Understanding Manipulation of Voting Schemes: Insights and Examples

1. Manipulation of Voting Schemes: A General Result By Allan Gibbard Presented by Rishi Kant

2. Roadmap • Introduction • Definition of terms 3 • Brief overview 4 • Importance 10 • Discussion • Definition of terms 13 • Properties 14 • Proof of statement 16 • Conclusion

3. Definition of terms • Voting scheme – a decision making system that depends solely on the preferences of participants, and leaves nothing to chance • Dictatorial – no matter what the other participants’ preferences are, the outcome is always decided by the preference given by the dictator • True preference – the player’s preference if he were the only participant / dictator • Non-trivial voting scheme – a voting scheme in which not every player has a dominant strategy

4. Problem • Can one design a voting scheme whose outcome is solely based on the true preference of each participant ? • Answer: Not unless the game is dictatorial or has less than 3 outcomes

5. Formal statement • “Any non-dictatorial voting scheme with at least 3 possible outcomes is subject to individualmanipulation” • Interpretation: Given a voting scheme (and certain circumstances) it is possible for an individual to force his desired outcome by disguising his true preference

6. Example • 4 contestants – w, x, y, z • 3 voters – a, b, c • Each voter ranks contestants (as i j k l) according to his/her preference • 1st gets 4 points, 2nd gets 3 … • Whoever has most points wins

7. Example Let the true preference of each voter be: a => w x y z b => w x y z c => x w y z If every voter put down his/her true preference then w would win [11 points]

8. Example However, for the given situation c can force the winner to be x by pretending that his preference order is different a => w x y z b => w x y z c => x w y z  c => x y z w x will now win with 10 points

9. Notes • Point to note: c could influence the voting scheme only due to the given circumstances • If a and b had slightly different orderings e.g. a => w y z x, then c would not be successful • Thus, subject to individual manipulation means that there is at least one scenario for which an individual can force the outcome that he wants => voting scheme is not totally tamper proof

10. Importance • No non-trivial decision making system that depends on informed self-interest can guarantee that the outcome was based on the true preferences of the participants • Informed self-interest => everyone knows everyone else’s true preference and will act in their own best interest

11. Importance • With respect to Mechanism design, this result deals with the question: “Would an agent reveal his/her true preference to the principal?” The answer: Only for binary or dictatorial choice schemes => only binary or dictatorial choices are DOM-implementable

12. Roadmap • Introduction • Definition of terms 3 • Brief overview 4 • Importance 10 • Discussion • Definition of terms 13 • Important properties 14 • Proof of statement 16 • Conclusion

13. Definition of terms • Game form – Any decision making system in which the outcome depends upon the individual actions (strategies) • Dominant strategy – a strategy that gives the best possible outcome to a player no matter what strategies others choose • Straightforward game – a game in which everyone has a dominant strategy

14. Properties • Properties of game forms • Game forms leave nothing to chance • Players in game forms may or may not have “honest” strategies • Game forms always have a single outcome – there are no ties • Game forms may be used to characterize any non-chance decision making system

15. Properties • Properties of voting schemes • Voting schemes are a special case of game forms in which the players’ preferences are their strategies • Every player in a voting schemes has a true preference (honest strategy) • Voting schemes do not have to be democratic or count all individuals alike • Voting schemes must always have an outcome, even if the outcome is inaction

16. Intuitive proof • Given a non-dictatorial voting scheme with more than 3 outcomes • Assume theorem: Every straightforward game form with at least 3 possible outcomes is dictatorial • Non-dictatorial => not straightforward => not every player / agent has a dominant strategy • No dominant strategy => true preference cannot be dominant • True preference not dominant => possible for a different preference to give a better outcome • Voting scheme cannot guarantee true preference for all players and can thus be manipulated

17. Formal approach used Proving theorem: “Every straightforward game form with at least 3 possible outcomes is dictatorial” is equivalent to proving theorem: “Any non-dictatorial voting scheme with at least 3 possible outcomes is subject to individual manipulation” as shown by previous slide

18. Formal approach used • Proved by invoking Arrow Impossibility Theorem • Arrow Impossibility Theorem states: “Every social welfare function violates at least one of Arrow’s conditions” where Arrow’s conditions are: • Scope • Unanimity • Pair wise determination • Non-dictatorship

19. Formal approach used • A social welfare function is generated from a straightforward game form with 3+ outcomes • The social welfare function is shown to conform to the first 3 Arrow conditions – Scope, Unanimity, Pair wise determination • Thus, the function must violate the non-dictatorial condition => it must be dictatorial • The dictator of the social welfare function is proven to be the dictator of the game form • Hence the theorem is proved

20. Roadmap • Introduction • Definition of terms 3 • Brief overview 4 • Importance 10 • Discussion • Definition of terms 13 • Important properties 14 • Proof of statement 16 • Conclusion

21. Conclusion • Results proved in the paper: • “Every straightforward game form with at least 3 possible outcomes is dictatorial” • “Any non-dictatorial voting scheme with at least 3 possible outcomes is subject to individual manipulation”

22. Conclusion • Comments about the paper: • The paper is written in a self-contained fashion i.e. one does not need to refer to other sources to decipher the content • The paper is well-structured • The paper leaves the rigorous math proof to the end making it easy to follow • The paper could elaborate on the implications of the result a bit more

23. Thank you End