Outline

• Introduction 
• Model and main results 
• Spatial interpretation 
• Conclusion

Introduction

• The paper studies competition in the market for news (political information) 
• Main results: 
  • Competition leads to informational specialization 
  • Competition makes every agent worse off 
• Why should we care? It is important to understand
  • how competition affects the strategic needs of information providers 
  • The possible consequences of competition on the political process 

Example

• A new health-care bill is under discussion (details not fully known to the public) 
  • promote overall healthcare quality 
  • expand the budget deficit 
  • induce more redistribution 
• Voters acquire information from the media before they approve/disapprove the policy 
• News outlets compete for profits by allocating their limited resources (journalists, airtime, etc.) 
• greater consensus \(\to\) more likely to get implemented 

Model

• Uncertain policy \(\omega = (\omega_0, \omega_1, \omega_2), \; \omega_k \stackrel{i.i.d.}{\sim} \mathcal{N}(0, 1)\) for \(k \in \{0, 1, 2\}\)
  • One vertical component - valence   \(\omega_0\)
  • Two horizontal components - ideological   \(\omega_1, \omega_2\)
• Policy implemented with probability = approval rate 
• Finite number of identical, non-partisan firms \(n \in N\)
  • Firm \(n\) chooses editorial strategy \( b_n = (b_{n, 0}, b_{n, 1}, b_{n, 2})\)
  • Firm \(n\) charges \(p_n(\theta_i)\)
• Finite number of heterogenous, Bayesian agents \(i \in I\)
  • Agent \(i\) has payoff type \(\theta_i =  (\theta_{i, 0}, \theta_{i, 1}, \theta_{i, 2}) \) independently drawn from uniform distribution \(\mathcal{F}\)

Assumptions explained

• Agent \(i\)'s payoff type: \(\theta_i =  (\theta_{i, 0}, \theta_{i, 1}, \theta_{i, 2}) \)
  • ​Agents have identical preferences over \(\omega_0 \implies \\ \theta_{i, 0} = 1 \; \forall i\)
  • Agents have heterogenous preferences over horizontal components \( \omega_1, \omega_2 \implies \) \( \theta_{i, 1}^2 + \theta_{i, 2}^2 = 1 \)
• ​Firm \(n\)'s editorial strategy \(b_n\) faces constraint \(\lVert b_n \rVert = \sqrt{b_{n, 0}^2 + b_{n, 1}^2 + b_{n, 2}^2} \leq 1\)

Model Timeline

Firms and agents interact over 3 consecutive stages: 

Firms choose editorial strategies \((b_n)_{n=1}^N\)

Agents' types \(\theta_i\) revealed

Agents acquire information, pay price \(p_n(\theta_i)\), privately observe signal \( s_i (\omega, b_n) = \omega \cdot b_n + \epsilon_i \), \( \epsilon_i \sim \mathcal{N}(0, 1)\)

Approve/disapprove policy;

payoff: \(u(\omega, \theta_i) = \omega \cdot \theta_i\) or 0

\(\theta_i =  (\theta_{i, 0}, \theta_{i, 1}, \theta_{i, 2}) \)

\( \theta_{i, 0} = 1, \; \theta_{i, 1}^2 + \theta_{i, 2}^2 = 1 \)

Firms observe each other's strategies and agents' types 

Firms set prices \(p_n(\theta_i)\)

Stage 3

Recall that an agent faces two choices:

1. where to acquire information, and
2. conditioning on obtained information, whether to approve the policy or not 

When would the agent approve the policy, given the signal she obtained? 

• Lemma 1. Equilibrium approval strategy: approves if and only if \( \mathbb{E}_\omega (u(\omega, \theta_i) | s_i(\omega, b_n)) \geq 0 \)

Stage 3 - value of information

Recall that an agent faces two choices:

1. where to acquire information, and
2. conditioning on obtained information, whether to approve the policy or not

Information has instrumental value! 

Lemma 2. Value of Information is defined as 

\[ v(b_n | \theta_i) = \frac{\lvert \theta_i \cdot b_n \rvert }{I\sqrt{2\pi (1 + \lVert b_n \rVert^2 )}} \]

- difference between agent's expected equilibrium payoff associated with observing a signal from a firm \(n\) and the one associated with observing no signal at all 

Stage 3 conclusion

At equilibrium, agent \(i\) has

• information acquisition strategy: choose the firm with the highest \(v(b_n | \theta_i)\) net price \(p_n(\theta_i)\).
  • Formally,  

    For type \(\theta_i\), given \( (b_n, p_n(\theta_i))_{n=1}^N \), agent will choose firm \(n\) such that

    \[ v(b_n|\theta_i) - p_n(\theta_i) \geq v(b_m|\theta_i) - p_m(\theta_i) \;\; \forall m \in [N] \]

• approval strategy: approve policy if and only if \( \mathbb{E}_\omega (u(\omega, \theta_i) | s_i(\omega, b_n)) \geq 0 \)

Stage 2 & 1

Recall that a firm faces two choices: 

1. Stage 1: Editorial strategy \(b_n\) 
2. Stage 2: What price to charge for the information it provides, observing other firms' editorial strategies and agents' types 

Stage 2

Recall that a firm faces two choices: 

1. Stage 1: Editorial strategy \(b_n\) 
2. Stage 2: What price to charge for the information it provides, observing other firms' editorial strategies and agents' types 

What we know so far: agents will choose the firms with the highest value of information net price 

\(\to\) What kind of prices will prevail in equilibrium? 

Stage 2 - Example

• Suppose two firms compete for type \(\theta_i\). 
• Suppose that for editorial strategies \(b_1, b_2 \), \(v(b_1|\theta_i) > v(b_2|\theta_i) \)
  • Firm 1's information has more value for \(\theta_i\)
• The lowest price that firm 2 can charge to be competitive: \( p_2(\theta_i) = 0 \)
• When firm 2 charges 0, agent \(i\) will still choose firm 1 at equilibrium as long as \[v(b_1|\theta_i) - p_1(\theta_i) > v(b_2|\theta_i) - \underbrace{p_2(\theta_i)}_{=0} \]

​\[\implies p_1(\theta_i) < v(b_1|\theta_i) - v(b_2 | \theta_i)  \]

Stage 2 - Generalization

• For \(N=2\), firm 1 "wins" by setting price ​\(p_1(\theta_i) < v(b_1|\theta_i) - v(b_2 | \theta_i)  \)
• Generalizing to \(N > 2\), firm \(n\) wins type \(\theta_i\) if and only if \[ v(b_n|\theta_i) \geq \max_{m \neq n} v(b_m|\theta_i) \]
• The equilibrium profit for firm \(n\) from type \(\theta_i\) is \[ \max_m v(b_m|\theta_i) - \max_{m \neq n} v(b_m | \theta_i) \geq 0 \] 
• Firm \(n\)'s total profit: \[ \sum_{i=1}^I \max_m v(b_m|\theta_i) - \max_{m\neq n} v(b_m | \theta_i) \]

Stage 1

• Recall: firms choose their editorial strategies before observing agents' types 
• Stage 2: Firm \(n\)'s total profit: \[ \sum_{i=1}^I \max_m v(b_m|\theta_i) - \max_{m\neq n} v(b_m | \theta_i) \]
• Agent's types are i.i.d according to uniform distribution 
• Firm \(n\)'s expected profit is \[ \Pi_n (b_n) = I \mathbb{E}_{\theta_i} \left(  \max_m v(b_m | \theta_i) - \max_{m \neq n} v(b_m | \theta_i) \right)\]

Choosing an editorial strategy

• Transform the problem of choosing an editorial strategy \(b_n\) into an equivalent location problem on a disk 

Recall: 

Firms choose \( b_n = (b_{n, 0}, b_{n, 1}, b_{n, 2})\) given constraint \(\lVert b_n \rVert \leq 1 \) for \(k \in \{0, 1, 2\}\)

Agent \(i\)'s type \(\theta_i =  (\theta_{i, 0}, \theta_{i, 1}, \theta_{i, 2}) \) with \( \theta_{i, 0} = 1, \; \theta_{i, 1}^2 + \theta_{i, 2}^2 = 1 \)

We can transform any type \(\theta_i\) uniquely to \(\theta_i = (1, \cos(t_i), \sin(t_i) )\), where \( t_i \in T = [-\pi, \pi]\)

We can transform all \(b_n\) such that \(\lVert b_n \rVert = 1\) to \( b_n = (\sqrt{x_n},  \sqrt{1 - x_n} \cos(t_n), \sqrt{1-x_n} \sin(t_n) ) \) with unique pair \( (x_n, t_n) \in [0, 1] \times T \)

Choosing an editorial strategy

• Agent \(i\)'s "new" type: \(t_i \in [-\pi, \pi]\)
  • \( \theta_i = (1, \cos(t_i), \sin(t_i) )\)
• Firm \(n\)'s "new" editorial strategy: \( (x_n, t_n) \in [0, 1]\times [-\pi, \pi] \) 
  • \(b_n = (\sqrt{x_n}, \sqrt{1-x_n}\cos(t_n), \sqrt{1-x_n}\sin(t_n))\)
• Interpretation 
  • \(t_i\) is a location on a circle; drawn uniformly from \(T\)
  • \(x_n\) captures how generalist the firm is 
  • \(t_n \in T\) is the firm's target type
    • ​note this is not the same as \(t_i\)
  • ​\( (x_n, t_n)\) corresponds to a location on the disk 
     

Choosing an editorial strategy

Where would a maximally generalist firm locate at? 

\(b_n = (1, 0, 0)\)

"new" value of information

Lemma 2. Value of Information is defined as 

\[ v(b_n | \theta_i) = \frac{\lvert \theta_i \cdot b_n \rvert }{I\sqrt{2\pi (1 + \lVert b_n \rVert^2 )}} \]

This can be rewritten as 

Recall: \( \theta_i = (1, \cos(t_i), \sin(t_i))\)

\( b_n = (\sqrt{x_n}, \sqrt{1-x_n}\cos(t_n), \\\sqrt{1-x_n}\sin(t_n)) \)

Interpretation: value of information is the sum of two terms

• \(\sqrt{x_n}\) is the valence component, which all agents care about
• \( \sqrt{1-x_n}(\cos(t_i - t_n))\) is the ideological component 

"new" value of information

Eq.2 Value of Information is defined as 

\[ v(b_n | \theta_i) = \frac{1}{2I\sqrt{\pi}}\cdot \left| \sqrt{x_n} + \sqrt{1-x_n} \cos(t_i-t_n) \right| \]

This clarifies the tradeoff when choosing editorial strategies: 

• more generalist \(\to\) high value even when agent's type is far away from target type 
• more specialist \(\to\) high value when agent's type is ideologically close to target type 

Example

target type \(t_n = 0\)

Going back to choosing a strategy 

Recall: 

\[ v(b_n | \theta_i) = \frac{1}{2I\sqrt{\pi}}\cdot \left| \sqrt{x_n} + \sqrt{1-x_n} \cos(t_i-t_n) \right| \]

Firm \(n\)'s first-best editorial strategy for an agent of type \(t_i\):

• choose \(t_n = t_i \to \) this maximizes \( \cos(t_i - t_n) = 1\)
• \( \sqrt{x_n} + \sqrt{1-x_n}\) is maximized when \(x_n = 1/2\), i.e. assign equal weight to valence and ideology 

First-best value: \( \bar{\mathcal{V}} = v((\frac{1}{2}, t_i )| t_i) \)

Competition leads to informational specialization

Firms specialize by providing relatively less information on the valence component and relatively more information on the ideological components 

Competition leads to informational specialization

Theorem 2: In equilibrium, firm's expected readership is an arc of length \(2\pi/N\) centered around firm's target type \(t_n\) 

As the number of firms \(N\) increases, expected readership \(2\pi/N\) decreases. 

Decreased readership is more ideologically homogeneous. 

\(\to\) Firms specialize (\(x_n\) decreases) 

Competition leads to informational specialization

• How do firms differentiate when they sell information? 
  • Increase the relative informativeness of private-interest components at the expense of the common-interest components 

• Equilibrium interactions between vertical and horizontal competition: 
  • As competition increases, firms disinvest from vertical features (beneficial to all consumers) and instead focus on horizontal features (beneficial only to a segment) 

Competition amplifies social disagreement

• Type \(t_i\)'s equilibrium opinion of policy: \(z_i (t_i) = \mathbb{E}_\omega [u(\omega, t_i) | s_i^*(\omega)] \)
  • By Lemma 1, agent will only approve policy when her opinion is non-negative 
• Social agreement: expected correlation in the opinions of two agents, \(i\) and \(j\)
  \[\mathcal{S}(N) = \mathbb{E}_{t_i, t_j} [\text{Corr}(z_i(t_i), z_j(t_j))]\]
• \(\mathcal{S}\) is strictly decreasing in \(N\). 

Example 

• Consider agents \(i, j\) with \(t_i = 0, t_j = \pi/2\)
  • \( \theta_1 = (1, 1, 0), \theta_2 = (1, 0, 1)\) 
  • \(i\) cares about \(\omega_1\), \(j\) cares about \(\omega_2\)
• Low competition \(\to\) less specialization, \(x^*(N) \) takes higher value 
  • signals are highly informative about \(\omega_0\) even if obtained from different firms 
  • higher correlation between opinions \(z_i(t_i)\) and \(z_j(t_j)\)
• Takeaway: competition pushes profit-maximizing firms to provide information about dimensions that agents disagree more  

Welfare implications of increased disagreement

• Agent's expected welfare decreases as competition increases 
• Agent \(i\)'s expected welfare:
  \[ \mathcal{U}(N) = \mathbb{E}_{\omega, t} [ A^*(\omega, t) u(\omega, t_i) -p^*(t_i) ]  \]
• Expected welfare can be decomposed to
  \[ \mathcal{U}(N) = \mathcal{V}(N) + \mathcal{G}(N) - \mathcal{P}(N)\]
  where \(\mathcal{V}(N)\) is expected value of information
  \(\mathcal{G}(N)\) is the impact of other's approval decisions on \(i\)'s utility
  \(\mathcal{P}(N)\) is expected price for acquired information 

Welfare implications of increased disagreement

\[ \mathcal{U}(N) = \mathcal{V}(N) + \mathcal{G}(N) - \mathcal{P}(N)\]

• \(\mathcal{V}(N)\): information's direct effect on agent's welfare
  • measures how an agent values the information that she personally acquires
• \(\mathcal{G}(N)\): information's indirect effect on agent's welfare
  • measures how an agent values the information that other agents acquire 

Welfare implications of increased disagreement

\[ \mathcal{U}(N) = \mathcal{V}(N) + \mathcal{G}(N) - \mathcal{P}(N)\]

• \(\mathcal{V}(N)\): information's direct effect on agent's welfare
• \(\mathcal{G}(N)\): information's indirect effect on agent's welfare

As competition increases, \(\mathcal{V}(N)\) increases as it's more specific to agent's taste.

Social disagreement \(\to\) \(\mathcal{G}(N)\) decreases 

When the society is large enough, \(\mathcal{V}(N)+\mathcal{G}(N)\) decreases. 

• increase in \(\mathcal{V}\) cannot compensate for decrease in \(\mathcal{G}\)

Welfare implications of increased disagreement

• Political information has value because it allows agents to influence outcomes in a way that aligns with their preferences. 
• However, outcomes have consequences for all members of the society. 
• Individual information acquisition strategies (acquire information with highest value net price) have social externalities on others. 

Conclusion

• Information has social externalities 
• Increased competition between profit-maximizing firms lead to informational specialization 
• Increased specialization of firms lead to greater social disagreement 

Thank you for listening! 

Unused slides from here on 

Motivating example

Stage 3

Recall that an agent faces two choices:

1. where to acquire information, and
2. conditioning on obtained information, whether to approve the policy or not 

Equilibrium behavior: 

• Suppose agent \(i\) chooses to acquire information from firm \(n\)
• Recall agent \(i\)'s payoff is \(u(\omega, \theta_i) \) if policy approved; 0 otherwise
•  

Proof of Lemma 2

Define \(A_{-i}(\omega, \theta_{-i}) \) to be the approval rate excluding type \(\theta_i\). 

Expected utility if \(\theta_i\) does not receive any information:

Since \( \omega_k \sim \mathcal{N}(0, 1)\), \(\mathbb{E}_\omega [u(\omega, \theta_i)] = 0\); original equation becomes 

Proof of Lemma 2

Expected utility when \(\theta_i\) observes signal induced by \(b_n\):

by the law of iterated expectations, 

Recall: \( u(\omega, \theta_i) =\omega\cdot\theta_i \sim \mathcal{N}(0, \lVert \theta_i \rVert ^2 )\)

\( s_i(\omega,b_n) = b_n \cdot \omega + \epsilon_i \sim \mathcal{N}(0, 1 + \lVert b_n \rVert ^2) \)

\(\to\) this part cancels out with the first part