Multilateral Market Power Analysis

• Multilateral market power is the strategic influence arising from interactions among multiple market participants rather than a single seller, impacting prices, allocations, and surplus distribution.
• This concept is applied in contexts like transmission-constrained electricity networks and firm-to-firm bargaining, where network constraints, residual capacity, and outside options are key determinants.
• Practical implications include guiding mechanism design and policy evaluation to mitigate pricing distortions and ensure stable market equilibria in complex, interconnected markets.

Searching arXiv for relevant papers on multilateral market power and closely related formulations. Multilateral market power denotes strategic influence over prices, allocations, or surplus that arises when outcomes are determined by interactions among multiple firms, coalitions, or trading links rather than by a single seller facing atomistic demand. In transmission-constrained electricity networks it appears when multiple generators at the same node compete jointly and local market structure governs withholding; in multilateral collaboration markets it is the ability of a coalition of agents to reshape the allocation by jointly signing a contract; in firm-to-firm trade and production networks it is embedded in bargaining relationships and in firms’ simultaneous ability to affect prices in input and output markets (Lin et al., 2017, Huang, 2024, Vanessa et al., 17 Jul 2025, Bizzarri, 23 Mar 2026). Taken together, these formulations suggest that market power in networked and coalition-based environments is governed by residual capacity, outside options, complementarities, and platform rules, not by size alone.

1. Definitions and analytical scope

A central distinction in the recent literature is between unilateral and multilateral market power. In production networks, multilateral market power means that a firm can affect prices in all markets it participates in—both its output market and its input markets—whereas unilateral market power treats the firm as strategic only on one side, typically the output side, while input prices are taken as given (Bizzarri, 23 Mar 2026). In multilateral collaboration with transferable utility, the corresponding concept is not bilateral bargaining power but the ability of a coalition of agents to reshape the allocation by jointly signing a contract; disagreement is resolved at the coalition level rather than at the level of isolated pairs (Huang, 2024).

The same idea appears in networked bargaining models of firm-to-firm trade. Prices are negotiated bilaterally, but each buyer’s and supplier’s fallback payoff depends on other trading links, so bargaining power is shaped by the broader supply-chain structure. In that setting, market power is neither purely oligopoly nor purely oligopsony; it is two-sided and network-dependent (Vanessa et al., 17 Jul 2025). In electricity networks, the multilateral feature is local but network-mediated: generators compete node by node, yet nodal quantities are coupled through transmission constraints, so the ability to mark up price depends on both a generator’s own market share and the residual local capacity available after accounting for rivals (Lin et al., 2017).

The scope of the concept is broader than goods markets narrowly defined. In oligopolistic risk-sharing, a finite number of agents strategically choose security payoffs, pricing kernels, or demand functions; the Nash equilibrium is suboptimal because agents submit different risk exposures than their true endowments, and agents with sufficiently lower risk aversion act as predatory traders (Anthropelos, 2012). This suggests that multilateral market power is a general property of thin markets in which each participant internalizes how its own action shifts the terms faced by others.

2. Transmission-constrained electricity markets

A canonical formulation is the transmission-constrained electric power network with perfectly inelastic demand and capacity-constrained generators competing in scalar-parameterized supply functions. The feasible injection set is

$$\mathcal{P}=\{y\in \mathbb{R}^n \mid \mathbf{1}^\top y=0,\; Hy\le c\},$$

the social optimum solves

$$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$

and each generator chooses a scalar bid parameter $\theta_j\ge 0$ defining the supply function

$S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$

The independent system operator clears the market under locational marginal pricing using surrogate costs

$$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$

so larger $\theta_j$ corresponds to more aggressive economic withholding (Lin et al., 2017).

The model admits a nodal decomposition. At node $i$, if total nodal bid weight is positive, the locational marginal price is

$$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$

and the resulting production split is explicit:

$$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$

This makes the mechanism of local market power transparent: nodal prices rise as the node approaches exhaustion of local capacity, while transmission constraints determine how tightly each node is disciplined by the rest of the network (Lin et al., 2017).

The paper’s structural market-power condition is the absence of pivotal suppliers. If

$$q_i^{\max}:=\sup\{q_i\mid q\in \mathbb{R}^n_+,\; q-d\in \mathcal{P}\},$$

then for generator $$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$0,

$$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$1

A generator is pivotal if $$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$2, and the analysis assumes $$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$3 for all $$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$4. Under convex costs and this no-pivotal-supplier condition, the game admits at least one pure-strategy Nash equilibrium; the equilibrium production profile $$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$5 and nodal supply profile $$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$6 are unique even though the bid profile $$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$7 need not be unique. Strategic behavior appears as a distortion of true costs, and the distortion is magnified when residual local capacity is small (Lin et al., 2017).

The efficiency and markup implications are explicit. The price of anarchy satisfies

$$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$8

and the Lerner index

$$\min_{x\in \mathbb{R}^N}\sum_{j=1}^N C_j(x_j)\quad \text{s.t.}\quad Ax-d\in\mathcal{P},\qquad 0\le x_j\le X_j,$$9

obeys, when $\theta_j\ge 0$0,

$\theta_j\ge 0$1

If $\theta_j\ge 0$2 is differentiable and the generator is exactly at the maximum nodal supply boundary $\theta_j\ge 0$3, the bound is tight:

$\theta_j\ge 0$4

The same framework yields a two-node Braess-like paradox: strengthening transmission capacity can increase the total cost of generation at Nash equilibrium exactly when the low-cost node is priced higher than the high-cost node (Lin et al., 2017).

3. Production networks, bargaining, and two-sided strategic power

In firm-to-firm trade, multilateral market power is modeled as bilateral bargaining embedded in a network of relationships. For a buyer $\theta_j\ge 0$5 and supplier $\theta_j\ge 0$6, the bargaining problem is a Nash-in-Nash solution,

$\theta_j\ge 0$7

where $\theta_j\ge 0$8 is the importer’s bargaining leverage and the outside-option payoffs depend on other trading links. The negotiated price takes the form $\theta_j\ge 0$9, with a pair-specific markup

$S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$0

so supplier concentration creates oligopoly power, buyer concentration creates oligopsony power, and observed prices reflect both (Vanessa et al., 17 Jul 2025).

Two bilateral shares summarize the local structure of power:

$S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$1

Here $S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$2 is the supplier’s share in the buyer’s total foreign input purchases, and $S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$3 is the buyer’s share in the supplier’s total output. The pass-through elasticity

$S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$4

decomposes into a markup channel $S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$5 and a cost channel $S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$6, with

$S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$7

Empirically, the paper estimates strong importer bargaining power, $S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$8 to $S_j(p;\theta_j)=X_j-\frac{\theta_j}{p},\qquad p>0.$9, a preferred estimate around $$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$0, and a returns-to-scale parameter $$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$1; pass-through of the 2018 tariffs is incomplete, around $$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$2–$$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$3 in repeated firm-to-firm relationships and around $$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$4–$$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$5 in the aggregate model-implied measure, primarily because dominant buyers induce exporter cost reductions (Vanessa et al., 17 Jul 2025).

A broader production-network formulation treats firm-to-firm trade as a system of coupled double auctions or supply-and-demand function competition. Each firm submits supply and demand schedules, market prices solve a unique clearing system $$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$6, and the firm’s price impact matrix is

$$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$7

A Supply and Demand Function Equilibrium is a Nash equilibrium in slopes $$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$8, with best reply

$$\widehat C_j(x;\theta_j)=\theta_j \log\!\left(\frac{X_j}{X_j-x}\right),$$9

The signed markup-markdown vector is

$\theta_j$0

so market power is a vector of markups and markdowns rather than a single scalar distortion (Bizzarri, 23 Mar 2026).

This framework yields a sharp comparison with models that impose input-market price-taking. Under unilateral or local market power, equilibrium slopes are higher, aggregate price impacts are smaller, final prices are underpredicted, and upstream surplus is overestimated. The same bias affects merger analysis: in a two-layer vertical merger example, there exists a range of market sizes where a merger is welfare-improving under multilateral market power but welfare-decreasing under unilateral market power, and the same can hold relative to sequential Cournot (Bizzarri, 23 Mar 2026). A plausible implication is that empirical and policy analyses that omit input-side strategic behavior mismeasure both incidence and welfare.

4. Coalitions, platforms, and mechanism design

In multilateral collaboration with transferable utility, any group of agents can sign a contract $\theta_j$1, where $\theta_j$2 is a primitive contract and transfers satisfy $\theta_j$3. Utilities are quasilinear,

$\theta_j$4

and an outcome is stable if it is individually rational and not blocked by any $\theta_j$5. The dynamic auction posts balanced prices $\theta_j$6 with $\theta_j$7 for each primitive contract, agents demand

$\theta_j$8

and equilibrium prices minimize the Lyapunov function $\theta_j$9 over balanced prices. Under integer-valued valuations, gross complements for all agents, and truthful reporting, the auction converges in finitely many steps to a stable outcome; the paper also notes that there is no incentive-compatible mechanism that always selects a stable outcome in this setting (Huang, 2024).

The mechanism relies on the notion of a complements chain,

$i$0

along which the auction moves one unit of price from one participant to another within each primitive contract. This preserves internal balance while decreasing the Lyapunov function by one unit. The paper’s three equivalent gross-complements conditions—supermodularity, antitone demand, and the gross-complements property itself—supply the structure required for finite convergence. In this setting, multilateral market power is most severe when bargaining over one term affects the attractiveness of other terms in the coalition (Huang, 2024).

Mechanism design for networked procurement reaches a similar conclusion from a different direction. In a finite network market with piecewise linear convex costs and quadratic losses $i$1, producers possess market power because production and flows at one node affect feasibility and cost elsewhere through network losses. The principal’s direct mechanism chooses production $i$2, payments $i$3, and flows $i$4 to minimize expected payments subject to network feasibility, incentive compatibility, and participation. The original mechanism problem is equivalent to minimizing expected virtual production cost,

$i$5

and the paper provides a monotone fixed-point algorithm to solve the principal allocation problem (Heymann et al., 2019).

Decentralized electricity trading platforms supply a further mechanism-design environment. A platform maps aggregate imports and exports into internal buy and sell prices $i$6 and $i$7, with period payment

$i$8

Because the producer price is decreasing in aggregate exports, strategic prosumers withhold supply and underutilize storage relative to the price-taking benchmark. In the baseline calibration, strategic play raises grid settlement cost by about $i$9 relative to price-taking. The distortion depends strongly on platform design: some designs can largely eliminate strategic incentives, while increased competition in storage ownership sharply reduces withholding, with most of the distortion disappearing once storage is split across more than three owners (Eschenbaum et al., 20 Mar 2026).

5. Equilibrium existence, multiplicity, and instability

A recurring result across multilateral market-power models is that equilibrium structure itself becomes an object of study. In spatial partial equilibrium models of natural gas markets with conjectural variations, the market reduces to a linear complementarity problem

$$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$0

for which existence is guaranteed but uniqueness is generally not, because the matrix is not a $$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$1-matrix. The full solution set is a convex polyhedron, and the paper shows that trader-to-consumer gas flow $$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$2 is unique whenever $$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$3, even though many infrastructure flows are not unique. The practical conclusion is that selecting one arbitrary equilibrium can yield erroneous substantive conclusions (Baltensperger et al., 2015).

In electricity markets with an oligopoly and a competitive fringe, endogenous investment intensifies the same issue. Standard mixed complementarity models and conjectural variations become myopic because strategic firms do not correctly anticipate the fringe’s equilibrium dispatch and investment response. The proposed Equilibrium Problem with Equilibrium Constraints models each price-making firm as solving an MPEC with the fringe’s KKT system embedded, and the resulting EPEC finds multiple equilibria for investment decisions and firms’ profits. Across those equilibria, firm 1 profits range from $$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$4 to $$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$5M, firm 2 profits range from $$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$6 to $$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$7M, and consumer costs remain only about $$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$8–$$p_i(q,\theta)=\frac{\sum_{j\in \mathcal{N}_i}\theta_j}{\left(\sum_{j\in \mathcal{N}_i}X_j\right)-q_i},$$9 above perfect competition. The same model shows that price-making firms may rationally sell electricity below marginal cost to de-incentivize fringe investment (Devine et al., 2020).

Two-stage electricity markets with system-level market power mitigation display a more severe form of instability. When generators bid supply functions and loads bid quantities, a real-time MPM policy can lead to equilibrium loss: the paper’s key theorem is that the Nash equilibrium in a two-stage market with a real-time MPM policy does not exist. By contrast, a day-ahead MPM policy yields a Stackelberg-Nash game with loads as leaders and generators as followers. The paper emphasizes that the timing of mitigation is decisive: superficially similar policies can generate either equilibrium loss in real time or a stable but strategically restructured market in day ahead (Bansal et al., 2023). A plausible implication is that multilateral market power is not only about markups or withholding; it can also determine whether a stable market outcome exists at all.

6. Measurement, mitigation, and welfare consequences

In transmission-constrained electricity markets, ex ante monitoring focuses on local market structure. The key structural indices are market share and the residual supply index, and the theory shows that a low RSI predicts a high markup. The same paper reports that Great Britain pool data show that the theoretical bound tracks observed spot-price–RSI relationships reasonably well, especially for the dominant producer (Lin et al., 2017). This use of RSI is not ad hoc: it follows directly from the equilibrium markup bound.

A different measurement strategy appears in differentiated-product Bertrand markets. There the demand Jacobian $$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$0 is interpreted as a network matrix and diagonalized as

$$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$1

with eigenvectors $$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$2 called eigenbundles. A tax shock aligned with one eigenbundle affects only that eigenbundle’s price component, with pass-through

$$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$3

For large risk aversion, the optimal budget-balanced policy taxes eigenbundles with low pass-through and subsidizes ones with high pass-through; Pigouvian leverage depends only on the dispersion of eigenvalues,

$$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$4

The paper’s bottom line is that market power is multidimensional, and the best policy exploits the market’s own interaction geometry (Galeotti et al., 2021).

Hydrothermal electricity markets add intertemporal and stochastic dimensions. In a long-term bid-based simulator with multiple strategic hydro owners, rising concentration increases spot prices, captured revenue, and spillage, while reservoir levels are roughly maintained. The paper explicitly identifies excess spillage as a key signature of market power abuse. Forward contracts substantially reduce spot prices; $$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$5 contracting is already very effective, and $$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$6 contracting nearly reproduces the centralized benchmark (Garcia et al., 2024). In mitigation-aware strategic bidding models, residual market power survives even when conduct and impact tests are imposed: a strategic generator can choose bids that stay just inside mitigation thresholds, raise profit, and reduce social welfare, especially in congested areas (Wu et al., 2022).

The welfare consequences extend beyond short-run price distortions. In production networks, assuming input-market price-taking systematically underestimates the final price and overestimates the surplus going upstream, which changes merger predictions (Bizzarri, 23 Mar 2026). In decentralized electricity trading, the platform remains highly valuable overall, reducing a passive consumer’s annual electricity bill by roughly $$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$7 relative to exclusive grid settlement, but strategic behavior claws back only about $$x_j(q_i,\theta)=X_j-\frac{\theta_j}{\sum_{k\in \mathcal{N}_i}\theta_k}\left(\sum_{k\in \mathcal{N}_i}X_k-q_i\right).$$8 of that saving (Eschenbaum et al., 20 Mar 2026). These results suggest that mitigation and market design should be evaluated not only by whether they suppress observed markups, but also by how they alter pass-through, surplus division, intertemporal incentives, and equilibrium selection.