# Chapter 17. Network Effects

[TOC]

## Preliminaries
* **Externality**: The welfare of an individual is affected by the actions of other individuals.
* A consumer's willingness to pay is determined by two things:
* Intrinsic interest
* `The number of other people using the good` (externality)

## 17.1 The Economy without Network Effects

### Reservation Prices.

:::info
**Notations & Presumptions**:
* ***reservation price***: Maximum amount a consumer is willing to pay for one unit of good
* We arrange consumers in the interval between 0 and 1 in order of decreasing intrinsic interest, i.e. if consumer $x$ has a intrinsic interest price than $y$, then $x < y$
* Denote $r(x)$ as the intrinsic interest(reservation price) of consumer $x$
* We assume that function $r$ is continuos and one-to-one.
* Denote market price for a unit of the good is $p$
:::

When $p$ lies between r(1) and r(0), since $r$ is continuos and one-to-one, </br>there must be a unique number $x$ such that $r(x) = p$


* All consumers between 0 and $x$ buy the product, while all consumers above $x$ don't
* $x$ fraction of the population buys the product.

### The Equilibrium Quantity of the Good.

Suppose the good can be produced at a constant cost of $p^*$ per unit, </br> and the producers supply the good at a price no less than $p^*$


* The profit to a producer with the price above $p^*$ would be driven to zero by competition from other producers. Thus we can assume a market price of $p^*$
* We can find a unique $x^*$ such that $r(x^*) = p^*$
* $x^*$ represents the equilibrium quantity
* upward pressure : less than an $x^*$ of the population purchased the good
* downward pressure : more than an $x^*$ of the population purchased the good

## 17.2 The Economy with Network Effects

:::info
**Notations & Presumptions**:
* Denote $r(x)f(z)$ as reservation price
* where $r(x)$ is the intrinsic interest
* where $f(z)$ measures the benefit to consumer from having a $z$ fraction of population using the good
* Function $f$ is increasing and continous
:::

### Equilibria with Network Effects.

***self-fulfilling expectations equilibrium***:
If each of consumer expects $z$ fraction of the population using the product
and the fraction of population who actually purchased is $z$

e.g.
If price $p^* > 0$, and everyone expects $z=0$, then reservation price of each consumer $x$ is $r(x)f(0) = 0$, which is below $p^*$, no one wants to purchase and hence $z = 0$ has been fulfilled.

:::success
**Claim.**
*If the price $p^*$ and quantity $z$ form self-fulfilling expectations equilibrium, then $p^* = r(z)f(z)$*

**Explain.**
Since distribution of consumer is based on intrinsic interest in decreasing order, and each consumer has same expectation on $z$, so the price $p^*$ which make exactly fraction $z$ of people purchase is the lowest reservation price $r(z)f(z)$
:::

## 17.3 Stability, Instability, and Tipping Points


Suppose $z$ fraction of the population were to purchase the good

* If $0 < z < z'$ or $z'' < z < 1$, since $r(z)f(z) < p^*$, the purchaser will value the good at less than $p^*$, and hence will wish they had not bought it, then there is *downward pressure*
* if $z' < z < z''$ since $r(z)f(z) > p^*$, the consumer have not purchased but will wish they had, then there is *upward pressure*

:::info
* ***Stability***:
* If $z$ is slightly larger than $z''$, it'll be pushed back to $z''$
* If $z$ is slightly smaller than $z''$, it'll be pulled back to $z''$
* ***Instability***:
* If $z$ is slightly larger than $z'$, it'll be pulled to $z''$
* If $z$ is slightly smaller than $z'$, it'll be pushed to $0$
* ***tipping point(critical point)***:
$z'$ is also a tipping point,
if $z$ is near and not equal to $z'$, then $z$ will be driven to $0$ or $z''$
:::


Define a function $\hat{z} = g(z)$:
if each consumers expects a $z$ fraction of the population to purchase the good, </br> then in fact a $g(z)$ fraction will do so.

## 17.4 A Dynamic View of the Market

If everyone believes a $z$ fraction of the population will use the product,
then consumer $x$ will purchase if $r(x)f(z) \geq p^*$.
Hence the set of people who will purchase is between 0 and $\hat{z}$, where $\hat{z}$ satisfy $r(\hat{z})f(z)=p^*$
Equivalently,
* $r(\hat{z}) = \frac{p^*}{f(z)}$
* $\hat{z}=r^{-1}(\frac{p^*}{f(z)})$


### The Dynamic Behavior of the Population.

`Long story short...`


## 17.5 Industries with Network Goods
Some qualitative level observations from the model **with network effects**:
1. About marketing a product
- The marketing will not succeed unless the producer can get past the **tipping point**
- **_How to?_** Set an initial low price to pass the tipping point more easily. Then raise the price gradually to overcome the initial losses.
2. About social optimality
- The equilibria are typically not optimal
- **_Why?_** Consider this:
- The consumers between $z^*$ and $z^*+c$ (for some small constant $c>0$) do not want to buy originally
- However, if they all did the purchase, then the value of the product to each consumer between 0 and $z^*$ would increase from $r(x)f(z^∗)$ to $r(x)f(z^∗+c)$
3. About product competition
- Being the first to reach the tipping point is very important

## 17.6 Mixing Individual Effects with Population-Level Effects
:::info
**Reminder**
- $r(x)$: **intrinsic interest** of consumer $x$, strictly decreasing
- $f(z)$: **benefit to each consumer** from having a $z$ fraction of the population use the good, strictly increasing
- $r(x)f(z)$ is the **reservation price** of consumer $x$ when a $z$ fraction of the population is using the good
- $p^*$: the price per unit of product
- $g(z)$: a function that gives the outcome (**true fraction** $\hat{z}$) in terms of the **expected fraction** $z$
$$
g(z) = \begin{cases}r^{-1}(\frac{p^*}{f(z)}) & \text{, if}\ \frac{p^*}{f(z)}\leq r(0)\ \text{holds}\\ 0 & \text{, otherwise}\end{cases}
$$
:::
- In previous sections, we focus on the case with assumption $f(0)=0$. Now we consider a more general model: $f(0)>0$.

### A concrete model
- $f(z) = 1+az^2$ for a constant $a>0$
- $r(x) = 1-x$
- $r(x)f(z) = (1-x)(1+az^2)$
- Assume $0<p^*<1$
- The outcome $\hat{z} = g(z) = r^{-1}(\frac{p^*}{f(z)}) = 1-\frac{p^*}{1+az^2}$, since $\frac{p^*}{f(z)}\leq r(0)$ always holds in this example
> $r(0)=1, f(z)\geq 1, p^*<1 \ \Rightarrow \ \frac{p^*}{f(z)}\leq r(0)$

|$f(0)=0$|$f(0)>0$|
|---|---|
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### Growing an audience from zero
- With $f(0)>0$, $z=0$ is no longer an equilibrium
- Apply the dynamics that were defined in Section 17.4, a stable equilibrium $(z^*,z^*)$ is reached

### Bottlenecks and Large Changes
- There is a much higher stable equilibrium $(z^{**},z^{**})$ that would be much more desirable if only it could be reached
- $(z^*,z^*)$ is a **bottleneck**
- If the producer lowers the price $p^*$, a larger $g(z)$ is derived.

|Original|After $p^*$ gets smaller enough|
|---|---|
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## 17.7 Advanced Material: Negative Externalities and the El Farol Bar Problem
### El Farol Bar problem
- Scenario:
- A bar with seating for 60 people has live music every Thurday evening
- There are 100 people each week who are interested in going to the bar
- Showing up for the music is enjoyable only when at most 60 people do so
- With more than 60 people in attendance, it would be preferable to have stayed home
- The problem describes a situation with a **negative externality**
- The payoffs to going to the bar decrease as the number of participants increases

### Nash equilibria
:::info
**Game formulation of El Farol Bar problem**
- Imagine **only a single concert** is hosted, so the game is only played once
- Players: 100 people who are interested in the concert
- Strategies: **_Go_** or **_Stay_**
- Payoffs: $$ \begin{cases}0 & \text{, if Stay is chosen}\\ x>0 & \text{, if Go is chosen and at most 60 people choose Go}\\ -y<0 & \text{, if Go is chosen and more than 60 people choose Go}\end{cases} $$
:::
- There is no equilibrium in which all players use the same pure strategy
- If **heterogeneous strategies** are allowed, any outcome in which exactly 60 people choose Go and 40 people choose Stay is a pure strategy Nash equilibrium
- If **mixed strategies** are allowed, in which each player chooses **_Go_** with the same probability $p$, we need to choose $p$ so that each player is indifferent between choosing **_Go_** and choosing **_Stay_**
:::success
**Goal:** expected payoff of **_Go_** = expected payoff of **_Stay_**
$$
\begin{aligned}
\ & x\cdot Pr[at\ most\ 60\ go] - y\cdot Pr[more\ than\ 60\ go] = 0 \\
\Rightarrow\ & x\cdot Pr[at\ most\ 60\ go] - y\cdot (1-Pr[at\ most\ 60\ go]) = 0 \\
\Rightarrow\ & Pr[at\ most\ 60\ go] = \frac{y}{x+y}
\end{aligned}
$$
When $x = y$, choosing $p = 0.6$ will work
:::

### Analogy
- Consider the **two-player version** of the El Farol Bar problem, each player wants to attend the bar as long as the other player doesn’t
- It is a **Hawk-Dove game**

|Two-player El Farol Bar problem|Hawk-Dove game|
|---|---|
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### Repeated El Farol problems
- Back to the original scenario: a live music is held every Thursday, so the game is played **repeatedly**
- How to study such problems? The textbook gives two approaches:
1. view the full sequence of Thursdays as a **dynamic game**
2. apply a **forecasting rule**
- A forecasting rule is any function that maps the past history of play to a prediction about the actions in the future
- Each forecasting rule produces a number between 0 and 99. One goes to the bar if his forecasting rule produces a number that is at most 59, and stays home otherwise
- If everyone uses the same forecasting rule for the audience size, then everyone will make very bad predictions
- The optimal result is that roughly 60% of the agents produce a forecast that causes them to go the bar. A long line of research has tried to understand whether the system converges to such a state

## Additional: Positive Externalities and the Social-networking Problem
### Social-networking problem
- Scenario:
- 100 employees are encouraged to use a particular corporate social-networking site
- Each employee wants to use the social-networking site if at least 60 other employees do so as well; otherwise, the effort required would not be worth it
- The problem describes a situation with a **positive externality**

### Nash equilibria
:::info
**Game formulation of Social-networking problem**
- Players: 100 empolyees who are encouraged to use the site
- Strategies: **_Join_** or **_Not Join_**
- Payoffs: $$ \begin{cases}0 & \text{, if Not Join is chosen}\\ -x<0 & \text{, if Join is chosen and at most 60 people choose Join}\\ y>0 & \text{, if Join is chosen and more than 60 people choose Join}\end{cases} $$
:::
- There are just two pure-strategy equilibria:
1. Everyone chooses **_Join_**
2. Everyone chooses **_Not Join_**
- The same mixed-strategy equilibrium that applied to the El Farol Bar problem also holds
:::success
**Goal:** expected payoff of **_Join_** = expected payoff of **_Not Join_**
$$
-x\cdot Pr[at\ most\ 60\ join] + y\cdot Pr[more\ than\ 60\ join] = 0
$$
We get the same value of $p$ that we did in the El Farol Bar problem
:::

### Analogy
- Consider the **two-player version** of the social-networking problem, the two players try to make their actions the same
- It is a **Coordination game**

|Two-player social-networking problem|Coordination game|
|---|---|
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