Notes on equilibrium and mechanisms

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Introduction

An interior-point solver computes Walrasian (CEEI) prices and allocations from reported data but, taken alone, is not a mechanism—it is a centralized algorithm. By contrast, a (Walrasian) tâtonnement or auction process, in which prices are iteratively posted and agents submit demands that guide price updates, can be viewed as a decentralized mechanism whose limiting outcome is a Walrasian equilibrium (when it converges and agents are effectively price takers).

Hurwicz Impossibility (informational decentralization).

Hurwicz (1972) showed that for certain environments (e.g., with detrimental externalities, infinite horizons, nonconvexities) no finite–dimensional informationally decentralized process can be guaranteed to achieve Pareto efficiency for all economies—even if agents follow the rules truthfully. In such settings, coordinating solely via a bounded set of signals (“prices”) is fundamentally inadequate; some objectives cannot be reached by any finite‐message decentralized mechanism. ¹

Interpretation.

An informationally decentralized mechanism lets each agent communicate only limited, local information (messages drawn from a finite‐dimensional space) rather than full preference/technology data. The impossibility says that, in the cited environments, any rule that always yields a Pareto‐satisfactory outcome must, somewhere, require unbounded (infinite) message dimension—or else fail to attain the goal. ¹

Link to incentives.

Subsequent work used revelation principles to extend Hurwicz’s logic: when agents hold private information, classical Pareto optimality generally cannot be attained by incentive‐compatible (dominant‐strategy) direct mechanisms over rich domains. One must relax goals (e.g., approximate efficiency) or allow richer institutions ^{1 2}

Incentives in Walrasian / CEEI Mechanisms.

• Finite markets. Walrasian (CEEI) direct‐revelation mechanisms are generally manipulable: agents with nonnegligible size can misreport demands to move prices; thus they are not strategy‐proof (dominant‐strategy incentive compatible). ¹

• Large markets (price‐taking limit). In replica / approximately competitive economies each participant’s price impact vanishes, making truthful reporting an $\varepsilon$‐best response. Roberts and Postlewaite show vanishing gains from manipulation in large exchange economies; Azevedo and Budish formalize strategy‐proofness in the large (SP‐L); Budish’s A‐CEEI for indivisible goods satisfies SP‐L ^{3 2 4}

Mechanism Design and Implementation

Mechanism design & implementation in an exchange market

Baseline economy.

Let $E = (I,{(X_i,\succeq_i,\omega_i)}_{i\in I})$. A Walrasian equilibrium is $(p^\ast,x^\ast)$ such that $\,x_i^\ast \in \arg\max{u_i(x_i): p^\ast!\cdot x_i \le p^\ast!\cdot\omega_i}$ for all $i$ and $\sum_i x_i^\ast=\sum_i\omega_i$.

Mechanism design.

• Players privately know $u_i$.

• Designer specifies

  • message space $M_i$ (e.g. reported utilities or demand schedules),

  • outcome rule $g!:M\to \mathbb R_+^L \times \mathbb R_+^L$ that returns $(p,x)$.

• Objective: choose $g$ so that truth telling is (approximately) optimal and the induced outcome is efficient/fair/feasible.

Implementation question.

Does there exist a mechanism $g$ such that every equilibrium outcome equals the Walrasian correspondence $F(E)={\,(p,x)\text{ Walrasian in }E\,}$ under a chosen solution concept (dominant strategy, Nash, Bayesian)?

Comparison table.

Key facts.

• Hurwicz (1972) shows impossibility of dominant‐strategy implementation of Walrasian allocations on rich domains.

• In large replica markets agents become price takers; Walrasian mechanisms achieve strategy‐proofness in the large (Azevedo–Budish, 2019).

What happens in a large exchange economy?

$\mathbf{x}= \mathcal{X}(\mathbf{p})$

• Direct-revelation version. Each agent $i$ reports her utility $u_i$. A central solver computes Walrasian prices $p^\ast$ and bundles $x_i^\ast$. Truthful reporting becomes an $\varepsilon$-best response as the replica size $m!\to!\infty$ (strategy-proofness in the large, Azevedo–Budish 2019).

• Replica: Given $E=(I,{(X_i,\succeq_i,\omega_i)}{i\in I})$, define its $m$-replica $E^m=(I\times{1,\dots,m},{(X_i,\succeq_i,\omega_i)}{i\in I,\ \ell=1}^m)$, i.e., $m$ identical copies of each trader type. Aggregate endowments scale by $m$, type proportions are preserved. The Debreu–Scarf core-convergence theorem studies the sequence ${E^m}_{m\ge1}$ and shows that allocations in the core of every replica must be Walrasian; thus, as $m\to\infty$, bargaining power vanishes and agents behave as price takers. ^{5 6 7}

• Demand-revelation (tâtonnement) version. The mechanism posts a tentative price vector $p^t$. Agents best-respond with demands $z_i(p^t)$; the auctioneer updates prices, e.g. $p^{t+1}=p^t+\alpha^t\sum_i z_i(p^t)$. In the large-market limit each agent’s price impact $\to 0$, so sending her true demand is again an $\varepsilon$-best reply.

• Is this a posted-price mechanism? Not exactly. A posted-price mechanism fixes prices once and agents decide quantities (take-it-or-leave-it). Tâtonnement adjusts prices based on revealed excess demand; it is a dynamic price-discovery mechanism. Only when it converges do the final prices play the role of posted prices.

• Truthfulness hinge. With finitely many non-negligible traders, either variant is manipulable (Hurwicz impossibility). As the market becomes large, individual manipulation gains shrink to $0$, so both the direct-revelation and the demand-revelation (tâtonnement) mechanisms attain approximate incentive compatibility.

References