In this appendix, we present some theoretical aspects of game theory that would be followed by players in a restructured energy market.

Transcription

1 Market Operatons n Electrc Power Systes: Forecastng, Schedulng, and Rsk Manageentg Mohaad Shahdehpour, Hat Yan, Zuy L Copyrght 2002 John Wley & Sons, Inc. ISBNs: (Hardback); X (Electronc) Appendx E Gae Theory Concepts In ths appendx, we present soe theoretcal aspects of gae theory that would be followed by players n a restructured energy arket. E. EQUILIBRIUM IN NON-COOPERATIVE GAMES The noral for of an N-player gae [Fer96a] conssts of a set of N players, N strategy sets X, =,...,N and the N-tuple payoff functon G(X,...,X N ). The value G (x,...,x N ) s the payoff functon of player when player plays the xed strategy x X,..., player N plays the strategy x N X N. We only consder pure strateges. Each player can only choose one strategy. An N-tuple of strateges x,, x N s an equlbru N-tuple f G (x,...,y,..., x N ) G (x,..., x N ) for all and for all strateges y for player. Hence, the player departng fro the xed strategy n the N-tuple at least does no better. In a two-player gae, the two-densonal set G ={G (X,X 2 ), G 2 (X,X 2 )} s called the non-cooperatve payoff regon [Fer96a]. The ponts n G are called payoff pars. If (u,v) and (u',v') are two payoff pars, (u,v) donates (u',v') f u v and v v'. Payoff pars that are not donated by any other par are sad to be Pareto optal. Clearly, ratonal cooperatng players n a power pool would never play so that ther payoffs are not Pareto optal. 483

2 484 APPENDIX E E.2 CHARACTERISTIC FUNCTION The ax-n value or characterstc functon gves each player a pessstc estate of how uch payoff can be expected. In a two-player gae, the ax-n value gves player (the row player) the expected payoff by assung that player 2 (the colun player) wll act to nze player s payoff. v = ax n G ( X, X 2 ) X X 2 where X and X 2 range over all xed strateges for players and 2 respectvely. The ax-n value for player 2 s defned n a slar way. The characterstc functons of the coalton and counter-coalton are coputed as follows: Let j be an eleent of the row player s payoff atrx. The row player s proble s to copute the row value v r = n j = p and an optal xed strategy [Fer96a] p = (p,..., p ) wth Σp =. A constant c s added to all entres large enough that j + c > 0 for all,j. The proble s forulated now as a functon of varable y = p / v for ; the proble can be forulated as a lnear progra: n s.t. y = j y y j for j n The colun s player proble n the non-cooperatve gae s coputed n the sae way after transposng the colun player s payoff atrx. In a two-player gae, the assupton that player 2 wll try to nze s payoff s probably false. Player 2 wll try nstead to axze hs payoffs; however, v gves player a lower bound (pessstc estate) on hs payoff.

3 APPENDIX E 485 E.3 N-PLAYERS COOPERATIVE GAMES As ndcated above, a coalton s a subset of players that s fored n order to coordnate strateges and to agree on how the total payoff s to be dvded aong ts ebers. The set of all N players s denoted by P. Gven a coalton S P, the counter-coalton to S s S c = P S = {P P : P S } In general, n a gae wth N players, there are 2 N coaltons. The gae between any coalton S and ts counter-coalton S c s a non-cooperatve gae. The rows n the atrx correspond to the set of strateges avalable to S and the coluns correspond to those avalable to S c. The ax-n value for the coalton s called the characterstc functon of S and s denoted v(s). When a coalton s fored, the payoff s dstrbuted aong players n the coalton. The aounts gong to each player for an N-tuple X of real nubers. Two condtons are usually requred fro an N-tuple n order to be lkely to actually happen n the gae [Fer96a]: Indvdual ratonalty x v({p }) Collectve ratonalty N = x = v(p ) Indvdual ratonalty requres that the coalton gve any player a hgher payoff than what the player can obtan on hs own. A N-tuple of payents that satsfes both of these condtons s called an putaton. The core of a gae conssts n all putatons that are not donated by any other putaton through any coalton. Hence, f an putaton X s at the core, there s no group of players whch has a reason to for a coalton and replace X wth a dfferent putaton. Every putaton n the core of the gae s an effcent allocaton. The standard defnton of effcent allocaton s Pareto optalty. If the allocaton s effcent, there s no player who can do better wthout akng soe other player worse off. Matheatcally, X s n the core f and only f

4 486 APPENDIX E for every coalton S. S P x v( S) The concept of a core of the gae has an portant ltaton: the core of the gae ay be of any sze; t ay even be epty, eanng that there are no stable coaltons. Whatever coalton s fored, there s soe ncentve for a subgroup to desert the coalton. In a restructured power syste as presented n Chapter 6, the objectve fro the pool coordnator s perspectve should be to ake the dstrbuton of savngs aong players at the core of the gae n the case of the grand coalton. E.4 GAMES WITH INCOMPLETE INFORMATION One ay dstngush between gaes wth coplete nforaton (c-gaes) and gaes wth ncoplete nforaton (-gaes). In an -gae, soe of the players are not certan of the characterstcs of soe of the other partes; players ay lack nforaton about the other player s payoff functons, strateges avalable to other players, the aount of nforaton that other players have on varous aspects of the gae and so on. That s to say, players would lack full nforaton on the atheatcal structure of the gae. Incoplete nforaton s consdered by odelng the player s unknown characterstcs as types of player. The type of a player ebodes any nforaton that s not coon to all players. An -gae ay be used n stuatons where each player s uncertan about other players payoff, but also n stuatons where each player s uncertan about other players knowledge. In any gven play of an -gae each player knows hs type. However, a player searchng for hs best strategy ay need to deterne the actons of the other players for other states, snce he ay be snfored about the state of the gae. Another dstncton s ade between gaes wth perfect nforaton and gaes wth perfect nforaton. Ths dstncton s based on the aount of nforaton that players have about the oves ade n earler stages of the gae. In gaes wth perfect nforaton for nstance, chess all players have full nforaton on all oves ade n early stages. In

5 APPENDIX E 487 gaes wth perfect nforaton players only have partal nforaton about the oves ade n earler stages. Equlbru of -gaes ay be derved fro Nash equlbru condtons. In Nash equlbru, each player chooses the best acton avalable to h gven the (ncoplete) nforaton that he receves. A proposed soluton of -gaes n [Fer98] s to transfor the orgnal -gae nto a c-gae wth perfect nforaton. Hence, Nash equlbru s appled to the transfored gae. Player s optal bds are derved for the equlbru condton.