Information Games: Strategic Crowding and Firm Repositioning in Language-Model Space

AI summary

Overview

Research area: Computational finance (q-fin.CP) at the intersection of industrial organization, asset pricing, and text-as-data economics — specifically, using distributions of corporate-news embeddings to measure how firms reposition themselves relative to competitors.

Technical level: Advanced. The paper combines a proportional-share multi-contest model, variational-inequality equilibrium arguments, optimal-transport (Wasserstein) measurement, and a multi-agent Monte Carlo study.

Scope: The paper builds an economic model in which firms reallocate operating capacity across competing activities, derives an observable implication linking competitive repositioning to financial returns and movement in language-model "information space," and evaluates a Wasserstein-distance-based estimator in simulations and in Nasdaq corporate news.

What This Paper Is About

When a firm expands into an opportunity, it changes the margins available to its rivals, and those rivals respond in turn. The paper asks whether this strategic crowding and gradual repositioning can be detected in the text of financial news: specifically, whether a firm's news distribution moves toward the news distributions of competitors that subsequently earn higher relative returns. To answer this, the authors extend a multi-contest model of resource allocation to firms choosing capacity across activities, add costly adjustment over time, and connect the resulting operating paths to both valuations and news coverage.

Key Contributions

1. A capacity-allocation model of competitive repositioning. The authors adapt the multi-contest framework of Xu et al. (2022) so that firms allocate operating capacity across activities serving overlapping customer demands, with proportional-appeal market shares, an outside alternative, firm-specific capability centers, and quadratic costs. They prove existence of a unique Nash equilibrium and show that identical firm primitives imply identical equilibrium capacity vectors.

2. A directional prediction linking returns to movement in information space. In a restricted two-opportunity economy, firms with greater initial exposure have larger return responses to a positive free-cash-flow shock, and firms expanding toward the improving opportunity move closer to the initial news distributions of more-exposed competitors. Proposition 2 states that, under common reporting and stated timing and error conditions, the expected local covariance between article-weighted excess returns and approach in Wasserstein distance is nonnegative.

3. A distribution-based estimator with timing weights. The paper defines approach as the year-over-year reduction in squared Wasserstein distance between a target firm's annual news distribution and each reference firm's distribution fixed at its 2018 position. Reference returns are weighted by the fraction of the target's articles published after each return date, and the resulting moments are centered within target-year and pooled into a single coefficient. The abstract describes this as a novel sandwich estimator.

4. Simulation and empirical evaluation. A 100-firm multi-agent economy with known population coefficients is used to separate economic approximation error from article-sampling error, and the estimator is then applied to news and returns for 100 Nasdaq-listed firms from 2018 to 2022.

Main Findings

• Unique equilibrium and a symmetry boundary. Under Assumption 1 (positive outside appeal, nonnegative opportunity values and appeal coefficients, and cost matrices bounded below by a common positive-definite term), the proportional-share capacity game has a unique Nash equilibrium on the nonnegative capacity space. Firms with identical primitives choose identical capacity vectors, so crowding alone does not generate heterogeneous business mixes; capability differences do.

• Exposure orders return responses. In the restricted two-opportunity economy, the return response takes the form g(p) = G(2p − 1)/[1 + (2p − 1)²] for G > 0, which increases with the firm's initial exposure p. Firms with greater initial exposure therefore have larger return responses to a positive exogenous free-cash-flow shock.

• Repositioning is visible even when averages are not. For the unit-axis component laws used in the paper, every mixture has zero centroid, yet the squared Wasserstein distance satisfies W₂²(C(p), C(q)) = 2|p − q|. Changes in distribution shape therefore reveal movement that a single average embedding vector would miss.

• Sign prediction, not magnitude. Proposition 2 gives a nonnegative expected local association between relative financial returns and decreases in distance to reference firms, and fixed nonnegative pooling preserves the sign. The theory does not claim the association is large.

• Small operating movements in the simulated economy. In the first responding month after the first innovation, absolute changes in normalized operating shares range from 0.031571363773115306 to 0.050227558934379246 percentage points across firms and histories. The largest absolute departure from an origin share over the complete paths is 0.20343851335852259 percentage points.

• Observed coverage recovers the sign only about half the time. Under common reporting, the mean population target is 0.60022980702219875 and 53.0859375% of article samples at observed coverage yield a numerically resolved positive estimate (stratified Monte Carlo standard error 0.98678805234686562 percentage points; absolute RMSE 11.269559609434811 coefficient units). The population coefficient is positive in each of the sixteen simulated news histories, so this frequency also measures correct-sign recovery.

• More articles improve recovery but do not eliminate measurement error. Correct-sign recovery rises to 55.546875% at fourfold coverage (standard error 0.97767723327016032; absolute RMSE 6.0127198700272606) and to 59.84375% at sixteenfold coverage (standard error 0.95310365413255727; absolute RMSE 3.0352587480546047). Each percentage uses 2560 attempted redraws, with 160 independent article panels per history.

• Reporting changes can mimic operating changes. The stationary-reporting control, with a mean target of 0, yields 49.6875% positive estimates (standard error 0.98726933295829733; absolute RMSE 11.639921096739526). The "reporting only" counterfactual, which holds capacities fixed and uses reporting amplitude 0.005, has a mean target of 0.68648889175149885 and yields 52.6171875% positive estimates (standard error 0.98852733243661661; absolute RMSE 11.386767683607664). A positive coefficient therefore cannot by itself be read as evidence of operating repositioning.

• Empirical association is positive but small. Across four specifications using 100 Nasdaq-listed firms from 2018 to 2022, movements relative to fixed 2018 reference distributions are positively associated with reference firms' relative article-weighted excess returns, although the paper characterizes the relationship as small. The authors describe the evidence as descriptive and do not test the null hypothesis β ≤ 0 in the corporate archive, because the proposition's conditions are not verified there and no established design samples independent firm histories. Specific coefficient magnitudes are not stated in the available text.

Methodology in Plain English

The authors start from a stylized market in which each firm chooses how much capacity to place in each of several activities. An activity's effectiveness at serving a customer segment declines with distance in an abstract activity space. A firm's share of each demand opportunity depends on its own appeal relative to rivals' appeal plus an outside alternative, so expansion by one firm takes margin away from others. Costs have three parts: a linear term reflecting each firm's comparative advantage, a congestion term that penalizes expanding several related activities at once, and a firm-specific penalty for locating far from its capability center.

Firms choose capacity simultaneously, taking rivals as given. The paper proves this game has one equilibrium and then adds a quadratic cost for changing inherited capacity. Each period's choice is again a Nash equilibrium conditional on the previous allocation, so adjustment unfolds gradually. Investors value the resulting stream of cash flows at a constant discount factor and immediately price in the anticipated adjustment path when news of an opportunity shock arrives.

To connect this to text, the authors assume a firm's news distribution is a mixture of activity-specific distributions, weighted by capacity times a reporting intensity. When reporting intensity is equal across activities, the news mix equals the business mix; when it is not, reported composition can change while capacity stays fixed. This distinction drives the later simulation controls.

Measurement uses squared Wasserstein distance, which finds the cheapest way to match probability mass between two article distributions and therefore captures compositional change that average embeddings miss. For each target firm and year, the estimator compares the target's distribution with each competitor's distribution frozen at 2018, computes how much closer the target moved, weights each competitor's daily excess returns by the fraction of target articles published later in the year, centers both measures across competitors within the target-year, and pools the resulting moments into one coefficient.

The authors then build a 100-firm simulation with two activities, two opportunities, baseline shares p_i = i/101, total capacity one, D = 51, cost curvature ν = 1/(4D²), and adjustment weight κ = d_perp = ν + 1/D². Sixty monthly ticks include independent permanent innovations of ±0.001 at ticks 18, 30, 42, and 54; all sixteen sign histories are evaluated with equal weight, monthly discounting is δ = 0.95^(1/12), and the operating engine solves every joint Nash stage at actual opportunity levels. Article panels are drawn at one, four, and sixteen times observed counts, reusing each solved operating path so that article-sampling variation is isolated from economic variation.

Why This Matters

Impact on research. The paper gives an economic foundation for treating firms as distributions rather than as points or average vectors. It links a measurable object — movement in Wasserstein distance between news distributions — to a specific competitive mechanism, while being explicit that reporting intensity is a confound. It also extends the multi-contest framework of Xu et al. (2022) to a dynamic, valuation-relevant setting with adjustment costs, and it supplies a simulation design that separates population-model error from finite-article measurement error.

Real-world applications (bullets):

• Competitive intelligence: tracking whether a company's disclosed activity mix is converging on that of higher-performing peers, using news and filings rather than waiting for segment reporting.
• Systematic equity signals: constructing text-based repositioning measures as candidate features, with the paper's simulation results as a warning about how much coverage is needed before the signal is reliable.
• Antitrust and market-structure analysis: quantifying crowding as firms move into overlapping activities, which bears on how competitive pressure transmits across markets.
• Investor information and disclosure policy: understanding that changes in media attention alone can produce apparent strategic shifts, which matters for anyone inferring firm strategy from coverage.

Industry relevance. For quantitative asset managers and data vendors, the paper's central practical message is cautionary: at the article coverage typical of a 100-firm Nasdaq sample, the estimator recovered the correct sign only 53.0859375% of the time, and changes in reporting alone produced positive alignment at fixed capacities. Any product built on "strategic repositioning" scores from news text needs to account for coverage intensity, not just textual similarity.

Future Directions

• Disentangling reporting from operations. The "reporting only" counterfactual shows that coverage changes can generate the same observed movement as capacity reallocation. Developing designs or instruments that separate the two, perhaps using disclosure events or publisher-level attention shocks, is a direct open problem.

• Improving measurement at realistic coverage. Recovery improved from 53.0859375% to 59.84375% as article counts rose from one to sixteen times observed levels, with absolute RMSE falling from 11.269559609434811 to 3.0352587480546047. Whether richer text models, pooling across years, or alternative transport costs can deliver comparable gains at observed coverage is unresolved.

• Extending beyond the restricted economy. The sign result relies on a two-opportunity construction, unit-axis component laws, common reporting, and a list of timing and zero-cross-moment restrictions. Generalizing the prediction to many activities, heterogeneous reporting intensities, and boundary equilibria would test how far the mechanism travels.

• Inference for independently sampled firm histories. The paper notes that no established design samples independent firm histories in the corporate archive, and Appendix E examines a variance approximation using a complete-triple representation. Valid standard errors and a genuine hypothesis test in the corporate data remain to be established.

Target Audience

This paper is best suited to financial economists and computational-finance researchers working on text-as-data asset pricing, industrial organization researchers interested in multi-market competition and dynamic repositioning, and quantitative researchers at asset managers or data providers who build signals from corporate news. Readers need comfort with equilibrium models, optimal transport, and simulation-based evaluation; the empirical application is relatively accessible, but the theoretical core is not introductory.

Authors’ abstract

When firms respond to new market opportunities, their capacity choices alter the future discounted free cash flows available to both competitors and investors. In competitive markets, these investment decisions publicized in news articles and analyst reports, measuring the changes in capital structure, product mix, firm leadership, risk exposure and operating procedure made by the firm. Modern search engines rely on Large Language Models to measure the relevancy of source documents to given search query. These measurements let researchers represent these news articles as clouds of dense vectors whose information geometry reflects the dissimilarity of firms over time. Building on Xu et al. (2022), we develop a model of costly competitive adjustment in which firms allocate capacity in which firms make firm capacity adjustment towards free cash flow shocks. Using a novel sandwich estimator, we analyse the sign and magnitude of this predicted adjustment using multi-agent simulation, and validate our findings on a sample of news and analyst reports from Nasdaq-listed companies between 2018-2022.