Keyword: counterexample

Theorem Condorcet Paradox staged

Theorem First-Price Fails DSIC proved

Example Mixed Extension Pathology admitted

Example Monotone Family Assumption Counterexamples admitted

Theorem No Deterministic Constant Competitive Ratio proved

Example Sion Boundary Counterexample admitted

Example Sion-Wolfe Game Has No Mixed Value admitted

Example Strict Value Comparison Breaks the Competitive Guarantee proved

Example Three-Player Minimax Failure formalized

Example Weak Value Comparison Degrades to 1/n on the Needle Profile proved