Game theory and market power

Transcription

1 Game theory and market power Josh Taylor Section 6.1.3, 6.3 in Convex Optimization of Power Systems. 1 Market weaknesses Recall Optimal power flow: minimize p,θ subject to λ i : χ ij 0 : f i (p i ) i p i = b ij (θ i θ j ) j b ij (θ i θ j ) s ij p i p i p i Prices: λ i : the price at node i. Agent i solves: minimize p i f i (p i ) λ i p i When does nodal pricing / microeconomics fail? Nonconvexity - the power flow equations, unit commitment. Bounded rationality - agent i has limited time, information, computing power - can t find optimal p i. Price-taker assumption: agent i oblivious to their influence on λ i. 2 Real-world examples 2.1 Enron Scandal, late 1990 s to 2001 Overview: 1

2 Energy trading, building power plants, natural gas Posterchild for electricity/energy markets Very shady accounting practices - see Wikipedia Gov. Gray Davis ruined political career California Electricity Crisis: Making power seem to be from out of state (where does your power come from?) Blocking transmission lines (over scheduling) to raise nodal prices Overall bad planning/market design Rolling blackouts in 2000, 2001, prices increase by factor of JPMorgan, Manipulative bidding strategies... JPMorgan pays $410 million in FERC settlement, Why? Lot s of markets have problems (healthcare, computer OS, diamonds) Failures of price-taker assumption in power Should we have power markets? Probably, but cautiously... Were Enron, JPMorgan too smart? Strategy: Markets need physically rigorous design Game theory helps identify vulnerabilities mathematically. Page 2 of 9

3 3 Game theory Regular optimization: Game theory: min f(x). x X min x f(x, y), min y g(x, y) Two players, know all about each other. 3.1 Example: prisoner s dilemma Setup Two players, caught criminals Two actions: silence, or betray partner Made simultaneously (like rock paper scissors) Payoffs: Both silent: both serve 1 year Both betray: both serve 3 years 1 silent, 1 betrays: silent 4 years, betrayer 0 years. Anticipatory decisions: Both silent... either improves by betraying 1 silent, 1 betrays... silent improves by betraying Both betray... no improvement for either Nash equilibrium: Both players betray Stable under unilateral actions Worse than both silent Page 3 of 9

4 Strategic form games Players, i = 1,..., n Pure player strategies, S i. Player utility function u i (s), s S = i S i Ordinary optimization with just one player s is a Pure Nash Eq. (PNE) if u i (s) u i (t, s i ) for all t S i. PNE exists if u i (s) convex in s i, continuous in s i S i convex and compact Mixed strategy: M i set of PDFs over S i Mixed strategy m i M i Expected payoff over m: u i (m) = S u i (s)m(s)ds. MNE if u i (m) u i (t, m i ) for all t M i (S i works) Discussion PNE often don t exist. Uniqueness uncommon when it does exist. MNE describe real situations like sales. MNE almost always exist. Game theory PPAD complete - easier than NP-complete, still bad. Page 4 of 9

5 3.2 Bertrand competition Demand: d Prices: λ i All demand goes to lowest price. Equilibrium: If λ 1 = λ 2 > 0, λ 1 ɛ is profitable for λ 1. If λ 1 > λ 2, λ 1 = λ 2 ɛ is profitable for λ 1. Nash Eq: λ 1 = λ 2 = 0 (silly) 4 Load shifting with storage Time-varying, inelastic load δ(t), t = 1,..., T Generation cost f(p) = a 2 p2 + bp Market clearing price: λ = df(p) dp = ap + b N storages inject/extract s i (t) - arbitrage Net load: δ(t) N i=1 s i(t) Centralized problem, Optimal solutions: minimize s subject to ( f δ(t) s i (t) = 0 ) N s i (t) i=1 ( ) s c i(t) = γ i δ(t) δ N γ i = 1 i=1 Page 5 of 9

6 where the average demand is Net load curve: δ γ which storage allocation δ = 1 T δ(t). Remove price-taker assumption. Market price: ( ) N λ(t) = a δ(t) s i (t) + b. Storage payoffs: maximize s i subject to i=1 ( ( a δ(t) s i (t) = 0 Coupling... N-player game Quantity competition - Cournot PNE: ) ) N s i (t) + b s i (t) i=1 s g i (t) = 1 ( ) δ(t) δ N + 1 Flatter, but less so than centralized. Efficiency loss: Φ = = T (δ(t) f ) N i=1 sc i (t) T (δ(t) f ) N i=1 sg i (t) T a 2 T ( a δ bδ ) ( 1 N+1 δ(t) + N N+1 δ) 2 ( + b 1 N+1 δ(t) + N N+1 δ). Letting N, the efficiency loss vanishes, i.e. Φ 1. Price of anarchy Page 6 of 9

7 δ(t) Net load, optimal Net load, game t Figure 1: The nominal load without storage, δ, and the net load in the centralized and game outcomes with N = 3. Worst case - duopoly (only monopoly worse) Game shows variations allowed to persist to preserve arbitrage More participants flattens net load, approaches true optimum. 5 Bertrand-Edgeworth competition Firms submit prices, λ i, i = 1, 2 (n player exists) Demand d Firm capacities: c 1, c 2 Reservation utility: λ r Payoffs: u i (λ) = { λ i min{c i, d}, λ i < λ i λ i min{c i, (d c i ) + }, λ i > λ i Expected payoff: ρ i (m) = u i (m) = u i (λ)m(λ)dλ = 0 λ λ r m 1 (λ 1 )m 2 (λ 2 )u i (λ) 0 λ λ r Case 1: d c 1, c 2 : Always left out player, PNE is λ i = 0 and ρ i (c) = 0. Standard Bertrand. Case 2: d c 1 + c 2 : Both fill up regardless, PNE is λ i = λ r, ρ i (c) = λ rc i. Page 7 of 9

8 Case 3: c j < d < c 1 + c 2 for some j. An MNE exists in which player i s profit is ρ i (c) = λ rmin {c i, d} min {c max, d} (d + c max c 1 c 2 ), where c max = max i c i... holding sales. ρ (c) d Figure 2: The expected profits of each player, ρ i (c), in a capacitated price duopoly for λ r = 1, c 1 = 1/5, c 2 = 3/5, and d [0, 1]. 5.1 Application to transmission 2 nodes, same demand d Max price λ r Generation p 1, p 2 Line capacity s Profits: λ i p i p 12 s d, p 1 d, p 2 Figure 3: Market power in a congested two-node transmission network. Page 8 of 9

9 Payoff function: λ i Equivalent to BE comp. with Demand 2d capacities d + s Case-wise Analysis d + min {d, s} if λ i < λ i max {d s, 0} if λ i > λ i. d if λ i = λ i d s: Undercutting, PNE λ 1 = λ 2 = 0. s = 0: Two monopolies, PNE λ i = λ r. 0 < s < d: No PNE. MNE: µ(λ i ) = λ r (d s) 2sλ 2, λ i i Expected profits: λ r (d s). [ λr (d s) d + s, λ r ], 6 Conclusion Other popular formats: Supply function competition - hard to analyze Complementarity models - tractable but crude Takeaways Game theory limited tractability More market participants means better competition. Transmission congestion local monopolies (e.g., Enron)! Physics matter! Initial market designs ignored transmission. References Page 9 of 9