Wireless Network Pricing Chapter 6: Oligopoly Pricing Jianwei Huang & Lin Gao Network Communications and Economics Lab (NCEL) Information Engineering Department The Chinese University of Hong Kong Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 1 / 29
The Book E-Book freely downloadable from NCEL website: http: //ncel.ie.cuhk.edu.hk/content/wireless-network-pricing Physical book available for purchase from Morgan & Claypool ( http://goo.gl/JFGlai ) and Amazon ( http://goo.gl/JQKaEq ) Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 2 / 29
Chapter 6: Oligopoly Pricing Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 3 / 29
Section 6.1 Theory: Game Theory Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 6 / 43
Extensive Form Game In extensive form games, players make decisions sequentially. Our focus is on the multi-stage game with observed actions where: I All previous actions (called history) are observed, i.e., each player is perfectly informed of all previous events; I Some players may move simultaneously within the same stage. Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 21 / 43
Market Entry • Firm 1 is considering entering a market that currently has an incumbent (firm 2). ! • Firm 1 can choose “In” or “Out”. ! • If “Out”, firm 1 gets nothing, and firm 2 enjoys monopoly. ! • If “In”, firm 2 can choose “Accept” or “Fight”. ! • If firm 2 accepts, then firm 1 gets a larger market share due to a newer technology. ! • If firm 2 fights, then there is a price war and both firms get negative profits.
Market Entry Out 0, 2 Firm 1 In Firm 2 Accept Fight 2, 1 -3, -1
Market Entry Firm 2 Accept Fight Out 0, 2 0, 2 Firm 1 2, 1 -3, -1 In
Market Entry Firm 2 Accept Out 0, 2 Firm 1 2, 1 In
Market Entry Firm 2 Fight Out 0, 2 Firm 1 -3, -1 In
Market Entry Firm 2 Accept Fight Out 0, 2 0, 2 Firm 1
Market Entry Firm 2 Accept Fight Firm 1 2, 1 -3, -1 In
Market Entry Firm 2 Accept Fight Out 0, 2 0, 2 Firm 1 2, 1 -3, -1 In
Market Entry Firm 2 Accept Fight Out 0, 2 0, 2 Firm 1 2, 1 -3, -1 In
Market Entry Out Firm 1 0, 2 • Consider the Nash equilibrium In (Out, Fight if entry occurs). ! Firm 2 • Firm 1 chooses to stay Out because of firm 2’s threat of Accept Fight Fight. 2, 1 -3, -1
Non-credible Threat Out Firm 1 0, 2 • However, if firm 1 chooses In, In then firm 2 will actually choose to Accept instead. ! Firm 2 • Hence Fight is a non-credible Accept Fight threat . 2, 1 -3, -1
Equilibrium Refinement • Principle of sequential rationality : an equilibrium strategy should be optimal at every point of the game tree. ! • Examine each subgame through backward induction .
Subgame Analysis Out 0, 2 Firm 1 In Firm 2 Accept Fight 2, 1 -3, -1
Subgame Analysis Firm 2 Accept Fight 2, 1 -3, -1
Subgame Analysis Firm 2 Accept Fight 2, 1 -3, -1
Subgame Analysis Out 0, 2 Firm 1 In Firm 2 Accept Fight 2, 1 -3, -1
Subgame Analysis Out 0, 2 Firm 1 In Firm 2 Accept Fight 2, 1 -3, -1
Subgame Analysis Out 0, 2 Firm 1 In 2, 1
Equilibrium Out 0, 2 Firm 1 In Firm 2 Accept Fight 2, 1 -3, -1
Subgame Perfect Nash Equilibrium • A strategy profile is a subgame perfect Nash equilibrium (SPNE) if it is a Nash equilibrium of every subgame of the original game. ! • For market entry game, the unique SPNE is ! (In, Accept if entry occurs).
Credible Threat • How to make credible threat? ! • Eliminate choices.
Dr. Strangelove Country A Not Attack 0, 0 Attack Country B Not Counter-Attack Counter-Attack - , - 100, -200 ∞ ∞
Dr. Strangelove Country A Not Attack 0, 0 Attack Country B Counter-Attack - , - ∞ ∞
Dr. Strangelove Country A Not Attack 0, 0 Attack Country B Counter-Attack - , - ∞ ∞
Dr. Strangelove Country A Not Attack 0, 0 Attack - , - ∞ ∞
Dr. Strangelove Country A Not Attack 0, 0 Attack Country B Counter-Attack - , - ∞ ∞
SPNE • The unique SPNE of the Dr. Strangelove game is (Not Attack, Counter-Attack if Country A attacks).
Extensive Form Game Definition (Extensive Form Game) An extensive form game consists of four main elements: A set of players I = { 1 , 2 , ..., I } ; The history h k = ( s 0 , ..., s k − 1 ) at stage k (after stage k − 1), where s t = ( s t i , ∀ i ∈ I ) is the action profile at stage t ; Each pure strategy for player i is defined as a contingency plan for every possible history after each stage; Payo ff s are defined on the outcome after the last stage. Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 22 / 29
Extensive Form Game Important Notations I h k = ( s 0 , ..., s k − 1 ): the history at stage k (after stage k − 1); I H k = { h k } : the set of all possible histories after stage k − 1; I S i ( h k ): the set of actions available to player i under a particular history h k at stage k ; h k ∈ H k S i ( h k ): the set of actions available to player i under I S i ( H k ) = S all possible histories at stage k ; i : H k → S i ( H k ): a mapping from every possible history in H k (after I a k stage k − 1) to an available action of player i in S i ( H k ); I s i = { a k i } ∞ k =0 : the pure strategy of player i . Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 23 / 29
First Mover Advantage • Let us look at how the first mover can have an advantage.
Battle of Sexes Wife Football Ballet Football Husband 4, 2 0, 0 Ballet 0, 0 2, 4
Battle of Sexes Wife Football Ballet Football Husband 4, 2 0, 0 Ballet 0, 0 2, 4
Sequential Battle of Sexes Husband Football Ballet Fo Wife Wife Football Ballet Football Ballet 4, 2 0, 0 0, 0 2, 4
Backward Induction Husband Football Wife Football Ballet 4, 2 0, 0
Backward Induction Husband Football Ballet Wife Football Ballet 0, 0 2, 4
Sequential Battle of Sexes Husband Football Ballet Fo Wife Wife Football Ballet 4, 2 2, 4
Sequential Battle of Sexes Husband Football Ballet Fo Wife Wife Football Ballet Football Ballet 4, 2 0, 0 0, 0 2, 4
Sequential Battle of Sexes • Unique subgame perfect Nash equilibrium is (Football, (Football if Husband chooses Football, Ballet if Husband chooses Ballet)). ! • Although the equilibrium path will be Husband picking Football and Wife picking Football, we need to specify how the Wife will pick if the Husband picks Ballet. ! • SPNE is a contingency plan that specifies the action at every point in the game tree.
Sequential Battle of Sexes Husband Football Ballet Fo Wife Wife Football Ballet Football Ballet 4, 2 0, 0 0, 0 2, 4
First-Mover Advantage • Husband makes a firm commitment by moving first. ! • Wife thus will follow to maximize her payoff. ! • In other games, second mover may have an advantage, if she can take advantage of the efforts of the first mover (such as the R&D cost of entering a new market).
More Than Two Stages • Next we look at a game with more than two stages.
Nuisance Suit • The police confirms the defendant a small offense. ! • The police can choose to do nothing, or sue the defendant and ask for a small penalty. ! • If the defendant accepts and pays the penalty, the case is settled. The police incur some cost and receive the payment. ! • If the defendant refuses to pay, the police can give up and bare the cost, or go to court. ! • When in court, the police incur additional cost but also get benefit by winning the case. The defendant incurs additional cost.
Nuisance Suit Do Nothing 0, 0 Police Ask for Penalty Accept 7, -10 Defendant Reject Give Up -3, 0 Police Go to Court -8, -20
Backward Induction Do Nothing 0, 0 Police Ask for Penalty Accept 7, -10 Defendant Reject Give Up -3, 0 Police Go to Court -8, -20
Nuisance Suit Do Nothing 0, 0 Police Ask for Penalty Accept 7, -10 Defendant Reject Give Up -3, 0 Police
Nuisance Suit Do Nothing 0, 0 Police Ask for Penalty Defendant Reject Give Up -3, 0 Police
Nuisance Suit Do Nothing 0, 0 Police Ask for Penalty Accept 7, -10 Defendant Reject Give Up -3, 0 Police Go to Court -8, -20
Nuisance Suit • Unique subgame perfect Nash equilibrium is ((Do Nothing, Give Up if Defendant rejects), Reject if Police ask for penalty). ! • As the police want to avoid the additional cost of going to court, the defendant will take advantage of this and refuses to pay the penalty. Hence the police is better off by just doing nothing in the first place.
Make Threat Credible • Police make the threat of Go to Court credible by paying the cost of going to court (such as the lawyer fee) even without going to court. ! • As this cost becomes sunk before the last stage, the police is determined to go to court.
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