Liquid Welfare Price of Anarchy (POA)

• Liquid Welfare Price of Anarchy (POA) measures inefficiency as "frustration," the shortfall between fair entitlements (like rights) and realized allocations in systems where agent contributions are capped by liquidity constraints.
• Hybrid mechanisms combining fairness-based rights allocation with decentralized market trading can significantly lower POA, improving outcomes compared to pure market approaches.
• Empirical findings show that hybrid systems, especially those using efficient market clearing algorithms, reduce average frustration by protecting agents with fewer resources and providing opportunities to utilize fair entitlements.

The Liquid Welfare Price of Anarchy (POA) measures the degradation in efficiency due to strategic, decentralized agent behavior in systems where welfare—by construction or necessity—is "liquid": each agent's contribution is capped by liquidity constraints such as budgets, buying rights, or other fairness-based entitlements, rather than simply their full potential utility or value. In the setting of networked distribution of critical resources during crises, as modeled by the Crisdis system, the POA is particularly interpreted in terms of "frustration": the gap between what agents are entitled to by a fair allocation and what they ultimately receive through unregulated market mechanisms. This represents a conceptual shift from classical economic or network liquid welfare to a frustration-centric, fairness-aware metric tuned for crisis response or other critical allocation environments.

1. Definition and Context of POA in Crisdis

In the Crisdis (Critical Distribution) framework, the Price of Anarchy is defined not as the loss in total utility or surplus, but as the average normalized shortfall—"frustration"—relative to fair entitlements. Formally, if $R_b$ is the number of buying rights (allocated by a fair mechanism) to buyer $b$, and $\mu_b^G$ is the actual amount of goods acquired in the equilibrium outcome of market trading, frustration is

$$f_b = \max\left\{ \frac{R_b - \mu_b^G}{R_b}, 0 \right\}.$$

Aggregated over all buyers $\mathcal{B}$ and trading periods $\tau$, the POA is

$$PoA^\tau = \frac{1}{\tau|\mathcal{B}|} \sum_{i=1}^\tau \sum_{b \in \mathcal{B}} f_b,$$

capturing, on average, how much less agents receive compared to fair allocation, due to strategic or budget-constrained trading. This definition generalizes to any situation where liquid welfare is naturally measured by ability or rights to pay, rather than total subjective utility.

2. Mechanism: Structure of Crisdis and Market Allocation

The Crisdis system is designed as a double-sided market with integrated fairness:

1. Rights Allocation: A central planner assigns buying rights to agents based on fairness mechanisms (e.g., the Aumann–Maschler contested garment solution), reflecting needs rather than budgets.
2. Trading Market: Agents trade both goods and rights in a decentralized (market-based) game. Each buyer cannot purchase more goods than rights held.
3. Clearing Mechanisms: Various market-clearing implementations are considered—ranging from random and greedy protocols to network flow and LP-based solutions that maximize total trades each round.
4. Iterations: The allocation/trading cycle repeats over multiple periods, modeling ongoing coordination in crisis settings.

This hybrid arrangement ensures that initial entitlements are decoupled from wealth, but trading friction and competition may still result in outcomes deviating from the ideal fairness.

3. Frustration Metric and POA Calculation

Frustration quantifies per-buyer deviation from entitlement: it measures the fraction of allocated rights not translated into goods due to the outcome of market trading. A system with low frustration—hence low POA—is one where market and allocation mechanisms jointly enable most agents to realize their fair share, regardless of their initial wealth.

The POA in Crisdis thus represents the average shortfall per agent (and per period) in the realized allocation, compared to what fairness dictates. This metric is critical for crisis scenarios where satisfying minimum needs is more important than maximizing aggregate surplus or efficiency.

4. Empirical Findings: POA in Hybrid Market-Fairness Mechanisms

Empirical analysis demonstrates several key results regarding POA in Crisdis:

• Rights Allocation Sharply Reduces POA: Introducing centrally assigned rights dramatically lowers average frustration compared to pure, unregulated trading. This reduction is significant for both "poor" and "rich" agents, with the former especially protected against deprivation in the hybrid regime.
• Multiple Trading Rounds Further Improve POA: Allowing more than one market-clearing round per period (e.g., $k=2$ rounds) gives buyers more opportunities to utilize their rights, reducing POA even further.
• Market Mechanism Matters: Max-clearing (network flow or LP-based) market mechanisms outperform greedy or random approaches, yielding lower POA by achieving closer-to-optimal matching between rights and realized allocations.
• Scalability: As system size (numbers of buyers and sellers) grows, the relative benefit of the hybrid (rights+market) system over a purely budget-driven market persists.
• Quantitative Effect: In well-configured hybrid systems, POA can be reduced by up to a factor of 6 relative to a laissez-faire market.

This suggests that layered mechanism design—combining fairness-based entitlements with efficient market trading—substantially mitigates the inefficiencies and inequalities intrinsic to decentralized allocation under scarcity.

5. Interpretation in the Liquid Welfare Paradigm

Liquid Welfare in mechanism design typically addresses the scenario where agents' value extracted from an allocation is limited by their budget or similar liquidity constraints: $LW(x) = \sum_{i} \min\{v_i(S_i), B_i\}.$ In the Crisdis context, buying rights function much like budgets, but are allocated according to fairness rather than revealed willingness-to-pay. The POA then captures the systemic loss in potential (fair) welfare caused by both market frictions and the limitations of rights trading, abstracting away from agent wealth.

The implications are direct:

• Hybrid Mechanism Design Outperforms Pure Market: Introducing fair, ex-ante allocation of "liquid entitlements" (budgets/rights), in combination with open markets, leads to consistently lower POA—greater realized liquid welfare.
• Mitigation of Market Power Effects: Even buyers with little monetary power acquire rights based on needs, safeguarding against complete exclusion.
• Practical Blueprint for Crisis Allocation: When distributing critical, scarce goods (e.g., medical supplies in emergencies), such mechanisms achieve both rapid delivery (via market) and equity (via rights), with minimal inefficiency.

6. Broader Implications and Applications

The POA framework developed in Crisdis highlights the relevance of fairness-aware, liquid welfare mechanisms well beyond crisis management:

• Resource Allocation Under Scarcity: Any system where agent budgets poorly reflect needs can benefit from fairness-based rights layered over market allocation.
• Budgeted Markets in Standard Mechanism Design: The analysis confirms and extends classical results on liquid price of anarchy, demonstrating that regulatory interventions calibrated to needs (not wealth) can robustly improve outcomes.
• Policy Guidance: Mechanism design for health, humanitarian, or utility allocation in emergencies should prioritize hybrid models that guarantee low POA in liquid/fairness terms, rather than relying solely on market-based distribution.

Summary Table: Core Definitions and Metrics

Concept	Formula	Context/Meaning
Frustration	$f_b = \max\{ (R_b - \mu_b^G)/R_b,\, 0 \}$	Individual agent's allocation shortfall
POA in Crisdis	$$PoA^\tau = \frac{1}{\tau|\mathcal{B}|}\sum_{i=1}^\tau\sum_{b\in\mathcal{B}} f_b$$	Average frustration (game-wide inefficiency)
Rights Allocation	Contested garment (Aumann–Maschler) bankruptcy algorithm	Needs-based, fair, ex-ante entitlements
Max-clearing Market	Flow or LP-based allocation during trading	Mechanism type with lowest frustration/POA

---

The paper of POA in Crisdis demonstrates quantitatively and conceptually that combining fairness-driven liquid allocations and market mechanisms yields low-frustration, efficient, and equitable outcomes in crisis resource distribution. This supports broader use of hybrid designs in critical settings where both equity and efficiency are essential.