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

2 Roadmap Introduction Introduction Definition of terms3 Definition of terms3 Brief overview4 Brief overview4 Importance10 Importance10 Discussion Discussion Definition of terms13 Definition of terms13 Properties14 Properties14 Proof of statement16 Proof of statement16 Conclusion Conclusion

3 Definition of terms Voting scheme – a decision making system that depends solely on the preferences of participants, and leaves nothing to chance 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 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 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 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 ? 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 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 individual manipulation” “Any non-dictatorial voting scheme with at least 3 possible outcomes is subject to individual manipulation” Interpretation: 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 4 contestants – w, x, y, z 3 voters – a, b, c 3 voters – a, b, c Each voter ranks contestants (as i j k l) according to his/her preference Each voter ranks contestants (as i j k l) according to his/her preference 1 st gets 4 points, 2 nd gets 3 … 1 st gets 4 points, 2 nd gets 3 … Whoever has most points wins 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 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. 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 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 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 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: 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 Introduction Definition of terms3 Definition of terms3 Brief overview4 Brief overview4 Importance10 Importance10 Discussion Discussion Definition of terms13 Definition of terms13 Important properties14 Important properties14 Proof of statement16 Proof of statement16 Conclusion Conclusion

13 Definition of terms Game form – Any decision making system in which the outcome depends upon the individual actions (strategies) 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 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 Straightforward game – a game in which everyone has a dominant strategy

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

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

16 Intuitive proof 1. Given a non-dictatorial voting scheme with more than 3 outcomes 2. Assume theorem: Every straightforward game form with at least 3 possible outcomes is dictatorial 3. Non-dictatorial => not straightforward => not every player / agent has a dominant strategy 4. No dominant strategy => true preference cannot be dominant 5. True preference not dominant => possible for a different preference to give a better outcome 6. 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 Proved by invoking Arrow Impossibility Theorem Arrow Impossibility Theorem states: Arrow Impossibility Theorem states: “Every social welfare function violates at least one of Arrow’s conditions” where Arrow’s conditions are: 1. Scope 2. Unanimity 3. Pair wise determination 4. Non-dictatorship

19 Formal approach used 1. A social welfare function is generated from a straightforward game form with 3+ outcomes 2. The social welfare function is shown to conform to the first 3 Arrow conditions – Scope, Unanimity, Pair wise determination 3. Thus, the function must violate the non- dictatorial condition => it must be dictatorial 4. The dictator of the social welfare function is proven to be the dictator of the game form 5. Hence the theorem is proved

20 Roadmap Introduction Introduction Definition of terms3 Definition of terms3 Brief overview4 Brief overview4 Importance10 Importance10 Discussion Discussion Definition of terms13 Definition of terms13 Important properties14 Important properties14 Proof of statement16 Proof of statement16 Conclusion Conclusion

21 Conclusion Results proved in the paper: Results proved in the paper: 1. “Every straightforward game form with at least 3 possible outcomes is dictatorial” 2. “Any non-dictatorial voting scheme with at least 3 possible outcomes is subject to individual manipulation”

22 Conclusion Comments about the paper: 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 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 is well-structured The paper leaves the rigorous math proof to the end making it easy to follow 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 The paper could elaborate on the implications of the result a bit more

23 Thank you End