The Dimensions of Consensus

Transcription

1 The Dimensions of Consensus Alex Gershkov Benny Moldovanu Xianwen Shi Hebrew and Surrey Bonn Toronto February 2017 The Dimensions of Consensus February / 1

2 Introduction Legislatures/committees vote on multi-dimensional issues issues on ballot are often endogenous The 1974 Congressional Budget and Impoundment Control U.S. Congress budgeting process: bottom up top down for better or for worse? How to structure issues on ballot to improve consensus? The Dimensions of Consensus February / 1

3 Introduction Legislatures/committees vote on multi-dimensional issues issues on ballot are often endogenous The 1974 Congressional Budget and Impoundment Control U.S. Congress budgeting process: bottom up top down for better or for worse? How to structure issues on ballot to improve consensus? The Dimensions of Consensus February / 1

4 Introduction Legislatures/committees vote on multi-dimensional issues issues on ballot are often endogenous The 1974 Congressional Budget and Impoundment Control U.S. Congress budgeting process: bottom up top down for better or for worse? How to structure issues on ballot to improve consensus? The Dimensions of Consensus February / 1

5 Single-Peaked Preferences on a Line Single-peaked preferences to sidestep Gibbard-Satterthwaite Impossibility Theorem Black (1948): mechanism that selects median peak is Pareto optimal, anonymous, and dominant strategy incentive compatible (DIC) Moulin (1980): any Pareto optimal, anonymous, and DIC mechanism is a generalized median The Dimensions of Consensus February / 1

6 Generalized Median: One Dimension Generalized median prior to voting, add (n 1) phantom ballots to the n real voters ballots and then pick the median Example there are n voters and K alternatives {1,..., K} DIC mechanism: pick the lowest of agents preferred alternatives generalized median with (n 1) phantoms placed at alternative 1 Gershkov, Moldovanu and Shi (2016) all generalized medians can be implemented by sequential voting derive welfare maximizing mechanism under single-crossing preferences and one-dimensional types efficient allocation can be attained in the limit as population grows The Dimensions of Consensus February / 1

7 Generalized Median: One Dimension Generalized median prior to voting, add (n 1) phantom ballots to the n real voters ballots and then pick the median Example there are n voters and K alternatives {1,..., K} DIC mechanism: pick the lowest of agents preferred alternatives generalized median with (n 1) phantoms placed at alternative 1 Gershkov, Moldovanu and Shi (2016) all generalized medians can be implemented by sequential voting derive welfare maximizing mechanism under single-crossing preferences and one-dimensional types efficient allocation can be attained in the limit as population grows The Dimensions of Consensus February / 1

8 Generalized Median: One Dimension Generalized median prior to voting, add (n 1) phantom ballots to the n real voters ballots and then pick the median Example there are n voters and K alternatives {1,..., K} DIC mechanism: pick the lowest of agents preferred alternatives generalized median with (n 1) phantoms placed at alternative 1 Gershkov, Moldovanu and Shi (2016) all generalized medians can be implemented by sequential voting derive welfare maximizing mechanism under single-crossing preferences and one-dimensional types efficient allocation can be attained in the limit as population grows The Dimensions of Consensus February / 1

9 This Paper Study multi-dimensional voting in a standard spatial model majority winner may not exist (Plott 1967, Kramer 1973) Simple majority voting, dimension by dimension voting equilibrium generally exists (Kramer 1972) structure induced equilibrium (Shapsle 1979) Our innovation: dimensions are endogenously chosen Main idea: improvements can be obtained by redefining and optimizing the dimensions on which voting takes place The Dimensions of Consensus February / 1

10 Model Setup Two public goods (X and Y), n (odd) voters Voter i has an ideal level t i = (x i, y i ) R 2, i = 1,..., n Voters have Euclidean preferences: v t i 2 v is the chosen levels of provision, v R 2 is the standard Euclidean norm t i = x 2 i + y 2 i Individual peaks {t i } are private: distribution F (x, y) with density f Utilitarian planner solves min E v n v t i 2 i=1 Without IC constraints, v = t = 1 n n i=1 t i is the ex post first best The Dimensions of Consensus February / 1

11 Rotation in the Plane Fix coordinates; rotating point (x, y) counter-clockwise is represented by multiplying a rotation matrix R (θ) ( ) ( ) ( ) ( z cos θ sin θ x x cos θ y sin θ = = z + sin θ cos θ y x sin θ + y cos θ }{{} R(θ) Equivalent to rotating coordinates clockwise with angle θ ) The Dimensions of Consensus February / 1

12 Voting Mechanisms Marginal median mechanisms first rotate coordinates and then choose coordinate-wise medians in R k, rotation matrices are k k orthogonal matrices with determinant +1 (special orthogonal group) Kim and Roush (1984) and Peters et al. (1992) in R 2, every Pareto optimal, anonymous and DIC mechanisms can be obtained by picking the marginal (coordinate-wise) medians for some arbitrary system of orthogonal coordinates for R k with k > 2, Pareto optimal, anonymous and DIC mechanisms do not exist The Dimensions of Consensus February / 1

13 Voting Mechanisms Marginal median mechanisms first rotate coordinates and then choose coordinate-wise medians in R k, rotation matrices are k k orthogonal matrices with determinant +1 (special orthogonal group) Kim and Roush (1984) and Peters et al. (1992) in R 2, every Pareto optimal, anonymous and DIC mechanisms can be obtained by picking the marginal (coordinate-wise) medians for some arbitrary system of orthogonal coordinates for R k with k > 2, Pareto optimal, anonymous and DIC mechanisms do not exist The Dimensions of Consensus February / 1

14 Design Problem The planner chooses rotation θ to solve where min E R (θ) t i ϕ θ (t 1, t 2,..., t n ) 2 θ [0,2π] ϕ θ (t 1, t 2,..., t n ) = (m (θ, t 1, t 2,..., t n ), m + (θ, t 1, t 2,..., t n )) m (θ, t 1, t 2,..., t n ) = median (x 1 cos θ y 1 sin θ,..., x n cos θ y n sin θ) m + (θ, t 1, t 2,..., t n ) = median (x 1 sin θ + y 1 cos θ,..., x n sin θ y n cos θ) The Dimensions of Consensus February / 1

15 Aside: Why Separable Preferences? Voter 2 : ( x x 2 y y 2 ) ( 1 ρ ρ 1 ) ( x x2 y y 2 ) The Dimensions of Consensus February / 1

16 Key Idea: Illustration Suppose we have three voters: A, B, C The Dimensions of Consensus February / 1

17 Key Idea: Illustration Now suppose we rotate the coordinates clockwise Mean: invariant to rotating coordinates Median: marginal (coordinate-wise) median, not invariant The Dimensions of Consensus February / 1

18 Finite Population The Dimensions of Consensus February / 1

19 Continuum of Population By central limit theorem and Bahadur (1966), if X 1,..., X n are I.I.D. with finite mean and variance, then n ( X µx ) d N ( 0, σ 2 ) ( ) ( ) d 1 n Xn/2:n m X N 0, 4f 2 (m X ) The design problem becomes ( min µ (θ) m (θ) ) 2 ( + µ+ (θ) m + (θ) ) 2 + σ 2 X + σ 2 Y θ [0,2π] where µ (θ), µ + (θ), m (θ) and m + (θ) are means and medians of Z (θ) = X cos θ Y sin θ, and Z + (θ) = X sin θ + Y cos θ The Dimensions of Consensus February / 1

20 Continuum of Population By central limit theorem and Bahadur (1966), if X 1,..., X n are I.I.D. with finite mean and variance, then n ( X µx ) d N ( 0, σ 2 ) ( ) ( ) d 1 n Xn/2:n m X N 0, 4f 2 (m X ) The design problem becomes ( min µ (θ) m (θ) ) 2 ( + µ+ (θ) m + (θ) ) 2 + σ 2 X + σ 2 Y θ [0,2π] where µ (θ), µ + (θ), m (θ) and m + (θ) are means and medians of Z (θ) = X cos θ Y sin θ, and Z + (θ) = X sin θ + Y cos θ The Dimensions of Consensus February / 1

21 Sufficient Condition for Sub-Optimality Without loss, normalize µ X = µ Y = 0 to obtain equivalent problem: Necessary first-order condition: min θ [0,2π] m2 (θ) + m 2 + (θ) m (θ) m (θ) + m + (θ) m + (θ) = 0 Sufficient second-order condition for sub-optimality of rotation θ: ( m (θ) ) 2 + ( m + (θ) ) 2 + m (θ) m (θ) + m + (θ) m + (θ) < 0 The Dimensions of Consensus February / 1

22 Voting on Independent Issues Is Not Optimal Theorem Assume X and Y are unimodal, independent and satisfy for X : M X m X µ X or µ X m X M X for Y : M Y m Y µ Y or µ Y m Y M Y where M, m, µ are mode, median, mean, respectively. Then θ = 0 is a local utility minimum. Hold more generally: m X f X (m X ) 0, m Y f Y (m Y ) 0, m 2 X + m 2 Y 0 The Dimensions of Consensus February / 1

23 Graphical Illustration Suppose m X, m Y 0. Small rotation improves welfare if triangle ABE contains more probability mass than BCD, which is true if f Y (m Y) > 0. The Dimensions of Consensus February / 1

24 Identical Marginals: 45-Degree Rotation Note Z (π/4) = (X Y) / 2 is symmetric! Conflict between efficiency and incentive compatibility vanishes in one dimension π/4-rotation better than 0-rotation if m 2 X + m2 Y m2 X+Y /2 (i.e., median function is sub-additive) The Dimensions of Consensus February / 1

25 Identical Marginals: 45-Degree Rotation Theorem If X and Y are identically distributed, the π/4 rotation is welfare superior to the zero rotation if m X < (>) 0, m X+Y < (>) 0 and m X + m Y < (>) m X+Y. (*) If X and Y are I.I.D., the π/4 rotation is a critical point, and a sufficient condition for (*) is F X (m X + ε) + F X (m X ε) ( ) 1 for all ε > 0 (**) Condition (**) implies M X < m X < µ X (µ X < m X < M X ) and is implied by concavity (convexity) of F X The Dimensions of Consensus February / 1

26 Identical Marginals: 45-Degree Rotation Theorem If X and Y are identically distributed, the π/4 rotation is welfare superior to the zero rotation if m X < (>) 0, m X+Y < (>) 0 and m X + m Y < (>) m X+Y. (*) If X and Y are I.I.D., the π/4 rotation is a critical point, and a sufficient condition for (*) is F X (m X + ε) + F X (m X ε) ( ) 1 for all ε > 0 (**) Condition (**) implies M X < m X < µ X (µ X < m X < M X ) and is implied by concavity (convexity) of F X The Dimensions of Consensus February / 1

27 Welfare Simulations Exponential: f (x) = e x, x 0 Gamma: f (x) = xe x, x 0 Pareto: f (x) = x 2, x 1 Rayleigh: f (x) = 2xe x2, x 0 The Dimensions of Consensus February / 1

28 Relative Efficiency The first-best expected utility is E (X µ X, Y µ Y ) 2 = ( σ 2 X + σ 2 ) Y The expected utility under rotation θ is E (X cos θ Y sin θ m (θ), X sin θ + Y cos θ m + (θ)) 2 = ( σ 2 X + σ 2 ) ( Y µ (θ) m (θ) ) 2 ( µ+ (θ) m + (θ) ) 2 Relative efficiency of rotation θ is EF (θ) = σ 2 X + σ2 Y σ 2 X + σ2 Y + ( µ + (θ) m + (θ) ) 2 + ( µ (θ) m (θ) ) 2 1 The Dimensions of Consensus February / 1

29 General Bounds on Relative Efficiency Hotelling-Solomons (1932) inequality µ X m X σ X imply EF (θ) σ2 X + σ2 Y 2 ( σ 2 X + σ2 Y) = 1 2 I.I.D. case: ( π ) EF = 4 2σ 2 X 2σ 2 X + ( µ + (π/4) m + (π/4) ) 2 2σ 2 X 2σ 2 X + σ2 X = 2 3 The Dimensions of Consensus February / 1

30 General Bounds on Relative Efficiency Hotelling-Solomons (1932) inequality µ X m X σ X imply EF (θ) σ2 X + σ2 Y 2 ( σ 2 X + σ2 Y) = 1 2 I.I.D. case: ( π ) EF = 4 2σ 2 X 2σ 2 X + ( µ + (π/4) m + (π/4) ) 2 2σ 2 X 2σ 2 X + σ2 X = 2 3 The Dimensions of Consensus February / 1

31 The Logconcave Case (1/2) Consider a logconcave density on R 2 with marginals f X and f Y Prekopa (1973): convolutions of f X and f Y are logconcave 1 Bobkov and Ledoux (2014): f 2 (m) 1 12σ 2 2σ 2 Ball and Boroczky (2010): f (m) m µ ln ( ) e 2 These two inequalities yield (m µ) 2 1 f 2 (m) ln2 ( ) ( ) e e 12σ 2 ln The efficiency bound for the I.I.D. case becomes ( π ) EF 4 2σ 2 X 2σ 2 X + 12σ2 X ln2 ( e 2 ) The Dimensions of Consensus February / 1

32 The Logconcave Case (2/2) Caplin and Nalebuff (1988, 1991) logconcave density, no IC constraints, no explicit voting mechanism main result: the mean cannot be beaten by another alternative if a super-majority of 64% (i.e., 1 1/e) is required beautiful and early application of concentration inequality for logconcave (and ρ-concave) densities Our results the mean is replaced by rotated median incentive compatibility can be implemented by simple majority voting in each dimension costs no more than 12% of maximal achievable expected utility The Dimensions of Consensus February / 1

33 The Logconcave Case (2/2) Caplin and Nalebuff (1988, 1991) logconcave density, no IC constraints, no explicit voting mechanism main result: the mean cannot be beaten by another alternative if a super-majority of 64% (i.e., 1 1/e) is required beautiful and early application of concentration inequality for logconcave (and ρ-concave) densities Our results the mean is replaced by rotated median incentive compatibility can be implemented by simple majority voting in each dimension costs no more than 12% of maximal achievable expected utility The Dimensions of Consensus February / 1

34 Conclusion Voting by simple majority on each dimension + judicious choice of issues constitute an incentive compatible, highly efficient aggregation mechanism in a complex multi-dimensional framework Bundling issues allows finding better consensus among ex-ante conflicting interests Generalization to higher dimensions: efficiency bounds get closer to 1 as the number of dimensions increases The Dimensions of Consensus February / 1