Capacity Expansion Games with Application to Competition in Power May 19, Generation 2017 Investmen 1 / 24

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1 Capacity Expansion Games with Application to Competition in Power Generation Investments joint with René Aïd and Mike Ludkovski CFMAR 10th Anniversary Conference May 19, 017 Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen 1 / 4

2 Introduction Renewable vs Coal-fired Power Plants Carbon emission price puts opposite effects upon the profit of investing in renewable and coal-fired power plants Modeling the carbon emission price: OU process dx t = µ (θ X t ) dt + σdw t. µ represents the strength of the political will to enforce a carbon price of θ Different market participants/sectors: Firm 1: nuclear power generator, prefers high X. Firm : coal-fired plant investor, prefers low X. Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen / 4

3 Introduction Duopoly Capacity Expansion Game Analyzing the competition between renewable and coal-fired plant industries? Combine real option frameworks with game theory. One-shot decision to invest irreversibly with fixed cost K. Expected revenue is related to a stochastic state process (X t ). Two firms interact through aggregate supply Q that drives market price. In our work: Asymmetric producers. Extend to a multi-stage game whose evolution is fully endogenized. Aim: Determine the optimal expansion strategies of these firms. Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen 3 / 4

4 Introduction Literature Review Capacity Expansion: Aïd et al (015): Impulse Control; Steg (01): Singular Control; Hamadène and Jeanblanc (007): Optimal Switching, etc. Game theory in real options literature: Pioneers: Smets (1993), Williams (1993) Leader/follower setting: Dixit and Pindyck (1994), Huisman and Kort (015) Review paper: Azevedo and Paxson (014) Related works: Grasselli, Leclere and Ludkovski (013): symmetric players and coordination games. Leung and Li (015): Optimal mean reversion trading. de Angelis, Ferrari and Moriarty (015): Nash equilibria for nonzero-sum Dynkin games. Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen 4 / 4

5 Problem Formulation X t and Game Stages X t denotes the spread between the firms revenue: dx t = b(x t )dt + σ(x t )dw t, x D. The game stage at time t is N t {(Nt 1, Nt ) t 0 : Nt i = 0, 1, }. N i t: number of expansion options remaining for firm i = 1,. Each coordinate of N t is piecewise constant, non-increasing and fully endogenized e.g. Nt : (, ) (1, ) (1, 1) (0, 1) (0, 0). Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen 5 / 4

6 Problem Formulation Net Present Values Given game policies of the firms, the total net present value of future profits (NPV) of firm i is { Jn i 1,n (x; α 1, α + n ) := E e rs π i N 1 (Xs)ds 1 K i s 0,N s }{{} j=1 Future Cashflows e rii j } {{ } Investment Costs Ij i : the j-th investment time. : the profit rate in terms of game stage, e.g. π i N 1 t,n t π 1 n 1,n (X t ) = (P n1,n C 1 +ρ 1 X t )Q 1 n 1,n π n 1,n (X t ) = (P n1,n C ρ X t )Q n 1,n. X 0 = x, N } 0 = (n 1, n ). Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen 6 / 4

7 Problem Formulation Game Nash Equilibria Decisions of one firm affect the other through the joint dependence on N t Capacity expansion becomes a nonzero-sum stochastic game. Use the standard concept of Nash equilibrium. Definition (Nash Equilibrium) Let J i (x, ) denote the NPV received by firm i with X 0 = x. A set of actions α = (α 1,, α, ) A is said to be a Nash equilibrium of the game, if for i {1, }, β i A i : J i (x, α i, β i ) J i (x, α ) := V i (x). Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen 7 / 4

8 Problem Formulation Feedback Game Policies Assume actions of firms to be of Feedback Perfect State or Markovian type A i = {α ( i := α i X t, N )} t Time-homogeneous strategies: dynamic programming Definition (Game Policies) The set of actions of firm 1 (resp. ) consists of finite F-adapted stopping times: A 1 = { α 1 := ( τ 1 n 1,n ) n 1 > 0, n }, A = { α := ( τ n 1,n ) n > 0, n 1 }. Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen 8 / 4

9 Problem Formulation Characterizing Equilibrium Policies Dynamic programming method: looking at τ i n 1,n recursively X t x 1 S, S, 1 S 1, S 1, 1 S 1,1 S 1,1 S 0, t Figure: Sample Trajectory of X t and N t Fixed point characterization: τ i, n 1,n is the best-response to τ i, n 1,n Induction on discrete stages (n 1, n ): Patch local equilibria to construct a global one. Boundary stages n = 0 (n 1 = 0): Reduces to single-agent optimization problem. Capacity Expansion Games with Application to Competition in Power May 19, Generation 017 Investmen 9 / 4

10 Problem Formulation Solving for the best-response Fixing τ n 1,n, firm 1 solves Ṽ 1 n 1,n (x, τ n 1,n ) = sup τ T [ τ τn 1,n E x e rs πn 1 1,n (X s ) ds 0 } {{ } before an expansion + e rτ ( 1 {τ<τ n1,n } V 1 n1 1,n (X τ ) K 1) ) + e rτ n 1,n 1 {τ>τ n1,n }{{} }Vn 1 1,n 1 (X τ n1,n }{{} firm 1 invests first: first-mover firm invests first: second-mover Assume strategy of firm are threshold-type stopping times: τ n 1,n = inf{t 0 : X t s }, where s corresponds to the investment threshold. ]. Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

11 Equilibrium Strategy and Game Value Methodology: Smallest Concave Majorant Abstract optimal stopping problem: { V (x) = sup E x 1{τ<τR }e rτ h(x τ ) + 1 {τ>τr }e rτ R l(x τr ) }. τ T τ R is the exit time of the interval R = (s, + ). h( ): first-mover payoff; l( ): second-mover payoff. The smallest concave majorant method for the absorbed process on R: Dayanik and Karatzas (003), De Angelis et al. (015). Returns the value function and structure of the optimal stopping region directly. Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

12 Equilibrium Strategy and Game Value Smallest Concave Majorant Illustration 1,s H 1,1 1,s WH 1,1 ψ(b 1 ψ(s ) 1,1 ) 0 ~* y = ψ(s 1 ) y Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

13 Equilibrium Strategy and Game Value Threshold-type best-response If h 1 (s ) < l 1 (s ): firm 1 benefits from having firm invest. The best-response of firm 1: Threshold-type stopping time Similar arguments for firm. τ 1 n 1,n (s ) = inf{t 0 : X t S 1 n 1,n (s )} Equilibrium policies correspond to crossing points of the best-response curves. Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

14 Equilibrium Strategy and Game Value Equilibrium Policies s 1 5 S,P,* 4 3 S 1,P,* S 1 (s ) S (s 1) Equilibrium s s 1 5 S,P,* 4 3 S 1,P,* S 1 (s ) S (s 1) Threshold type 1 0 Preemptive s s S 1,P,* S,P,* S 1 (s ) S (s 1) Preemptive Equilibrium 4 0 s Crossing points: threshold-type equilibrium strategies. Equilibrium selection: logic of Stackelberg game. Existence is not guaranteed in some cases. Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

15 Equilibrium Strategy and Game Value Preemptive best-response & equilibrium If h 1 (s ) > l 1 (s ): firm 1 is incentivized to preempt when L 1 < x s. stopping at s is too late: Right before No corresponding optimal stopping time. Exist a unique preemptive equilibrium: τ 1,e, = inf{t 0 : L 1 < X t L or X t S 1,e, } Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

16 Equilibrium Strategy and Game Value Equilibrium Policies s 1 5 S,P,* 4 3 S 1,P,* S 1 (s ) S (s 1) Equilibrium s s 1 5 S,P,* 4 3 S 1,P,* S 1 (s ) S (s 1) Threshold type 1 0 Preemptive s s S 1,P,* S,P,* S 1 (s ) S (s 1) Preemptive Equilibrium 4 0 s marks a unique preemptive equilibrium. No threshold-type = a unique preemptive equilibrium. No preemptive = existence of threshold-type equilibria. Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

17 Results Loss due to competition Apples-to-apples quantification for: the value of being a guaranteed first-mover; a cooperative solution: central planner case. Firms act preemptively and over-invest Expansion is accelerated by competition. Loss in terms of aggregate profit compared to cooperative action: V 1 + V vs. maximizing i πi Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

18 Results Effect of Market Fluctuation Parameterized by the volatility σ Under competition, facing a more volatile market: firms become more aggressive and expand capacity sooner their game values decline, highlighting the effects of competition threshold-type equilibria disappear for σ large A guaranteed first-mover will wait longer and reap higher reward Consistent with classical real options where volatility is good Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

19 Results Expansion Flexibility Interpret flexibility as splitting a large-scale investment e.g.: sequentially build small plants rather than one large one Both firms have two options (expansion size=0.5) Compare to both firms have only one option ( Q i = 0.5). Both firms improve their game values First expansion is sooner; second expansion is later The game lasts longer Firm 1: two options, Q 1 = 0.5; Firm : one option, Q = 0.5. Firm 1 obtains higher game value as expected. Firm receives higher game value as well! Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

20 Results Application to Power Generation Carbon emission price: dx t = µ (θ X t ) dt + σdw t. µ: political will to enforce a carbon price of θ. Firm 1: nuclear power generator, prefers high X Firm : coal-fired plant investor, prefers low X Figure: Price of the one year-ahead emission allowance of the EU-TS Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

21 Results Parameter Value Unit Private discount rate r 10% Investment cost K USD/kWh Investment cost K USD/kWh P, 4 USD/MWh P 1,0 10 USD/MWh P 0,1 10 USD/MWh P 0,0 8 USD/MWh Weight ρ 0.5 Long-run carbon price θ 30 USD/MWh Political will µ [0.1, 0.5] Initial carbon price 5 USD/tCO Table: Parameter values Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

22 Results Effect of Political Will µ % % Probability Prob 0, Prob 1,1 Prob,0 15% % µ * µ µ Low µ one small coal-fired plant is built instantly. Strong political will µ guides the market to exclusively green power plants. Aggregate profit loss due to competition can be as high as 40%. Capacity Expansion Games with Application to Competition inmay Power 19, Generation 017 Investmen / 4

23 Results Resulting Equilibrium Types Impact of µ is non-monotone. One industry is clearly ahead in the short-term Equal-strength competition raises benefits of preemption σ µ Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4

24 References References R Aïd, L Li, and M Ludkovski. Capacity expansion games with application to competition in power generation investments. SSRN, preprint, 016. Alcino Azevedo and Dean Paxson. Developing real option game models. European Journal of Operational Research, 37(3):909 90, 014. Tiziano De Angelis, Giorgio Ferrari, and John Moriarty. Nash equilibria of threshold type for two-player nonzero-sum games of stopping. arxiv preprint arxiv: , 015. MR Grasselli, V Leclere, and M Ludkovski. Priority option: the value of being a leader. International Journal of Theoretical and Applied Finance, 16(01): , 013. Thank You! Capacity Expansion Games with Application to Competition inmay Power 19, Generation Investmen / 4