Discarding Information

Diagram

Overview

Discarding information selectively filters data by projecting onto subsets. The discard operation models privacy, access control, and information asymmetry by limiting what downstream games observe, creating partial information scenarios essential for strategic reasoning.

Mathematical Structure

The discard operation projects products onto components:

• Projection π₁: X×Y→X drops the Y component
• Covariant lifting (π₁,1): (X×Y,1)→(X,1) as strategically trivial open game
• Contravariant form (1,π₁): (1,X)→(1,X×Y) "forgets" in reverse direction
• Reduces information content irreversibly

Key Properties

• Selective removal: Choose what to hide
• Irreversible: Cannot recover discarded data
• Information reduction: Decreases available information
• Asymmetry creation: Different agents see different subsets

Role in Composition

Discarding creates information asymmetry in games. Enables modeling of private information, hidden actions, and partial observability. Combined with copying, allows fine-grained control over information flow—copy broadcasts to all, discard hides from some.

Example

Sealed-bid auction with private bids:

Full information: All bids (b₁, b₂, ..., bₙ) Discard: Hide others' bids from each agent Result: Agent i observes only own bid bᵢ Enables: Private value revelation without coordination risk Each bidder acts without knowing competitors' valuations

This models the essential feature of sealed bids: information asymmetry that prevents strategic manipulation based on others' values.