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Social Welfare in Battery Charging Games

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Pith's one-line read The paper claims that in a capacitated electricity network, selfish battery charging decisions can make the socially useful energy flow arbitrarily small relative to the optimum, for every price schedule, even when small coalitions of agent

desk verdict A genuine new model with strong negative results; the central unbounded-PoA theorem survives, but the upper-bound proof in Theorem 9 needs a rewrite before publication. read the letter →

arxiv 2508.06320 v1 pith:4WXEAW6S submitted 2025-08-08 cs.GT

Simon Krogmann , Pascal Lenzner , Alexander Skopalik , Tobias Sträubig This is my paper · ORCID
classification cs.GT MSC 91A1091A80
keywords batterycharginggamespriceofanarchystabilityNashequilibriumk-strongenergynetworksrenewablemaximumflow
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Household batteries are spreading fast, and who charges or discharges them is a strategic decision, not a grid command. This paper introduces a flow-based game in which battery agents buy and sell against a fixed per-time-step price, and grid feasibility is enforced by declaring any trading plan invalid if some maximum flow through the network would not need every planned transaction. The central message is that this selfish behavior can destroy the network's ability to use excess renewable energy: for more than two time steps, the $k$-strong price of anarchy is infinite for every price schedule when coalitions are limited to $k \le T-2$ agents, and cooperation by at least $T-1$ agents is needed to cap the loss at a factor of $T$. Along the way the paper shows that equilibrium existence itself depends on the pricing: with two time steps every price admits a Nash equilibrium, while with more steps ascending prices can admit none. A sympathetic reader should take away that price incentives alone, of the time-of-use kind, may not be enough to steer distributed storage toward grid-efficient charging.

What carries the argument

The central object is the time-expanded energy network $G$ built from $T$ copies of the grid graph, battery nodes and storage edges for each agent, and bidirectional transaction edges where each agent's charging strategy sets the capacity. The load-bearing defined mechanism is admissibility (Definition 1): a transaction edge is admissible if every maximum flow in the induced graph saturates it, and a violating agent receives utility $-\infty$. This rule turns the grid's max-flow structure into a feasibility test for strategies and is what makes the zero strategy always safe and the deviations in the theorems profitable. Social welfare is the maximum flow value $W(s)$ under the strategy-induc

What would settle it

Take the four-time-step instance constructed with prices $p=(1,11,12,13)$ and run an exhaustive best-response search: Theorem 5 asserts that no pure Nash equilibrium exists, so finding any equilibrium profile would refute that claim. To target the headline result, search small capacitated networks with $T>2$ and $k \le T-2$ for a single $k$-strong equilibrium with finite welfare ratio; Theorem 8 asserts the ratio is infinite for all such equilibria, so one finite case refutes it.

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Extended reading notes

Core claim

The paper defines a charging game on a time-expanded capacitated graph: each agent chooses a charge/discharge vector, transaction edges receive those capacities, and social welfare is the value of the maximum flow from surplus to deficit nodes. A profile is admissible only if every maximum flow saturates every transaction edge; otherwise the agent gets utility $-\infty$. For $T=2$, Nash equilibria always exist, with price of stability $1$ for ascending and uniform prices and infinite price of stability for descending prices. For $T>2$, uniform prices always admit an equilibrium, ascending prices may admit none, and if they do the best equilibrium loses at least $\lfloor T/2 \rfloor$; supply-

Load-bearing premise

The results rest on the rule that a charging plan is legal only if every maximum flow through the grid saturates every planned buy or sell transaction; if legality required only that some maximum flow saturate them, the equilibrium set and efficiency bounds could differ.

Editorial extensions

If this is right

  • With two time steps and rising prices, every instance has a Nash equilibrium and the best equilibrium achieves optimal flow; the price of anarchy is exactly $2$, so some inefficiency is unavoidable.
  • For more than two time steps, uniform pricing guarantees an equilibrium at full efficiency, but natural ascending prices can have no equilibrium at all.
  • Demand-based (supply-sign) price signals can be infinitely bad even for the best equilibrium, so pricing at the aggregate surplus or deficit of each time step does not align battery incentives with grid flow.
  • If at most $T-2$ agents are allowed to coordinate, the $k$-strong price of anarchy is infinite for any price schedule: no price system can prevent arbitrarily large waste without near-horizon-scale cooperation.
  • Raising the cooperation threshold from $T-2$ to $T-1$ agents jumps the price of anarchy from unbounded to at most $T$, so the same network can be arbitrarily bad or only linearly bad depending on coalition size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If admissibility were relaxed to require only that some maximum flow saturates a transaction edge, the zero-strategy safety and the deviation arguments could fail; recomputing the bounds under that relaxation would isolate how much of the negative result comes from the all-maximum-flows rule.
  • The threshold at $T-1$ cooperating agents suggests that the effective unit for congestion management is a coordinated block at least as large as the number of trading periods, not pairwise or small-group coordination.
  • A testable real-world analogue: using day-ahead prices and a feeder's residual capacities, compute the maximum-flow utilization achieved by profit-maximizing battery agents; the model predicts utilization far below the max-flow optimum, with the shortfall growing with the time horizon.
  • The unbounded price of anarchy for every price profile implies that a price-setting Stackelberg leader cannot steer storage by time-of-use prices alone; if the result carries to realistic grids, capacity reservations or quotas would be needed alongside prices.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper introduces a noncooperative game model of battery charging in a capacity-constrained energy network over T time steps. Each agent chooses a charge/discharge vector; a strategy is admissible only if every maximum flow in the induced network saturates the agent's transaction edges, otherwise utility is -∞. A leader sets time-dependent prices. The authors study existence and efficiency (PoS, PoA, and k-strong PoA) of pure Nash equilibria. For T=2 they prove equilibrium existence for all prices, PoS=1 for ascending/uniform, PoS=∞ for descending, and PoA≥2 for all prices. For T>2 they show uniform pricing always admits an equilibrium but ascending prices need not; they give low welfare bounds for ascending and supply-sign pricing; and for k≤T−2 the k-strong PoA is infinite for any price scheme. If at least T−1 agents cooperate, they claim an upper bound of T on the PoA for ascending prices. The paper concludes that selfish battery charging can severely reduce renewable-energy utilization even with extensive cooperation.

Significance. The paper addresses a timely and underexplored interface of algorithmic game theory and energy networks. Its main negative result, Theorem 8, is striking: for any price schedule and any coalition size up to T−2, there are instances where the k-strong price of anarchy is infinite. The construction is simple and the proof is convincing. The paper is self-contained and does not rely on free parameters or black-box prior results. The model is stylized and the admissibility assumption is strong, but the central unbounded-efficiency result is robust to the natural weakening of that assumption in the constructed instance. If the upper-bound result (Theorem 9) is made rigorous, the contrast between low and high cooperation is a valuable conceptual message. The paper will likely stimulate further work on market design for distributed storage.

major comments (3)
  1. [Section 5.3, Theorem 9] The proof is not rigorous. It asserts that any augmentation of f along a single path must decrease flow on some battery edge, but does not prove this from the residual-graph perspective, nor does it formalize the 'infinitesimally small augmentations' decomposition. The bound |f_diff| ≤ |f|(T−1) is also asserted without derivation. Please replace this sketch with a complete proof: decompose f_OPT − f into paths/cycles, argue for each s-t path why a path that does not decrease a battery edge would yield an improving coalition of at most T−1 agents under ascending prices, and bound the total battery-flow decrease by (T−1)|f|. As written, the theorem's proof cannot be checked.
  2. [Definition 1] The admissibility condition that every maximum flow must saturate a transaction edge is central and strong. It is this condition that makes the zero strategy safe and that justifies the deviation arguments in Theorems 2, 4, 5, and 8. The paper should (i) give a fuller physical justification for penalizing an agent whenever some maximum flow does not use her edge, and (ii) discuss the robustness of the results to a weaker 'some maximum flow saturates' condition. The main negative result appears to survive in the particular construction of Theorem 8, but the existence and welfare statements for other sections may not. Since the contribution is framed as applying to arbitrary price schedules, this modeling choice needs explicit defense.
  3. [Section 5.1, Theorem 5] The non-existence proof is under-specified and, as written, difficult to verify. The variables za, zc, ya, yc do not uniquely determine the strategy profile; the claims that total time-1 charge is 1 and total time-4 discharge is 3/2 in every Nash equilibrium are not proven; and equations (1)–(5) do not transparently follow from the utility definition with the price profile p=(1,11,12,13). For instance, the utility of the deviation s'_a given in the proof does not obviously equal the expression in (3). Please rewrite the proof with explicit strategy tuples and a step-by-step utility calculation.
minor comments (5)
  1. [Theorem 9 statement] The theorem title says 'Any Prices' but the statement restricts to ascending price functions. Please adjust the title to avoid confusion.
  2. [Definition 1] The notation 'e ∈ C' is introduced without defining C. Please define the set of transaction edges explicitly.
  3. [Observation 2] The argument that uniform prices make all admissible profiles equilibria should explicitly note that any admissible strategy has net zero charge/discharge over all time steps, so the utility is -p·0 = 0.
  4. [Theorem 1] In the proof, 'lowering its capacity by ε > 0' should be stated as 'for some sufficiently small ε > 0' to be mathematically precise.
  5. [Section 5.2, Theorem 6] The claim that there is a unique Nash equilibrium in the constructed instance needs more justification. It is not immediately obvious that no other admissible profile with lower profit but higher flow could be an equilibrium.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all results follow from explicit model definitions; no fitted inputs and no load-bearing self-citations.

full rationale

This paper's derivation is self-contained. The model definitions (capacities, transaction edges, admissible strategies, maximum-flow welfare) are stated as assumptions, and the theorems are proved directly from these definitions using standard max-flow and equilibrium arguments. No fitted parameter is renamed as a prediction; the PoA/PoS results are worst-case constructions over the defined instance classes, not empirical claims. The only self-citations (e.g., [19], [26]) appear in related-work context and are not load-bearing. The strongest claimed result, Theorem 8, is built on an explicit path instance in which no coalition of size ≤ T−2 can create any feasible flow, so the infinite k-strong PoA follows from the definitions rather than from an imported theorem. The flagged soft spot—Theorem 9's flow-augmentation argument—is a rigor gap in an upper bound, not a circular reduction: it does not assume the conclusion. No circular step was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The model rests on a small set of stated modeling choices: the network is a capacitated directed graph with feasible flows, the operator's objective is maximum flow value, agents are price-takers with identical buy/sell prices, and admissibility punishes any transaction edge not saturated by every maximum flow. These are explicit in Section 2.

assumptions (5)
  • standard math Max-flow min-cut theorem and flow decomposition for real-valued flows in directed capacitated graphs.
    Used to characterize admissibility via saturated cuts (Theorem 2) and to define welfare as maximum flow value throughout Section 2.
  • domain assumption The electricity grid is a directed graph with capacities, supplies, and demands; electricity flows are feasible single-commodity flows, ignoring losses and AC physics.
    Stated in Section 2 'Electricity Flow': 'ignoring transmission and conversion losses and the actual physics of power flows.' This defines the welfare measure and the admissibility check.
  • domain assumption The operator's objective is to maximize utilization of excess energy, operationalized as the maximum flow value.
    Stated after Definition 1: 'Using maximum flows as the set of acceptable flows is based on the assumption that the operator wants to maximize the usage of excess energy that otherwise would be wasted.'
  • domain assumption Agents are price-takers: they buy and sell at the same exogenous per-time-step price p_t, with no transaction costs.
    Section 2 'Prices and Payoffs': utility is -sum s_{b,t} p_t for the same p_t on both directions, and prices are fixed per time step.
  • ad hoc to paper A transaction edge is admissible only if every maximum flow saturates it; otherwise the agent receives -∞.
    Definition 1 and the utility definition in Section 2. This nonstandard strong requirement is load-bearing for all equilibrium existence and inefficiency proofs.

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Pith. "Pith review of Social Welfare in Battery Charging Games." pith.science (2026). https://pith.science/paper/4WXEAW6S

@misc{pith2026250806320,
  author       = {Pith},
  title        = {Pith review of: Social Welfare in Battery Charging Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4WXEAW6S}},
  note         = {Machine review of arXiv:2508.06320}
}
read the original abstract

The recent rise of renewable energy produced by many decentralized sources yields interesting market design challenges for electrical grids. Balancing supply and demand in such networks is both a temporal and spatial challenge due to capacity constraints. The recent surge in the number of household-owned batteries, especially in regions with rooftop solar adoption, offers mitigation potential but often acts misaligned with grid-level objectives. In fact, the decision to charge or discharge a household-owned battery is a strategic choice by each battery owner governed by selfish incentives. This calls for an analysis from a game-theoretic point of view. We initiate this timely research direction by considering a game-theoretic setting where selfish agents strategically charge or discharge their batteries to increase their profit. In particular, we study a Stackelberg-like market model where a third party introduces price incentives, aiming to optimize renewable energy utilization while preserving grid feasibility. For this, we study the existence and the quality of equilibria under various pricing strategies. We find that the existence of equilibria crucially depends on the chosen pricing and that the obtained social welfare varies widely. This calls for more sophisticated market models and pricing mechanisms and opens up a rich field for future research in Algorithmic Game Theory on incentives in renewable energy networks.

Figures

Figures reproduced from arXiv: 2508.06320 by the authors.

Figure 1
Figure 1. An instance of the charging game with two time steps. In this case, H has two (unnamed) vertices connected by two edges. The supplies and demands are given inside the vertices. There are two battery agents a and b. The vertical edges between the time steps are their battery edges and the network and battery capacities induced by κ are given next to the edges. Strategies. For each time step t ∈ [T], each agent decide… view at source ↗
Figure 2
Figure 2. For a graph G, the subfigures contain the graphs Gs for several strategy profiles s and the corresponding maximum flows. 4 Two Time Steps As a warm-up, we investigate the case of two time steps. This is a relatively simple case because there are only three price functions to look at: Ascending, descending and equal prices over the two steps. The absolute difference does not matter, because it only scales the utiliti… view at source ↗
Figure 3
Figure 3. The graph Gs where only agent c buys/sells electricity with capacity 1 on all edges. The thick red edges mark the maximum flow in Gs. Profile s is a Nash equilibrium with social welfare 1. However, an alternative strategy profile enabling the dashed blue flow has a social welfare of 2 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Graph G of an instance with no Nash equilibrium, all capacities are 2. In Theorem 5, we consider equilibrium candidates in which za (or zc) is discharged in time step 2 by agent a (or agent c) and ya (or yc) is charged in time step 3, visualized with its intended flow …
Figure 5
Figure 5. Figure 5: (a) The strategy profile with the maximum flow value of  T 2  . (b) Gs of a Nash equilibrium s with flow value 1. All capacities are 1. The strategy profile in (a) is not a Nash equilibrium under any ascending price function. We also show that the natural class of pr…
Figure 6
Figure 6. Figure 6: In this instance, with a supply-sign pricing function, the strategy profile given by the thick red transaction edges produces a flow value of 1. However, it yields negative utility for agent a and thus is not a Nash equilibrium. edge capacities are 1 for all edges of t…
Figure 7
Figure 7. Figure 7: This is graph Gs for the graph in Theorem 8 with T = 4 with the empty strategy profile s, which is a Nash equilibrium. All black edges have capacity 1, the red edges have capacity 0. This graph needs cooperation of at least T − 1 = 3 agents to escape from strategy prof…

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Works this paper leans on

31 extracted references · 25 canonical work pages

  1. [1]

    Cheng, J., Chen, G., Wu, Z., Mu, Y.: An optimal pricing formula for smart grid based on Stackelberg game (2024),https://arxiv.org/abs/2407.09948

    arXiv 2024

  2. [2]

    IEEE Access7, 25727–25762 (2019).https://doi.org/10

    Cheng, L., Yu, T.: Game-theoretic approaches applied to transactions in the open and ever-growing electricity markets from the perspective of power demand response: An overview. IEEE Access7, 25727–25762 (2019).https://doi.org/10. 1109/ACCESS.2019.2900356

    work page arXiv 2019

  3. [3]

    Automatica96, 150–158 (2018), https://doi.org/10.1016/j.automatica.2018.06.043

    Deori, L., Margellos, K., Prandini, M.: Price of anarchy in electric vehicle charging control games: When Nash equilibria achieve social welfare. Automatica96, 150–158 (2018), https://doi.org/10.1016/j.automatica.2018.06.043

    work page doi:10.1016/j.automatica.2018.06.043 2018

  4. [4]

    IEEE Access12, 152908–152920 (2024)

    Dorahaki, S., Muyeen, S.M., Amjady, N.: A Stackelberg game theory model for integrated community energy storage systems. IEEE Access12, 152908–152920 (2024). https://doi.org/10.1109/ACCESS.2024.3481155

    work page arXiv 2024

  5. [5]

    Journal of Polit- ical Economy 65(2), 135–150 (1957),https://www.jstor.org/stable/1827369

    Downs, A.: An economic theory of political action in a democracy. Journal of Polit- ical Economy 65(2), 135–150 (1957),https://www.jstor.org/stable/1827369

    work page arXiv 1957

  6. [6]

    Dürr, C., Thang, N.K.: Nash equilibria in voronoi games on graphs. In: ESA. pp. 17–28 (2007). https://doi.org/10.1007/978-3-540-75520-3_4

    work page doi:10.1007/978-3-540-75520-3_4 2007

  7. [7]

    ZEW Discussion Paper (22-060) (2022)

    Ehrhart, K.M., Eicke, A., Hirth, L., Ocker, F., Ott, M., Schlecht, I., Wang, R.: Congestion management games in electricity markets. ZEW Discussion Paper (22-060) (2022). https://doi.org/10.2139/ssrn.4300874

    work page doi:10.2139/ssrn.4300874 2022

  8. [8]

    Eurostat: Renewable energy statistics (2024),https://ec.europa.eu/eurostat/ statistics-explained/index.php?title=Renewable_energy_statistics

    work page 2024

Show all 31 references
  1. [9]

    In: AAAI

    Feldman, M., Fiat, A., Obraztsova, S.: Variations on the hotelling-downs model. In: AAAI. pp. 496–501 (2016).https://doi.org/10.1609/aaai.v30i1.10054

    2016 doi

  2. [10]

    Fochesato, M., Cenedese, C., Lygeros, J.: A Stackelberg game for incentive-based demand response in energy markets. In: CDC. pp. 2487–2492 (2022).https://doi. org/10.1109/CDC51059.2022.9993196

    2022

  3. [11]

    Gao, B., Liu, X., Wu, C., Tang, Y.: Game-theoretic energy management with storage capacity optimization in the smart grids. J. Mod. Power Syst. Clean Energy 6(4), 656–667 (2018).https://doi.org/10.1007/s40565-017-0364-2

    2018 doi

  4. [12]

    In: ACM e-Energy

    Hekkelman, B., La Poutré, H.: Fairness in power flow network congestion manage- ment with outer matching and principal notions of fair division. In: ACM e-Energy. p. 106–115 (2020).https://doi.org/10.1145/3396851.3397701

    2020

  5. [13]

    In: ACM e-Energy

    Hekkelman, B., La Poutré, H.: Fairness vs welfare: a hybrid congestion aftermarket. In: ACM e-Energy. p. 93–104 (2022).https://doi.org/10.1145/3538637.3538843

    2022

  6. [14]

    In: ISGT-Europe (2019)

    Hekkelman, B., La Poutré, H.: Fairness in smart grid congestion management. In: ISGT-Europe (2019). https://doi.org/10.1109/ISGTEurope.2019.8905496

    2019

  7. [15]

    CIRED 2017(1), 1629–1633 (2017).https://doi.org/10.1049/oap-cired.2017.0340 18 Krogmann, Lenzner, Skopalik, Sträubig

    Hoogsteen, G., Molderink, A., Hurink, J.L., Smit, G.J., Kootstra, B., Schuring, F.: Charging electric vehicles, baking pizzas, and melting a fuse in lochem. CIRED 2017(1), 1629–1633 (2017).https://doi.org/10.1049/oap-cired.2017.0340 18 Krogmann, Lenzner, Skopalik, Sträubig

    2017

  8. [16]

    The Economic Journal 39(153), 41–57 (1929)

    Hotelling, H.: Stability in competition. The Economic Journal 39(153), 41–57 (1929). https://doi.org/10.2307/2224214

    1929 doi

  9. [17]

    Frontiers in Energy Research10 (2022)

    Huang, W., Li, H.: Game theory applications in the electricity market and renewable energy trading: A critical survey. Frontiers in Energy Research10 (2022). https: //doi.org/10.3389/fenrg.2022.1009217

    2022

  10. [18]

    In: WINE

    Kiyomi, M., Uno, T., Matsui, T.: Efficient algorithms for the electric power transaction problem. In: WINE. pp. 602–611 (2005).https://doi.org/10.1007/ 11600930_60

    2005

  11. [19]

    In: IJCAI

    Krogmann, S., Lenzner, P., Molitor, L., Skopalik, A.: Two-stage facility location games with strategic clients and facilities. In: IJCAI. pp. 292–298 (2021).https: //doi.org/10.24963/ijcai.2021/41

    2021 doi

  12. [20]

    IEEE Trans

    Liu, N., Yu, X., Wang, C., Wang, J.: Energy sharing management for microgrids with PV prosumers: A Stackelberg game approach. IEEE Trans. Ind. Informatics 13(3), 1088–1098 (2017).https://doi.org/10.1109/TII.2017.2654302

    2017

  13. [21]

    In: IJCAI

    Manvi, B., Chandlekar, S., Subramanian, E.: Optimizing prosumer policies in periodic double auctions inspired by equilibrium analysis. In: IJCAI. pp. 2931–2939 (2024). https://doi.org/10.24963/ijcai.2024/325

    2024 doi

  14. [22]

    Thesis, Chapter 4 (2024),https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37379051

    Mibuari, E.M.: Learning to Mitigate AI Collusion in Electricity Markets. Thesis, Chapter 4 (2024),https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37379051

    2024

  15. [23]

    Sustainability11(10) (2019)

    Naz, A., Javaid, N., Rasheed, M.B., Haseeb, A., Alhussein, M., Aurangzeb, K.: Game theoretical energy management with storage capacity optimization and photo- voltaic cell generated power forecasting in micro grid. Sustainability11(10) (2019). https://doi.org/10.3390/su11102763

    2019 doi

  16. [24]

    International Journal on Game Theory47(3) (2018)

    Peters, H., Schröder, M., Vermeulen, D.: Hotelling’s location model with negative network externalities. International Journal on Game Theory47(3) (2018). https: //doi.org/10.1007/S00182-018-0615-0

    2018 doi

  17. [25]

    IEEE Trans

    Rahi, G.E., Etesami, S.R., Saad, W., Mandayam, N.B., Poor, H.V.: Managing price uncertainty in prosumer-centric energy trading: A prospect-theoretic Stackelberg game approach. IEEE Trans. Smart Grid10(1), 702–713 (2019).https://doi.org/ 10.1109/TSG.2017.2750706

    2019

  18. [26]

    In: ICALP

    Schmand, D., Schröder, M., Skopalik, A.: Network investment games with wardrop followers. In: ICALP. pp. 151:1–151:14 (2019).https://doi.org/10.4230/LIPIcs. ICALP.2019.151

    2019 doi

  19. [27]

    SolarPower Europe: European market outlook for battery storage 2025-2029 (2025), https://www.solarpowereurope.org/insights/outlooks/european-market- outlook-for-battery-storage-2025-2029

    2025

  20. [28]

    The Netherlands Authority for Consumers and Market: Algorithmic trading in wholesale energy markets: Key findings of an exploratory market study by the acm (2024), https://www.acm.nl/en/publications/acm-market-study- algorithmic-trading-wholesale-energy-markets

    2024

  21. [29]

    AIP Advances14(1), 010701 (2024).https://doi.org/10.1063/ 5.0165108

    Ullah, K., Ishaq, M., Albalwi, M.D., Bonyah, E., Hussain, N.F., Rasool, T., Rasool, A., Ahmad, H., Ahmad, Z.: Application of game theory in modern electrical power system (a review). AIP Advances14(1), 010701 (2024).https://doi.org/10.1063/ 5.0165108

    2024

  22. [30]

    Springer (2010).https: //doi.org/10.1007/978-3-642-12586-7

    Von Stackelberg, H.: Market structure and equilibrium. Springer (2010).https: //doi.org/10.1007/978-3-642-12586-7

    2010 doi

  23. [31]

    IEEE Transactions on Power Systems28(2), 884–892 (2013)

    Yang, P., Tang, G., Nehorai, A.: A game-theoretic approach for optimal time-of-use electricity pricing. IEEE Transactions on Power Systems28(2), 884–892 (2013). https://doi.org/10.1109/TPWRS.2012.2207134

    2013

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