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Managing Price Uncertainty in Prosumer-Centric Energy Trading: A Prospect-Theoretic Stackelberg Game Approach
Georges El Rahi, S. Rasoul Etesami, Walid Saad, Narayan Mandayam, H. Vincent Poor
TL;DR
The paper addresses energy trading under uncertain future prices when prosumers’ subjective perceptions depart from fully rational classical-game assumptions. It combines a Stackelberg game with prospect theory, analyzes equilibrium behavior, and finds that reference points and loss aversion affect grid load while ignoring subjective behavior can reduce company profits.
Problem
Prior energy-management studies largely assume fully rational consumers and do not jointly model hierarchical prosumer–power-company interaction with uncertainty from variable pricing.
Method
The paper formulates energy trading as a Stackelberg game and uses prospect-theory utility framing to model prosumers’ subjective responses to uncertain future prices, with distributed equilibrium algorithms.
Results
Prosumer reference points and loss aversion significantly change grid load, while ignoring subjective behavior reduces the power company’s profits by up to 15% at a reference point of $2.
Takeaways & Limitations
Accounting for prosumers’ subjective perceptions is important for pricing and energy-trading decisions, because behavioral parameters materially affect consumption and company profits.
Takeaways & Limitations
The Stackelberg formulation is static, although the solution uses repeated interactions and is presented as a step toward dynamically changing multi-stage smart-grid games.
Abstract
from arXiv · showhide
In this paper, the problem of energy trading between smart grid prosumers, who can simultaneously consume and produce energy, and a grid power company is studied. The problem is formulated as a single-leader, multiple-follower Stackelberg game between the power company and multiple prosumers. In this game, the power company acts as a leader who determines the pricing strategy that maximizes its profits, while the prosumers act as followers who react by choosing the amount of energy to buy or sell so as to optimize their current and future profits. The proposed game accounts for each prosumer's subjective decision when faced with the uncertainty of profits, induced by the random future price. In particular, the framing effect, from the framework of prospect theory (PT), is used to account for each prosumer's valuation of its gains and losses with respect to an individual utility reference point. The reference point changes between prosumers and stems from their past experience and future aspirations of profits. The followers' noncooperative game is shown to admit a unique pure-strategy Nash equilibrium (NE) under classical game theory (CGT) which is obtained using a fully distributed algorithm. The results are extended to account for the case of PT using algorithmic solutions that can achieve an NE under certain conditions. Simulation results show that the total grid load varies significantly with the prosumers' reference point and their loss-aversion level. In addition, it is shown that the power company's profits considerably decrease when it fails to account for the prosumers' subjective perceptions under PT.
I. INTRODUCTION
The paper addresses energy trading among prosumers and a power company under uncertain future prices, where conventional models assume rational consumers. It combines Stackelberg pricing with prospect-theoretic framing to represent prosumers’ subjective valuation of gains and losses.
- Prosumers can generate and store energy, enabling demand-side management that can reduce costly peak consumption and match demand to intermittent renewable supply.
- Prior work studies storage scheduling, wind generation, demand response, and Stackelberg interactions between power companies and consumers.
- Existing demand-side models mainly assume fully rational consumers, although behavioral studies show deviations from classical game theory under probabilistic uncertainty.
- Earlier prospect-theoretic energy studies generally omit the hierarchical prosumer–power-company interaction and uncertainty from variable or dynamic pricing.
- The paper models energy trading as a single-leader, multiple-follower Stackelberg game with a distributed equilibrium algorithm and prospect-theoretic framing.
- The pricing model sets unit price as ρ = ρbase + α Σ_n x_n, with positive bids representing purchases and negative bids representing sales.
- Future energy prices are modeled as uncertain, with ρf assumed uniformly distributed over [ρmin, ρmax], while prosumer bids are bounded by generation, storage, and load constraints.
A. Stackelberg game formulation
The Stackelberg formulation makes the power company choose a base price first, after which prosumers play a noncooperative energy-trading game. The model is static, while repeated play motivates extensions toward dynamic stochastic games.
- A. Stackelberg game formulation: The power company acts as leader by selecting ρbase, and prosumers then respond through a noncooperative game whose final price depends on total grid load.
- A. Stackelberg game formulation: The followers’ actions are constrained to their feasible energy-trading intervals.
- A. Stackelberg game formulation: Under classical game theory, prosumers maximize expected profits given other prosumers’ actions and the company’s pricing decision.
- A. Stackelberg game formulation: A Stackelberg equilibrium requires prosumers to solve their follower problems and the company to optimize its pricing decision accordingly.
- U CGT: The formulation is static, but repeated interaction is used to interpret the solution as a stationary condition of a more general dynamic stochastic game.
III. GAME SOLUTION UNDER CGT
The classical-game solution treats prosumers as expected-utility maximizers and characterizes their best responses. A distributed relaxation algorithm uses these responses to approach the unique pure-strategy Nash equilibrium.
- The CGT analysis seeks a solution satisfying both the prosumers’ follower problems and the power company’s leader problem.
- The best response of each prosumer is defined as the solution of its constrained optimization problem given the other prosumers’ strategies.
- The best-response characterization is supported by a proof in Appendix I.
- The individual best responses are rewritten in combined matrix form using an interaction matrix with zero diagonal and negative off-diagonal entries.
- The analysis uses projection onto the feasible strategy cube to maintain bounded prosumer actions.
- The closed-form best-response representation supports the subsequent best-response dynamics.
- Stationary strategies are interpreted as optimal actions for a state regardless of game history or time, linking the repeated analysis to stochastic-game stationarity.
- The relaxation learning algorithm updates each action using the current action and the corresponding best response at each time step.
A. Existence and uniqueness of the followers’ NE under CGT
Under CGT, the prosumers’ game admits a unique pure-strategy Nash equilibrium, and a fully distributed relaxation algorithm converges to it at a polynomial rate.
- The prosumers’ game admits a unique pure-strategy Nash equilibrium under CGT.
- Algorithm 1 uses each prosumer’s current action and best response to update strategies in a distributed manner.Each prosumer needs only its own actions and best-response function, without tracking other prosumers’ actions or action history.
- The Nikaido-Isoda function links best-response maximization to Nash equilibrium characterization.For a best-response action profile, the function equals zero if and only if the current profile is a pure-strategy Nash equilibrium.
- After t steps, Algorithm 1 produces an ϵ-NE with ϵ = O(t^-1).
- The polynomial convergence rate follows from the structured utility functions of this concave game.The formulation remains computationally tractable with polynomial-time distributed algorithms regardless of the number of players.
C. Finding the Stackelberg Nash equilibrium under CGT
Under CGT, the full Stackelberg equilibrium is obtained by combining the followers’ unique Nash equilibrium with the power company’s optimization over its pricing action.
- The power company first selects its optimal pricing action using the followers’ unique equilibrium response x∗(ρbase).The equilibrium response is treated as a well-defined continuous function of ρbase.
- The followers’ equilibrium is characterized by the fixed-point condition x∗ = ΠΩ[a + Ax∗].This condition is equivalent to each prosumer choosing a best response within the feasible action set.
- Solving the primal-dual KKT system yields the unique pure-strategy Nash equilibrium of the followers’ game.The system contains 3n equations with 3n variables comprising the action profile and dual variables.
- A second method avoids solving the nonlinear inequalities and is used to construct an approximate Stackelberg equilibrium efficiently.The method evaluates discretized power-company actions and uses approximate follower equilibria for each announced price.
2) Method 2:
Method 2 constructs an ϵ-Stackelberg equilibrium by discretizing the power company’s pricing interval, repeatedly computing prosumers’ approximate equilibria, and selecting the most profitable action.
- The company repeats this process at most 1/ϵ times and chooses the action that maximizes its utility.
- The method is extended beyond CGT because prosumers may have subjective valuations of uncertain energy-trading payoffs.The extension uses prospect theory to model behavior under unknown future energy prices and stored-energy values.
A. Energy Trading Analysis through Utility Framing
Prospect-theoretic utility framing evaluates uncertain prosumer payoffs relative to individual reference points. Under specified concavity conditions, the followers’ game has a pure-strategy Nash equilibrium; otherwise, a sequential best-response method is used.
- A. Energy Trading Analysis through Utility Framing: A prosumer perceives an uncertain payoff as a gain or loss relative to an individual reference point.The reference point represents anticipated profits derived from past experience and future aspirations.
- A. Energy Trading Analysis through Utility Framing: The prospect-theoretic utility uses a framing value function with diminishing sensitivity and loss aversion.The parameters satisfy β−, β+ ∈ (0, 1] and λ ≥ 1; losses have a larger slope than gains.
- A. Energy Trading Analysis through Utility Framing: Theorem 4 guarantees at least one pure-strategy Nash equilibrium under any of three specified conditions on the utility and action bounds.
- A. Energy Trading Analysis through Utility Framing: Under Theorem 4’s conditions, the Stackelberg equilibrium can be obtained using the CGT procedure because the prosumers’ game is concave.Algorithm 1 is combined with either of the methods used for the CGT game.
- A. Energy Trading Analysis through Utility Framing: When concavity cannot be guaranteed, sequential best responses are used; convergence is guaranteed to reach an NE when it occurs.The simulations reported convergence and a pure-strategy NE for all simulated scenarios, but an analytical existence/convergence proof was challenging.
V. SIMULATION RESULTS AND ANALYSIS
Simulations show that prosumers’ reference points and loss aversion substantially affect energy consumption, grid load, and power-company profits. The PT model also supports convergence of the followers’ best-response algorithm as the number of prosumers increases.
- Loss aversion: Up to 14%: increasing the loss multiplier λ from 2 to 6 decreases total purchased energy under a fixed power-company strategy.The loss multiplier captures prosumers’ loss aversion; stronger loss aversion leads them to purchase less current energy.
- Power-company profits: 15%: the power company’s profit decrease peaks when the reference point is $2 if it assumes rational prosumers while they actually behave irrationally.The pricing strategy is no longer optimal when it fails to account for prosumers’ subjective behavior.
- Grid scale: 100 kWh: the consumption difference between rational prosumers and subjective prosumers with Rn = $1 reaches this value for 50 prosumers.The difference becomes more pronounced as the number of prosumers in the grid increases.
- Heterogeneous groups: Prosumer groups with different reference points begin reducing consumption at different ρbase values, with Rn = $1 responding before rational prosumers and Rn = $3 prosumers.The reported thresholds are −5 cents for Rn = $1, 2 cents for rational prosumers, and 5 cents for Rn = $3.
- Algorithm convergence: The best-response algorithm converges to a followers’ NE for the tested prosumer counts, with reasonable iteration counts as the population rises from 10 to 70.The simulations use N = 9 prosumers unless otherwise stated and set λ = 2.25 by default.
APPENDIX I PROOF OF THEOREM 1
The prosumers’ follower game has concave utilities over closed convex action sets, ensuring existence of a pure-strategy Nash equilibrium. Strict diagonal concavity then establishes uniqueness.
- Each prosumer’s utility is quadratic and concave in its own action, while each action set is closed and convex.
- These properties guarantee at least one pure-strategy Nash equilibrium for the prosumers’ game.
- Strict diagonal concavity is verified using a weighted gradient mapping with r_j = 1 for every prosumer.
- The matrix K = −α(I + J) is negative definite because I + J is positive definite and −α < 0.
- The negative-definiteness condition implies that the prosumers’ Nash equilibrium is unique.
[U EUT
The Nikaido-Isoda function is bounded, convex in its first argument, and strongly concave in its second argument. These properties support the equilibrium analysis.
- The Nikaido-Isoda function is bounded by K∥x − y∥ for some constant K > 0.
- Its Hessian with respect to x is positive, showing that the function is convex in x.
- The equality condition establishes that the Nikaido-Isoda function is strongly concave with respect to y.
APPENDIX IV PROOF OF THEOREM 3
The distributed iterative procedure converges to a pure-strategy Nash equilibrium of the prosumers’ game. Its approximation error decreases polynomially with the number of iterations.
- The algorithm generates action profiles x(t) that converge to a pure-strategy Nash equilibrium x* as t tends to infinity.
- The distance between an action profile and its best response decreases as the iteration count grows and reaches zero at the limit.
- The proof establishes the convergence rate through recursive inequalities involving the projection update and the Nikaido-Isoda function.
- Ψ(x(t), x^r(t)) tends to zero, which is equivalent to convergence of the iterates to a pure-strategy Nash equilibrium.
- After t iterations, the action profile is an ε-NE with ε = O(t^-1The supplied passage continues the exponent across adjacent fragments.
APPENDIX V PROOF OF THEOREM 4
The PT expected utility is analyzed under three reference-point cases. In each case, conditions are derived that ensure concavity of the prosumer’s utility.
- The analysis first identifies conditions for uniformity over each prosumer’s action space and then imposes an additional condition for strict concavity.
- Case 1: When R_n < ρ_min^c + d across all actions, the PT utility simplifies to the CGT expected utility minus R_n and is concave.
- Case 2: When R_n > ρ_max^c + d across all actions, the PT expected utility also simplifies and is strictly concave when λ is strictly positive.
- Case 3: For ρ_min^c + d < R_n < ρ_max^c + d, concavity is assessed from the second derivative of the middle PT utility expression.
- The case analysis determines parameter ranges under which the PT utility retains the concavity needed for equilibrium analysis.