Multilevel Pricing Schemes in a Deregulated Wireless Network Market Yezekael Hayel CERI/LIA University of Avignon Workshop on New Avenues for Network Models IISc - 15/01/2014 1 / 35
Plan Introduction to Hierarchical Game Theory principles 1 Application: Spectrum market in Wireless Networks 2 Conclusions 3 2 / 35
Plan Introduction to Hierarchical Game Theory principles 1 Application: Spectrum market in Wireless Networks 2 Conclusions 3 3 / 35
Introduction Introduction In a competition setting, there are several major concerns when some players may determine their actions after observing the actions of the other players. In game theory setting, we say that one of the players has the ability to enforce his strategy on the other players. 4 / 35
Definition Leader/Follower In a game theoretic formulation of a hierarchical competition, we define two types of players: Leaders: players that take their decisions first, Followers: players that take their decisions after observing the leader’s decisions. IMPORTANT: The leaders know, ex ante, that the followers observe their action. Stackelberg setting The Stackelberg competition model a is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially. It considers only two players in competition. a H. von Stackelberg, Market Structure and Equilibrium: 1st Edition Translation into English, Bazin, Urch & Hill, Springer 2011 (1934). 5 / 35
Definition Leader/Follower In a game theoretic formulation of a hierarchical competition, we define two types of players: Leaders: players that take their decisions first, Followers: players that take their decisions after observing the leader’s decisions. IMPORTANT: The leaders know, ex ante, that the followers observe their action. Stackelberg setting The Stackelberg competition model a is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially. It considers only two players in competition. a H. von Stackelberg, Market Structure and Equilibrium: 1st Edition Translation into English, Bazin, Urch & Hill, Springer 2011 (1934). 5 / 35
Basic formulation Utility functions As the leaders know that the followers observe their action, the follower’s actions can be seen as implicit functions in the leader’s utility. Stackelberg setting We consider two players with the utility functions U 1 ( a 1 , a 2 ) and U 2 ( a 1 , a 2 ) . Each player maximizes his utility function: max U 1 ( a 1 , a 2 ) and max U 2 ( a 1 , a 2 ) . a 1 a 2 6 / 35
Basic formulation Utility functions As the leaders know that the followers observe their action, the follower’s actions can be seen as implicit functions in the leader’s utility. Stackelberg setting We consider two players with the utility functions U 1 ( a 1 , a 2 ) and U 2 ( a 1 , a 2 ) . Each player maximizes his utility function: max U 1 ( a 1 , a 2 ) and max U 2 ( a 1 , a 2 ) . a 1 a 2 6 / 35
Basic formulation Simultaneous play If both player takes his action without observing the action of the other, we look for a standard Nash equilibrium of this non-cooperative game. Non-simultaneous play We assume that player 1 decides first his action and player 2 takes his action after observing player 1’s action. player 1 is the leader, player 2 is the follower, player 1 knows that player 2 observes his action before taking his own decision. 7 / 35
Basic formulation Simultaneous play If both player takes his action without observing the action of the other, we look for a standard Nash equilibrium of this non-cooperative game. Non-simultaneous play We assume that player 1 decides first his action and player 2 takes his action after observing player 1’s action. player 1 is the leader, player 2 is the follower, player 1 knows that player 2 observes his action before taking his own decision. 7 / 35
Basic formulation Maximization problem For each action a 1 of the leader, we denote by BR 2 ( a 1 ) the best-response function (or set) which determines the best action of the follower depending on the action of the leader, i.e. ∀ a 1 , BR 2 ( a 1 ) = arg max U 2 ( a 1 , a 2 ) . a 2 As the leader knows that the follower observes his action, the leader has the following utility function to maximize: max U 1 ( a 1 , BR 2 ( a 1 )) . a 1 IMPORTANT: This function depends only on his own action a 1 , through the best-response function of the follower. 8 / 35
Basic formulation Maximization problem For each action a 1 of the leader, we denote by BR 2 ( a 1 ) the best-response function (or set) which determines the best action of the follower depending on the action of the leader, i.e. ∀ a 1 , BR 2 ( a 1 ) = arg max U 2 ( a 1 , a 2 ) . a 2 As the leader knows that the follower observes his action, the leader has the following utility function to maximize: max U 1 ( a 1 , BR 2 ( a 1 )) . a 1 IMPORTANT: This function depends only on his own action a 1 , through the best-response function of the follower. 8 / 35
Basic formulation Maximization problem For each action a 1 of the leader, we denote by BR 2 ( a 1 ) the best-response function (or set) which determines the best action of the follower depending on the action of the leader, i.e. ∀ a 1 , BR 2 ( a 1 ) = arg max U 2 ( a 1 , a 2 ) . a 2 As the leader knows that the follower observes his action, the leader has the following utility function to maximize: max U 1 ( a 1 , BR 2 ( a 1 )) . a 1 IMPORTANT: This function depends only on his own action a 1 , through the best-response function of the follower. 8 / 35
Stackelberg solution Stackelberg equilibrium A Stackelberg equilibrium is a vector of actions ( a ∗ 1 , a ∗ 2 ) such that: a ∗ 1 = arg max U 1 ( a 1 , BR 2 ( a 1 )) , a 1 and a ∗ 2 = BR 2 ( a ∗ 1 ) . Figure: H. Von Stackelberg 9 / 35
Stackelberg model Subgame perfect Nash equilibrium The Stackelberg model can be solved to find the subgame perfect Nash equilibrium (SPNE), i.e. the strategy profile that serves best each player, given the strategies of the other player and that entails every player playing in a Nash equilibrium in every subgame . 10 / 35
Stackelberg solution In practice using Backward induction In order to determine a Stackelberg equilibrium solution, there are three steps: To compute the best-response function of the follower, for each action of 1 the leader. To solve the optimization problem for the leader, using the best-response 2 function of the follower. To determine the best action of the follower, using the best-response 3 function, when the leader takes his decision determined in step 2. 11 / 35
Stackelberg solution In practice using Backward induction In order to determine a Stackelberg equilibrium solution, there are three steps: To compute the best-response function of the follower, for each action of 1 the leader. To solve the optimization problem for the leader, using the best-response 2 function of the follower. To determine the best action of the follower, using the best-response 3 function, when the leader takes his decision determined in step 2. 11 / 35
Stackelberg solution In practice using Backward induction In order to determine a Stackelberg equilibrium solution, there are three steps: To compute the best-response function of the follower, for each action of 1 the leader. To solve the optimization problem for the leader, using the best-response 2 function of the follower. To determine the best action of the follower, using the best-response 3 function, when the leader takes his decision determined in step 2. 11 / 35
Matrix game example First example (Normal Form Game) We consider the following matrix game: K U � � L ( 3 , 1 ) ( 1 , 3 ) . R ( 2 , 1 ) ( 0 , 0 ) The (pure) NE is the couple ( L , U ) . If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple ( L , U ) (resp. ( R , K ) ). The SE if the line player is the leader, is the couple ( R , K ) . The payoff of the leader is increased. 12 / 35
Matrix game example First example (Normal Form Game) We consider the following matrix game: K U � � L ( 3 , 1 ) ( 1 , 3 ) . R ( 2 , 1 ) ( 0 , 0 ) The (pure) NE is the couple ( L , U ) . If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple ( L , U ) (resp. ( R , K ) ). The SE if the line player is the leader, is the couple ( R , K ) . The payoff of the leader is increased. 12 / 35
Matrix game example First example (Normal Form Game) We consider the following matrix game: K U � � L ( 3 , 1 ) ( 1 , 3 ) . R ( 2 , 1 ) ( 0 , 0 ) The (pure) NE is the couple ( L , U ) . If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple ( L , U ) (resp. ( R , K ) ). The SE if the line player is the leader, is the couple ( R , K ) . The payoff of the leader is increased. 12 / 35
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