Value of Information in Social Learning (2503.05015v2)

Abstract: This study extends Blackwell's (1953) comparison of information to a sequential social learning model, where agents make decisions sequentially based on both private signals and the observed actions of others. In this context, we introduce a new binary relation over information structures: an information structure is more socially valuable than another if it yields higher expected payoffs for all agents, regardless of their preferences. First, we establish that this binary relation is strictly stronger than the Blackwell order. Then, we provide a necessary and sufficient condition for our binary relation and propose a simpler sufficient condition that is easier to verify.

• The paper demonstrates that an information structure is 'more socially valuable' if it leads to higher expected payoffs for all agents in a sequential decision-making framework.
• It characterizes necessary and sufficient conditions for social value, emphasizing the role of unbounded beliefs and the likelihood of obtaining conclusive signals.
• The analysis reveals that traditional Blackwell comparisons can fall short in social contexts, underscoring the need for refined criteria to evaluate sequential information sources.

This paper investigates how to compare the value of different information sources (information structures) within a sequential social learning model. In standard decision theory, Blackwell's theorem provides a way to compare information structures based on their value to a single decision-maker. However, in social learning settings, agents learn from both private signals and the actions of previous agents. This creates an information externality, as an agent's action informs subsequent agents. The paper introduces a new way to compare information structures in this context.

An information structure $\pi$ is defined as "more socially valuable" than another structure $\pi'$ (denoted $\pi \succsim_S \pi'$) if $\pi$ leads to higher expected payoffs for all agents in the sequence, regardless of their specific preferences (payoff functions) or the equilibrium strategy being played, compared to $\pi'$.

Key findings include:

1. Stronger than Blackwell Order: The "more socially valuable" relation ($\succsim_S$) is strictly stronger than the standard Blackwell order ($\succsim_B$) (Proposition 1). This means that if $\pi \succsim_S \pi'$, then $\pi \succsim_B \pi'$, but the reverse is not always true. The reason is that observing past actions is generally less informative than observing past signals directly. An information structure needs to be sufficiently informative to overcome this "garbling" effect of actions to be considered more socially valuable.
2. Characterization: An information structure $\pi$ is more socially valuable than $\pi'$ if and only if, for any decision problem and any equilibrium under $\pi$, the expected payoff for every agent $i$ is weakly higher than the maximum possible expected payoff agent $i$ could achieve under $\pi'$ if they could observe all past signals (not just actions) from $\pi'$ (Proposition 2). This is denoted as $$V_i^{\mathcal{D}(\pi,\bm{\sigma}^*)} \geq \overline{V}_i^{\mathcal{D}(\pi')}$$.
3. Necessary Condition: A necessary condition for $\pi \succsim_S \pi'$ (when $\pi'$ is not completely uninformative) is that $\pi$ must induce "unbounded beliefs" (Corollary 1). This means the private signals from $\pi$ must be capable, in principle, of pushing an agent's belief arbitrarily close to 0 or 1. Information structures that lead to information cascades (where agents eventually ignore their private signals) cannot be more socially valuable than other informative structures.
4. Sufficient Condition: A simpler, verifiable sufficient condition for $\pi \succsim_S \pi'$ is provided (Proposition 3). This condition relates the probability of obtaining "conclusive signals" (signals that perfectly reveal the state, leading to beliefs of 0 or 1) under $\pi$ to a measure of information overlap in $\pi'$. Specifically, if $\pi$ has a sufficiently high probability of revealing conclusive signals compared to the minimum overlap in signal probabilities across states in $\pi'$, then $\pi \succsim_S \pi'$. Mathematically:

   $$1 - \sum_{s \in \text{supp}(\pi')} \min\{\pi'(s|L), \pi'(s|H)\} \leq \min \{\pi(\mu=0|L), \pi(\mu=1|H)\}$$

5. Role of Simple Structures: The proofs rely heavily on analyzing information structures that are mixtures of "full information" (signals perfectly reveal the state) and "no information" (signals are uninformative). The sufficient condition (Proposition 3) is shown to be equivalent to the existence of such a mixture structure $\pi''$ that sits between $\pi$ and $\pi'$ in the Blackwell order ($\pi \succsim_B \pi'' \succsim_B \pi'$) (Lemma 1). If such a $\pi''$ exists, the paper shows $\pi \succsim_S \pi'' \succsim_S \pi'$ (Lemma 2). Under these simple mixture structures, the equilibrium payoffs in the social learning setting match the payoffs achievable if past signals were directly observable (Lemma 3).

The paper also discusses limitations. The definition of $\succsim_S$ requires the payoff improvement to hold for all equilibria, which is very strong. This leads to the counter-intuitive result that $\pi \succsim_S \pi$ only holds if $\pi$ is a mixture of full and no information (Proposition 4). A weaker relation, "weakly more socially valuable" ($\succsim_W$), is proposed, requiring only that there exists an equilibrium under $\pi$ that is better than all equilibria under $\pi'$. While $\pi \succsim_W \pi$ always holds, finding general conditions for $\succsim_W$ is challenging due to equilibrium selection issues.

Practical Implications:

• Evaluating information sources in social settings (e.g., online reviews, sequential decision-making in teams) requires more than just the standard Blackwell comparison.
• Simply providing more information (in the Blackwell sense) might not improve outcomes for everyone in the sequence due to the loss of information when observing actions instead of signals.
• Information structures that guarantee learning in the long run (inducing unbounded beliefs) are necessary candidates for being "more socially valuable".
• Information structures that frequently provide very strong (conclusive) evidence are more likely to be socially valuable, as they help overcome the noise introduced by inferring information from actions. This suggests a potential benefit for information systems that sometimes provide definitive signals, even if they are often uninformative, over systems that always provide moderately informative signals.
• The theoretical conditions highlight the difficulty in robustly improving social learning outcomes; specific equilibrium behaviors can lead to payoff reversals even when one information structure seems intuitively "better".