Clubs, Beliefs, and Entrapment Scott Duke Kominers Harvard - - PowerPoint PPT Presentation

Clubs, Beliefs, and Entrapment

Scott Duke Kominers

Harvard University

AMS Session on Behavioral Sciences Joint Mathematics Meetings January 7, 2009

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 1 / 6

Overview

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 2 / 6

Overview

Question

Why do people join clubs?

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 2 / 6

Overview

Question

Why do people join clubs they would rather not be in?

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 2 / 6

Overview

Question

Why do people join clubs they would rather not be in?

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 2 / 6

Overview

Question

Why do people join clubs they would rather not be in?

Classical Answer

Because they have to.

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 2 / 6

Overview

Question

Why do people join clubs they would rather not be in?

Subtle Answer

Because they have to.

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 2 / 6

Overview

Question

Why do people join clubs they would rather not be in?

Subtle Answer

Because they think they have to.

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 2 / 6

Overview

Question

Why do people join clubs they would rather not be in?

Subtle Answer

Because they think they have to.

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 2 / 6

Examples

Question

Why do people join clubs they would rather not be in?

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we tip in restaurants?

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we tip in restaurants?

Answer

Because we have to.

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we tip in restaurants?

Answer

Because we have to. But the first tippers. . .

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we tip in restaurants?

Answer

Because we have to. But the first tippers wanted to.

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we tip in restaurants?

Answer

Because we have to. But the first tippers wanted to.

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we tip in restaurants?

Answer

Because we have to. But the first tippers wanted to.

Positive utility from early adoption

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we tip in restaurants?

Answer

Because we have to. But the first tippers wanted to.

Positive utility from early adoption + network effect

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we tip in restaurants?

Answer

Because we have to. But the first tippers wanted to.

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question Answer

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we use VHS tapes, instead of Betamax?

Answer

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we use VHS tapes, instead of Betamax?

Answer

Because we have to.

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we use VHS tapes, instead of Betamax?

Answer

Because we have to. But the first VHS users. . .

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we use VHS tapes, instead of Betamax?

Answer

Because we have to. But the first VHS users had to hope that VHS would catch on.

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we use VHS tapes, instead of Betamax?

Answer

Because we have to. But the first VHS users had to hope that VHS would catch on.

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we use VHS tapes, instead of Betamax?

Answer

Because we have to. But the first VHS users had to hope that VHS would catch on since Betamax was “better”.

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we use VHS tapes, instead of Betamax?

Answer

Because we have to. But the first VHS users had to hope that VHS would catch on since Betamax was “better”.

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Examples

Question

Why do we use VHS tapes, instead of Betamax?

Answer

Because we have to. But the first VHS users had to hope that VHS would catch on since Betamax was “better”.

Positive utility from early adoption + network effect = entrapment!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 3 / 6

Math Model

Notation

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 4 / 6

Math Model

Notation

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 4 / 6

Math Model

Notation

N > 1 players, indexed i = 1, 2, . . . , N

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 4 / 6

Math Model

Notation

N > 1 players, indexed i = 1, 2, . . . , N

Must (simultaneously) choose

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 4 / 6

Math Model

Notation

N > 1 players, indexed i = 1, 2, . . . , N

Must (simultaneously) choose

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 4 / 6

Math Model

Notation

N > 1 players, indexed i = 1, 2, . . . , N

Must (simultaneously) choose Join Status Quo

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 4 / 6

Math Model

Notation

N > 1 players, indexed i = 1, 2, . . . , N

Must (simultaneously) choose Join Status Quo J(i, n) S(i, n)

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 4 / 6

Math Model

Notation

N > 1 players, indexed i = 1, 2, . . . , N

Must (simultaneously) choose Join Status Quo J(i, n) S(i, n) where n = number of others choosing Join

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 4 / 6

Math Model

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo J(i, n) S(i, n)

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo J(i, n) S(i, n)

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions

Players sorted in order of enthusiasm for Join: ∂A ∂i < 0, ∂S ∂i > 0

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo J(i, n) S(i, n)

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions

Players sorted in order of enthusiasm for Join: ∂A ∂i < 0, ∂S ∂i > 0 Positive network effect: ∂A ∂n > 0, ∂S ∂n < 0

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo J(i, n) S(i, n)

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions

Functional forms (following Dixit (2003)):

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo J(i, n) S(i, n)

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions

Functional forms (following Dixit (2003)): J(i, n) = β + γn − δi S(i, n) = σi − τn

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo J(i, n) S(i, n)

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions

Functional forms (following Dixit (2003)): J(i, n) = β + γn − δi S(i, n) = σi − τn

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Dixit’s Assumptions Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Dixit’s Assumptions

Early adoption positive: β > δ + σ

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Dixit’s Assumptions

Early adoption positive: β > δ + σ Strong network effect: γ + τ ≥ δ + σ

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Dixit’s Assumptions

Early adoption positive: β > δ + σ Strong network effect: γ + τ ≥ δ + σ

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Dixit’s Assumptions

Early adoption positive: β > δ + σ Strong network effect: γ + τ ≥ δ + σ

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

N − 1 Join ⇒ N Join: β + (N − 1)(γ + τ) N > δ + σ

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions

Common ex ante belief

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Our Assumptions

Common ex ante belief: Prob(others Join) = q

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Key Condition Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Key Condition

Player i Joins ⇐ ⇒

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Key Condition

Player i Joins ⇐ ⇒ q(β + γ(N − 1) − δi) + (1 − q)(β − δi)

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Key Condition

Player i Joins ⇐ ⇒ q(β + γ(N − 1) − δi) + (1 − q)(β − δi) >

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Key Condition

Player i Joins ⇐ ⇒ q(β + γ(N − 1) − δi) + (1 − q)(β − δi) > q(σi − τ(N − 1)) + (1 − q)(σi)

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Key Condition

Player i Joins ⇐ ⇒ β + q(N − 1)(γ + τ) i > σ + δ

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

At least

• N
• β+q(N−1)(γ+τ)

β+(N−1)(γ+τ)

• players Join!

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

At least

• N
• β+q(N−1)(γ+τ)

β+(N−1)(γ+τ)

• players Join!

When β = 0 (no early adoption benefit), this is ⌊Nq⌋.

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

At least

• N
• β+q(N−1)(γ+τ)

β+(N−1)(γ+τ)

• players Join!

When β = 0 (no early adoption benefit), this is ⌊Nq⌋.

q = 1

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

At least

• N
• β+q(N−1)(γ+τ)

β+(N−1)(γ+τ)

• players Join!

When β = 0 (no early adoption benefit), this is ⌊Nq⌋.

q = 1 ⇒ all Join

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

At least

• N
• β+q(N−1)(γ+τ)

β+(N−1)(γ+τ)

• players Join!

When β = 0 (no early adoption benefit), this is ⌊Nq⌋.

q = 1 ⇒ all Join q = 0

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

At least

• N
• β+q(N−1)(γ+τ)

β+(N−1)(γ+τ)

• players Join!

When β = 0 (no early adoption benefit), this is ⌊Nq⌋.

q = 1 ⇒ all Join q = 0 ⇒ none Join

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

q = 1 ⇒ all Join q = 0 ⇒ none Join

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

q = 1 ⇒ all Join q = 0 ⇒ none Join convincing rumor

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

q = 1 ⇒ all Join q = 0 ⇒ none Join convincing rumor + network effect

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

q = 1 ⇒ all Join q = 0 ⇒ none Join convincing rumor + network effect entrapment!

Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Math Model

Results

q = 1 ⇒ all Join q = 0 ⇒ none Join convincing rumor + network effect entrapment!

QED Notation

N > 1 players i = 1, 2, . . . , N Join Status Quo β + γn − δi σi − τn

Our Assumptions

β+(N−1)(γ+τ) N

> δ + σ Prob(others Join) = q

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 5 / 6

Acknowledgments

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 6 / 6

Acknowledgments

• Prof. Avinash Dixit and Prof. Andrei Shleifer

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 6 / 6

Acknowledgments

• Prof. Avinash Dixit and Prof. Andrei Shleifer
• Mrs. Susan Schwartz Wildstrom

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 6 / 6

Acknowledgments

• Prof. Avinash Dixit and Prof. Andrei Shleifer
• Mrs. Susan Schwartz Wildstrom

Harvard College PRISE

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 6 / 6

Acknowledgments

• Prof. Avinash Dixit and Prof. Andrei Shleifer
• Mrs. Susan Schwartz Wildstrom

Harvard College PRISE Family

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 6 / 6

Acknowledgments

• Prof. Avinash Dixit and Prof. Andrei Shleifer
• Mrs. Susan Schwartz Wildstrom

Harvard College PRISE Family, friends

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 6 / 6

Acknowledgments

• Prof. Avinash Dixit and Prof. Andrei Shleifer
• Mrs. Susan Schwartz Wildstrom

Harvard College PRISE Family, friends, and you!

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 6 / 6

Acknowledgments

• Prof. Avinash Dixit and Prof. Andrei Shleifer
• Mrs. Susan Schwartz Wildstrom

Harvard College PRISE Family, friends, and you! (QED)

Scott Duke Kominers (Harvard) Clubs, Beliefs, and Entrapment January 7, 2009 6 / 6