Selling Information in Competitive Environments

Abstract

Data buyers compete in a game of incomplete information about which a single data seller owns some payoff-relevant information. The seller faces a joint information- and mechanism-design problem: deciding which information to sell, while eliciting the buyers' types and imposing payments. We derive the welfare- and revenue-optimal mechanisms for a class of games with binary actions and states. Our results highlight the critical properties of selling information in competitive environments: (i) the negative externalities arising from buyer competition increase the profitability of recommending the correct action to one buyer exclusively; (ii) for the buyers to follow the seller's recommendations, the degree of exclusivity must be limited; (iii) the buyers' obedience constraints also limit the distortions in the allocation of information introduced by a monopolist seller; (iv) as competition becomes fiercer, these limitations become more severe, weakening the impact of market power on the allocation of information.

Figures (11)

• Figure 1: Welfare-maximizing recommendation rule from \ref{['prop:second-best']} with two buyers, for $\alpha=2/3$ (left) and $\alpha=3/2$ (right). The label in each region indicates the set of buyers who are recommended the correct action ($A_i=\theta$)—buyers in the complement set are recommended the wrong action ($A_i=1-\theta$). The two states are equally likely ex ante, so $v^\star=F^{-1}(1/2)$ is the median of the type distribution—chosen to be a standard exponential here).
• Figure 2: Building blocks for the welfare-maximizing mechanism of \ref{['prop:second-best']}. Left: mechanism guaranteeing obedience at all types while minimizing externalities. Center and right: first-best mechanism (ignoring the obedience constraint) for $\alpha=2/3$ and $\alpha=3/2$.
• Figure 3: Revenue-maximizing recommendation rule from \ref{['prop:revenue-alpha']} for $\alpha=1/2$ (left) and $\alpha=2$ (right). Types are distributed exponentially, so that $\phi(v)=v-1$ and $v_0=\phi^{-1}(0) = 1$. The prior on $\theta$ is symmetric ($P_{\max}=1/2$), hence $v^\star=F^{-1}(1/2)=\ln 2<v_0$.
• Figure 4: Revenue-maximizing recommendation rule from \ref{['prop:revenue-alpha']} for $\alpha=1/2$ (left) and $\alpha=2$ (right). Types are distributed exponentially, so that $\phi(v) = v-1$ and $v_0=\phi^{-1}(0)=1$. The first row shows the first best mechanism. The second row is the second-best mechanism (subject to obedience) with a symmetric prior on $\theta$, for which $v^\star = F^{-1}(1/2)=\ln 2<v_0$. The third row is the second-best mechanism with an asymmetric prior ($p_{\max}=3/4$), for which $v^\star=\ln 4> v_0$.
• Figure 5: Payment as a function of a buyer's type, for different values of $P_{\max}$ with exponentially distributed types. Left panel: $\alpha=1/2$. Right panel: $\alpha=2$.

Theorems & Definitions (40)

• Example 2.1: Binary Product Choice
• Definition 3.1: Incentive Compatibility
• Definition 3.2: Obedience
• Definition 3.3: Truthfulness
• Definition 3.4: Individual Rationality
• Proposition 3.5: IC Characterization
• proof
• Proposition 3.6: Truthfulness Characterization
• proof
• Proposition 3.7: Obedience Characterization