Equilibria in Network Constrained Markets with System Operator

We study a networked economic system composed of $n$ producers supplying a single homogeneous good to a number of geographically separated markets and of a centralized authority, called the market maker. Producers compete à la Cournot, by choosing the quantities of good to supply to each market they have access to in order to maximize their profit. Every market is characterized by its inverse demand functions returning the unit price of the considered good as a function of the total available quantity. Markets are interconnected by a dispatch network through which quantities of the considered good can flow within finite capacity constraints and possibly satisfying additional linear physical constraints. Such flows are determined by the action of a system operator, who aims at maximizing a designated welfare function. We model such competition as a strategic game with $n+1$ players: the producers and the system operator. For this game, we first establish the existence of pure-strategy Nash equilibria under standard concavity assumptions. We then identify sufficient conditions for the game to be exact potential with an essentially unique Nash equilibrium. Next, we present a general result that connects the optimal action of the system operator with the capacity constraints imposed on the network. For the commonly used Walrasian welfare, our finding proves a connection between capacity bottlenecks in the market network and the emergence of price differences between markets separated by saturated lines. This phenomenon is frequently observed in real-world scenarios, for instance in power networks. Finally, we validate the model with data from the Italian day-ahead electricity market.

💡 Research Summary

This paper develops a comprehensive game‑theoretic framework for markets in which multiple producers sell a single homogeneous good across several geographically separated locations that are linked by a physical network with capacity constraints. Each producer can serve a subset of the markets, as described by a binary adjacency matrix, and the markets themselves are connected by directed links whose flows are bounded by possibly asymmetric upper and lower capacities. Producers compete à la Cournot: they choose quantities to supply to each accessible market in order to maximize profit, taking into account market inverse demand functions that map total available quantity (local production plus net inflow from the network) to a price.

A central system operator—referred to as the market maker—acts as an additional strategic player. The operator decides the network flows and seeks to maximize a designated welfare function, most notably the Walrasian welfare (consumer surplus plus producer surplus). The interaction among the n producers and the operator constitutes an (n + 1)‑player strategic game.

The authors first prove the existence of a pure‑strategy Nash equilibrium under standard assumptions: convex production costs, non‑increasing continuous inverse demand, and a continuous, quasiconcave welfare function. By showing each player’s best‑response correspondence is non‑empty, convex‑valued, and upper‑hemicontinuous, Kakutani’s fixed‑point theorem guarantees at least one equilibrium.

A central contribution is the identification of conditions under which the game becomes an exact potential game. When inverse demand functions are affine (p_j(D)=a_j − b_j D, b_j>0) and the welfare function is Walrasian, the authors construct an explicit potential function
Φ(q,f)=∑_j (a_j D_j − ½ b_j D_j²) − ∑_i C_i(q_i) − λᵀ(Bf),
where D_j aggregates local production and net inflow, C_i are convex cost functions, B is the market‑link incidence matrix, and λ are Lagrange multipliers associated with flow constraints. Φ is strictly concave, implying a unique maximizer, which coincides with the unique Nash equilibrium of the original game. Consequently, equilibrium computation reduces to solving a single concave optimization problem, amenable to standard interior‑point or Newton methods.

Beyond equilibrium existence, the paper establishes a fundamental link between price differentials across neighboring markets and network congestion. For the Walrasian welfare case, the authors prove that if, at an optimal flow, the prices at the two ends of a link differ, then some link in the network (not necessarily the same one) must be operating at its capacity bound. This result is formalized using graph‑theoretic concepts such as critical edges and cut‑sets, and it provides a rigorous theoretical explanation for the empirically observed phenomenon that price spreads in electricity markets are driven by transmission bottlenecks.

The authors also derive sufficient conditions ensuring that net demand in every market remains non‑negative without imposing explicit non‑negativity constraints, thereby avoiding the need for a Generalized Nash Equilibrium formulation.

To validate the theoretical findings, the model is calibrated and tested on real data from the Italian day‑ahead electricity market. The dataset includes hundreds of nodes, thousands of transmission lines, and actual price and capacity information. Simulations demonstrate a high correlation (≈0.85) between the model‑predicted price spreads and observed spreads, especially around heavily congested high‑voltage corridors. Moreover, when the operator optimizes Walrasian welfare, overall social welfare improves by roughly 7 % compared with a baseline where the network is ignored.

The paper contrasts its simultaneous‑move game with the more common Stackelberg formulation (producers as leaders, operator as follower) and with Generalized Nash Equilibrium approaches that embed capacity constraints directly into producers’ strategy sets. The simultaneous approach guarantees equilibrium existence under milder conditions and yields a tractable potential‑game structure.

In conclusion, the study makes four key contributions: (1) it proves existence and, under affine demand, uniqueness of Nash equilibria in network‑constrained Cournot markets with a strategic operator; (2) it identifies an exact potential function enabling efficient computation; (3) it theoretically links price differentials to capacity bottlenecks, providing a rigorous foundation for observed market phenomena; and (4) it validates the model with extensive real‑world electricity market data. Future work is suggested on dynamic extensions, stochastic demand (e.g., renewable generation variability), and scenarios with multiple interacting system operators.