On risk averse competitive equilibrium
OOn risk averse competitive equilibrium
Henri Gerard a,c , Vincent Leclere a , Andy Philpott b a Universit´e Paris-Est, CERMICS (ENPC), F-77455 Marne-la-Vall´ee, France b Electric Power Optimization Centre, The University of Auckland, New Zealand c Universit´e Paris-Est, Labex B´ezout, F-77455 Marne-la-Vall´ee, France
Abstract
We discuss risked competitive partial equilibrium in a setting in which agents are endowed with coherent risk measures. Incontrast to social planning models, we show by example that risked equilibria are not unique, even when agents’ objective functionsare strictly concave. We also show that standard computational methods find only a subset of the equilibria, even with multiplestarting points.
Keywords:
Stochastic Equilibrium, Stochastic Programming, Risk averse equilibrium, Electricity Markets
1. Introduction
Most industrialised regions of the world have over the lastthirty years established wholesale electricity markets that takethe form of an auction that matches supply and demand. The ex-act form of these auction mechanisms vary by jurisdiction, butthey typically require o ff ers of energy from suppliers at coststhey are willing to supply, and clear a market by dispatchingthese o ff ers in order of increasing cost. Day-ahead marketssuch as those implemented in many North American electric-ity systems, seek to arrange supply well in advance of its de-mand, so that thermal units can be prepared in time. Since thedemand cannot be predicted with absolute certainty, day-aheadmarkets must be accompanied by a separate balancing marketto deal with the variation in load and generator availability inreal time. These are often called two-settlement markets. Themarket mechanisms are designed to be as e ffi cient as possiblein the sense that they should aim to maximize the total welfareof producers and consumers.In response to pressure to reduce CO emissions and in-crease the penetration of renewables, electricity pool marketsare procuring increasing amounts of electricity from intermit-tent sources such as wind and solar. If probability distribu-tions for intermittent supply are known for these systems thenit makes sense to maximize the expected total welfare of pro-ducers and consumers in each dispatch. Then many repetitionsof this will yield a long run total benefit that is maximized.Maximizing expected welfare can be modeled as a two-stagestochastic program. Methods for computing prices and single-settlement payment mechanisms for such a stochastic marketclearing mechanism are described in a number of papers (seePritchard et al. [11], Wong and Fuller [16] and Zakeri et al.[17]). When evaluated using the assumed probability distribu-tion on supply, stochastic market clearing can be shown to bemore e ffi cient than two-settlement systems.If agents in these systems are risk averse then one might alsoseek to maximize some risk-adjusted social welfare. In this set- ting the computation of prices and payments to the agents be-comes more complicated. If agents use coherent risk measuresthen it is possible to define a complete market for risk in a pre-cise sense. If the market is complete then a perfectly competi-tive partial equilibrium will also maximize risk-adjusted socialwelfare, i.e. it is e ffi cient. On the other hand if the market forrisk is not complete, then perfectly competitive partial equilib-rium can be ine ffi cient. This has been explored in a number ofpapers (see e.g. de Maere d’Aertrycke et al. [4], Ehrenmannand Smeers [5] and Ralph and Smeers [12]).In this paper we study a class of stochastic dispatch and pric-ing mechanisms under the assumption that agents will attemptto maximize their risk-adjusted welfare at these prices. Agentshave coherent risk measures and are assumed to behave as pricetakers in the energy and risk markets. We aim at enlighteningsome di ffi culties that arise when risk markets are not complete.We describe a simple instance of a stochastic market that hasthree di ff erent equilibria. Two of these points are stable in thesense of Samuelson [13] and are attractors of tatˆonnement al-gorithms. The third equilibrium is unstable, yet is the solutionyielded by the well-known PATH solver in GAMS (See Ferrisand Munson [8]). Our example illustrates the delicacy of seek-ing numerical solutions for equilibria in incomplete markets.Since these are used for justifying decisions, the nonuniquenessof solutions in this setting is undesirable.The paper is laid out as follow. In Section 2 we present theequilibrium and optimization models we are going to study. InSection 3 we give links between equilibrium and optimizationproblems in the risk neutral and complete risk-averse cases. Fi-nally, in Section 4 we showcase a simple example with multipleequilibria in the incomplete risk-averse case. We use the following notation throughout the paper: [[ a ; b ]] isthe set of integers between a and b (included), random variablesare denoted in bold, Ω is a finite sample space, P is a probabil-ity distribution over Ω , E P is used to refer to expectation with Preprint submitted to Operations Research Letters June 27, 2017 a r X i v : . [ m a t h . O C ] J un espect to P , F is used to refer to a risk measure. We denote by x ⊥ y the complementarity condition x T y =
2. Statement of problem
Consider a two time-step single-settlement market for onegood. In a single-settlement market, the producer can arrangein advance for a production of x at a marginal cost cx as a first-step decision, and choose the value of a recourse variable x r incurring an uncertain marginal cost c r x r . We assume that thereare a finite number of scenarios ω ∈ Ω determining the coe ffi -cient c r ( ω ).The product is purchased in the second step by a consumerwith a utility function V ( ω ) y ( ω ) − r ( ω ) y ( ω ). The consumerhas no first-stage decision, and the amount purchased y ( ω ) de-pends on the scenario. Decisions x , x r ( ω ) and y ( ω ) can be made to maximize a so-cial objective. We denote by W p ( ω ) = − cx − c r ( ω ) x r ( ω ) , (1a)the welfare of the producer, and by W c ( ω ) = V ( ω ) y ( ω ) − r ( ω ) y ( ω ) , (1b)the welfare of the consumer where both these expressions ig-nore the price paid for the good in scenario ω . Then the welfareof the social planner can be defined by W sp = W p + W c .Optimization of the social objective requires us to aggregatethe uncertain outcomes from the scenarios. This can be doneby taking expectations with respect to an underlying probabilitymeasure P or using a more general risk measure. Endow the set of scenario Ω with a probability P , then a risk-neutral social planner might seek to maximize the expected totalsocial welfare under the constraint that supply equals demand.This problem is denoted by RnSp( P ) and readsRnSp( P ) :max x , x r , y E P [ W sp ] , (2a)s.t. x + x r ( ω ) ≥ y ( ω ) , ∀ ω ∈ Ω . (2b) Choosing expectation E P , assumes a risk-neutral point ofview, where two random losses with same expectation but dif-ferent variances are deemed equivalent. In practice a number ofagents are risk averse. To model risk aversion we generally usea risk measure F , that is a functional that associates to a ran-dom welfare its deterministic equivalent, i.e. the deterministicwelfare deemed as equivalent to the random loss. A risk-averse planner solves a maximization problemRaSp( F ) defined byRaSp( F ) :max x , x r , y F [ W sp ] , (3a)s.t. x + x r ( ω ) ≥ y ( ω ) , ∀ ω ∈ Ω . (3b)A risk measure F is said to be coherent (see Artzner et al. [2])if it satisfies four natural properties: monotonicity ( if X ≥ Y then F [ X ] ≥ F [ Y ]), concavity ( F is concave), translation-equivariance ( F [ X + c ] = F [ X ] + c with c ∈ R ) and positivehomogeneity ( F [ λ X ] = λ F [ X ], with λ ≥ F (cid:2) Z (cid:3) = min Q ∈ Q E Q (cid:2) Z (cid:3) ,where Q is a closed, convex, non-empty set of probability dis-tributions over Ω . If Q is a polyhedron defined by K extremepoints ( Q k ) k ∈ [[1; K ]] , then the risk measure is denoted ˇ F and saidto be polyhedral , with ˇ F [ Z ] = min Q ,..., Q K E Q k (cid:2) Z (cid:3) .Problem RaSp( ˇ F ) can be written as followsRaSp( ˇ F ) :max θ, x , x r , y θ (4a)s.t. θ ≤ E Q k (cid:2) W sp (cid:3) , ∀ k ∈ [[1; K ]] , (4b) x + x r ( ω ) ≥ y ( ω ) , ∀ ω ∈ Ω . (4c)In what follows we assume that all risk measures are coherent. By linearity of expectation we have E P [ W sp ] = E P [ W p ] + E P [ W c ] hence the criterion of the social planner is natural,which is not the case anymore with risk-aversion. The socialplanner criterion could be either F [ W sp ] or F [ W p ] + F [ W c ].Furthermore, by concavity and positive homogeneity, we have F [ W p + W c ] ≥ F [ W p ] + F [ W c ]. We now define a competitive partial equilibrium for ourmodel. This competitive equilibrium can be risk neutral or riskaverse. Definitions come from general equilibrium theory (SeeArrow and Debreu [1] or Uzawa [15]).
Given a probability P on Ω , a risk-neutral equilibrium RnEq( P ) is a set of prices (cid:8) π ( ω ) , ω ∈ Ω (cid:9) such that there existsa solution to the systemRnEq( P ) :max x , x r E P (cid:104) W p + π (cid:0) x + x r (cid:1)(cid:105) , (5a)max y E P (cid:2) W c − π y (cid:3) , (5b)0 ≤ x + x r ( ω ) − y ( ω ) ⊥ π ( ω ) ≥ , ∀ ω ∈ Ω . (5c)Here, the producer maximizes its expected profit (5a), theconsumer maximizes its expected utility (5b) and the market2lears with (5c) (which means that either prices are null or sup-ply equals demand). As the consumer has no first stage de-cision, she can optimize each scenario independently and soproblem (5b) can be replaced bymax y ( ω ) W c ( ω ) − π ( ω ) y ( ω ) , ∀ ω ∈ Ω . Given two risk measures F p and F c over Ω , a risk-averseequilibrium RaEq( F p , F c ) is a set of prices (cid:8) π ( ω ) : ω ∈ Ω (cid:9) suchthat there exists a solution to the following systemRaEq( F p , F c ) :max x , x r F p (cid:104) W p + π (cid:0) x + x r (cid:1)(cid:105) , (6a)max y F c (cid:2) W c − π y (cid:3) , (6b)0 ≤ x + x r ( ω ) − y ( ω ) ⊥ π ( ω ) ≥ , ∀ ω ∈ Ω . (6c)Since the coherent risk measure F c of the consumer is mono-tonic, and noting that she has no first-stage decision, she canoptimize scenario per scenario. Thus, she is insensitive to risk as any monotonic risk measure will lead to the same action (al-though not the same welfare). Since F p is also monotonic, wecan endow both agents with the same risk measure. In that case,we denote problem (6) by RaEq( F ).We now consider polyhedral risk measure ˇ F , using formula-tion (4), the equilibrium problem (6) readsRaEq( ˇ F ) :max θ, x , x r θ (7a)s.t. θ ≤ E Q k (cid:2) W p + π ( x + x r ) (cid:3) , ∀ k ∈ [[1; K ]] , max y ( ω ) W c ( ω ) − π y ( ω ) , ∀ ω ∈ Ω , (7b)0 ≤ x + x r ( ω ) − y ( ω ) ⊥ π ( ω ) ≥ , ∀ ω ∈ Ω . (7c) Until now, we have considered equilibrium problems in anincomplete market. Following the path of Philpott et al. [10],we complete the market using Arrow-Debreu securities.
Definition 1. An Arrow-Debreu security for node ω ∈ Ω is acontract that charges a price µ ( ω ) in the first stage, to receive apayment of 1 in scenario ω .The consumer now has a first-stage decision which is thenumber of contracts she buys, so the choice of the consumerrisk measure F c has now consequences. For convenience, thisrisk measure F c is chosen to be the same as that of the producer F p and will be denoted by F . Unless stated otherwise, from nowon we use polyhedral risk measures.Denote a ( ω ) (resp. b ( ω )) the number of Arrow-Debreu secu-rities bought by the producer (resp. the consumer). We denoteby µ ( ω ) the price of the Arrow-Debreu securities associatedwith scenario ω . In this case the producer pays (cid:80) ω ∈ Ω µ ( ω ) a ( ω )in the first stage, in order to receive a ( ω ) in scenario ω . As a ( ω ) + b ( ω ) represents excess demand, requiring that supply is greater than demand consists in requiring a ( ω ) + b ( ω ) ≤ { π ( ω ) , µ ( ω ) } ω ∈ Ω form a risk-trading equilibrium if thereexists a solution to:RaEq-AD( ˇ F ) :max θ, x , x r , a θ − (cid:88) ω ∈ Ω µ ( ω ) a ( ω ) (8a)s.t. θ ≤ E Q k (cid:104) W p + π ( x + x r ) + a (cid:105) , ∀ k ∈ [[1; K ]] , (8b)max φ, y , b φ − (cid:88) ω ∈ Ω µ ( ω ) b ( ω ) (8c)s.t. φ ≤ E Q k (cid:2) W c − π y + b (cid:3) , ∀ k ∈ [[1; K ]] , (8d)0 ≤ x + x r ( ω ) − y ( ω ) ⊥ π ( ω ) ≥ , ∀ ω ∈ Ω , (8e)0 ≤ − a ( ω ) − b ( ω ) ⊥ µ ( ω ) ≥ , ∀ ω ∈ Ω . (8f)
3. Some equivalences between social planner problems andequilibrium problems
We recall a trivial equivalence between problem RnSp( P )and problem RnEq( P ) before showing an equivalence betweenproblem RaSp( ˇ F ) and problem RaEq-AD( ˇ F ). Proposition 1.
Let P be a probability measure over Ω . The ele-ments x (cid:93) , x (cid:93) r and y (cid:93) are optimal solutions to RnSp ( P ) if and onlyif there exist equilibrium prices π (cid:93) for RnEq ( P ) with associatedoptimal decisions x (cid:93) , x (cid:93) r and y (cid:93) .Proof. As the producer and the consumer optimize over di ff er-ent uncoupled variables, it is equivalent to optimize their ob-jectives separately or jointly. Problem (5) is thus equivalent tomax x , x r , y E P (cid:2) W p + π ( x + x r ) (cid:3) + E P (cid:2) W c − π y (cid:3) , ≤ x + x r ( ω ) − y ( ω ) ⊥ π ( ω ) ≥ , ∀ ω ∈ Ω , which by linearity of the expectation is equivalent tomax x , x r , y E P (cid:2) W sp + π ( x + x r − y ) (cid:3) , ≤ x + x r ( ω ) − y ( ω ) ⊥ π ( ω ) ≥ , ∀ ω ∈ Ω . This is equivalent to the optimality conditions for problem (2a).Convexity and linearity of constraints ends the proof. (cid:3)
Corollary 2.
If both the producer’s and the consumer’s crite-rion are strictly concave and if P charges all ω , then RnSp ( P ) admits a unique solution and RnEq ( P ) admits a unique equilib-rium.Proof. The probability distribution P charges all ω . Then bystrict concavity, RnSp( P ) has a unique solution. If RnEq( P ) hastwo di ff erent solutions ( x , x r , y ) and ( x , x r , y ) with π and π respectively then, by Proposition 1, x = x , x r = x r , and y = y . Since (5b) implies π ( ω ) = V ( ω ) − r ( ω ) y ( ω ), we have π = π which gives the result. (cid:3) .2. Equivalence in the risk-averse case The following proposition is an extension of Theorem 7of Ralph and Smeers [12], to a model with producers and con-sumers, in the special case of a finite number of scenarios withpolyhedral risk measures.
Proposition 3.
Let π and µ be equilibrium prices such that (cid:0) x (cid:93) , x (cid:93) r , y (cid:93) , a , b , θ, ϕ (cid:1) solves RaEq-AD ( ˇ F ) . Then(i) µ is a probability measure, and x (cid:93) , x (cid:93) r , y (cid:93) solves the risk-neutral social planning problem when evaluated using proba-bility µ , RnSp ( µ ) .(ii) x (cid:93) , x (cid:93) r , y (cid:93) solves the risk-averse social planning problem,RaEq-AD ( ˇ F ) with worst case measure µ .Proof. (i) Each agent problem is convex with linear con-straints. Hence the optimal solution satisfies for each problemthe Karush-Kuhn-Tucker (KKT) conditions. The Lagrangian ofthe producer problem reads L p = θ − (cid:88) ω ∈ Ω µ ( ω ) a ( ω ) + (cid:88) k λ k (cid:16) E Q k (cid:2) W p + π ( x + x r ) + a (cid:3) − θ (cid:17) , where λ k is the multiplier associated to constraint (8b). Then,the KKT conditions imply that (cid:80) k λ k =
1, and µ = (cid:80) k λ k Q k . Inparticular, µ is a probability measure in Q . Furthermore ( x (cid:93) , x (cid:93) r )maximizes (cid:80) ω ∈ Ω µ ( ω ) (cid:0) W p ( ω ) − π ( ω )( x + x r ( ω )) (cid:1) which is therisk-neutral producer objective evaluated with measure µ .Similarly, looking at the consumer problem with multiplier σ k associated to constraint (8d), we obtain (cid:80) k σ k = µ = (cid:80) k σ k Q k . Hence, the consumer maximizes her risk-neutralobjective under the same probability µ as the producer.Since by hypothesis the solutions satisfy (8e) we have that (cid:0) x (cid:93) , x (cid:93) r ( ω ) , y (cid:93) ( ω ) (cid:1) solves RnSp( µ ).(ii) Observe that complementary slackness gives λ k (cid:16) E Q k (cid:2) W (cid:93) p + π (cid:0) x (cid:93) + x (cid:93) r (cid:1) + ¯ a ) (cid:3) − ¯ θ (cid:17) = ,σ k (cid:16) E Q k (cid:2) W (cid:93) c − π y (cid:93) + ¯ b (cid:3) − ¯ ϕ (cid:17) = , where W (cid:93) p and W (cid:93) c are defined by (1) in terms of x (cid:93) , x (cid:93) r and y (cid:93) .Summing over k , and leveraging (8f) gives¯ θ + ¯ ϕ = E µ [ W (cid:93) p + π (cid:0) x (cid:93) + x (cid:93) r (cid:1) + ¯ a ] + E µ [ W (cid:93) c − π ¯ y + ¯ b ] , = E µ [ W (cid:93) p + W (cid:93) c ] . (11)However as¯ θ + ¯ ϕ = min Q ∈ Q E Q [ W (cid:93) p + π (cid:0) x (cid:93) + x (cid:93) r (cid:1) + ¯ a ] + min Q (cid:48) ∈ Q E Q (cid:48) [ W (cid:93) c − π y (cid:93) + ¯ b ] , ≤ min Q ∈ Q E Q [ W (cid:93) p + W (cid:93) c + ¯ a + ¯ b ] , ≤ min Q ∈ Q E Q [ W (cid:93) p + W (cid:93) c ] . (12)Combining (11) and (12) and observing that µ ∈ Q , we have E µ [ W (cid:93) p + W (cid:93) c ] = min Q ∈ Q E Q [ W (cid:93) p + W (cid:93) c ] . (13) To complete the proof, consider any feasible x , x r ( ω ) , y ( ω ). Bypart (i) and µ ∈ Q , we have E µ [ W (cid:93) p + W (cid:93) c ] ≥ E µ [ W p + W c ] ≥ min Q ∈ Q E Q [ W p + W c ] , where W p and W c are defined by (1). Thus (13) givesmin Q ∈ Q E Q [ W (cid:93) p + W (cid:93) c ] ≥ min Q ∈ Q E Q [ W p + W c ] . This shows that (cid:0) x (cid:93) , x (cid:93) r , y (cid:93) (cid:1) ∈ arg max x , x r , y min Q ∈ Q E Q [ W p + W c ] , as required. (cid:3) Remark 1.
Note that an equilibrium of RaEq-AD ( ˇ F ) consistsof a price vector π , giving one price per scenario, and a proba-bility µ that is seen by both the producer and the consumer as aworst-case probability for the welfare plus trade evaluation. ♦ Remark 2.
In Section 4 we give an example of three riskedequilibrium without Arrow-Debreu securities, each corre-sponding to a risk-neutral equilibrium with di ff erent measure µ ( ω ) . However if Arrow-Debreu securities are included thentwo of these equilibria are no longer equilibria in a risk-aversesetting. The risk-averse consumer, who without Arrow-Debreusecurities had no mechanism to alter his outcomes will tradethese securities to improve their risk-adjusted payo ff . ♦ Remark 3.
Consider a set of prices π that gives a risked equi-librium in which agent i has payo ff W i ( π ) and risked payo ff F i (cid:2) W i ( π ) (cid:3) . Suppose that there exists a probability measure Q ∗ such that F i (cid:2) W i ( π ) (cid:3) = E Q ∗ (cid:104) W i ( π ) (cid:105) . Observe that this does notimply that choosing actions x to maximize E Q ∗ [ W i ( π )] will give max x F i (cid:2) W i ( π ) (cid:3) . This is because x ∗ solves max x F i (cid:2) W i ( π ) (cid:3) = max x min Q ∈ Q E Q (cid:2) W i ( π ) (cid:3) , and not max x E Q ∗ (cid:2) f i ( x , π ) (cid:3) , since Q ∗ depends on x. ♦ Remark 4.
Proposition 3 is easily extended to the case wherethe agents have di ff erent risk measures F p and F c with non-disjoint risk set. In this case, (12) becomes ¯ θ + ¯ ϕ = min Q p ∈ Q p E Q p [ π (cid:0) x (cid:93) + x (cid:93) r (cid:1) + W (cid:93) p + ¯ a ] + min Q c ∈ Q c E Q c [ W (cid:93) c − π y (cid:93) + ¯ b ] , ≤ min Q ∈ Q p ∩ Q c E Q [ W (cid:93) c + W (cid:93) p ] , (14) and the social planner uses a risk measure with Q = Q p ∩ Q c. ♦ The following proposition (Theorem 11 Philpott et al. [10])stands as a reverse statement for Proposition 3.4 roposition 4.
Let the elements x (cid:93) , x (cid:93) r and y (cid:93) r be optimal solu-tions to RaSp ( ˇ F ) , with associated worst case probability mea-sure µ . Then there exists prices π such that the couple ( π , µ ) forms a risk trading equilibrium for RaEq-AD ( ˇ F ) with associ-ated optimal solutions ( x (cid:93) , x (cid:93) r , y (cid:93) ) . Combining Proposition 3 and Proposition 4, we are able tostate the following result of uniqueness of equilibrium.
Corollary 5.
If both the producer’s and consumer’s crite-rion are strictly concave, and if each of the extreme points Q k charges all ω , then RaSp ( ˇ F ) admits a unique solution ( x (cid:93) , x (cid:93) r , y (cid:93) ) . Furthermore RaEq-AD ( ˇ F ) admits unique optimaldecisions ( x (cid:93) , x (cid:93) r , y (cid:93) ) . If, in addition, solving RaSp ( ˇ F ) admita unique worst case probability measure µ , then equilibriumprices ( π , µ ) are unique.Proof. As each of the extreme points Q k charges all ω , the riskaverse social planner problem is strictly convex with linear con-straints. Thus there exists a unique solution ( x (cid:93) , x (cid:93) r , y (cid:93) ) attainedfor a worst case probability µ . Applying Proposition 4, weknow that there exists π such that ( π , µ ) forms a risk tradingequilibrium. Suppose now that there exists two risk-tradingequilibria ( π , µ , x , x r , y ) and ( π , µ , x , x r , y ). Then, byProposition 3, they both solve RaSp( ˇ F ) which admits a uniquesolution. Consequently, we have x (cid:93) = x = x , x (cid:93) r = x r = x r and y (cid:93) = y = y .If in addition, µ = µ , then by Corollary 2, we deduce that π = π which ends the proof. (cid:3) We have shown a first equivalence between RnSp( P ) andRnEq( P ) and a second one between RaSp( ˇ F ) and RaEq-AD( ˇ F ).These equivalences lead to uniqueness of equilibrium if there isuniqueness of the solution of the social planner. A natural ques-tion arises: if RaSp( ˇ F ) has a unique solution, is there a uniqueequilibrium for RaEq( ˇ F )? The next section provides a simplecounterexample.
4. Multiple risk averse equilibrium
In this section, we present a toy problem where RaSp( ˇ F ) hasa unique optimum but there are three di ff erent equilibria forRaEq( ˇ F ). They are first found numerically using classical meth-ods (PATH solver and a tˆatonnement algorithm), then derivedanalytically. An interesting point is that the equilibrium foundby PATH is unstable.Let Ω = { , } and Q = conv (cid:8) ( , ) , ( , ) (cid:9) . For simplicityof notation index by i ∈ { , } the realization of each randomvariable. We choose the following parameters: V = V = , c = , c = c = , r = r = First we look for equilibrium using GAMS with the solverPATH in the EMP framework (SeeBrook et al. [3], Ferris et al.[7] and Ferris and Munson [8]). We have run GAMS from dif-ferent starting points defined by a grid 100 ×
100 over the square [1 . . × [2 .
05; 2 . π = ( π , π ) = (1 . . , leading to risked adjusted welfare (2 . . We now compute the equilibrium using a tˆatonnement algo-rithm (See Uzawa [15]).
Data:
MAX-ITER, ( π , π ) , τ for k from to MAX-ITER do Compute an optimal decision for each player givena price : x , x , x ∈ arg max F (cid:2) W p + π ( x + x r ) (cid:3) ; y , y ∈ arg max F [ W c − π y ]; Update the price : π = π − τ max (cid:8) y − ( x + x ) (cid:9) ; π = π − τ max (cid:8) y − ( x + x ) (cid:9) ; end return ( π , π ) Algorithm 1:
Walras tˆatonnementRunning algorithm 1 starting from (1 .
25; 2 . .
22; 2 . .
1, wefind two new equilibria: π = (1 . . π = (1 . . , leading to risked-adjusted welfare for producer and consumerrespectively (2 . . . . We now compute the three equilibrium analytically. Detailsof the computation are in Appendix A.Consider two probabilities ( p , − p ) and ( ¯ p , − ¯ p ) Givenprices 0 < π < π , we solve the producer (resp. consumer)optimization problem. Optimal decisions are derived in Ap-pendix A.1.4 and summed up in Table 1 where x c is given by x c ( π ) = π − π ) π c − π c . We see that there are three regimes, depending only on theprices ( π , π ), of optimal first stage solutions. Case a) (resp.case c)), corresponds to a set of prices such that E ¯ p [ W p ] < E p [ W p ] (resp. E ¯ p [ W p ] > E p [ W p ]), and the optimal decisioncorresponds to an optimal risk-neutral decision with respect toone of the two extreme points of Q . On the other hand, case b)corresponds to a set of prices such that the expected welfare isequivalent for all probability in Q , i.e. E ¯ p [ W p ] = E p [ W p ]. InFigure 1, the red area corresponds to case a), the blue to case5ondition x (cid:93) x (cid:93) i y (cid:93) i case a) x c ≤ E ¯ p (cid:2) π (cid:3) c E ¯ p (cid:2) π (cid:3) c π i c i V i − π i r i case b) E ¯ p (cid:2) π (cid:3) c ≤ x c ≤ E p (cid:2) π (cid:3) c x c π i c i V i − π i r i case c) E p (cid:2) π (cid:3) c ≤ x c E p (cid:2) π (cid:3) c π i c i V i − π i r i Table 1: Optimal control for producer and consumer problems b) and the red to case c), separated by black lines of equations E ¯ p [ π ] c = x c ( π ) and E p [ π ] c = x c ( π ) respectively.We are now looking for prices ( π , π ) such that the comple-mentarity constraints are satisfied. For strictly positive prices,these constraints can be summed up as z i ( π ) = x (cid:93) ( π ) + x (cid:93) i ( π ) − y (cid:93) i ( π ) = , i ∈ { , } . Accordingly we define excess supply functions z li for case l ∈ { a , b , c } , and i ∈ { , } . The red, blue and green lines corre-sponds to manifolds of null excess supply function for scenario i , that is of prices such that z li ( π , π ) =
0. When the linescross we have z l = z l =
0, and thus we have candidate equilib-rium. If the lines cross in the area of the same color we have anequilibrium. This is the case with the parameters chosen, andequilibrium can be derived in exact arithmetic.We end with a few remarks derived from this example.
Remark 5.
The PATH solver finds the blue equilibrium, Algo-rithm 1 finds the green and the red equilibrium as illustrated byFigure 2. Interestingly it can be shown that the blue equilib-rium is unstable in the sense that the dynamical system drivenby π (cid:48) = z ( π ) is not locally stable (see [13]) around the blueequilibrium (see Appendix B). ♦ Remark 6.
No equilibrium dominates another: if going fromone equilibrium to another increases the (risk-adjusted) welfareof one agent, then it decreases the (risk-adjusted) welfare of theother. ♦ Remark 7.
Using the analytical results we check that thereexists a set of non-zero Lebesgue measure of parametersV , V , c , c , c , r , and r (albeit small), that have three distinctequilibria with the same properties. ♦ Remark 8.
We can show that the blue equilibrium is a convexcombination of red and green equilibrium, illustrated on Fig-ure 1 by the dashed blue line. ♦ Acknowledgments
The first-named author want to thank French ambassy ofNew-Zealand for their administrative help and for the financialsupport thanks to France–New-Zealand friendship fund.The authors want to thank PGMO programs for their finan-cial support. Figure 1: Null excess function per scenario manifold for V = V = , c = , c = c = , r = r = References [1] Arrow, K. J., Debreu, G., 1954. Existence of an equilibrium for a compet-itive economy. Econometrica: Journal of the Econometric Society, 265–290.[2] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1999. Coherent measuresof risk. Mathematical Finance 9 (3), 203–228.[3] Brook, A., Kendrick, D., Meeraus, A., 1988. GAMS, a user’s guide. ACMSignum Newsletter 23 (3-4), 10–11.[4] de Maere d’Aertrycke, G., Ehrenmann, A., Smeers, Y., 2017. Investmentwith incomplete markets for risk: The need for long-term contracts. En-ergy Policy 105, 571–583.[5] Ehrenmann, A., Smeers, Y., 2011. Generation capacity expansion in arisky environment: a stochastic equilibrium analysis. Operations Re-search 59 (6), 1332–1346.[6] Facchinei, F., Kanzow, C., 2007. Generalized Nash equilibrium problems.4OR: A Quarterly Journal of Operations Research 5 (3), 173–210.[7] Ferris, M. C., Dirkse, S. P., Jagla, J.-H., Meeraus, A., 2009. An ex-tended mathematical programming framework. Computers & ChemicalEngineering 33 (12), 1973–1982.[8] Ferris, M. C., Munson, T. S., 2000. Complementarity problems in GAMSand the PATH solver. Journal of Economic Dynamics and Control 24 (2),165–188.[9] Mattheij, R., Molenaar, J., 2002. Ordinary di ff erential equations in theoryand practice. SIAM.[10] Philpott, A., Ferris, M., Wets, R., 2016. Equilibrium, uncertainty and riskin hydro-thermal electricity systems. Mathematical Programming 157 (2),483–513.[11] Pritchard, G., Zakeri, G., Philpott, A., 2010. A single-settlement, energy-only electric power market for unpredictable and intermittent participants.Operations Research 58, 1210–1219.[12] Ralph, D., Smeers, Y., 2015. Risk trading and endogenous probabilities ininvestment equilibria. SIAM Journal on Optimization 25 (4), 2589–2611.[13] Samuelson, P. A., 1941. The stability of equilibrium: comparative staticsand dynamics. Econometrica: Journal of the Econometric Society, 97–120.[14] Shapiro, A., Ruszczynski, A., Dentcheva, D., 2014. Lectures on Stochas-tic Programming: Modeling and Theory. SIAM.[15] Uzawa, H., 1960. Walras’ tatonnement in the theory of exchange. TheReview of Economic Studies 27 (3), 182–194.[16] Wong, S., Fuller, J., 2007. Pricing energy and reserves using stochasticoptimization in an alternative electricity market. IEEE Transactions onPower Systems 22 (2), 631–638.[17] Zakeri, G., Pritchard, G., Bjorndal, M., Bjorndal, E., 2016. Pricing wind:A revenue adequate cost recovering uniform price for electricity marketswith intermittent generation. NHH Dept. of Business and ManagementScience. .223 1.224 1.225 1.226 1.227 1.228 (a) around green equilibrium. (b) around blue equilibrium. (c) around red equilibrium.Figure 2: Representation of vector field π (cid:48) = z ( π ) Appendix A. Analytical results
We first analyses the best responses of the producer and theconsumer given a price π . Then, we deduce conditions on theprice and find equilibrium prices. Appendix A.1. Parametric solution with respect to π Assume without loss of generality that 0 < π < π . Appendix A.1.1. Statement of consumer’s problem
The consumer solves one problem per scenario ω i , i = , V , V , r and r be strictly positive constants. The con-sumer problem for ω i ismin y i π i y i − V i y i + r i y i . Appendix A.1.2. Statement of producer’s problem
The risk aversion of the producer is represented by a coherentrisk measure F with risk set Q . Then the producer problemreads min x ≥ , x r ≥ F (cid:104) C ( x ) + C r ( x r ) − π ( x + x r ) (cid:105) . Note that in the case of two outcomes the probability P mea-sure can be defined by P ( ω ), which we denote p . Hence theprobability set P can be described by an interval [ p , ¯ p ].Then the producer problem readsmin x ≥ , x ≥ , x ≥ cx + max p ∈ [ p , ¯ p ] (cid:40) p (cid:16) c x − π ( x + x ) (cid:17) + (1 − p ) (cid:16) c x − π ( x + x ) (cid:17)(cid:41) (A.1) Appendix A.1.3. Statement of complementary constraints
The complementary constraint states that a feasible solutionis a solution where production is greater than demand for eachscenario ω ∈ Ω . Moreover, we want equality between produc-tion and demand at equilibrium. These constraints are written0 ≤ ( x + x r ( ω )) − y ( ω ) ⊥ π ( ω ) ≥ . (A.2) Appendix A.1.4. Analytic solution of the producer’s problem
Focusing on the second stage problem of (A.1) we have Q ( π ) ( x ) = max p ∈ [ p , ¯ p ] p min x ≥ (cid:110) c x − π ( x + x ) (cid:111) + (1 − p ) min x ≥ (cid:110) c x − π ( x + x ) (cid:111) . Note that for i ∈ { , } c i >
0, hence we have x (cid:93) i = π i c i whichin turn gives Q ( π ) ( x ) = max p ∈ [ p , ¯ p ] − p (cid:16) π c + π x (cid:17) − (1 − p ) (cid:16) π c + π x (cid:17) (A.4) = max p ∈ [ p , ¯ p ] p (cid:16)(cid:16) π c − π c (cid:17) + (cid:0) π − π (cid:1) x (cid:17) − (cid:16) π c + π x (cid:17) . Defining x c ( π ) = − π − π (cid:16) π c − π c (cid:17) , (A.5)we see that the worst case probability is given by p (cid:93) ( π ) = ¯ p if x > x c ( π ) , p if x < x c ( π ) , any p ∈ [ p , ¯ p ] if x = x c ( π ) , and thus Equation (A.4) yields Q ( π ) ( x ) = − E ¯ p (cid:104) π c r + π x (cid:105) if x ≥ x c ( π ) , − E p (cid:104) π c r + π x (cid:105) if x < x c ( π ) . Now the first stage problem (Problem (A.1)) readsmin x ≥ cx − E ¯ p (cid:104) π c r + π x (cid:105) x ≥ x c − E p (cid:104) π c r + π x (cid:105) x < x c . We havemin x ≥ x c cx + Q ( π ) ( x ) = − c E ¯ p (cid:2) π (cid:3) − E ¯ p (cid:104) π c r (cid:105) if x c ≤ E ¯ p (cid:2) π (cid:3) c , cx c − E ¯ p (cid:104) π c r + π x c (cid:105) if E ¯ p (cid:2) π (cid:3) c ≤ x c , x (cid:93) x (cid:93) i y (cid:93) i case a) x c ≤ E ¯ p (cid:2) π (cid:3) c E ¯ p (cid:2) π (cid:3) c π i c i V i − π i r i case b) E ¯ p (cid:2) π (cid:3) c ≤ x c ≤ E p (cid:2) π (cid:3) c x c π i c i V i − π i r i case c) E p (cid:2) π (cid:3) c ≤ x c E p (cid:2) π (cid:3) c π i c i V i − π i r i Table A.2: Optimal control for producer and consumer problems attained at E ¯ p (cid:2) π (cid:3) c and x c respectively.If x c > ≤ x ≤ x c cx + Q ( π ) ( x ) = cx c − E ¯ p (cid:104) π c r + π x c (cid:105) if x c ≤ E p (cid:2) π (cid:3) c , − c E p (cid:2) π (cid:3) − E p (cid:104) π c r (cid:105) if E p (cid:2) π (cid:3) c ≤ x c , attained at x c and E p (cid:2) π (cid:3) c respectively. If x c ≤ E p (cid:2) π (cid:3) ≤ E p (cid:2) π (cid:3) , thus the optimal solution can besummed up in Table A.2 Appendix A.2. Finding price equilibrium
Looking at Table A.2 we see that there are three regimes,depending only on the prices ( π , π ) of optimal first stage so-lutions. We are now looking for prices ( π , π ) such that thecomplementarity constraint (A.2) is satisfied. For strictly posi-tive prices, this constraint can be summed up as z i ( π ) = x (cid:93) ( π ) + x (cid:93) ( π ) − y (cid:93) i ( π ) = , i ∈ { , } . (A.6)To go further we are going to split cases by defining the aux-iliary excess demand function z ai ( π ) = E ¯ p (cid:2) π (cid:3) c + π i c i − V i − π i r i , z bi ( π ) = x c ( π ) + π i c i − V i − π i r i , z ci ( π ) = E p (cid:2) π (cid:3) c + π i c i − V i − π i r i , such that we have z = z a cx c ( π ) ≤ E ¯ p (cid:2) π (cid:3) + z b E ¯ p (cid:2) π (cid:3) ≤ cx c ( π ) ≤ E p (cid:2) π (cid:3) + z c E p (cid:2) π (cid:3) ≤ cx c ( π ) . Appendix A.2.1. Case a and c
The set of prices such that z ai ( π ) = π = cc V − (cid:0) c r ¯ p + c ( r + c ) (cid:1) π c r (1 − ¯ p ) ,π = cc V − c r ¯ pc r (1 − ¯ p ) + c ( r + c ) , and the equilibrium can be found by solving the linear sys-tem. Case c is similar, subtituting ¯ p by p . Appendix A.2.2. Case b
The set of prices such that z bi ( π ) = π − π (cid:16) π c − π c (cid:17) + π c − V − π r = , π − π (cid:16) π c − π c (cid:17) + π c − V − π r = , whose a ffi ne equations read π c − (cid:16) c + r (cid:17) π π + (cid:16) r + c (cid:17) π + ( π − π ) V r = , (cid:16) r + c (cid:17) π − (cid:16) c + r (cid:17) π π + c π − ( π − π ) V r = . Appendix B. Unstability of equilibriumDefinition 2.
Let π ( t ) be the general solution of the di ff erentialequation π (cid:48) = z ( π ) , (B.1)such that π (0) = π An equilibrium π (cid:93) such that z ( π ) = locally stable if for all (cid:15) >
0, there exists δ > (cid:107) π − π (cid:93) (cid:107) < δ ⇒ (cid:107) π ( t ) − π (cid:93) (cid:107) < (cid:15) , ∀ t > . (B.2)Using classical results from the field of Ordinary Di ff erentialEquations (see Mattheij and Molenaar [9]), the local stabilitycan be determined from studying the linearization of the systemaround the equilibrium point. Proposition 6.