A Brief Introduction to Game Theory for Marketing · 2017-02-13 · Game: A situation or context in...

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© Arvind Rangaswamy 2017, All Rights Reserved

February 14-21, 2017

MKTG 555: Marketing Models

A Brief Introduction to Game Theory for Marketing

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Basic Definitions

Game: A situation or context in which players (e.g., consumers, firms)

make strategic decisions that take into account each other’s actions

and responses.

Strategy: A rule or plan of action for playing a game. Each player has

a strategy 𝒔𝒊.

Strategy profile/vector: A rule or plan of action for all players in the

game (s1, s2, s3, si, …..sN).

Payoff: The value (e.g., utility, dollars) associated with a possible

outcome in the game. Each player (i) has his/her own payoff in a

game.

Optimal strategy: For each player, the strategy (𝒔𝒊∗) that maximizes

that player’s expected payoff.

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Some Types of Non-Cooperative Games

Static Games with Complete Information

Dynamic Games with Complete Information

Static Games with Incomplete (Private) Information

Simple Dynamic Games with Incomplete Information

(Signaling Games)

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(Signaling Games)

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Concept of Dominant Strategy

Dominant Strategy for firm A: Advertise

Dominant Strategy for firm B: Advertise

Payoff Matrix for an Advertising Game

Firm B Advertise Don’t Advertise

Firm A Advertise (10, 6) (15, 1) Don’t Advertise (7, 8) (10, 3)

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Nash Equilibrium

Strategy for firm A: ?

Strategy for firm B: ?

Payoff Matrix for a Second Advertising Game



A strategy profile (𝒔𝟏∗ , 𝒔𝟐

∗ , … . , 𝒔𝒏∗ ) is a Nash equilibrium, if for each

player i, his choice 𝒔𝒊∗ is the best response to the other players

choices, 𝒔−𝒊∗ .

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Nash Equilibrium



Payoff Matrix for a Third Advertising Game



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Nash Equilibrium



Payoff Matrix for a Fourth Advertising Game



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Nash Equilibrium



Payoff Matrix for a Fifth Advertising Game



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(Signaling Games)

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Subgame-Perfect Nash Equilibrium (from Gibbons 1997)

Two potential Nash equilibria: (L, L′) and (R, R′). Which one will be played?

A Game that Relies on a Noncredible Threat (Gibbons)

L′ R′

L (1, 2) (1, 2) R (0, 0) (2, 1)

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Players: 2 firms selling identical products

Strategies: Quantities produced of identical products (substitutes) are 𝒒𝒊 𝐚𝐧𝐝 𝒒−𝒊. Players make their moves simultaneously.

Payoffs:

Cost of production is c per unit

Price: 𝒑 = 𝒂 − 𝒃 𝒒𝒊 + 𝒒−𝒊

𝑷𝒓𝒐𝒇𝒊𝒕𝒊(𝒒𝒊, 𝒒−𝒊) = (𝒑 − 𝒄)𝒒𝒊

𝑷𝒓𝒐𝒇𝒊𝒕−𝒊(𝒒𝒊, 𝒒−𝒊) = (𝒑 − 𝒄)𝒒−𝒊

Cournot Duopoly

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a Marginal revenue curve for Monopolist (slope -2b)

c

𝒒𝑴∗

Demand curve (slope –b)

𝒑𝑴∗

𝒒𝑪∗

Basic Econ

𝒒𝑴∗ : 𝐐𝐮𝐚𝐧𝐭𝐢𝐭𝐲 𝐩𝐫𝐨𝐝𝐮𝐜𝐞𝐝 𝐛𝐲 𝐚 𝐦𝐨𝐧𝐨𝐩𝐨𝐥𝐢𝐬𝐭

𝒒𝑪∗ : 𝐐𝐮𝐚𝐧𝐭𝐢𝐭𝐲 𝐩𝐫𝐨𝐝𝐮𝐜𝐞𝐝 𝐮𝐧𝐝𝐞𝐫 𝐩𝐞𝐫𝐟𝐞𝐜𝐭 𝐜𝐨𝐦𝐩𝐞𝐭𝐢𝐭𝐢𝐨𝐧

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𝒂 − 𝒄

𝟐𝒃

𝒒𝑴∗ =

𝒂 − 𝒄

𝟐𝒃

𝒒𝑪∗ =

𝒂 − 𝒄

𝒃

𝒂 − 𝒄

𝒃

𝒒𝒊

𝒒−𝒊

𝑩𝑹𝒊−𝟏(𝒒𝒊) =𝒂−𝒄

𝟐𝒃−

𝒒𝒊

𝟐

𝑩𝑹𝒊 𝒒−𝒊 =𝒂−𝒄

𝟐𝒃−

𝒒−𝒊

𝟐

Quantities that produce monopoly profits

Cournot-Nash Equilibrium

Cournot Duopoly

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Economic Implications of Different Market Structures

Marginal revenue curve for Monopolist (slope -2b)

Demand curve (slope –b)

𝒂 − 𝒄

𝟐𝒃

𝒒𝑴∗

𝒒𝑪∗

𝒂 − 𝒄

𝒃

𝒒𝒊

𝒒−𝒊

𝑩𝑹𝒊−𝟏 𝒒𝒊 =𝒂−𝒄

𝟐𝒃−

𝒒𝒊

𝟐

𝑩𝑹𝒊 𝒒−𝒊 =𝒂−𝒄

𝟐𝒃−

𝒒−𝒊

𝟐


c

𝒒𝒊∗

𝒒−𝒊∗

𝒒𝒊∗ = 𝒒−𝒊

∗ =𝒂 − 𝒄

𝟑𝒃

𝒒𝑴∗ < 𝒒𝒊

∗ + 𝒒−𝒊∗ < 𝒒𝑪

∗

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Players: 2 firms selling identical products

Strategies: Quantities produced of identical products (substitutes) are 𝒒𝒊 𝐚𝐧𝐝 𝒒−𝒊. Player 1 makes the first move, and player 2 then follows (Leader-Follower game)

Payoffs:


Price: 𝒑 = 𝒂 − 𝒃 𝒒𝒊 + 𝒒−𝒊

𝑷𝒓𝒐𝒇𝒊𝒕𝒊(𝒒𝒊, 𝒒−𝒊) = (𝒑 − 𝒄)𝒒𝒊

𝑷𝒓𝒐𝒇𝒊𝒕−𝒊(𝒒𝒊, 𝒒−𝒊) = (𝒑 − 𝒄)𝒒−𝒊

Stackelberg Duopoly

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𝒒𝑴∗ =

𝒂 − 𝒄

𝟐𝒃

𝒂 − 𝒄

𝟐𝒃

𝒒𝑪∗ =

𝒂 − 𝒄

𝒃

𝒒𝒊

𝒒−𝒊

𝑩𝑹𝒊−𝟏 𝒒𝒊 =𝒂−𝒄

𝟐𝒃−

𝒒𝒊

𝟐


Dynamic Game Backward Induction

Will the Stackelberg Nash equilibrium be the same as the Cournot equilibrium?

Will 𝒒𝒊∗ here be higher than in the Cournot

equilibrium?

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Players: 2 (or more) firms selling identical products

Strategies: Price charged 𝒑𝒊 𝐚𝐧𝐝 𝒑−𝒊.

Payoffs:


Demand Q = − p; 𝒑 ≤ 𝜶, 𝒑 > 𝟎 𝟎 𝒑 > 𝜶

where: 𝒑 = 𝑴𝒊𝒏(𝒑𝒊, 𝒑−𝒊)

Profiti =

− 𝒑𝒊 𝒑𝒊 − 𝒄 𝒊𝒇 𝒑𝒊 < 𝒑−𝒊 (𝜶−𝒑)

𝟐𝒑 − 𝒄 𝒊𝒇 𝒑𝒊 = 𝒑−𝒊 = 𝒑

𝟎 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆

Bertrand Competition

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c

𝒑𝑯

𝒑𝒊

𝒑−𝒊

𝒑𝑴 𝒑𝑯 =

𝒑𝑴

Bertrand Nash equilibrium

Reaction Functions in Bertrand Price Competition

𝒑𝑴: 𝐌𝐨𝐧𝐨𝐩𝐨𝐥𝐲 𝐩𝐫𝐢𝐜𝐞

𝒑𝑯: 𝐇𝐢𝐠𝐡𝐞𝐬𝐭 𝐟𝐞𝐚𝐬𝐢𝐛𝐥𝐞 𝐩𝐫𝐢𝐜𝐞

c: Unit cost

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In an imperfect information game, the players are unaware of the actions chosen by other players. However they know who the other players are, what their possible strategies/actions are, and their preferences/payoffs, i.e., information about the other players in imperfect information is complete.

In incomplete information games, players may or may not know some information about the other players, e.g., their “type”, their strategies, payoffs or their preferences.

Imperfect Vs. Incomplete Games

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Balasubramanian Equilibrium Results

(N Retailers + 1 Direct Marketer)

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Balasubramanian Equilibrium Results

(N Retailers + 1 Direct Marketer)

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Equilibrium Results for Two-Stage Game

Coverage () + Price Competition

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Equilibrium Results for Two-Stage Game

Coverage () + Price Competition

A Brief Introduction to Game Theory for Marketing · 2017-02-13 · Game: A situation or context in...

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Transcript of A Brief Introduction to Game Theory for Marketing · 2017-02-13 · Game: A situation or context in...