Wireless Network Pricing Chapter 6: Oligopoly Pricing

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1 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 November 15, / 69

2 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 ( and Amazon ( Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

3 Section 6.2 Theory: Oligopoly Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

4 Oligopoly In this part, we consider three classical strategic form games to formulate the interactions among multiple competitive entities (Oligopoly): I The Cournot Model I The Bertrand Model I The Hotelling Model Our purpose in this part is to illustrate I (a) Game Formulation: the translation of an informal problem statement into a strategic form representation of a game; I (b) Equilibrium Analysis: the analysis of Nash equilibrium when a player can choose his strategy from a continuous set. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

5 The Cournot Model The Cournot model describes interactions among firms that compete on the amount of output they will produce, whichtheydecide independently of each other simultaneously. Key features I At least two firms producing homogeneous products; I Firms do not cooperate, i.e., there is no collusion; I Firms compete by setting production quantities simultaneously; I The total output quantity a ects the market price; I The firms are economically rational and act strategically, seekingto maximize profits given their competitors decisions. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

6 The Cournot Model Example: The Cournot Game I Two firms decide their respective output quantities simultaneously; I The market price is a decreasing function of the total quantity. Game Formulation I The set of players is I = {1, 2}, I The strategy set available to each player i 2I is the set of all nonnegative real numbers, i.e., q i 2 [0, 1), I The payo received by each player i is a function of both players strategies, defined by i (q i, q i )=q i P(q i + q i ) c i q i F F The first term denotes the player i s revenue from selling q i units of products at a market-clearing price P(q i + q i ); The second term denotes the player i s production cost. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

7 The Cournot Model Consider a linear cost: P(q i + q i )=a (q i + q i ) Equilibrium Analysis I Given player 2 s strategy q2,thebestresponse of player 1 is: I q1 = B 1 (q 2 )= a q 2 c 1, 2 Given player 1 s strategy q 1,thebest response of player 2 is: I q2 = B 2 (q 1 )= a q 1 c 2, 2 A strategy profile (q1, q 2 )isannashequilibriumifevery player s strategy is the best response to others strategies: q 1 = B 1 (q 2 ), and q 2 = B 2 (q 1 ) I Nash Equilibrium: q 1 = a + c 1 + c 2 3 c 1, q 2 = a + c 1 + c 2 3 c 2 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

8 The Cournot Model Illustration of Equilibrium I Geometrically, the Nash equilibrium is the intersection of both players best response curves. q 2 a c (a c 2) B 1 (q 2 ) Nash Equilibrium B 2 (q 1 ) 0 1 a c 2 q 1 2 (a c 1) Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

9 The Bertrand Model The Bertrand model describes interactions among firms (sellers) who set prices independently and simultaneously, underwhichthe customers (buyers) choose quantities accordingly. Key features I At least two firms producing homogeneous products; I Firms do not cooperate, i.e., there is no collusion; I Firms compete by setting prices simultaneously; I Consumers buy products from a firm with a lower cost (price). F If firms charge the same price, consumers randomly select among them. I The firms are economically rational and act strategically, seekingto maximize profits given their competitors decisions. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

10 The Bertrand Model Example: The Bertrand Game I Two firms decide their respective prices simultaneously; I The consumers buy products from a firm with a lower price. Game Formulation I The set of players is I = {1, 2}, I The strategy set available to each player i 2I is the set of all nonnegative real numbers, i.e., p i 2 [0, 1), I The payo received by each player i is a function of both players strategies, defined by i (p i, p i )=(p i c i ) D i (p 1, p 2 ) F F c i is the unit producing cost; D i (p 1, p 2)istheconsumers demandtoplayeri: (i) D i (p 1, p 2)=0ifp i > p i ;(ii)d i (p 1, p 2)=D(p i )ifp i < p i ; (iii) D i (p 1, p 2)=D(p i )/2 ifp i = p i. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

11 The Bertrand Model Equilibrium Analysis I Given player 2 s strategy p2,thebestresponse of player 1 is to select a price p 1 slightly lower than p 2 under the constraint that p 1 c 1 : I I p 1 = B 1 (p 2 )=max{c 1, p 2 } Given player 1 s strategy p 1,thebest response of player 2 is to select a price p 2 slightly lower than p 1 under the constraint that p 2 c 2 : p 2 = B 2 (p 1 )=max{c 2, p 1 } Both players will gradually decrease their prices, until one player gets to his producing cost. Therefore, the Nash equilibrium is 8 >< p1 = c 2, p2 = c 2, if c 1 < c 2 p1 = c 1, p2 = c 1, if c 1 > c 2 >: p1 = p2 = c, if c 1 = c 2 = c F Here we assume that the demand is always inelastic, henceeachseller wants to set the price as high as possible. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

12 The Bertrand Model Illustration of Equilibrium I Geometrically, the Nash equilibrium is the intersection of both players best response curves. p 2 c 2 B 2 (p 1 ) Nash Equilibrium B 1 (p 2 ) c 1 0 p 1 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

13 The Hotelling Model The Hotelling model studies the e ect of locations on the price competition among two or more firms. Key features I Two firms at di erent locations sell the homogeneous good; I The customers are uniformly distributed between two firms. I Customers incur a transportation cost as well as a purchasing cost. I The firms are economically rational and act strategically, seekingto maximize profits given their competitors decisions. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

14 The Hotelling Model Example: The Hotelling Game I Two firms at di erent locations decide their respective prices simultaneously; I The consumers buy products from a firm with a lower total cost, including both the transportation cost and the purchasing cost. Game Formulation I The set of players is I = {1, 2}, each locating at one end of the interval [0, 1]; I The strategy set available to each player i 2I is the set of all nonnegative real numbers, i.e., p i 2 [0, 1); I The payo received by each player i is a function of both players strategies, defined by i (p i, p i )=(p i c i ) D i (p 1, p 2 ) F F c i is the unit producing cost; D i (p 1, p 2)istheratioofconsumerscomingtoplayeri. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

15 The Hotelling Model Consumer Demand: D i (p 1, p 2 ) I Under price profile (p1, p 2 ), the total cost of a consumer at location x 2 [0, 1] buying products from player 1 or 2 is C 1 (x) =p 1 + w x, and C 2 (x) =p 1 + w (1 x) I Under (p1, p 2 ), two players receive the following consumer demand: D 1 (p 1, p 2 )= p 2 p 1 + w, D 2 (p 1, p 2 )= p 1 p 2 + w 2w 2w Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

16 The Hotelling Model Equilibrium Analysis I Given player 2 s strategy p2,thebestresponse of player 1 is I p 1 = B 1 (p 2 )= p 2 + w + c 1 2 Given player 1 s strategy p1,thebest response of player 2 is I Nash Equilibrium: p 2 = B 2 (p 1 )= p 1 + w + c 2 2 p 1 = 3w + c 1 + c c 1 3, p 2 = 3w + c 1 + c 2 + c Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

17 The Hotelling Model Illustration of Equilibrium I Geometrically, the Nash equilibrium is the intersection of both players best response curves. p 2 B 1 (p 2 ) Nash Equilibrium w+c 1 2 B 2 (p 1 ) 0 w+c 2 p 1 2 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69