Equilibria in Network Constrained Markets with Market Maker

TL;DR

The paper develops a networked Cournot framework with a centralized market maker on a constrained market network with $n$ producers, $m$ markets, and $l$ links, and proves existence of Nash equilibria under general concavity assumptions. It identifies conditions under which the game is an exact potential game when the market maker maximizes Walrasian welfare with affine price functions, yielding a (essentially) unique equilibrium that can be computed via a concave program. A key theoretical contribution shows that, under Walrasian welfare, price differences across markets imply saturated or empty cuts in the network, linking capacity bottlenecks to price dispersion observed in power systems. The authors validate the model with Italian day-ahead electricity market data, demonstrating qualitative agreement between predicted and observed price groupings and saturated links, and discuss implications for market design and network policy.

Abstract

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. Such flows are determined by the market maker, who aims at maximizing a designated welfare function. We model such competition as a strategic game with $n+1$ players: the producers and the market game. For this game, we first establish the existence of Nash equilibria under standard concavity assumptions. We then identify sufficient conditions for the game to be potential with an essentially unique Nash equilibrium. Next, we present a general result that connects the optimal action of the market maker 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.

Figures (12)

• Figure 1: The network of Example \ref{['example:1']}. The labels on the links represent their capacities.
• Figure 2: On the left: the market network of Example \ref{['example:basic']}. On the right: an equivalent network for the special case (b) of Example \ref{['example:basic']}.
• Figure 3: The best response set \ref{['eq:BR0']} for the market maker in the NCGMM of Example \ref{['example:basic']} in four special cases: (i) for $x_{11}- 2x_{12}+1\le-3\chi$, $\mathcal{B}_0(x)$ is the grey bullet point; (ii) for $-3\chi<x_{11}- 2x_{12}+1\le0$, $\mathcal{B}_0(x)$ is the grey segment; (iii) for $0<\chi<x_{11}- 2x_{12}+1\le3\chi$, $\mathcal{B}_0(x)$ is the black segment; (iv) for $x_{11}- 2x_{12}+1\ge3\chi$, $\mathcal{B}_0(x)$ is the black bullet point.
• Figure 4: A cut in a network.
• Figure 5: Production quantities, flows, prices and bottlenecks at equilibrium. The grey numbers represent the total production of each producer, the colored numbers next to each market are the respective equilibrium prices and the numbers close to each link are the actual flow at equilibrium. In red, we highlighted the saturated link.

Theorems & Definitions (28)

• Example 1
• Remark 1
• Example 2
• Remark 2
• Definition 1
• Definition 2
• Remark 3
• Example 3
• Definition 3
• Lemma 1