Data as Commodity: a Game-Theoretic Principle for Information Pricing

TL;DR

The paper addresses the challenge of valuing information in digital markets by modeling a Bayesian game with $N \ge 3$ players who bet on an underlying stochastic process using partially informative data strings. A data owner (seller) may either exploit private information or transfer part of her data to a less informed buyer, and equilibrium analysis yields natural price bounds $\Psi_{\min}$ and $\Psi_{\max}$ that delineate mutually beneficial trades. The authors derive detailed Nash equilibrium characterizations for volatility-neutral and volatility-averse settings, revealing non-intuitive regimes such as symbiotic data sharing and the blessing of ignorance, and they show how the number of players and data lengths affect outcomes. This framework provides a principled valuation principle for intangible data in dynamic, asymmetric-information markets and suggests broad avenues for extending the model to richer processes and networks.

Abstract

Data is the central commodity of the digital economy. Unlike physical goods, it is non-rival, replicable at near-zero cost, and traded under heterogeneous licensing rules. These properties defy standard supply--demand theory and call for new pricing principles. We propose a game-theoretic approach in which the value of a data string emerges from strategic competition among N players betting on an underlying stochastic process, each holding partial information about past outcomes. A better-informed player faces a choice: exploit their informational advantage, or sell part of their dataset to less-informed competitors. By analytically computing the Nash equilibrium of the game, we determine the price range where the trade is beneficial to both buyer and seller. We uncover a rich landscape of market effects that diverge from textbook economics: first, prospective sellers and buyers can compete or jointly exploit the less informed competitors depending on the quality of data they hold. In a symbiotic regime, the seller can even share data for free while still improving her payoffs, showing that losing exclusivity does not necessarily reduce profit. Moreover, rivalry between well-informed players can paradoxically benefit uninformed ones, demonstrating that information abundance does not always translate to higher payoffs. We also show that the number of players influences the competition between informed parties: trades impossible in small markets become feasible in larger ones. These findings establish a theoretical foundation for the pricing of intangible goods in dynamically interacting digital markets, which are in need of robust valuation principles.

Figures (7)

• Figure 1: Schematic representation of the information levels held by different players. All players share the publicly available string $\vec{s}_0$ of past outcomes. P1 is the most informed player and holds a longer string $\vec{s}_1$. In scenario (ii) she may sell a portion of her extra string to P2, who therefore can base her betting strategy on the string $\vec{s}_2$. The length of $\vec{s}_2$ is intermediate between $\vec{s}_1$ and the common string $\vec{s}_0$.
• Figure 2: Equilibrium strategy $p_\text{eq}$ as a function of $\hat{\rho}$, for different values of $N$, in the case of equally informed players.
• Figure 3: Payoff and strategy of P1 before transaction as a function of $\mathop{\mathrm{\hat{\rho}}}\nolimits_1$ and $\mathop{\mathrm{\hat{\rho}}}\nolimits$, for $N=5$. The heatmap shows the expected payoff $\mathop{\mathrm{\mathcal{U}}}\nolimits_1(\vec{p}^{\,\text{pre}},\mathop{\mathrm{\hat{\rho}}}\nolimits_1)$. In the squares delimited by the white lines, P1's payoff is zero, since she is playing the same strategy as other players. Above the magenta dashed line, P1 plays according to the pure strategy Heads, below the line she plays with pure strategy Tails.
• Figure 4: Post-transaction payoffs and strategy as a function of $\mathop{\mathrm{\hat{\rho}}}\nolimits_1,\mathop{\mathrm{\hat{\rho}}}\nolimits_2$ for $\mathop{\mathrm{\hat{\rho}}}\nolimits=0.5$ and $N=5$. In panels (a),(b),(c) the black lines separate the region with positive payoff from that with negative payoff. (a) P1's payoff. (b) P2's payoff, computed by P1. (c) Less informed players' payoff, computed by P1. (d) Phase diagram showing the strategies played by P1,P2. Every color corresponds to a different combination of strategies, indicated by the pair of arrows. The left (right) arrow indicate the strategy of P1 (P2), with $\uparrow,\downarrow, \rightarrow$ corresponding respectively to pure Heads, pure Tails, and mixed strategies. The dashed black line separates the upper region, wheree P1 plays Heads from the lower region where she plays Tails.
• Figure 5: Prices and transaction feasibility as a function of $\hat{\rho}_1$ and $\hat{\rho}_2$ for $\hat{\rho}=0.5$ and $N=5$ fixed throughout all panels. (a) Heatmap of the minimum price $\Psi_\text{min}$. (b) Heatmap of the maximum price $\Psi_\text{max}$. (c) Transaction feasibility in the $\hat{\rho}_1,\hat{\rho}_2$ plane: yellow regions indicate where the transaction is possible, dark blue regions where it is not. (d)$\Psi_\text{min}$ and $\Psi_\text{max}$ as functions of $\hat{\rho}_1$ for fixed $\hat{\rho}_2=0.25$. This corresponds to a vertical slice of the heatmaps. The background color matches panel (c), indicating transaction feasibility. Different strategic regions can be identified: (1) Cooperative, $\Psi_{\max}>\Psi_{\min}>0$ --- trade is possible at a strictly positive price; (2) Competitive, $\Psi_{\min}>\Psi_{\max}>0$ --- trade is not feasible; (3) Cooperative --- the possibility of a trade reopens; (4) Symbiotic, $\Psi_{\min}<0<\Psi_{\max}$ --- trade would be possible also at zero price; (5) Exploitative cooperation, $\Psi_{\min}<\Psi_{\max}<0$ --- trade would be feasible only if the seller compensates the buyer; (6) No-deal, $\Psi_{\max}<\Psi_{\min}<0$ --- no mutually beneficial agreement exists.

Theorems & Definitions (6)

• Lemma A.1
• proof
• Lemma A.2
• proof
• Lemma A.3
• proof