Price Competition and the Use of Consumer Data

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# Price Competition and the Use of Consumer Data
### Brian C. Albrecht<br> <span style="font-size: 50%;"> University of Minnesota<br> Department of Economics</span>
### July 6, 2020

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Consumer Data and Price Discrimination
====================================

.left-column[
-  Firms are collecting more and more data on consumers

-  That data/information   is vital to price discrimination

-  Monopoly: information raises total profit
  -  Information is valuable for single receiver: Blackwell (1951), Bergemann, Brooks, and Morris (2015)
    
-  Unless output increases, price discrimination **lowers consumer surplus**
]

.right-column[
  ![](./images/pricecomp/pricing.gif)
]

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Competition and Price Discrimination
====================================
-  But one role of price discrimination is to lure buyers from other sellers

-  Studying this aspect of price discrimination requires an explicit model of competition

-  We know price discrimination can **raise consumer surplus**
    - *e.g.* Holmes (1989)
    
--

- However, most models only consider special information structures
  - *e.g.* Full information vs. no information

- Goal: build a model that does not rely on specific details of the data
  - **Robust mechanism design** (Bergemann & Morris 2013)

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My Dissertation Chapter
====================================

.left-column[
- Fix the underlying valuations, i.e. demand curve in the aggregate market

- Vary the information to each seller, allowing price discrimination

- Not all firms need to have the same information `$\implies$` higher order beliefs matter

- *e.g.* Target doesn't know my exact address, but Target knows that Amazon knows...
]
.right-column[
  ![](./images/pricecomp/demand.png)
  ]
  
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Environment Considered
==================

- 2 sellers of differentiated goods, produced at constant marginal cost (normalized to zero)

- 3 types of buyers who are willing to pay at most 1

-  Loyal to seller 1, only buy from 1
    
    -  Loyal to seller 2, only buy from 2
    
    -  Indifferent between seller 1 and seller 2, buy at cheapest price

-  Definition: a market for seller is a distribution of types of buyers

- Example: `$\left(\frac{1}{4}, \frac{1}{6}, \frac{7}{12}\right)$`

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Main Results
====================================

-  New construction of  equilibrium, requires price dispersion

-  Price discrimination can help or hurt consumers even when output is constant
    - Consumer surplus under complete public information `$>=$` consumer surplus under no information

- **Bounds**
  - Max consumer surplus: complete public data
  - Min consumer surplus with public data: nested markets
  - Lower consumer surplus with private data: (conjecture a specific bound, do not have proof yet)

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Road Map for Today's Talk
================
1. Introduction: Consumer Data and Competition

2. **Environment and Baseline Information**
  - Complete Information
  - No Information

3. General Public Information
  - Consumer Surplus Maximizing
  - Profit Maximizing

4. General Public + Private Information

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Basic Game: Consumer Types
============================

- 2 sellers with differentiated goods that cost 0 to produce

- Each consumer's type is a pair of valuations

`$$v = (v_1,v_2) \in \{(1,0),(0,1),(1,1)\} = V$$`

- Continuum of non-strategic consumers each have unit demand for total consumption

- Summarized through a residual demand curve

- Common prior: `$(m_{10}, m_{01}, m_{11})$`

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Information Structure
============================

- An information structure is a set of signals for each seller `$S_i$`,

- And a probability distribution which maps the profile of the consumer's values to the profile of signals:

`$$\pi~:~ V \to \Delta (S).$$`

- The utility functions and the information structure `$(S, \pi)$` are the parameters for a game of incomplete information

- Define the rest of the game fixing `$(S, \pi)$`

- Most price discrimination models fix an implicit information structure

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Strategies for Fixed `$(S, \pi)$`
============================

- Fix `$(S, \pi)$`

- Seller `$i$` observes a signal `$s_i \in S_i$`

- Pure strategy for seller `$i$` is a price `$\{p_i\}_{s_i} \in \mathbb{R}_+^{|S_i|}$` and

- Discontinuity of payoffs requires mixed prices (price dispersion)

- Mixed strategy, `$F_i(p|s_i)$`, is the probability that `$p_i\leq p$` given receiving a signal `$s_i$`

- Buyers: choose which seller (or no seller) to maximize value minus price

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Equilibrium
============================

-  For a given `$(S, \pi)$`, a strategy profile is a Bayes Nash equilibrium (BNE) if `$f_i ( p_i | s_i)$` is not defined (*i.e.* `$p_i$` is played with positive probability) or `$f_i ( p_i | s_i) > 0$` implies
`\begin{align*}
&p_i \in \underset{p_i'}{\text{arg max}}  &p_i' \underbrace{\mathbb{E}[v_i=1,v_j=0| s_i]}_\text{Loyal Buyers}  + p_i'\underbrace{\mathbb{E}\left[\left(1 - F_j(p_i)\right),v_i=1,v_j=1| s_i\right]}_\text{Indifferent Buyers},
\end{align*}`

given `$F_j(p)$`, for all `$s_i,s_j, i, j$`.

- A strategy profile is a **Bayes correlated equilibrium (BCE)** if it is a BNE for some information structure (Bergemann & Morris 2016)

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Road Map for Today's Talk
================
1. Introduction: Consumer Data and Competition

2. Environment and Baseline Information
  - **Complete Information**
  - No Information

3. General Public Information
  - Consumer Surplus Maximizing
  - Profit Maximizing

4. General Public + Private Information

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Complete Information: Perfect PD
============================

- One case is complete information: `$s_1 = s_2 = v$`

- Three markets:

- Loyal to Seller 1 market: price = 1
  
  - Loyal to Seller 2 market: price = 1
  
  - Indifferent customer market: price = 0

- If `$v_i=0$`, seller `$j$` sets monopoly price of 1

- If `$v_1=v_2=1$`, both sellers set competitive price of 0

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background-image: url(./images/pricecomp/pricing_example1.png)
background-size: contain

---
Road Map for Today's Talk
================
1. Introduction: Consumer Data and Competition

2. Environment and Baseline Information
  - Complete Information
  - **No Information**

3. General Public Information
  - Consumer Surplus Maximizing
  - Profit Maximizing

4. General Public + Private Information

---

background-image: url(./images/pricecomp/market2.png)
background-size: contain

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No Information: No PD
============================

- Conditional on `$v_1 =1$`, seller 1 assigns probability `$\frac{m_{10}}{m_{10} + m_{11}}$` to being the monopolist

- Regardless of what seller 2 does, seller 1 will never set a price below `$\underline{p} =\frac{m_{10}}{m_{10} + m_{11}}$`

- Neither will seller 2

--
 
- Both firms will randomize between `$[\underline{p}, 1]$`

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Lemma (Narasimhan 1988)
===========================
Let `$m_{10} \geq m_{01}$`. The unique BNE profit is `$m_{10}$` for seller 1 and `$\frac{m_{10}}{m_{10} + m_{11}}(1-m_{10})$` for seller 2. The unique strategies are given by

`$$F^*_1(p) = \begin{cases} \begin{aligned} & 0 & &p< \underline{p}&\\ & 1 - \frac{\underline{p}(m_{11} + m_{01}) - p m_{01}}{pm_{11}} & &p \in \left[\underline{p}, 1\right)&\\ & 1 & &p\geq 1 & \end{aligned}\end{cases}$$`

and

`$$F^*_2 (p) = \begin{cases} \begin{aligned} &0 & &p< \underline{p}\\ &1 - \frac{m_{10}(1 - p)}{pm_{11}}& &p \in \left[\underline{p}, 1\right], \end{aligned}\end{cases}$$`
???
The construction relies on simple observations of the each seller's best-response when facing a residual demand curve

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background-image: url(./images/pricecomp/tikz_A.png)
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background-image: url(./images/pricecomp/tikz_C.png)
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background-image: url(./images/pricecomp/tikz_D.png)
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background-image: url(./images/pricecomp/tikz_E.png)
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background-image: url(./images/pricecomp/tikz_F.png)
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background-image: url(./images/pricecomp/tikz_G.png)
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background-image: url(./images/pricecomp/pricing_example2.png)
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Road Map for Today's Talk
================
1. Introduction: Consumer Data and Competition

2. Environment and Baseline Information
  - Complete Information
  - No Information

3. **General Public Information**
  - Consumer Surplus Maximizing
  - Profit Maximizing

4. General Public + Private Information

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General Markets
==================

- The lemma applies to any market, not just the "aggregate" market

- Total industry profits:

`$$m_{10} + \frac{m_{10}}{m_{10} + m_{11}}(1-m_{10})$$`

- Consumer surplus; `$$1 - \left(m_{10} + \frac{m_{10}}{m_{10} + m_{11}}(1-m_{10})\right)$$`

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background-image: url(./images/pricecomp/cs_simplex.png)
background-size: contain

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Corollaries
=============
1. Consumer surplus under perfect price discrimination is weakly higher than consumer surplus under no price discrimination.

2. Total profits under perfect price discrimination are weakly lower than total profits under no price discrimination.

3. The relationships are strict if `$m_{10} \neq m_{01}.$`

- We can also consider all intermediate levels of information

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Concavification
=========================
- Bayes' Law only restricts the ex-post distributions of types to sum to the prior

- We can maximize/minimize over all feasible markets by "concavification" of the consumer surplus function (Kamenica and Gentzkow 2011)

- Max feasible consumer surplus is equal to lowest concave envelope of the consumer surplus function at the prior (aggregate market)

- **Proposition**: With only public data, consumer surplus is maximized under perfect price discrimination.

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background-image: url(./images/pricecomp/cs_simplex.png)
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background-image: url(./images/pricecomp/cs_concave.png)
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Nested Markets Maximize Profit
=====================
- Definition: a **nested market** is a market where one firm's potential customers are a subset of the other firm's (Armstrong and Vickers 2019)

- That is, it is a market on the two sides of the simplex (not all loyal)

- Profit is just one minus consumer surplus

- **Proposition**: With only public data, total profits are maximized with nested markets.

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background-image: url(./images/pricecomp/profit_concave.png)
background-size: contain

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background-image: url(./images/pricecomp/pricing_example3.png)
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Road Map for Today's Talk
================
1. Introduction: Consumer Data and Competition

2. Environment and Baseline Information
  - Complete Information
  - No Information

3. General Public Information
  - Consumer Surplus Maximizing
  - Profit Maximizing

4. **General Public + Private Information**

---
Private Data
=================

- Most models of market competition assume firms only have public information

- However, firms often collect their own information 
  - Therefore, I am a different person according to Amazon vs. Target, depending on the data they collect
  
- But Amazon's data collection may affect Target's pricing strategy in equilibrium

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Baseline Private Information
==================
- To see how this works, consider a baseline form of private information
  - Firm 1: complete information
  - Firm 2: no information
  
- No longer one objective market for both firms
  
- Higher-order beliefs matter for pricing and we need to specify those

- Consider when first-order information is common knowledge

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background-image: url(./images/pricecomp/market_firm1_pd1.png)
background-size: contain

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One Firm Private Data
====================

- We can then compare this information structure to no information for either firm

- Profits: `$$m_{10} + \left(1-m_{10}\right)  \underbrace{\left(\frac{m_{01}}{m_{01}+m_{11}} + \frac{m_{01}}{m_{01} + m_{11}}(1-m_{01})\right)}_{\text{Profit from Lemma 1}}$$`

- In general, profits/consumer surplus can go up or down

`$$m_{10} +m_{01} + m_{01}(1-m_{01}) \lessgtr m_{10} + \frac{m_{10}}{m_{10} + m_{11}}(1-m_{10})$$`

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background-image: url(./images/pricecomp/pricing_example4.png)
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background-image: url(./images/pricecomp/pricing_example5.png)
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Symmetric Markets
=========

- If we are interested in symmetric markets, we can prove more

- **Proposition**: For any symmetric interior market, perfect price discrimination by only one firm strictly decreases consumer surplus relative to no price discrimination.

- Proof is just from comparing profit functions

- It is a simple comparison of profit functions
  - Not sure of interesting comparisons???
  - This is why bounding all information structures is more relevant

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Correlated Information
=========
- We now want to find the maximum feasible profit for any information structure

- For now, I just have a weaker proposition

- **Proposition**: Imperfectly correlated data can strictly increase profits relative to any public data.

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Construction
=======

- Imagine an information designer who reveals consumer data to the firms and recommends an incentive-compatible price

- The designer's problem is choose information and allocations to maximize project, subject to the allocation is a BNE given the information

- Suppose the designer commits the following information for the indifferent consumers, only partially revealing the indifferent consumers

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Proof
============

- First, fix the information: `$\alpha_1$` and `$\alpha_2$`

- Consider when firm 1 receives a signal and knows the consumer is indifferent

- Prices will not be driven to zero because firm 1 does not know whether firm 2 knows that the consumer is indifferent

- With probability `$\alpha_1$`, firm 2 did not receive a signal and will, therefore, follow the designer's recommend strategy and set a price of 1.

- Otherwise, firm 2 will price according to a distribution over `$[\underline{p},1]$`.

- How high can the designer raises firm 2's pricing distribution and still remain incentive compatible?

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background-image: url(./images/pricecomp/tikz_L.png)
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Proof Cont.
============

- We have constructed incentive compatible strategies for when the firm receives no signal

- Now we need to do the same for after a signal

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background-image: url(./images/pricecomp/tikz_P.png)
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Incentive Constraints
============

- Therefore, we have found an incentive compatible allocation, given the information

- The relevant ICs
`$$\begin{align}    &\underline{p} \geq \frac{\alpha_1}{1-\alpha_2}\quad \text{and}\\ &\underline{p} \geq \frac{\alpha_2}{1-\alpha_1},\end{align}$$`
and 
`$$\begin{align}  &m_{10} \geq \underline{p} \left(m_{10} + \alpha_2 m_{11} \right) ~\quad \text{and}\\ &m_{01} \geq \underline{p} \left(m_{01} + \alpha_1 m_{11} \right)   \end{align}$$`
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Designer's Problem
=========================

- The designer's problem is to maximize
`$$\text{Industry Profit } = \alpha_1 + \frac{\alpha_1}{1-\alpha_2}
\left(1 - \alpha_1\right) + m_{10} + \frac{\alpha_1}{1-\alpha_2} \left(m_{01} + \alpha_1 m_{11}\right)$$`

- Subject to

`$$\begin{align}    &\underline{p} \geq \frac{\alpha_1}{1-\alpha_2},\\ &\underline{p} \geq \frac{\alpha_2}{1-\alpha_1},\\  &m_{10} \geq \underline{p} \left(m_{10} + \alpha_2 m_{11} \right) ~\quad \text{and}\\ &m_{01} \geq \underline{p} \left(m_{01} + \alpha_1 m_{11} \right)   \end{align}$$`

- In general, first and third ICs (for firm 1) will bind

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background-image: url(./images/pricecomp/pricing_example6.png)
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background-image: url(./images/pricecomp/surplus.png)
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Conclusion
==================

- I argue studying price discrimination should involve an explicit model of competition

- Results of monopoly price discrimination can be completely reversed under competition
  - I prove perfect price discrimination is optimal for consumers
  
- However, results may be sensitive to exact informational assumptions

- Bound equilibrium outcomes
  - Max consumer surplus: complete public data
  - Min consumer surplus with public data: nested markets
  - Lower consumer surplus with private data