Week 3 - Strategy proof mechanisms and Gibbard-Satterthwaite Theorem

Question 1

Social choice function

Answer 1

• For each reported joint preference, the SCF produces one alternative, which is society’s top choice

The social choice function takes in everyone’s reported preferences (complete orderings over alternatives) and produces a single alternative
f : Ln → A

Question 2

What properties ensure that reported preferences are ture preferences? That is when is SCF strategy proof?

Answer 2

SCF is strategy proof when every individual preferes to report their true preference ranking, given the reports supported by remaining individuals.

Question 3

The Gibbard-Satterthwaite Theorem

Answer 3

• If |A| ≥ 3 and f is strategy-proof and surjective then f is a dictatorial social choice function
• Any function that takes reported preferences and chooses an action can be manipulated by misreporting the true preferences, unless there is a single individual whose most preferred alternative is chosen regardless of
  everyone else’s preferences.
• Step 1 demonstrates that if an SCF is strategy-proof and surjective, it is also monotonic and unanimous. Step 2 establishes that if an SCF is unanimous and monotonic, it must be dictatorial.