# Game Theory & Firms. Jacob LaRiviere & Justin Rao April 20, 2016 Econ 404, Spring 2016

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1 Game Theory & Firms Jacob LaRiviere & Justin Rao April 20, 2016 Econ 404, Spring 2016

2 What is Game Theory?

3 Game Theory Intuitive Definition: Theory of strategic interaction Technical Definition: Account for agents optimizing behavior by using first order conditions to solve for equilibrium Normal form representation of a game Defining terms Players: agents in a game Actions: what players can do Payoffs: what players get given a set of actions Information States: what players know Strategy: Actions which relate to different states of the world Best Response: action which gives player highest payoff taking actions of the other players as given. Nash Equilibrium (NE): occurs when all players in a game are best responding to each other simultaneously Player A Player B Left Right Up (2,1) (0,0) Down (0,0) (1,2) NOTE: In a NE, no player has an incentive to change their action or deviate from that equilibrium

4 Game Theory Intuitive Definition: Theory of strategic interaction Technical Definition: Account for agents optimizing behavior by using first order conditions to solve for equilibrium Normal form representation of a game Defining terms Players: agents in a game Actions: what players can do Payoffs: what players get given a set of actions Left Player B Right Information States: what players know Strategy: Actions which relate to different states of the world Best Response: action which gives player highest payoff taking actions of the other players as given. Player A Up (2,1) Down (0,0) (0,0) (1,2) Nash Equilibrium (NE): occurs when all players in a game are best responding to each other simultaneously NOTE: In a NE, no player has an incentive to change their action or deviate from that equilibrium

5 Game Theory Intuitive Definition: Theory of strategic interaction Technical Definition: Account for agents optimizing behavior by using first order conditions to solve for equilibrium Normal form representation of a game Defining terms Players: agents in a game Actions: what players can do Payoffs: what players get given a set of actions Left Player B Right Information States: what players know Strategy: Actions which relate to different states of the world Best Response: action which gives player highest payoff taking actions of the other players as given. Player A Up (2,1) Down (0,0) (0,0) (1,2) Nash Equilibrium (NE): occurs when all players in a game are best responding to each other simultaneously NOTE: In a NE, no player has an incentive to change their action or deviate from that equilibrium In this game, there are two Pure Strategy Nash Equlibria

6 Game Theory Intuitive Definition: Theory of strategic interaction Technical Definition: Account for agents optimizing behavior by using first order conditions to solve for equilibrium Defining terms Players: agents in a game Actions: what players can do Payoffs: what players get given a set of actions Information States: what players know Strategy: Actions which relate to different states of the world Best Response: action which gives player highest payoff taking actions of the other players as given. Nash Equilibrium (NE): occurs when all players in a game are best responding to each other simultaneously Normal form representation of a game Battle of the Sexes : Coordination game Ann Baseball Game Ballet Bob Baseball Game (2,1) (0,0) Ballet (0,0) (1,2) NOTE: In a NE, no player has an incentive to change their action or deviate from that equilibrium In this game, there are two Pure Strategy Nash Equlibria

7 Game Theory Concept 1: Cournot

8 q1* Lets solve firm 1 s problem taking firm 2 s actions as given: max q1 π 1 q1 = (A b q 1 + q 2 )q 1 c 1 q 1 Using Game Theory to Understand Firms F.O.C.: π 1 q 1 = A b 2q 1 + q 2 c 1 = 0 Cournot Competition Two firms (firm 1 and firm 2) competing selling perfectly substitutable (e.g., homogeneous) goods. Simplify use algebra to get solution function for q 1. NOTE: Solution function is just q 1 solved in terms of the parameters. Firm 1 s solution function: A bq 2 c 1 2b = q 1 Inverse demand curve is assumed to be NOTE: q 1 < 0 for q 2, c 1 and b and q 1 > 0 for A (this should make sense) P = A b(q 1 + q 2 ) Firm 1's "Best Response" c1 A b Assume constant MC = c j for firms j = 1, Firm 1's "Best Response" q2

9 A b A bq 2 c 1 c 2b 2 2b = q 2 A + bq 2 + c 1 2c 2 = 4bq 2 Using Game Theory to Understand Firms Firm 2 has the exact same maximization problem; only difference is - must use firm 2 s own marginal cost. - firm 2 chooses their own quantity. A + c 1 2c 2 = 3bq 2 A + c 1 2c 2 3b A + Σ j i c j Nc 2 (N + 1)b = q 2 = q i Firm 2 s solution function: A bq 1 c 2 2b = q 2 Firm 1 and 2 have identical solution functions (with numbers switched). What is industry Q & P? Recall that firm 1 s solution was A bq 2 c 1 2b This is a system of two equations and two unknowns. Can sub in to solve. = q 1 Q = A + c 1 2c 2 3b P = A b 2A c 2 c 1 3b + A + c 2 2c 1 3b = A 3 + c 2 + c 1 3 = 2A c 2 c 1 3b = = N(A c) ҧ (N + 1)b 1 N + 1 A + Ncҧ N + 1

10 q1 Cournot Best Response Functions Insights from Cournot: 1) P N = 1 N+1 N ҧ A + c N lim N 1 N + 1 A + Ncҧ N + 1 = c ҧ = Ave MC So Cournot has perfect competition as a limiting case (recall we assumed constant MC) Firm 1's "Best Response" Firm 2's "Best Response" 2) This shows that firms behavior can be modeled as an equilibrium concept. - Can imagine think about consumer s responses to offerings (e.g., cannibalization/substitution within firm, churn/substitution across firms, etc.) similarly q2 3) Can think about Cournot using a figure - Equilibrium is where curves cross. 4) Bottom line: If a firm tries to reduce quantity to make more on intensive margin, competitors will partially offset that.

11 Game Theory Concept 2: Bertrand Whereas Cournot has firms choosing quantities and implicitly letting market determine price, Bertrand has them choosing price.

12 As before, lets solve firm 1 s problem taking firm 2 s actions as given: Bertrand Competition max P1 π 1 P 1 = ቊ (P 1 c j )M if P 1 < P 2 0 else Assume firms randomly allocated in event of tie (e.g., each firm gets.5m). Really a cautionary tale about why product differentiation is so important The moral of the story will be don t try to make money in a commoditized market. Q: What is best response of firm 1 given that firm 2 charges P 2? A: Set P 1 = P Q: Now how does firm 2 respond? A: Set P 2 = P Game environment: Assume constant MC = c j for firms j = 1,2. Firms choose P 1 and P 2. Entire market (size M) purchases from low cost firm. This continues until one firm hits their MC at MC = max(c 1, c 2 ). At that price (P=MC), the lower cost firm charges, the lower cost firm charges (P=MC-.01). Since no incentive for either firm to deviate from that equilibrium and lower cost firm takes the market.. Until a lower cost firm comes along. These are bad markets to be in.

13 Game Theory Concept 3: Hotelling Line Horizontal Differentiation: simplest way to introduce heterogeneity

14 Game Theory Concept 3: Hotelling Line Horizontal Differentiation: simplest way to introduce heterogeneity

15 Hotelling Line Assume that substitutes exist but they are imperfect and therefore costly. - ex: Stevens Pass versus Crystal Mountain Idea: How much should firms charge given where they are in horizontal product space? Assume firms identical in every way but location (simplest possible model of differentiation. Model: 1) two firms at each end of a linear city 2) travelling is costly (decreases utility by t) 3) N consumers uniformly distributed between firms (on unit interval ) with common valuations v. v tx Pi Firm i Purchase -> Utility: U i = ቐv t 1 x Pn Neighbor firm purchase 0 No Purchase Assuming marginal consumer defined by: v tx Pi = v t 1 x P n Pi+t 2t Pn = x, Di P n, P i = Nx NOTE: Might not be profit maximizing to sell to entire market (e.g., high MC) and lower t (e.g., more substitutable) means lots of competition.

16 Hotelling Line So firm i s maximization problem is max Pi (Pi c)nx = P i c N P n Pi + t 2t 1 F.O.C: P i c N + N P n Pi +t = 0 2t 2t = P n 2P i +t + c P n +t+c 2 = P i Insight: travel costs are no different than any other cost of production. Costs are costs.

17 Game Theory Concept 4: Vertical Differentiation Horizontal Differentiation: simplest way to introduce heterogeneity

18

19

20 VR headsets: different strengths and weaknesses Differentiated on embedded 3D display, graphics quality, number of uses, comfort to user, etc

21 A consumer buys at a price p if θs > p given that consumer s taste parameter θ. D p = N(1 F(p/s)) = N(1 F(θ )) Quality Competition (lifted from Tirole 88) Assume there is a clear order of preferences on quality (e.g., high resolution better than low res) Caveat: How could we identify this taste parameter empirically? Need random variation in prices for products of different quality and to observe consumption decisions. Now assume there are two goods: p 1, s 1, p 2, s 2, s 2 > s 1, p 2 > p 1 Define quality as s and taste for quality as θ -> Utility: U i = ቐ θs p Purchase 0 No Purchase Assume that tastes are distributed throughout the population according to a cumulative distribution function (CDF) F(θ) You can show that if s 2 > s 1 then all consumers purchase p 2 p 1 high quality good if they purchase. (HW problem) What if s 2 > s 1, p 2 > p 1, s 2 p 2 < s 1 p 1? Now what gives a higher utility is a function of θ. θ s 2 p 2 = θ s 1 p 1 θ = (p 2 p 1 )/(s 2 s 1 ) conditional on purchasing

22 Game Theory Concept 5: Entry and Exit Fixed costs & Entry and Exit.

23 Cournot and Firm Entry/Exit Answer is that a marginal firm should be just indifferent between entering and not entering the market accounting for F. P N c q = F defines N NOTE 1: Assume all firms have same MC to makes things easier NOTE 2: Technically it is simply the NPV of profits: Recall first insight from Cournot: P N = 1 N+1 lim N 1 N + 1 A + N ҧ A + c N+1 Ncҧ N + 1 = c ҧ = Ave MC Assume there are fixed costs of entry F like permits, licenses, factory infrastructure, entering a new market, etc What is equilibrium N? σ t=1 δ t P N ci q i = F NOTE 3: Idea is that some firms can earn rents from being in industry already. NOTE 4: F not always fixed over time. For example, cloud computing has turned fixed IT costs to marginal costs. NOTE 5: Can actually identify fixed costs of empirically if you are clever enough (e.g., identify MC, observe P, know demand to get quantity).

24 Empirical Exercise 6: Entry into Collaborations When should firms partner versus go it alone?

25 Research Collaboration: Research Joint Venture. Firms work together on research (e.g., share costs) and share rewards. Some of these collaborations are funded by the US Government. Question: What is stronger? Drive to leverage or drive to appropriate government funds.

26 Research Collaboration: Research Joint Venture. Firms work together on research (e.g., share costs) and share rewards. Some of these collaborations are funded by the US Government. Question: What is stronger? Drive to leverage or drive to appropriate government funds. Funding and partnering could be complements (leveraging is better) or substitutes (appropriating is better). Your assignment will have you investigate.

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