a ) := x Γ( t, x, µ, q , a ) + Λ 1 ( t, x, µ, q , a ) − h ( t, x ) = k 0 Z A ˜ a dq t (˜ a ) · (1 + ϕ ) X 0 − 2 ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 − 1 2       a  1 + ϕ 2 2 ϕ − 1  2 +      k 0  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2      2      − h ( t, x ) , f 2 ( t, x, µ, q , a ) := x Γ( t, x, µ, q , a ) + Λ 2 ( t, x, µ, q , a ) − h ( t, x ) = k 0 Z A ˜ a dq t (˜ a ) · (1 + ϕ ) X 0 − 2 ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  2  X 0 − ϕ Z t 0 Z A ˜ a dq s (˜ a ) ds  2 − a  1 + ϕ 2 2 ϕ − 1  k 0 ( X 0 + T M ) 4 − h ( t, x ) . Both functions induc e MFG pr oblems that satisfy c onditions (S.1) – (S.5) . Pr o of. Conditions (S.1) and (S.2) remain v alid since they do not inv olv e Λ . Condition (S.3) follo ws b ecause b oth Λ 1 and Λ 2 are w ell-defined for ev ery ( t, x ) , hence progressively measurable, and contin uous as functions of a . F or (S.4) it suffices to b ound | Λ 1 | and | Λ 2 | : | Λ 1 | ≤ M 2     1 − 1 + ϕ 2 2 ϕ     2 + k 2 0 ϵ 8 0 , | Λ 2 | ≤ M     1 + ϕ 2 2 ϕ − 1     k 0 ( X 0 + T M ) 4 . Both b ounds are constants, so the condition follows. Finally , it is clear that f 1 and f 2 admit a separation of the terms in volving the control a from those inv olving the control distribution q , so (S.5) holds for b oth. In the remainder of this section we v erify conditions (C) and (E) required by Theo- rem (Existence of MFG solution). Corollary 4.3. Condition (C) holds for f 1 and f 2 . Pr o of. As sho wn in [ 2 ], the condition holds whenev er b is affine in a and f 1 , f 2 are concav e in a . The first part follows from b ( a ) = a . F or the second, we compute the second deriv ativ es 12 with resp ect to a : ∂ f 1 ∂ a = − a  1 + ϕ 2 2 ϕ − 1  2 = ⇒ ∂ 2 f 1 ∂ a 2 = −  1 + ϕ 2 2 ϕ − 1  2 < 0 , ∂ f 2 ∂ a = −  1 + ϕ 2 2 ϕ − 1  k 0 ( X 0 + T M ) 4 = ⇒ ∂ 2 f 2 ∂ a 2 = 0 . Hence f 1 is strictly concav e and f 2 is affine (and thus conca ve) in a . W e state the next condition without pro of, as the argumen t is en tirely analogous to that in [ 2 ]. Corollary 4.4. The mo dels induc e d by the auxiliary functions f 1 and f 2 satisfy c ondition (E) . In voking Theorem (Existence of MFG solution), w e conclude: Prop osition 4.5. Ther e exists an MFG solution for the pric e-imp act mo dels induc e d by the utility functions f 1 and f 2 . 5 Bac k to the Original Problem In this section we return to the mean field game gov erned by the original utilit y function f , whic h fails to satisfy hypothesis (S.5) due to the coupling b etw een the con trol a and the con trol distribution q . Despite this structural limitation, we sho w that f can b e sandwiched b et w een the tw o auxiliary cost functions f 1 and f 2 in tro duced earlier. By comparing the v alue functions associated with f 1 , f , and f 2 , and appealing to stabilit y argumen ts, we deriv e explicit bounds for the original problem. This bridging approac h yields meaningful information ab out the original system even when standard existence results do not apply directly . Lemma 5.1. L et f 1 , f , f 2 : [0 , T ] × C × P ψ ( C ) × P ( A ) × A → R b e me asur able functions satisfying the p ointwise ine quality f 1 ( t, x, µ, q , a ) ≤ f ( t, x, µ, q , a ) ≤ f 2 ( t, x, µ, q , a ) for al l ( t, x, µ, q , a ) . (16) Supp ose that f 1 , f 2 , and f al l satisfy hyp othesis (S.4) . Then the optimal values satisfy V f 1 ≤ V f ≤ V f 2 , wher e V ∗ denotes the optimal inventory value (11) asso ciate d with e ach of the functions f , f 1 , f 2 . Pr o of. The inequalit y (16) implies J f 1 ( α ) ≤ J f ( α ) ≤ J f 2 ( α ) ∀ α. (17) 13 Moreo ver, hypothesis (S.4) ensures that V f 1 , V f , V f 2 < ∞ . The inequality (17) must pass to the suprema, giving V f 1 ≤ V f ≤ V f 2 . W e v erify this by con tradiction. Without loss of generality , supp ose V f > V f 2 . Then there exists ϵ > 0 small enough that V f − ϵ > V f 2 . By definition of the suprem um applied to V f , there exists a control α ϵ suc h that J f ( α ϵ ) > V f − ϵ, whic h implies J f ( α ϵ ) > V f 2 ≥ J f 2 ( α ϵ ) . This contradicts (17), so the result follows. In practice, one can compute the optimal con trols ˆ α 1 and ˆ α 2 for the auxiliary MF G problems induced by f 1 and f 2 , resp ectively . If the corresp onding v alues J f 1 ( ˆ α 1 ) and J f 2 ( ˆ α 2 ) are sufficiently close — for instance, if J f 2 ( ˆ α 2 ) − J f 1 ( ˆ α 1 ) < ε, — then eith er control provides an ε -approximate solution to the original problem go verned b y f . In this sense, ˆ α 1 and ˆ α 2 act as ε -Nash equilibria of the original game. This is made precise in the following prop osition. Prop osition 5.2 ( ε -Nash equilibrium via b ounding problems) . L et f 1 and f 2 b e c ost func- tions such that f 1 ≤ f ≤ f 2 p ointwise, and supp ose that the me an field games asso ciate d with f 1 and f 2 admit solutions ˆ α 1 and ˆ α 2 , r esp e ctively. Supp ose further that V f 2 − V f 1 < ε, wher e as b efor e V f 1 = J f 1 ( ˆ α 1 ) and V f 2 = J f 2 ( ˆ α 2 ) . Then, for any c ontr ol α ∈ { ˆ α 1 , ˆ α 2 } , V f − J f ( α ) < ε. In p articular, α is an ε -Nash e quilibrium for the original pr oblem governe d by f . Pr o of. By Lemma 5.1, V f 1 ≤ V f ≤ V f 2 , whic h gives V f − V f 1 ≤ V f 2 − V f 1 . Moreo ver, V f 1 ≤ J f ( α 1 ) . Therefore, V f − J f ( α ) ≤ V f − V f 1 ≤ V f 2 − V f 1 < ε. 14 Remark. This r esult pr ovides a pr actic al way to c onstruct appr oximate solutions for a me an field game when the c ost function f fails to satisfy the structur al c ondition (S.5), pr ovide d it c an b e sandwiche d b etwe en two auxiliary c ost functions f 1 and f 2 that do satisfy the standar d hyp otheses. By exploiting the existenc e of exact solutions for the games asso ciate d with f 1 and f 2 , and quantifying the gap b etwe en the induc e d optimal values, one obtains an explicit ε -Nash e quilibrium for the original pr oblem. This metho d is p articularly useful when the dep endenc e on the c ontr ol distribution c annot b e cle anly sep ar ate d fr om the c ontr ol itself, but such a sep ar ation is r e c over e d in suitable appr oximations. The pr op osition ther efor e serves as a fundamental to ol for stability and appr oximation ar guments in extende d me an field game fr ameworks. Although f 1 and f 2 satisfy condition (S .5) and therefore admit approximate Nash equi- libria via Theorem (Nash Equilibrium Appro ximation), the original function f do es not. A ccordingly , no direct appro ximation result for f is currently av ailable, and w e conclude our analysis with the sandwich inequality of Lemma 5.1, whic h nonetheless provides useful information ab out the range of ac hiev able costs. Corollary 5.3 (Reco very of the zero-fee mo del as ϕ → 1 ) . As ϕ → 1 (i.e., τ → 0 ), the gap b etwe en the auxiliary functions f 1 and f 2 and the original function f tends to zer o. Mor e pr e cisely: 1. The pr oblematic term Λ satisfies Λ → 0 as ϕ → 1 , sinc e 1 + ϕ 2 2 ϕ − 1 = (1 − ϕ ) 2 2 ϕ ϕ → 1 − − → 0 . 2. Conse quently, f 1 , f , f 2 → x Γ − h , which c oincides with the r ewar d function of the zer o-c ost mo del in [ 2 ]. 3. (Conje ctur e) The sandwich gap V f 2 − V f 1 → 0 , and the solution to the original pr oblem c onver ges to the solution of the zer o-c ost mo del. Pr o of. P oin t (1) is a direct computation. F or (2), when ϕ = 1 the function f reduces to f ( t, x, µ, q , a ) = x · k 0 Z A ˜ a dq t (˜ a ) · 2 X 0 − 2 Z t 0 Z A ˜ a dq s (˜ a ) ds  X 0 − Z t 0 Z A ˜ a dq s (˜ a ) ds  3 − h ( t, x ) , whic h coincides with the rew ard function of the zero-cost mo del (up to algebraic simplification of the numerator). F or p oin t (3): one can verify directly that f 1 and f 2 con verge uniformly to f as ϕ → 1 , since the difference f 2 − f 1 is prop ortional to Λ 2 − Λ 1 → 0 (see p oint (1)). How ev er, passing from uniform con vergence of cost functions to conv ergence of optimal v alues V f i → V f requires additional arguments — in particular, v erifying the hypotheses of Berge’s maximum theorem (compactness of the admissible control set and contin uit y of the constrain t corresp ondence). This is not established in the present work, so p oint (3) is stated as a natural conjecture, strongly suggested by the uniform conv ergence. 15 Remark 5.4 (Practical justification of the ε -Nash equilibrium) . The pr op osition on ε -Nash e quilibria via b ounding pr oblems pr ovides a c oncr ete c omputational p ath for assessing the quality of the appr oximation: one ne e d only solve the MFG pr oblems asso ciate d with f 1 and f 2 (which satisfy al l standar d hyp otheses) and c omp ar e the r esulting optimal values. If the gap V f 2 − V f 1 is smal l, either optimal c ontr ol of the auxiliary pr oblems is a go o d appr oximation to the e quilibrium of the original pr oblem. In p articular, the most c ommon fe e in curr ent AMM pr oto c ols is of the or der τ = 0 . 3% (Uniswap v3), giving ϕ = 0 . 997 and 1 + ϕ 2 2 ϕ − 1 = (1 − ϕ ) 2 2 ϕ ≈ 4 . 5 × 10 − 6 . This me ans the sc aling factor of Λ is of or der 4 . 5 × 10 − 6 , suggesting that the p erturb ation intr o duc e d by tr ansaction c osts is smal l r elative to the frictionless c ase, in the sense that f 2 − f 1 is pr op ortional to this factor. However, quantifying the effe ctive gap V f 2 − V f 1 r e quir es numeric al c omputation, sinc e it dep ends on the ful l structur e of f 1 and f 2 — including the quadr atic terms in a and the denominators involving the r eserves — and not only on the sc aling factor of Λ . Such quantific ation is identifie d as futur e work (se e R emark 5.5). Remark 5.5 (F uture direction: n umerical quantification of the gap) . A natur al dir e ction for futur e work is to develop a numeric al solver for the pr oblems asso ciate d with f 1 and f 2 , and to quantify the gap V f 2 − V f 1 explicitly for various values of ϕ . This would al low a c omputational verific ation that the gap is ne gligible for typic al r e al-pr oto c ol fe es, and would pr ovide numeric al validation c omplementary to the analytic al estimate of Cor ol lary 5.3. Conclusion In this pap er we extend the mean field game framework developed in [ 2 ] to a more realistic setting in whic h traders operate in a decen tralized liquidity po ol sub ject to transaction costs. By explicitly mo delling the impact of p o ol fees on price dynamics and on traders’ in ven tories, w e deriv e a modified price pro cess and a corresp onding ob jective functional. Although the introduction of transaction costs complicates the game-theoretic structure — in particular, it breaks the separabilit y condition (S.5) required b y standard existence results — we establish the existence of mean field equilibria for t w o auxiliary cost functions that sandwic h the original one from ab o ve and b elow. The main con tribution of this w ork is a sandwic h-type result showing that the v alue function of the original game is enclosed b et ween those of the t w o auxiliary games. Ev en though the original ob jectiv e function do es not admit an approximate Nash equilibrium via standard theory due to its non-separable structure, our result pro vides a meaningful c haracterisation of the optimal cost. This op ens the do or to future research on existence and appro ximation results that go b eyond classical structural assumptions, or to numerical metho ds tailored to these non-standard settings. F urthermore, Corollary 5.3 sho ws that as ϕ → 1 (i.e., as the fee τ → 0 ) the gap b etw een the auxiliary problems and the original one tends to zero, and the mo del recov ers the results of the zero-cost case of [ 2 ]. F or the fee lev els t ypical of curren t AMM proto cols, of order 16 τ = 0 . 3% , the p erturbation factor is of order 4 . 5 × 10 − 6 , suggesting that the appro ximation is quan titatively very accurate. The n umerical quan tification of the gap V f 2 − V f 1 is left for future work. This pap er also lays the groundw ork for future extensions that incorp orate the full ecosys- tem of decen tralized mark ets. In ongoing work we include liquidit y pro viders and arbitrageurs to obtain a more complete game-theoretic representation of p o ol dynamics, and dev elop the corresp onding mo dels with transaction costs. These extensions are key to bridging the theory of mean field games with real-w orld applications in decen tralized finance. 17 References [1] R. Carmona and D. Lack er. A probabilistic w eak formulation of mean field games and applications. The A nnals of Applie d Pr ob ability , 25(3):1189–1231, 2015. [2] A. Muñoz Gonzalez, J.I. Sequeira, and R. Orive Illera. Liquidit y p o ols as mean field games: A new framework. 2025. [3] V. Mohan. Automated market maker and decen tralized exc hanges: a defi primer. RMIT Blo ckchain Innovation Hub , 2021. 18