Transcript Lecture 29 : Direct Benefit Effects

Lecture 29 : Direct
Benefit Effects
JOEL OREN
What are the factors that govern our
valuations?
•Intrinsic: we have our own intrinsic
valuation for a product.
•External: we care about how many
people are using the product when
considering buying it.
•Questions:
1. How many people buy the product?
2. At what price?
3. What are the dynamics that govern
these outcomes?
$50
$28
?
Warmup: No Network Effects
•A huge population – one’s decision is negligible
w.r.t. entire population.
𝑟(𝑥)
Continuous, monotone,
market demand curve
•If a significant group of people act – their
collective action carries weight.
•Each person’s name – some 𝑥 ∈ 0,1 .
•Reservation price: 𝑟(𝑥) – maximal price at
which 𝑥 buys.
•Assume: 𝑥 < 𝑦 ⇒ 𝑟 𝑥 > 𝑟(𝑦)
0
1
𝑥
Price to demand correspondence
𝑟(𝑥)
•By continuity: for every price r 1 < 𝑝 < 𝑟(0)
there is a fraction of the market that will buy
the item at price 𝑝.
𝑟(0)
𝑝
𝑟(1)
0
𝑟 −1 (𝑝)
All buy
1
𝑥
Markets with Multiple Competing Producers
•Multiple producers, producing the same
product.
•All have identical minimum production cost
𝑝∗ .
•What will be resulting equilibrium? (price)
• 𝑝∗ !
•Why?
Blackberry Samsung LG
⋯
∗
Fixed (equilibrium price) of 𝑝 and
resulting demand
•𝑝∗ determines a unique 𝑥 ∗ such that 𝑟(𝑥 ∗) =
𝑝∗ .
•𝑥 ∗ is an equilibrium. Why?
𝑟(𝑥)
𝑟(0)
𝑝∗
•A word on social welfare: (total valuation) –
(total payments).
𝑟(1)
0
𝑥 ∗ = 𝑟 −1 (𝑝)
1
𝑥
Back to Markets with Direct Benefit effects
•Direct benefit: the more people, the more beneficial it is for me  my
valuation goes up.
•Our model: if a consumer 𝑥 believes that a 𝑧 fraction of population buys
product, valuation is scaled by 𝑓(𝑧): x buys iff 𝑓 𝑧 ⋅ 𝑟 𝑥 ≥ 𝑝∗ .
•Assume:
• 𝑓(⋅) – is continuous, monotone increasing
• 𝑓 0 = 0, 𝑓(1)
•Consumer 𝑥 buys product if 𝑝∗ ≤ 𝑓 𝑧 𝑟(𝑥)
Self-Fulfilling Convergence Equilibirum
• All consumers share belief about 𝑧 – all believe that a 𝑧 fraction will buy the
product.
• Real world examples?
• Self-fulfilling convergence equilibrium: if the fraction of buying consumers ends
up being the predicted fraction 𝑧.
• If 𝑥′ buys at price 𝑝∗ , and 𝑥 < 𝑥′  consumer 𝑥 buys also (𝑟 𝑥 > 𝑟 𝑥 ′ ).
Selling price is exactly 𝑝∗ = 𝑓 𝑧 𝑟(𝑧)
•Note: if 𝑧 = 0, no one buys (𝑓 0 = 0).
Concrete Example: r x = 1 − x, 𝑓 𝑧 = 𝑧
• Three equilibria:
• z=0
• 𝑧’, 𝑧’’ such that 𝑝∗ = 𝑧 ′ 1 − 𝑧 ′ , 𝑝∗ = 𝑧′′(1 − 𝑧 ′′ )
1
4
•Any price above leads to no one buying the
product
•As 𝑝∗ goes down, 𝑧′′ goes up, 𝑧′ goes down.
𝑝𝑟𝑖𝑐𝑒
1
4
𝑝∗
0
𝑧’
1
2
𝑧′′
𝑥
∗
′
Why 𝑝 and 0, 𝑧 , 𝑧′′ are in an
equilibrium
•For the price 𝑝∗ , the demand must be either 0, 𝑧′,
or 𝑧′′. Why?
• Case 1: 0 < 𝑧 < 𝑧′
• Case 2: 𝑧 ′ < 𝑧 < 𝑧′′
• Case 3: z ′′ < 𝑧 < 1
𝑝𝑟𝑖𝑐𝑒
• Important:
∗
𝑝
• 𝑧′′ is a stable: if 𝑧 is in region 2 or 3  dynamics
will push to z’’.
1
0
• 𝑧′ is unstable (“tipping point”).
𝑧’
2
3
𝑧′′𝑥
Iterative Demand Dynamics
•Scenario: the “product” is website. Certain fraction of population visit it
everyday.
•𝑝∗ -- Cost of visiting a website (e.g., time to read main headlines).
•People visit the website on day 𝑖 based on day 𝑖 − 1.
•Move away from self-fulfilling convergence equilibrium: fraction visited does
not equal prediction necessarily.
•For prediction 𝑧, 𝑥 visits only if 𝑟 𝑥 𝑓 𝑧 ≥ 𝑝∗
•If prediction is 𝑧, price is 𝑝∗ , and 𝑥 visits only
if 𝑟 𝑥 𝑓 𝑧 ≥ 𝑝∗  fraction of consumers visiting
is:
∗
𝑝
𝑧 = 𝑟 −1 (
)
𝑓 𝑧
•In our example: 𝑟 −1 𝑦 = 1 − 𝑦:
𝑧=1
𝑝∗
−
𝑧
Back to the iterative dynamics
•Based on fraction visited on day 𝑡 − 1 -- 𝑧𝑡−1 , (and price 𝑝∗ ) people will visit
website:
•𝑧𝑡−1 = 𝑔 𝑧𝑡−1
•We will now show convergence to one of three equilibria, based on 𝑧0 -initial belief.