Selling Information in Games with Externalities

TL;DR

The paper develops a joint information-design and mechanism-design framework in a two-player, binary-state game where a seller monetizes exposure to payoff-relevant information about the state. By modeling information as a product and introducing a competition-induced externality cost, the authors derive a profit-maximizing mechanism that screens buyer types via an informativeness profile $I(v_b)$ and transfers $t(v_b)$. They show that with binary types, payments incentivize information sharing, the optimal menu can benefit both parties, and the seller cannot steer actions at the expense of social welfare; as competition intensifies, it may become optimal to sell no information. Extending to continuous types, the problem reduces to virtual surplus optimization, with the result that partial information is rarely offered unless competition is moderate, and the optimal mechanism often collapses to a binary, threshold structure. The findings illuminate how competition and externalities shape the value of information in analytics markets and provide guidance on when information sharing should be promoted or discouraged.

Abstract

A competitive market is modeled as a game of incomplete information. One player observes some payoff-relevant state and can sell (possibly noisy) messages thereof to the other, whose willingness to pay is contingent on their own beliefs. We frame the decision of what information to sell, and at what price, as a product versioning problem. The optimal menu screens buyer types to maximize profit, which is the payment minus the externality induced by selling information to a competitor, that is, the cost of refining a competitor's beliefs. For a class of games with binary actions and states, we derive the following insights: (i) payments are necessary to provide incentives for information sharing amongst competing firms; (ii) the optimal menu benefits both the buyer and the seller; (iii) the seller cannot steer the buyer's actions at the expense of social welfare; (iv) as such, as competition grows fiercer it can be optimal to sell no information at all.

Figures (11)

• Figure 1: Existing frameworks for sharing information. The blue, red and green arrows indicate monetary transactions, strategic actions, and information flows between the firms and the market, respectively. Whilst in each framework, Firm B shares information with Firm A, the setups differ as follows: in (a) information is shared freely between competitors, in (b) information is purchased yet Firm B is a third party that doesn't compete with Firm A, and in (c) information is purchased and both firms compete in the market.
• Figure 2: One-dimensional informativeness.
• Figure 3: Feasible region of $I_0 (b)$ and $I_1 (b)$. The ceiling of the feasible region is highlighted in red.
• Figure 4: Gain as a function and buyer types, as described in (\ref{['eq:gain-communication-rule-expanded']}), for $I=0$ (solid), $I=1/2$ (dashed) and $I=-1/2$ (dotted).
• Figure 5: A congruent binary type distribution. Without additional information, both types will take the same action, that is, $\sigma (v_b^l) = \sigma (v_b^h) = 0$.

Theorems & Definitions (25)

• Remark 2.1
• Remark 2.2
• Definition 3.1: Direct Communication Rule
• Proposition 3.2
• proof
• Definition 3.3: Individual Rationality
• Definition 3.4: Truthfulness
• Definition 3.5: Obedience
• Definition 3.6: Incentive Compatibility
• Proposition 4.1