Pay-as-Clear (PaC) in Electricity Markets

• Pay-as-Clear is a uniform-price marginal pricing mechanism where all accepted bids receive the same market-clearing price based on the highest winning bid.
• It sets the clearing price at the marginal bid necessary to meet inelastic demand, directly linking system costs to input fuel prices and strategic bidding behavior.
• Strategic bidding and volatile input costs can elevate system prices and disproportionately transfer rents to inframarginal units, affecting market efficiency.

Pay-as-Clear (PaC) is the standard uniform-price marginal pricing mechanism widely employed in European wholesale electricity markets, such as the Italian Day-Ahead Market (MGP) and the Euphemia-coupled EU markets. In PaC, all accepted supply offers receive the same market-clearing price, set by the highest accepted (marginal) bid required to meet the system’s inelastic demand. The methodology underlying PaC, its equilibrium properties, vulnerability to strategic bidding, and relationship to alternative market-clearing mechanisms are central to ongoing electricity market design debates (Caragiannis et al., 8 Jul 2025, Altamura et al., 1 Feb 2026).

1. Formal Definition and Clearing Rule

Let the market consist of a set of supply offers indexed by $j \in S$, where each offer has a price $p_j$ and a quantity $q_j$. Let total inelastic demand be $D>0$. The aggregate supply function is given by

$$S(p) = \sum_{j \in S} q_j \mathbf{1}_{\{p_j \leq p\}}.$$

The market-clearing price $P$ and accepted quantities $\{x_j\}$ satisfy

$P = \min\{p \in \mathbb{R} : S(p) \geq D\},$

with dispatch rule

$$x_j = \begin{cases} q_j, & p_j < P, \ D - \sum_{i: p_i < P} q_i, & p_j = P, \ 0, & p_j > P. \ \end{cases}$$

All accepted bids (i.e., those with $p_j < P$ or at $p_j = P$ up to $D$) are paid the uniform market-clearing price $P$ per unit, regardless of their actual bid or cost (Altamura et al., 1 Feb 2026).

In multi-agent game-theoretic settings, producers $i=1,\dots,n$ each have capacity $s_i \in (0,1]$ (with normalized total demand $D=1$) and private marginal costs $c_i \in \{0,1,\dots,M\}$, submitting bids $b_i \in \{0,1,\dots,M\}$. Producers are sorted in ascending order of $b$, and the pivotal agent $\tau(b)$ is the first whose cumulative capacity fulfills $D$: $\sum_{j \prec_b \tau(b)} s_j < 1$ and $\sum_{j \preceq_b \tau(b)} s_j \geq 1$. The market-clearing price is $q(b) := b_{\tau(b)}$, and the allocation $x_i(b)$ follows the above rules (Caragiannis et al., 8 Jul 2025).

2. Intramarginal Profits and Price Sensitivity

Profits for dispatched generators $i$ with marginal cost $c_i$ and dispatch $q_i$ are

$\pi_i = (P - c_i) q_i.$

Units with $c_i < P$ derive intramarginal (or "submarginal") rents: the difference between the market price and their production costs, multiplied by their quantity. This is particularly significant when the clearing price is set by gas-fired generators, as renewable units with $c_i \approx 0$ capture rents proportional to the full gap between $P$ and their negligible cost (Altamura et al., 1 Feb 2026).

The market-clearing price $P$ is highly sensitive to the marginal technology's cost, especially natural gas. If $c_g$ is the marginal gas price, then for a small increment $\Delta c_g$, $\Delta P = \Delta c_g$. Thus, volatility in natural gas costs translates directly to market price volatility for all consumers under PaC. Using the continuous supply curve $Q_s(p)$ and $Q_s(P) = D$,

$\sigma(P) \approx \sigma(c_g),$

as $Q_s'(P)$ is typically large, indicating a horizontal merit-order stack at $P$ (Altamura et al., 1 Feb 2026).

3. Game-Theoretic Equilibrium and Strategic Behavior

PaC induces a non-cooperative game in which producers select bids $b_i$ to maximize their profits $U_i(b) = (p_i(b) - c_i) x_i(b)$, where $p_i(b)$ is the payment resulting from the market-clearing rule. Nash equilibria, particularly in mixed strategies (MNE), are characterized by distributions $\sigma = (\sigma_1, \dots, \sigma_n)$ over bid profiles such that unilateral deviations yield no expected profit gain.

Key theoretical results include:

• No universal dominance: For any feasible and individually rational mechanism $\mathcal{M}$ (including PaC, Pay-as-Bid, or VCG), there exists a market instance and mixed NE where $\mathcal{M}$ achieves strictly lower unit cost than any alternative mechanism (Caragiannis et al., 8 Jul 2025).
• Worst-case performance: Pay-as-Bid (PaB) always yields a worst-case maximum price that is strictly lower than that under PaC for any market instance and at all mixed equilibria (Caragiannis et al., 8 Jul 2025). For PaC, the clearing price at NE can be maintained at a high threshold $B$, but PB equilibria are confined to $[B-1, B]$ (see section 5).

Simulations using no-regret learning (Hedge algorithm) confirm that under PaC, equilibrium prices quickly approach theoretical bounds; under PaB, prices typically oscillate around the lower bound, resulting in consistently lower long-run averages (Caragiannis et al., 8 Jul 2025).

4. Strategic Learning and Empirical Results

Reinforcement Learning (Q-Learning) models strategic producer behavior under incomplete information, framing each agent as an RL agent adapting markups over their offered portfolio. Empirically, two contrasting market scenarios were analyzed:

• Scenario 1 (“2030 PNIEC”): Two symmetric agents with 1 GW each, 40% renewables, 60% thermal.
• Scenario 2 (“10-Operator”): Realistic portfolios from GME 2024 offers; highly concentrated ownership (Op1: 44% capacity).

Summary of observed outcomes under PaC:

Scenario	Marginal Cost Bidding (€/MWh)	RL/Strategic Bidding (€/MWh)	Cost Increase
PNIEC	69	82.8	+20%
10-Operator	163.18	≈ 187	+15%

Strategic bidding significantly increases total system cost and clearing prices. Profits concentrate disproportionately in large operators and zero-cost units (renewables), confirming the vulnerability of PaC to price elevation through noncompetitive strategies (Altamura et al., 1 Feb 2026).

5. Comparative Mechanisms: PaC, PaB, and SPaC

Comparisons across mechanisms yield the following:

Mechanism	Clearing Price Rule	Average Profits/Costs	Strengths	Weaknesses
PaC	Marginal accepted offer	67% of cost (RL case, PNIEC)	Simplicity, short-term efficiency, uniform pricing	Large inframarginal rents, volatile, non-truthful
PaB	Each accepted gets own bid	58% of cost (RL case, PNIEC)	Minimal rents if bids = costs	Susceptible to strategic gaming; extreme price spikes
SPaC	Segmented uniform pricing	39% of cost (RL case, PNIEC)	Reduces volatility/rents, robust in oligopoly	Complex clearing, higher computational burden

SPaC divides the supply stack by technology, applying separate clearing prices per group, which halves both consumer price volatility and inframarginal rents, decoupling renewable compensation from gas-derived volatility. PaB, while theoretically robust to price manipulation in worst-case equilibrium, empirically enables large markups under strategic bidding, with consumer costs in some cases increasing by 149% (Altamura et al., 1 Feb 2026).

6. Incentive Compatibility and Market Design Implications

PaC is widely adopted for its transparency and settlement simplicity, providing clear investment and operational signals. It is not incentive-compatible in the VCG sense; producers retain incentives to inflate their bids if allocation is not impacted. In concentrated or small markets, PaC is structurally vulnerable to strategic price elevation, with the pivotal agent able to sustain clearing prices at or above threshold $B$ at equilibrium.

From a consumer protection perspective, while PaB offers stronger worst-case guarantees, its billing complexity and counterintuitive equilibrium bidding have limited regulatory traction. SPaC has emerged as a compromise, preserving uniform pricing within technological segments and reducing systemic risk to extreme price spikes and rent transfers, albeit with increased algorithmic complexity (Altamura et al., 1 Feb 2026).

7. Policy Recommendations and Future Perspectives

Empirical and theoretical analyses converge on the structural limitations of the PaC mechanism:

• Efficiency and uniform pricing are achieved at the cost of exposing consumers to input-fuel price volatility and potentially high rents for inframarginal (usually renewable) units.
• Policy recommendations from recent simulation studies highlight the potential for SPaC piloting, robust technological registries, real-time monitoring for manipulation, and integration of demand-side flexibility and hedging instruments to further insulate consumers from market shocks (Altamura et al., 1 Feb 2026).

Ongoing reforms consider the endogenous separation of demand for low- and high-cost technologies, with implications for price formation, volatility, and investment signals during the energy transition. A plausible implication is that transition-robust mechanisms must balance allocative efficiency, consumer risk, and dynamic investment incentives, leveraging market segmentation and enhanced monitoring in the presence of strategic generator behavior (Altamura et al., 1 Feb 2026, Caragiannis et al., 8 Jul 2025).