# Informational Substitutes

## TL;DR

This paper introduces formal definitions of informational substitutes and complements, exploring their implications in game theory, prediction markets, crowdsourcing, and optimization, and analyzing their structural properties and design considerations.

## Contribution

It provides a unified framework for understanding informational substitutes and complements across game-theoretic and algorithmic contexts, with new definitions and structural insights.

## Key findings

- Substitutes capture diminishing marginal value of information.
- Substitutes characterize optimal equilibria in prediction markets.
- Designing scoring rules can promote substitutability or complementarity.

## Abstract

We propose definitions of substitutes and complements for pieces of information ("signals") in the context of a decision or optimization problem, with game-theoretic and algorithmic applications. In a game-theoretic context, substitutes capture diminishing marginal value of information to a rational decision maker. We use the definitions to address the question of how and when information is aggregated in prediction markets. Substitutes characterize "best-possible" equilibria with immediate information aggregation, while complements characterize "worst-possible", delayed aggregation. Game-theoretic applications also include settings such as crowdsourcing contests and Q\&A forums. In an algorithmic context, where substitutes capture diminishing marginal improvement of information to an optimization problem, substitutes imply efficient approximation algorithms for a very general class of (adaptive) information acquisition problems. In tandem with these broad applications, we examine the structure and design of informational substitutes and complements. They have equivalent, intuitive definitions from disparate perspectives: submodularity, geometry, and information theory. We also consider the design of scoring rules or optimization problems so as to encourage substitutability or complementarity, with positive and negative results. Taken as a whole, the results give some evidence that, in parallel with substitutable items, informational substitutes play a natural conceptual and formal role in game theory and algorithms.