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 

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 - 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 $\mathbb{E}_\omega (u(\omega, \theta_i) | s_i(\omega, b_n)) \geq 0$

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

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) 

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

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

• $\mathcal{V}(N)$: 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. 

Unused slides from here on