Voting and Mechanism Design

Transcription

1 José M Vidal Department of Computer Science and Engineering, University of South Carolina March 26, 2010 Abstract Voting, Mechanism design, and distributed algorithmics mechanism design. Chapter 8.

2 Voting 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

3 Voting Why Vote? Common way of aggregating agents preferences. Well understood. But, centralized.

4 Voting The Problem 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

5 Voting The Problem The Voting Problem Beer Wine Milk Milk Wine Beer Plurality Runoff Pairwise Beer Wine Milk Wine Beer Milk

6 Voting The Problem The Voting Problem Beer Wine Milk Milk Wine Beer Beer Wine Milk Plurality Runoff Pairwise Wine Beer Milk

7 Voting The Problem The Voting Problem Beer Wine Milk Milk Wine Beer Beer Wine Milk Plurality Runoff 5,9 4 6,6 Pairwise Wine Beer Milk

8 Voting The Problem The Voting Problem Beer Wine Milk Milk Wine Beer Beer Wine Milk Plurality Runoff 5,9 4 6,6 Pairwise Wine Beer Milk

9 Voting Solutions 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

10 Voting Solutions Symmetry Reflectional symmetry: If one agent prefers A to B and another one prefers B to A then their votes should cancel each other out. Rotational symmetry: If one agent prefers A,B,C and another one prefers B,C,A and another one prefers C,A,B then their votes should cancel out.

11 Voting Solutions Symmetry Reflectional symmetry: If one agent prefers A to B and another one prefers B to A then their votes should cancel each other out. Rotational symmetry: If one agent prefers A,B,C and another one prefers B,C,A and another one prefers C,A,B then their votes should cancel out. Plurality vote violates reflectional symmetry, so does runoff voting. Pairwise comparison violates rotational symmetry.

12 Voting Solutions Borda Count 1 With x candidates, each agent awards x to points to his first choice, x 1 points to his second choice, and so on. 2 The candidate with the most points wins. Jean-Charles de Borda Borda satisfies both reflectional and rotational symmetry.

13 Voting Solutions Formalization There is a set of A agents, and O outcomes. Each agent i has a preference function > i over the set of outcomes. Let > be the global set of social preferences. That is, what we want the outcome to be.

14 Voting Solutions Definition (Desirable Voting Outcome Conditions) 1 > exists for all possible inputs > i

15 Voting Solutions Definition (Desirable Voting Outcome Conditions) 1 > exists for all possible inputs > i 2 > exists for every pair of outcomes

16 Voting Solutions Definition (Desirable Voting Outcome Conditions) 1 > exists for all possible inputs > i 2 > exists for every pair of outcomes 3 > is asymmetric and transitive over the set of outcomes

17 Voting Solutions Definition (Desirable Voting Outcome Conditions) 1 > exists for all possible inputs > i 2 > exists for every pair of outcomes 3 > is asymmetric and transitive over the set of outcomes 4 > should be Pareto efficient.

18 Voting Solutions Definition (Desirable Voting Outcome Conditions) 1 > exists for all possible inputs > i 2 > exists for every pair of outcomes 3 > is asymmetric and transitive over the set of outcomes 4 > should be Pareto efficient. 5 The scheme used to arrive at > should be independent of irrelevant alternatives.

19 Voting Solutions Definition (Desirable Voting Outcome Conditions) 1 > exists for all possible inputs > i 2 > exists for every pair of outcomes 3 > is asymmetric and transitive over the set of outcomes 4 > should be Pareto efficient. 5 The scheme used to arrive at > should be independent of irrelevant alternatives. 6 No agent should be a dictator in the sense that > is always the same as > i, no matter what the other > j are.

20 Voting Solutions Theorem (Arrow s Impossibility) There is no social choice rule that satisfies the six conditions. Kenneth Arrow

21 Voting Solutions Theorem (Arrow s Impossibility) There is no social choice rule that satisfies the six conditions. Kenneth Arrow Plurality voting relaxes 3 and 5. Adding a third candidate can wreak havoc. Pairwise relaxes 5. Borda violates 5.

22 Voting Solutions Borda Example 1 a > b > c > d 2 b > c > d > a 3 c > d > a > b 4 a > b > c > d 5 b > c > d > a 6 c > d > a > b 7 a > b > c > d

23 Voting Solutions Borda Example 1 a > b > c > d 2 b > c > d > a 3 c > d > a > b 4 a > b > c > d 5 b > c > d > a 6 c > d > a > b 7 a > b > c > d 1 c gets 20 points 2 b gets 19 points 3 a gets 18 points 4 d gets 13 points

24 Voting Solutions Borda Example Let s get rid of d. 1 a > b > c > d 2 b > c > d > a 3 c > d > a > b 4 a > b > c > d 5 b > c > d > a 6 c > d > a > b 7 a > b > c > d

25 Voting Solutions Borda Example Let s get rid of d. 1 a > b > c 2 b > c > a 3 c > a > b 4 a > b > c 5 b > c > a 6 c > a > b 7 a > b > c

26 Voting Solutions Borda Example Let s get rid of d. 1 a > b > c 2 b > c > a 3 c > a > b 4 a > b > c 5 b > c > a 6 c > a > b 7 a > b > c 1 a gets 15 points 2 b gets 14 points 3 c gets 13 points

27 Voting Summary 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

28 Voting Summary Voting Summary 1 Use Borda count whenever possible. 2 Practically, Borda requires calculating all preferences: often computationally hard.

29 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

30 Problem Description 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

31 Problem Description Painting the House Name Alice Bob Caroline Donald Emily Wants house painted? Yes No Yes Yes Yes

32 Problem Description Formal Definition Each agent i has a type θ i Θ i which is private.

33 Problem Description Formal Definition Each agent i has a type θ i Θ i which is private. θ = {θ 1, θ 2,..., θ A }.

34 Problem Description Formal Definition Each agent i has a type θ i Θ i which is private. θ = {θ 1, θ 2,..., θ A }. The protocol results in some outcome o O.

35 Problem Description Formal Definition Each agent i has a type θ i Θ i which is private. θ = {θ 1, θ 2,..., θ A }. The protocol results in some outcome o O. Each agent i gets a value v i (o, θ i ) for outcome o.

36 Problem Description Formal Definition Each agent i has a type θ i Θ i which is private. θ = {θ 1, θ 2,..., θ A }. The protocol results in some outcome o O. Each agent i gets a value v i (o, θ i ) for outcome o. The social choice function f (θ) tells us the outcome we want to achieve. For example, f (θ) = arg max o O n v i (o, θ i ) (1) i=1

37 Problem Description Painting the House Name Type (θ i ) v i (Paint, θ i ) v i (NoPaint, θ i ) Alice WantPaint 10 0 Bob DontNeedPaint 0 0 Caroline WantPaint 10 0 Donald WantPaint 10 0 Emily WantPaint 10 0

38 Problem Description Painting the House Name Type (θ i ) v i (Paint, θ i ) v i (NoPaint, θ i ) Alice WantPaint 10 0 Bob DontNeedPaint 0 0 Caroline WantPaint 10 0 Donald WantPaint 10 0 Emily WantPaint 10 0 Try this: Everyone votes Y/N. If majority votes Y then paint house. All pay 1/5 of cost (4 each).

39 Problem Description Painting the House Name Type (θ i ) v i (Paint, θ i ) v i (NoPaint, θ i ) Alice WantPaint 10 0 Bob DontNeedPaint 0 0 Caroline WantPaint 10 0 Donald WantPaint 10 0 Emily WantPaint 10 0 Try this: Everyone votes Y/N. If majority votes Y then paint house. All pay 1/5 of cost (4 each). Bob must pay for a paint job he didn t want.

40 Problem Description Painting the House Name Type (θ i ) v i (Paint, θ i ) v i (NoPaint, θ i ) Alice WantPaint 10 0 Bob DontNeedPaint 0 0 Caroline WantPaint 10 0 Donald WantPaint 10 0 Emily WantPaint 10 0 Try this: Everyone votes Y/N. Split cost among those who voted Y.

41 Problem Description Painting the House Name Type (θ i ) v i (Paint, θ i ) v i (NoPaint, θ i ) Alice WantPaint 10 0 Bob DontNeedPaint 0 0 Caroline WantPaint 10 0 Donald WantPaint 10 0 Emily WantPaint 10 0 Try this: Everyone votes Y/N. Split cost among those who voted Y. There is an incentive for all but Bob to lie.

42 Problem Description Definition (g Implements f ) A mechanism g : S 1 S A O implements social choice function f ( ) if there is an equilibrium strategy profile (S1 ( ),..., S A ( )) of the game induced by g such that g(s1 (θ 1),..., SA (θ A)) = f (θ 1,..., θ A ) for all θ Θ. Where we let S i (θ i ) be agent i s strategy given that it is of type θ i.

43 Problem Description Definition (Dominant Strategy Equilibrium) A strategy profile (S1 ( ),..., S A ( )) of the game induced by g is a dominant strategy equilibrium if for all i and all θ i, v i (g(s i (θ i ), s i ), θ i ) v i (g(s i, s i ), θ i ) for all s i S i and all s i S i.

44 Problem Description Definition (g Implements f ) A mechanism g : S 1 S A O implements social choice function f ( ) in dominant strategies if there is a dominant strategy equilibrium strategy profile (S1 ( ),..., S A ( )) of the game induced by g such that g(s1 (θ 1),..., SA (θ A)) = f (θ 1,..., θ A ) for all θ Θ.

45 Problem Description Definition (Strategy-Proof) The social choice function f ( ) is truthfully implementable in dominant strategies (or strategy-proof) if s i (θ i) = θ i (for all θ i Θ i and all i) is a dominant strategy equilibrium of the direct revelation mechanism f ( ). That is, if for all i and all θ i Θ i, v i (f (θ i, θ i ), θ i ) v i (f (ˆθ i, θ i ), θ i ) for all ˆθ i Θ i and all θ i Θ i.

46 Problem Description Theorem (Revelation Principle) If there exists a mechanism g that implements the social choice function f in dominant strategies then f is truthfully implementable in dominant strategies.

47 An Example Problem and Solution 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

48 An Example Problem and Solution Selling Example θ i R: types are the valuations.

49 An Example Problem and Solution Selling Example θ i R: types are the valuations. o {1,..., n}: index of agent who gets the item.

50 An Example Problem and Solution Selling Example θ i R: types are the valuations. o {1,..., n}: index of agent who gets the item. v i (o, θ i ) = θ i if o = i, and 0 otherwise.

51 An Example Problem and Solution Selling Example θ i R: types are the valuations. o {1,..., n}: index of agent who gets the item. v i (o, θ i ) = θ i if o = i, and 0 otherwise. f (θ) = arg max i (θ i )

52 An Example Problem and Solution Selling Example θ i R: types are the valuations. o {1,..., n}: index of agent who gets the item. v i (o, θ i ) = θ i if o = i, and 0 otherwise. f (θ) = arg max i (θ i ) Each agent gets a p i (o) so that u i (o, θ i ) = v i (o, θ i ) + p i (o).

53 An Example Problem and Solution Selling Example θ i R: types are the valuations. o {1,..., n}: index of agent who gets the item. v i (o, θ i ) = θ i if o = i, and 0 otherwise. f (θ) = arg max i (θ i ) Each agent gets a p i (o) so that u i (o, θ i ) = v i (o, θ i ) + p i (o). Set p(o) such that the agent who wins must pay a tax equal to the second highest valuation. No one else pays/gets anything. { θi max u i (o, θ i ) = j i θ j if o = i 0 otherwise.

54 An Example Problem and Solution Truth-Telling is Dominant in Vickrey Payments Example. 1 Let b i (θ i ) be i s bid given that his true valuation is θ i.

55 An Example Problem and Solution Truth-Telling is Dominant in Vickrey Payments Example. 1 Let b i (θ i ) be i s bid given that his true valuation is θ i. 2 Let b = max j i b j (θ j ) be the highest bid amongst the rest.

56 An Example Problem and Solution Truth-Telling is Dominant in Vickrey Payments Example. 1 Let b i (θ i ) be i s bid given that his true valuation is θ i. 2 Let b = max j i b j (θ j ) be the highest bid amongst the rest. 3 If b < θ i then any bid b i (θ i ) > b is optimal since u i (i, θ i ) = θ i b > 0

57 An Example Problem and Solution Truth-Telling is Dominant in Vickrey Payments Example. 1 Let b i (θ i ) be i s bid given that his true valuation is θ i. 2 Let b = max j i b j (θ j ) be the highest bid amongst the rest. 3 If b < θ i then any bid b i (θ i ) > b is optimal since u i (i, θ i ) = θ i b > 0 4 If b > θ i then any bid b i (θ i ) < b is optimal since u i (i, θ i ) = 0

58 An Example Problem and Solution Truth-Telling is Dominant in Vickrey Payments Example. 1 Let b i (θ i ) be i s bid given that his true valuation is θ i. 2 Let b = max j i b j (θ j ) be the highest bid amongst the rest. 3 If b < θ i then any bid b i (θ i ) > b is optimal since u i (i, θ i ) = θ i b > 0 4 If b > θ i then any bid b i (θ i ) < b is optimal since u i (i, θ i ) = 0 5 Since we have that if b < θ i then i should bid > b and if b > θ i then i should bid < b, and we don t know b then i should bid θ i.

59 The Groves-Clarke Mechanism 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

60 The Groves-Clarke Mechanism Theorem (Groves-Clarke Mechanism) If f (θ) = arg max o O then calculating the outcome using f ( θ) = arg max o O n v i (o, θ i ) i=1 n v i (o, θ i ) (where θ are reported types) and giving the agents payments of i=1 p i ( θ) = j i v j (f ( θ), θ j ) h i ( θ i ) (where h i (θ i ) is an arbitrary function) results in a strategy-proof mechanism.

61 The Groves-Clarke Mechanism Groves-Clarke Payments for House Painting Name v i (o, θ) v i (o, θ) + j i v j( θ) Alice = 5 Bob 0 0 = 0 Caroline = 5 Donald = 5 Emily = 5 Assuming all tell the truth.

62 The Groves-Clarke Mechanism Groves-Clarke Payments for House Painting Name v i (o, θ) v i (o, θ) + j i v j( θ) Alice = = 20 Bob 0 0 = = 20 Caroline = = 20 Donald = = 20 Emily = = 20 Assuming all tell the truth.

63 The Groves-Clarke Mechanism Groves-Clarke Payments for House Painting Name v i (o, θ) v i (o, θ) + j i v j( θ) Alice 0 0 = ( 10 3 Bob 0 0 = ( 10 3 Caroline = ( 10 3 Donald = ( 10 3 Emily = ( ) = 20 3) = 10 2) = 10 2) = 10 2) = 10 Alice lies.

64 The Vickrey-Clarke-Groves Mechanism 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

65 The Vickrey-Clarke-Groves Mechanism Theorem (Vickrey-Clarke-Groves (VCG) Mechanism) If f (θ) = arg max o O then calculating the outcome using f ( θ) = arg max o O n v i (o, θ i ) i=1 n v i (o, θ i ) (where θ are reported types) and giving the agents payments of i=1 p i ( θ) = j i v j (f ( θ i ), θ j ) j i v j (f ( θ), θ j ) (where h i (θ i ) is an arbitrary function) results in a strategy-proof mechanism.

66 The Vickrey-Clarke-Groves Mechanism VCG Payments for House Painting Name v i (o, θ) j i v j(f ( θ i ), θ j ) j i v j( ) j i v j( ) Alice = 5 ( ) 3 = = 5 Bob 0 0 = 0 ( ) 4 = = 0 Caroline = 5 ( ) 3 = = 5 Donald = 5 ( ) 3 = = 5 Emily = 5 ( ) 3 = = 5 Alice tells the truth.

67 Distributed Mechanism Design 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

68 Distributed Mechanism Design Distributed Algorithmic Mechanism Design Algorithmic mechanism design: make mechanism polynomial time Distributed algorithmic mechanism design: make mechanism distributed

69 Distributed Mechanism Design Inter-Domain Routing Problem Z c z = 4 D c d = 1 A c A = 5 Y c y = 3 X c x = 2 B c b = 2

70 Conclusion 1 Voting The Problem Solutions Summary 2 Problem Description An Example Problem and Solution The Groves-Clarke Mechanism The Vickrey-Clarke-Groves Mechanism 3 Distributed Mechanism Design 4 Conclusion

71 Conclusion Conclusion GC and VCG payment equations are useful ready-made solutions to many problems.

72 Conclusion Conclusion GC and VCG payment equations are useful ready-made solutions to many problems. But, we still need more research into how to distribute the mechanisms and how to make their calculation computationally tractable.