GAME THEORY By: Rishika a nd Nithya 12/04/13 Outline • 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
What is game theory? History of game theory Basic concepts of game theory Game theory and Informa8on Systems Defini8on of Games Nash Equilibrium Applica8on of game theory in Power pools The power transac8ons game Costs and benefits in power transac8on Conclusion References What is game theory? •  It is the formal study of decision-­‐making where several players must make choices that poten8ally affect the interests of the other players. History of Game Theory •  Earliest example-­‐study of duopoly by Antonie Cournot in 1838. •  In 1950-­‐1960’s it broadened and was applied to problems of war and poli8cs. It has found applica8ons in sociology and psychology, and established links with evolu8on and biology. •  Received special aWen8on in 1994 with the awarding of the Nobel prize in economics to Nash. •  1990’s-­‐applica8on of game theory has been the design of auc8ons. Basic Concepts of Game Theory Players Nodes Game Moves Pay-­‐offs •  Players-­‐Players are independent decision-­‐makers who influence the development of a game through their decisions. •  Nodes-­‐As a game proceeds, various states of the world it is describing can be achieved. Such states are also called nodes. •  Move-­‐A move allows the game to evolve from a well-­‐defined state to another and therefore may indicate sequen8ality . •  Pay-­‐offs-­‐A payoff is a development of the game that the players value for itself according to their respec8ve preferences. Game Theory and InformaLon Systems •  The internal consistency and mathema8cal founda8ons of game theory make it a prime tool for modeling and designing automated decision-­‐making processes in interac8ve environments. •  For example, one might like to have efficient bidding rules for an auc8on website. •  As a mathema8cal tool for the decision-­‐maker the strength of game theory is the methodology it provides for structuring and analyzing problems of strategic choice. DefiniLons of Games Coopera8ve Non-­‐
coopera8ve CooperaLve Game •  In game theory, a cooperaLve game is a game where groups of players ("coali8ons") may enforce coopera8ve behavior, hence the game is a compe88on between coali&ons of players, rather than between individual players. •  For example, Recrea8onal games are rarely coopera8ve. Non-­‐cooperaLve Game •  In game theory, a non-­‐cooperaLve game is one in which players make decisions independently. •  Examples include Auc8on theory, Strategic vo8ng. Extensive form game •  The extensive form, also called a game tree, is more detailed than the strategic form of a game. •  It is a complete descrip8on of how the game is played over8me and includes the order in which players take ac8ons. Example Strategic form game •  The strategic form game is usually represented by a matrix which shows the players, strategies, and pay-­‐offs. •  More generally it can be represented by any func8on that associates a payoff for each player with every possible combina8on of ac8ons Example Nash Equilibrium •  A set of strategies is a Nash equilibrium if no player can do beWer by unilaterally changing his or her strategy. •  Imagine that each player is told the strategies of the others Suppose then that each player asks himself or herself: "Knowing the strategies of the other players, and trea8ng the strategies of the other players as set in stone, can I benefit by changing my strategy?“ •  If any player would answer "Yes", then that set of strategies is not a Nash equilibrium. Structure of deregulated system ApplicaLon of Game Theory for pricing Electricity in deregulated power pools •  Independent System Operator receives bids by Pool par8cipants and defines transac8ons among par8cipants by looking for the minimum price that sa8sfies the demand in the Pool. •  As deregula8on evolves, pricing electricity becomes a major issue in the electric industry. •  In deregulated power systems, emphasis is given to benefit maximiza8on from the perspec8ve of par8cipants rather than maximiza8on of system-­‐wide benefits. The Pool Co •  Bids by generators as well as loads create a spot market for electricity. •  The spot price is set by the last generator dispatched by the IS0 to balance Pool Co.'s genera8on and demand. •  The objec8ve is to maximize each par8cipant’s benefits hence, earnings from transac8ons should be maximized. •  A seller has to evaluate the possibility of either selling more power at a low per unit price or selling less at a high price per unit of power. A similar analysis can be made for a buyer. •  Each par8cipant defines bids based on incomplete informa8on on other par8cipant’s bids. The Power TransacLons Game •  Game theory has been used in power systems to understand a par8cipant's behavior in deregulated environments and to allocate costs among Pool par8cipants. •  The game is assumed non-­‐coopera8ve and the Nash equilibrium idea for non-­‐coopera8ve games is used. •  In power transac8on game, par8cipants compete against each other to maximize their benefits. •  Economic benefits from transac8ons are the payoff of the game. •  The basic probability of the game is based on a random fuel price. Costs and Benefits in Power TransacLons •  Each par8cipant k has several generators and a total load Lk; in this case ƩPi0= Lk corresponds to the genera8on level before transac8ons are defined. •  The increment in cost incurred to change the genera8on level in is •  The marginal cost is given by •  The incremental cost and the marginal cost of generators are the linear func8ons of genera8on levels. •  The par8cipant’s bids are also linear func8ons of genera8on level. •  The IS0 receives par8cipants' bids as in equa8on and matches the lowest bid with the Pool Co.'s load. •  Power transac8on for par8cipant K is computed as •  The minimum price that matches genera8on and load is called the spot price of electricity ƿ. •  For a given spot price, the par8cipant’s benefit is •  Condi8on for benefit maximiza8on is • 
•  The par8cipant adjusts its genera8on so as in the absence of binding constraints: •  In a perfect compe00on, sellers and buyers are very small as compared with the market size; no par8cipant can significantly affect the exis8ng spot price. Conclusion •  In a perfect compe88on, the spot price in the Pool Co is essen8ally given and the op8mal price decision can be obtained without a game-­‐theore8cal approach. Condi8ons of perfect compe88on are altered by several factors including network constraints. •  Game theory approaches have been used to study imperfect compe88on in electricity markets. •  Pricing electricity in deregulated pool is becoming a major issue in the electric industry and addi8onal mathema8cal support is needed to define pricing strategies in this environment. •  If we analyze the problem without a game-­‐theore8cal approach, a par8cipant may obtain lower benefits than those obtained from the applica8on of the proposed method. References [1] Fudenberg, Drew and Tirole, Jean (1991), Game Theory. MIT Press, Cambridge, MA. [2] Gibbons, Robert (1992), Game Theory for Applied Economists. Princeton University Press, Princeton, NJ. [3] R. W. Ferrero, J. F. Rivera, and S. M. Shahidehpour, "Applica8on of games with incomplete informa8on for pricing electricity in deregulated power pools," Power Systems, IEEE Transac&ons on, vol. 13, pp. 184-­‐189, 1998. [4] J.P. Aubin, “Mathema&cal Methods of Game and Economic Theory,” Amsterdam: North-­‐Holland Publishing Company, 1982. [5] J . Harsanyi, “Games with Incomplete Informa8on,” The American Economic Review, Vol. 85, No. 3, June 1995, pp. 291-­‐303.