Autonomy-Complete Competitive Equilibrium

• Autonomy-Complete Competitive Equilibrium is defined as a framework that recovers the standard Walrasian equilibrium by making explicit autonomy-relevant margins in both adaptive information-theoretic and augmented welfare contexts.
• Its adaptive formulation employs limits such as infinite capacities and zero entropy to drive agents to frictionless, rational expectations, effectively eliminating stochasticity and adaptive frictions.
• The augmented welfare framework internalizes delegation, verification, and autonomy rights, ensuring equilibrium allocations achieve autonomy-Pareto efficiency under competitive market conditions.

Autonomy-Complete Competitive Equilibrium (ACCE) denotes a competitive-equilibrium construction in which the standard Walrasian benchmark is recovered only after autonomy-relevant margins are made explicit rather than treated as ambient background conditions. In one recent formulation, ACCE is the frictionless corner of an adaptive, information-theoretic economy, defined by $$\{h=0,\ \forall i: C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^{\ast},\ T=T^{\ast}\}$$; at that corner, agents are unconstrained optimizers with rational expectations on a fixed structure, and the rest state is Walrasian (Bhandari, 26 Jun 2026). In another formulation, ACCE is an augmented competitive-equilibrium tuple $(p,x,r,s,\sigma,\pi,V)$ that internalizes welfare-status assignment, delegation accounting, verification institutions, and autonomy-relevant rights; under stated regularity, internalization, and nonmanipulation assumptions, every such equilibrium is autonomy-Pareto efficient (Perrier, 23 Apr 2026). The term is not standard across all equilibrium literatures: in the fair-division literature on A-CEEI, it is explicitly stated not to be a synonym for “Approximate Competitive Equilibrium with Equal Incomes” (Othman et al., 2013).

1. Dual formalizations and scope

The contemporary literature represented here gives ACCE two distinct formal meanings. One meaning is a limiting construction inside a broader adaptive economy; the other is a welfare-theoretic augmentation of general equilibrium for post-AGI settings. A separate strand of work explicitly rejects identifying the term with A-CEEI.

Framework	Formal object	Competitive meaning
Adaptive information-theoretic economy	$$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$	Frictionless corner; rest state is Walrasian
Post-AGI welfare economics	$\mathcal E=(p,x,r,s,\sigma,\pi,V)$	Competitive equilibrium on an augmented commodity space
Fair division with indivisibilities	A-CEEI, not ACCE	Separate concept with approximate clearing and budget inequality

In the adaptive formulation, the point of the term is to identify the competitive canon as an infinite-capacity, zero-entropy, infinite-intensity rest point of a more general adaptive order (Bhandari, 26 Jun 2026). In the post-AGI formulation, the point is different: the equilibrium description must include who counts as a welfare subject, who actually chooses, whether delegation wedges are internalized, whether verification is institutionally consistent, and whether manipulation channels are priced, assigned, or governed (Perrier, 23 Apr 2026).

By contrast, the fair-division literature uses the standard term “Approximate Competitive Equilibrium from Equal Incomes,” abbreviated A-CEEI, for indivisible goods and combinatorial demands. That literature explicitly states that “Autonomy-Complete Competitive Equilibrium” does not appear in Budish’s work or in the relevant complexity and algorithmic papers, and is not a synonym for A-CEEI (Othman et al., 2013). A later practical A-CEEI paper repeats that the standard term is A-CEEI and that the queried phrase is not used there (Budish et al., 2023).

2. ACCE as the frictionless corner of an adaptive, information-theoretic economy

In "Equilibrium as a Limit: The Competitive Canon Nested in an Adaptive, Information-Theoretic Economy" (Bhandari, 26 Jun 2026), the economy is modeled as an asymptotically mean stationary (AMS) information source. Let $X=(X_t)_{t\in\mathbb Z}$ be the realized history on a probability space $(\Omega,\mathcal F,\mathbb P)$ with measurable shift $\tau$. AMS means that the time averages of shifted laws converge setwise to a stationary mean measure,

$$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$

and the source has entropy rate

$$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$

Here $h>0$ measures per-period novelty, while $(p,x,r,s,\sigma,\pi,V)$0 means that the present is asymptotically determined by the past.

The same framework equips the stationary mean $(p,x,r,s,\sigma,\pi,V)$1 with a partially identified operator of statistical dependence. On the Hilbert space $(p,x,r,s,\sigma,\pi,V)$2 of mean-zero square-integrable observables, contemporaneous dependence is encoded by a bounded linear operator $(p,x,r,s,\sigma,\pi,V)$3. Its self-adjoint and skew parts are

$(p,x,r,s,\sigma,\pi,V)$4

with $(p,x,r,s,\sigma,\pi,V)$5 admitting the spectral decomposition

$(p,x,r,s,\sigma,\pi,V)$6

From second-order comovement in $(p,x,r,s,\sigma,\pi,V)$7, the identified content is exactly the self-adjoint part $(p,x,r,s,\sigma,\pi,V)$8, its real spectrum $(p,x,r,s,\sigma,\pi,V)$9, the leading eigenvalue $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$0, and the leading eigenvector $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$1. The skew part, individual directed edges $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$2, and transpose-related operators are not identified. The paper also states a finite-sample certification gate: with dimension $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$3 and sample length $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$4, the leading eigenpair is recoverable only when $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$5.

Agents are modeled as finite-capacity information channels. If $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$6 denotes the signal and $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$7 the action, capacity is represented by

$$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$8

Equivalently, agent $$\{h=0,\ \forall i:C_i=\infty,\ \beta_i=\infty,\ \sigma=\infty,\ m=m^\ast,\ T=T^\ast\}$$9 solves the free-energy program

$\mathcal E=(p,x,r,s,\sigma,\pi,V)$0

whose unique solution is the Gibbs law

$\mathcal E=(p,x,r,s,\sigma,\pi,V)$1

The interpretation is explicit: $\mathcal E=(p,x,r,s,\sigma,\pi,V)$2 concentrates $\mathcal E=(p,x,r,s,\sigma,\pi,V)$3 on $\mathcal E=(p,x,r,s,\sigma,\pi,V)$4, while $\mathcal E=(p,x,r,s,\sigma,\pi,V)$5 collapses it to the prior $\mathcal E=(p,x,r,s,\sigma,\pi,V)$6.

The adaptive environment is completed by selection and learning. Population shares over structures obey the logit rule

$\mathcal E=(p,x,r,s,\sigma,\pi,V)$7

with $\mathcal E=(p,x,r,s,\sigma,\pi,V)$8 concentrating on the unique evolutionarily stable configuration under uniqueness and global attractor assumptions. Beliefs are updated by stochastic approximation,

$\mathcal E=(p,x,r,s,\sigma,\pi,V)$9

with Robbins–Monro gains satisfying $X=(X_t)_{t\in\mathbb Z}$0 and $X=(X_t)_{t\in\mathbb Z}$1. Expectational stability means that the rational-expectations fixed point $X=(X_t)_{t\in\mathbb Z}$2 is locally convergent under the learning rule.

3. Joint-limit recovery of the competitive canon

The central theorem of the adaptive formulation states that the competitive rational-expectations equilibrium is recovered exactly as a joint limit taken along an admissible scaling path (Bhandari, 26 Jun 2026). Let

$X=(X_t)_{t\in\mathbb Z}$3

with $X=(X_t)_{t\in\mathbb Z}$4, $X=(X_t)_{t\in\mathbb Z}$5, observational innovation scale $X=(X_t)_{t\in\mathbb Z}$6, and Robbins–Monro gains satisfying the standard summability conditions. Under four additional conditions—no persistent cycle, unique attracting structure, learnability, and structural fixity at the limit—the following hold jointly:

1. $X=(X_t)_{t\in\mathbb Z}$7, with innovations vanishing at observational resolution.
2. For all $X=(X_t)_{t\in\mathbb Z}$8, $X=(X_t)_{t\in\mathbb Z}$9 and $(\Omega,\mathcal F,\mathbb P)$0, so $(\Omega,\mathcal F,\mathbb P)$1 concentrates on $(\Omega,\mathcal F,\mathbb P)$2.
3. $(\Omega,\mathcal F,\mathbb P)$3, so the population concentrates on the unique ESS configuration $(\Omega,\mathcal F,\mathbb P)$4.
4. $(\Omega,\mathcal F,\mathbb P)$5, so beliefs coincide with the realized law of motion.
5. $(\Omega,\mathcal F,\mathbb P)$6, a fixed dependence structure with no co-evolution in the limit.

At this joint corner, the limiting object satisfies the axioms of the competitive canon: market clearing, frictionless optimization, fixed structure, and rational expectations. The paper’s formal definition of autonomy-complete competitive equilibrium is precisely that corner,

$(\Omega,\mathcal F,\mathbb P)$7

and the rest state is Walrasian.

The recovered Walrasian fixed-point conditions are standard. Market clearing is

$(\Omega,\mathcal F,\mathbb P)$8

with Walras’ law and homogeneity,

$(\Omega,\mathcal F,\mathbb P)$9

Rational expectations may be written as

$\tau$0

or equivalently as the fixed-point equation for the adaptive rest point induced by the update operator. For prices $\tau$1, each agent solves

$\tau$2

with feasibility $\tau$3.

The limiting construction is also path-independent in a precise sense. The three parameter limits commute, and the iterated limits equal the joint limit by uniform convergence in each coordinate together with a continuous rest-state map and Moore–Osgood. This excludes dependence on relative speeds along admissible scaling paths.

4. Erasure, indeterminacy, and relation to classical general equilibrium

The adaptive formulation is not merely a recovery theorem; it also characterizes what the competitive limit erases (Bhandari, 26 Jun 2026). Away from the corner, for $\tau$4, the economy is a genuine generator of novelty. The recovered operator $\tau$5 is non-degenerate and co-evolving, inducing nontrivial joint distributions over states, signals, actions, and outcomes. Directed dependence in the skew part and time-irreversibility through positive entropy production cannot be represented in the equilibrium primitive.

The paper gives an exact enumeration of erased contents at the corner. First, stochasticity is removed because $\tau$6 and $\tau$7 eliminate observational innovations. Second, informational frictions disappear because $\tau$8 and $\tau$9 eliminate mutual-information constraints and collapse stochastic choice to degenerate argmax. Third, selection noise vanishes because $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$0 removes mutation or softening. Fourth, co-evolution of structure disappears because $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$1 freezes the topology. Fifth, path dependence and adaptation are removed because convergence to the expectationally stable rest point eliminates learning dynamics.

This limiting analysis is paired with a result-by-result correspondence to standard general-equilibrium theory. Arrow–Debreu–McKenzie existence is inherited at the corner; the rest state of the limiting source is the Walrasian equilibrium whose existence is secured by the classical theorem. The framework also preserves the force of the Sonnenschein–Mantel–Debreu theorem. At the frictionless aggregate, only continuity, Walras’ law, and homogeneity of degree zero remain, so essentially any such excess-demand map can be realized. The paper is explicit that this indeterminacy is a property of the zero-friction corner, and that the recovered spectral content away from the corner is never used to infer forward-looking stability. Hence SMD’s negative results on uniqueness and stability remain intact at the limit.

The same section ties the construction to regular economies in the differentiable approach. Local manifold dimensionality and transversality carry over, and generic local uniqueness corresponds to local determinacy of adaptive dynamics under expectational stability. Stability and learnability, however, remain separate criteria. A central implication is that autonomy-completeness, understood as the frictionless corner, does not by itself guarantee uniqueness or global stability.

5. ACCE as an autonomy-qualified welfare theorem

In "Post-AGI Economies: Autonomy and the First Fundamental Theorem of Welfare Economics" (Perrier, 23 Apr 2026), ACCE is not a limit point inside adaptive dynamics but a competitive-equilibrium concept on an augmented commodity space. Let $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$2 be the set of humans, $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$3 the set of artificial systems, $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$4 the economically relevant entities, and $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$5 the set of physical goods. Private consumption is $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$6. The model augments this space with autonomy-relevant rights and attributes. Each entity has a rights vector $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$7, there is a public institutional state $$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$8, and welfare-status assignment

$$n^{-1}\sum_{t=0}^{n-1}\mathbb P\circ \tau^{-t}\Rightarrow \bar\mu,$$9

The welfare-bearing set is

$$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$0

For each $$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$1, welfare is autonomy-conditioned:

$$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$2

The private augmented bundle is $$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$3. Delegation is handled by a principal map $$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$4 from delegates to principals, a delegate objective $$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$5, and divergence

$$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$6

on the shared domain. Verification is modeled by an institution $$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$7 that maps claims, evidence, and institutional state into updated institutional state and, if desired, a procedurally determined status assignment. The model also defines admissible action sets $$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$8 to exclude manipulation channels that alter the welfare of others through unpriced or unassigned pathways.

The formal definition of ACCE is an equilibrium tuple

$$h=\lim_{n\to\infty}\frac{1}{n}H(X_0,\dots,X_{n-1}).$$9

with $h>0$0 and $h>0$1 such that five conditions hold. First, every welfare subject is a price-taker on the augmented bundle and maximizes $h>0$2 on the budget set. Second, tools are fixed by technology across feasible alternatives. Third, delegation accounting must either be faithful, $h>0$3, or fully internalize divergence through agency-cost terms priced at $h>0$4 and consolidated into principals’ budgets. Fourth, every welfare-relevant component of $h>0$5 must be priced, directly assigned, or institutionally protected in $h>0$6, and verification attributes must be priced or governed. Fifth, feasibility and market support require aggregate support by endowments and production so that the aggregate budget-cost inequality holds for all feasible alternatives.

The corresponding welfare notion is autonomy-Pareto efficiency. Holding $h>0$7 and $h>0$8 fixed, a feasible state $h>0$9 autonomy-Pareto dominates $(p,x,r,s,\sigma,\pi,V)$00 if every welfare-bearing entity weakly gains and some welfare-bearing entity strictly gains. The paper’s Autonomy-Qualified First Fundamental Theorem states that under five assumptions—regularity of welfare, delegation internalization, verification consistency and rights completeness, nonmanipulation, and competitive behavior with aggregate support—every ACCE is autonomy-Pareto efficient at $(p,x,r,s,\sigma,\pi,V)$01. The proof follows the standard budget-cost contradiction structure, but only after non-welfare entities, delegation wedges, and autonomy-relevant rights have been properly internalized inside the equilibrium description.

A further result is a low-autonomy reduction. If all artificial systems are tools, rights are fixed, verification is complete, and the institutional state is fixed, then the welfare-bearing set collapses to humans and the augmented bundle collapses to ordinary consumption. In that limit, ACCE induces a classical competitive equilibrium and the autonomy-qualified welfare theorem reduces to the standard First Fundamental Theorem of Welfare Economics.

6. Neighboring uses, interpretive extensions, and terminological disputes

Outside those two formal constructions, the phrase has been used more as an interpretive lens than as a settled technical term. In generalized proportional dynamics for linear Fisher markets with seller-side auto-bidding, a “fully decentralized, protocol-aligned dynamic” is described as an “autonomy-complete” pathway to competitive equilibrium: buyers autonomously update seller-level budgets proportionally to realized utility, sellers autonomously run multiplicative pacing, truthful implementation converges to competitive equilibrium with an ergodic $(p,x,r,s,\sigma,\pi,V)$02 rate in the Eisenberg–Gale objective, and trade occurs at every round (Li et al., 2024). That paper also shows a 2-approximation guarantee for buyer adherence to the proportional rule, a seller deviation bound $(p,x,r,s,\sigma,\pi,V)$03, a unique pure Nash equilibrium in the seller deviation game, and a fairness guarantee

$(p,x,r,s,\sigma,\pi,V)$04

This is not the same formal object as either of the two ACCE definitions above, but it uses the phrase to mark decentralized attainability of competitive equilibrium.

A similar interpretive use appears in traffic-network dynamics. There, routes are treated as commodities, traffic densities as prices, and decentralized driver responses generate the dynamic system

$(p,x,r,s,\sigma,\pi,V)$05

Under strong gross substitutes, homogeneity of degree zero, and existence of an interior equilibrium, there exists a unique equilibrium ray

$(p,x,r,s,\sigma,\pi,V)$06

and trajectories converge to that ray (Šiljak, 2017). The phrase “autonomy-complete competitive equilibrium” is used to characterize a decentralized, consensus-like market-clearing state with equalized normalized densities, again as an interpretation rather than a formal definition native to the original traffic model.

Dynamic multi-agent systems for decentralized resource allocation offer yet another nearby usage. There, competitive equilibrium is embedded in a finite- or infinite-horizon control problem with price trajectories, social shaping of utility parameters, and explicit affordability bounds. The synthesized presentation defines an “Autonomy-Complete Competitive Equilibrium” as a decentralized equilibrium in which agents solve their own dynamic optimization given prices, markets clear at every time, prices remain within socially acceptable bounds $(p,x,r,s,\sigma,\pi,V)$07, and the induced allocation maximizes social welfare (Salehi et al., 2022). In quadratic settings, the framework provides Riccati recursions, a constrained LQR formulation, and zero-price results for sufficiently small initial conditions or after finite time.

The most important terminological dispute remains the fair-division literature. Both the complexity paper on A-CEEI and the later practical algorithmic paper explicitly state that “Autonomy-Complete Competitive Equilibrium” is not the standard term there and is not a synonym for A-CEEI (Othman et al., 2013). A-CEEI concerns approximate market clearing and approximately equal incomes for indivisible goods, with parameters such as $(p,x,r,s,\sigma,\pi,V)$08 for clearing error and $(p,x,r,s,\sigma,\pi,V)$09 for budget inequality, whereas ACCE in the two formal senses above concerns either a frictionless corner of an adaptive information source or an augmented equilibrium that internalizes autonomy, delegation, verification, and welfare status (Budish et al., 2023).

Taken together, these usages show that ACCE is best understood not as a single settled doctrine but as a family of attempts to make explicit what ordinary competitive equilibrium leaves implicit: information-processing limits and adaptive frictions in one line of work, and autonomy-bearing rights, delegation, verification, and welfare status in another. The classical competitive benchmark is recovered in both cases only after those margins are either driven to a limiting corner or fully internalized within the commodity and institutional structure.