Global Optmzaton n Mult-Agent Models John R. Brge R.R. McCormck School of Engneerng and Appled Scence Northwestern Unversty Jont work wth Chonawee Supatgat, Enron, and Rachel Zhang, Cornell 11/19/2004 1
Outlne I. Problem Overvew II. Model III. Results V. Summary and Extensons 11/19/2004 2
General Issue Standard Equlbrum Results Concave utlty functons for agents Consstent nformaton sets Unque equlbrum wth strct concavty Realstc Markets Market mechansms (and other thngs) negate concavty assumptons Inconsstent and varyng nformaton sets Multple, dsconnected equlbra (or dsequlbrum) Goal: Fnd the Set of Equlbra (Worst Case?) Example: Electrc Power Market 11/19/2004 3
Compettve Electrc Power Markets N Supplers (bdders), Each submts bd prce and quantty Consumer Demand Supply bds Power Exchange Market 11/19/2004 4
Market Clearng Process Demand s 10 Suppler 1 : 5MWh @ $10 Suppler 2 : 10MWh @ $15 Suppler 3 : 10MWh @ $20 25 20 Demand MCP =15 Supply bd prce 15 10 5 0 1 2 3 0 5 10 15 20 25 30 total bd quantty Problem: fnd optmal bddng strateges and the resultng MCP 11/19/2004 5
Market Overvew Non-sealed bd, Mult-round Bdders can see each other s bds and can adjust ther prces as many tmes as they want Market s closed when no bdder wants to adjust hs/her bd prce Sellng at spot: All dspatched unts are traded at the same prce 11/19/2004 6
Model Demand D (or d) must be satsfed Bdder has unt cost c Sngle bd per bdder : (x, p ) x = bd quantty of bdder p = bd prce of bdder : (assumed dscrete) { l l = 0,1,..., O}, = 1 N p ε,..., b = [(x 1,p 1 ),, (x N,p N )] 11/19/2004 7
11/19/2004 8 Model Market clearng prce Dspatch quantty of Bdder * Dspatch of margnal bdders follows order of bd submsson tmes = < > = ), ( f ), ( ), ( f ), ( f 0 ), ( d MCP p d q d MCP p x d MCP p d q b b b b b = = } : { ) (, : mn ), ( ) ( j j I j j p p j I d x p d MCP b
Model Bdder s payoff f [( MCP( b, D) c ) q ( b, )] ( b) = E D D Objectve: Fnd Nash equlbrum {p *, = 1,, N} such that f ([ ] * * ( x ) 1, p1 ),..., ( x, p ),..., ( xn, pn ) f ([( x, p ),..., ( x, p ),..., ( x, p )] * * * ) 1 1 N N for all feasble p, for bdder, and all = 1,..., N 11/19/2004 9
Model Dstnct bdders: c c > 2ε, j j Fxed bd quantty: bdders can adjust only bd prce Gven a number x, x = max{ε ε x, =0,..,O} and x = mn{ε ε x, =0,..,O} 11/19/2004 10
Market Stablty Condton D x for j = 1,..., j N D s the hghest demand realzaton No one s guaranteed to be dspatched 11/19/2004 11
Results Multple equlbra Known demand: At the hghest MCP equlbrum pont, every bdder bds at c, except the margnal bdder j who bds at c j Unque margnal bdder f partally dspatched Stochastc demand: Sngle margnal At any demand, bds at p j -ε At the hghest demand, bds at c j Two margnal bdders,, j They bd just above the cost of the bdder wth a lower quantty p = p j = c j where x j x 11/19/2004 12
Hghest Equlbrum MCP Known demand: the hghest possble equlbrum MCP must be n the set { } c, =1,..., N Stochastc demand: the hghest possble equlbrum MCP must be n {, + ε, = 1 N} c c, c,..., 11/19/2004 13
Compettve Bdder Set (CBS) CBS: bdders wth the lowest costs and satsfy the market stablty condton D x for j = 1,..., N j Bdder set CBS 11/19/2004 14
Compettve Bdder Set (CBS) At an equlbrum pont, All bdders outsde the CBS are not dspatched When demand s known, at least one bdder n the CBS s not dspatched 2ε plus the hghest cost among the bdders n the CBS s an upper bound on the MCP 11/19/2004 15
Payoff functon Gven other bdders bd prces and demand Bdder s payoff (f ) (p 2 -c )x q = d-x 1 -x 2 q = d-x 1 -x 2 -x 3.... p 1 p 2 p 3 p 4 p 5 p 11/19/2004 16
Algorthm for Fndng the Hghest MCP Equlbrum Pont wth Determnstc Demand Constructng CBS Condton on each bdder to be margnal whle others bd at cost Fnd the optmal bd prce f... Pck the one wth the hghest optmal bd prce to be the margnal bdder; others bd at costs. c 1 c 2 c 3 c 4 p 11/19/2004 17
A Specal Case: Identcal Bd Quanttes Assume x = x for all =1,,N Equlbrum pont wth the hghest MCP p * MCP = c + = 1,..., N 1 = 1 c k where k s the frst undspatched bdder, gven all bdders bd at ther costs. 11/19/2004 18
Fndng Hghest MCP wth Stochastc Demand Search for possble soluton values Format: Search over potental margnal bdders at hghest demand level Start wth hghest cost bdder Bdder j can bd at prevous bds (p or p -ε) or c j or c j +ε Fnd combnaton for global maxmum 11/19/2004 19
A Numercal Example c x 1 1.01 5 2 6.01 5 3 7.01 1 4 9.01 1 5 10.51 11 ε = 0.01, D = 7 or 11 w.p. 0.5 11/19/2004 20
Comparson of Payoffs $ 45 40 35 30 25 20 15 10 5 Case 1: Algorthm (worst equlbrum), MCP = 9.75 Case 2: at next hgher bdder's cost, MCP = 8 Case 3: at cost, MCP =6.51 0 Bdder 1 Bdder 2 Bdder 3 Bdder 4 Bdder 5 11/19/2004 21
Comparson of Dspatch Quantty 5 4.5 4 3.5 3 Case 1: Algorthm (worst equlbrum) Case 2: at next hgher bdder's cost Case 3: at cost 2.5 2 1.5 1 0.5 0 Bdder 1 Bdder 2 Bdder 3 Bdder 4 Bdder 5 11/19/2004 22
Extensons Include start-up cost of generaton Analyze mult-perod problems: cost depends on the dspatch of the prevous perod Allow multple bds per bdder 11/19/2004 23
Summary Multple equlbra, non-convextes Demand s known MCP <= c N margnal at c j, others at c +ε Demand s stochastc MCP <= c N +ε margnal at p j -ε, c k, or c k +ε, others at c +ε Algorthm to fnd hghest MCP equlbrum pont Extensons for multple unts, perods and dfferent cost structures 11/19/2004 24
11/19/2004 25 Worst Equlbrum pont Worst equlbrum MCP = hghest possble bd prce ε ε ε = = = = = * * 1 1 * * 1 1,..., for N N N x O p d x N c p
Multple-Unt Dspatchng Suppose control of multple unts \n L What s the hghest possble equlbrum prce? 11/19/2004 26