Flexible Information Acquisition in the Kyle Model

1. Problem Statement

The paper investigates the optimal information acquisition strategy of an informed trader in the continuous-time Kyle (1985) model. Unlike traditional literature that restricts the informed trader to acquiring normally distributed signals about a normally distributed asset payoff, this study allows for:

• General Priors: The asset payoff ($\tilde{v}$) can have an arbitrary distribution (discrete, continuous, or non-normal).
• Flexible Signal Structures: The trader can design signals ($\tilde{s}$) with arbitrary marginal and joint distributions with the payoff.
• Cost Function: The cost of information acquisition is modeled using mutual information (entropy cost), following the Rational Inattention framework of Sims (1998, 2003). The cost is $\lambda I(\tilde{s}; \tilde{v})$, where $\lambda$ is the marginal cost of information.

The trader faces a trade-off between:

1. Profit Potential: Gaining a precise signal increases the variance of the posterior expected payoff, offering higher trading profits.
2. Information Leakage Cost: Aggressive trading based on precise signals reveals information to market makers, reducing profits. This cost is measured by the Wasserstein-2 distance between the distribution of the posterior expected payoff and the distribution of noise trades.
3. Acquisition Cost: Higher precision signals incur higher entropy costs.

2. Methodology

The authors utilize a combination of Optimal Transport Theory and Information Theory to solve the problem.

• Equilibrium Characterization: Building on Back (1992) and Back et al. (2021), the paper reformulates the Kyle equilibrium using optimal transport. The informed trader's ex-ante expected profit is expressed as:
  $$\text{Profit} = \frac{1}{2} \left( \text{Var}(E[\tilde{v}|\tilde{s}]) + E[Z_T^2] - W_2^2(F_s, G) \right)$$
  Where $F_s$ is the distribution of the posterior expected payoff, $G$ is the distribution of noise trades (Normal), and $W_2$ is the Wasserstein-2 distance.
• Equivalent Formulation: The problem is transformed into maximizing the expected profit net of the entropy cost. The authors show that the pricing rule depends only on the distribution of the posterior expected payoff, not the specific signal realization, provided the signal is monotone.
• Optimization via Sinkhorn Algorithm: The optimization problem is solved using the Sinkhorn problem (entropy-regularized optimal transport). The authors derive the optimal posterior probabilities using a logit-type structure (similar to the multinomial logit model but adapted for continuous signals).
• Duality Arguments: To prove the optimality of continuous signals for discrete priors, the authors employ duality arguments, showing that splitting signal states (increasing the number of signals) strictly improves the objective function due to the convexity of the log-sum-exp function.

3. Key Contributions and Results

A. Optimality of Continuous Signals

• Result: Regardless of whether the asset payoff prior is discrete or continuous, the optimal signal is always continuously distributed.
• Mechanism: Even with a discrete prior (e.g., a two-point distribution), the optimal posterior expected payoff has a continuous distribution. This occurs because the noise trades in the Kyle model are continuous, creating an infinite number of potential "actions" (trading strategies) for the informed trader. The trader's value is convex in the partition of the signal space, incentivizing a continuum of signals rather than a discrete set.
• Contrast: This differs from standard Rational Inattention literature (e.g., Matějka and McKay, 2015) where discrete signals are often optimal for discrete choice problems.

B. Indeterminacy of Signal Choice (The "Normal Signal" Result)

• Result: Any continuous signal that induces a strictly increasing mapping between the signal realization and the posterior expected payoff yields the same ex-ante value for the trader and the same distribution for the posterior expected payoff.
• Implication: The specific functional form of the signal does not matter for the equilibrium outcome. Consequently, assuming a normally distributed signal is without loss of generality (WLOG). This provides a rigorous theoretical justification for the standard assumption of normal signals in Kyle (1984) and Back and Pedersen (1998), even when the underlying asset payoff is non-normal.

C. Comparative Statics

The paper analyzes how the distribution of the posterior expected payoff changes with parameters:

1. Information Acquisition Cost ($\lambda$):
   • As $\lambda$ increases (information becomes more expensive), the signal becomes less precise.
   • The variance of the posterior expected payoff decreases.
   • The distribution of the posterior expected payoff increasingly resembles a Normal distribution (kurtosis converges to 3, skewness to 0). The trader sacrifices profit potential to minimize information leakage costs.
2. Noise Trade Volatility ($\sigma_Z$):
   • As noise trade volatility increases, the cost of information leakage decreases (trades are harder to detect).
   • The trader acquires a more precise signal.
   • The posterior expected payoff distribution becomes more similar to the prior distribution (which may be heavy-tailed or discrete), and the kurtosis increases.

D. Specific Cases

• Normal Prior: The optimal signal is normal. The Gelbrich bound confirms that a normal distribution minimizes the Wasserstein distance to the normal noise trades.
• Discrete Prior (Two-State): The optimal posterior follows a logit form: $P(v_n|z) \propto \exp((v_n z + \mu_n)/\lambda)$. The ratio $\lambda / \sigma_Z$ acts as a sufficient statistic for the distribution of the expected payoff in the symmetric case.
• Continuous Non-Normal Prior: The results extend naturally. For example, with a double-exponential prior, as $\lambda \to \infty$, the posterior distribution converges to normality.

4. Significance

• Theoretical Unification: The paper unifies the Kyle model with Rational Inattention and Optimal Transport. It demonstrates that the "Normal Signal" assumption is robust and not merely a mathematical convenience but an optimal outcome of flexible information acquisition.
• Resolution of Discreteness: It resolves the apparent paradox of why discrete priors lead to continuous trading strategies and signals in market microstructure models, attributing it to the continuous nature of noise trading.
• Policy and Modeling Implications: The findings suggest that in markets with high information acquisition costs or low noise volatility, asset prices and informed traders' expectations will naturally gravitate toward normality, even if the fundamental asset values are highly skewed or discrete. Conversely, in deep, noisy markets, non-normal features of the fundamental value are preserved in the trading strategies.
• Methodological Innovation: The application of the Sinkhorn problem and entropy-regularized optimal transport to a dynamic trading game offers a powerful new toolkit for analyzing information acquisition in economics.

In summary, Viswanathan and Xing demonstrate that the tension between profit maximization and information leakage, mediated by entropy costs, leads to a universal equilibrium structure where continuous signals are optimal, and the specific signal choice is indeterminate, allowing for the standard normal signal assumption to be used generally.