Power in voting games: axiomatic and probabilistic approaches

Transcription

1 : axiomatic and probabilistic approaches a a Department of Mathematics Technical Universit of Catalonia Summer School, Campione d Italia Game Theor and Voting Sstems

2 Outline Power indices, several interpretations 1 Power indices, several interpretations Probabilit distribution Barr s equation: Dube and Shaple s equation

3 Notion(s) of power What do ou want to measure?

4 Notion(s) of power What do ou want to measure? Depending on our answer we should look for a suitable tool.

5 Notion(s) of power What do ou want to measure? Depending on our answer we should look for a suitable tool. An open definition of power...

6 Notion(s) of power What do ou want to measure? Depending on our answer we should look for a suitable tool. An open definition of power... Roughl speaking... a power index is a numerical measure that estimates the a priori... of each voter in a simple game. What could the... be?

7 What do ou want to measure? Several options: :

8 What do ou want to measure? Several options: : divide a cake (P-power)

9 What do ou want to measure? Several options: : divide a cake (P-power) Decisiveness as influence:

10 What do ou want to measure? Several options: : divide a cake (P-power) Decisiveness as influence: a priori influence capacit to make decisions in a committee (I-power)

11 What do ou want to measure? Several options: : divide a cake (P-power) Decisiveness as influence: a priori influence capacit to make decisions in a committee (I-power) Success:

12 What do ou want to measure? Several options: : divide a cake (P-power) Decisiveness as influence: a priori influence capacit to make decisions in a committee (I-power) Success: expectation to achieve the desired result

13 What do ou want to measure? Several options: : divide a cake (P-power) Decisiveness as influence: a priori influence capacit to make decisions in a committee (I-power) Success: expectation to achieve the desired result Inclusiveness:

14 What do ou want to measure? Several options: : divide a cake (P-power) Decisiveness as influence: a priori influence capacit to make decisions in a committee (I-power) Success: expectation to achieve the desired result Inclusiveness: expectation to be part in the winning side

15 What do ou want to measure? Several options: : divide a cake (P-power) Decisiveness as influence: a priori influence capacit to make decisions in a committee (I-power) Success: expectation to achieve the desired result Inclusiveness: expectation to be part in the winning side Luckiness, etc.

16 Measuring decisiveness as a paoff: P-power The propert of efficienc ( i N ψ i = 1) for P-power seems inescapable. Thus, it is a requirement rather than an axiom. Axiomatic approach: Properties for a power index ψ : S N R n :

17 Measuring decisiveness as a paoff: P-power The propert of efficienc ( i N ψ i = 1) for P-power seems inescapable. Thus, it is a requirement rather than an axiom. Axiomatic approach: Properties for a power index ψ : S N R n : Efficienc, If i is a null voter then ψ i [W ] = 0, If i and j are equivalent voters then ψ i [W ] = ψ j [W ] If W and W are two games, consider W W and W W. Then ψ[w ] + ψ[w ] = ψ[w W ] + ψ[w W ] How man indices exist with these properties?

18 ...onl one! the Shaple Shubik index which is the restriction to simple games of the Shaple value for cooperative games. Formulation for simple games, W a are the winning coalitions SS a [W ] = S:S / W, S {a} W s!(n s 1)! n! ( S = s)

19 ...onl one! the Shaple Shubik index which is the restriction to simple games of the Shaple value for cooperative games. Formulation for simple games, W a are the winning coalitions SS a [W ] = S:S / W, S {a} W s!(n s 1)! n! ( S = s) Probabilistic approach: derivation of the model from a bargaining model

20 Shaple value bargaining model, v(s) v(s {a}) s s + 1 a s + 2 n Comment: Rationalit of plaers + superadditivit stimulate the formation of N and to divide revenues or costs according to the Shaple value. Question: Is the probabilistic model provided a convincing argument for cooperative games?

21 Shaple value bargaining model, v(s) v(s {a}) s s + 1 a s + 2 n Comment: Rationalit of plaers + superadditivit stimulate the formation of N and to divide revenues or costs according to the Shaple value. Question: Is the probabilistic model provided a convincing argument for cooperative games? es

22 Shaple value bargaining model, v(s) v(s {a}) s s + 1 a s + 2 n Comment: Rationalit of plaers + superadditivit stimulate the formation of N and to divide revenues or costs according to the Shaple value. Question: Is the probabilistic model provided a convincing argument for cooperative games? es and for the restriction to simple games?

23 Shaple value bargaining model, v(s) v(s {a}) s s + 1 a s + 2 n Comment: Rationalit of plaers + superadditivit stimulate the formation of N and to divide revenues or costs according to the Shaple value. Question: Is the probabilistic model provided a convincing argument for cooperative games? es and for the restriction to simple games?...mabe not

24 Standard formula for both S-value and SS-index 1 2 s! (n s 1)! s s + 1 a s + 2 n Well-known formula b taking common factors: The Shaple value φ is given b φ a (v) = s!(n s 1)! [v(s {a}) v(s)] n! S 2 N\{a} for an a N, where s = S.

25 The S&S bargaining model for the S&S index, Assume that all orderings of voters are equall probable. 2 Assume that everbod votes es in his/her turn. 3 A plaer is pivotal if the coalition of his/her predecessors in the queue is losing and his/her addition to it does the new coalition winning. The S&S index is the probabilit of being pivotal under de above scheme, or equivalentl it is the expected value of the marginal contributions under this scheme.

26 1 2 LOSING WINNING s s + 1 a s + 2 n Question: Is this the most natural probabilistic scheme for the index?

27 1 2 LOSING WINNING s s + 1 a s + 2 n Question: Is this the most natural probabilistic scheme for the index?...should it be possible for a voter to vote no?

28 Example Consider the 1958 EU voting sstem: [12; 4, 4, 4, 2, 2, 1] [6; 2, 2, 2, 1, 1, 0] B B B S S (we ignore the null voter because receives 0) a big countr is 4 times pivotal in the third position a big countr is 24 times pivotal in the fourth position. B

29 Example Consider the 1958 EU voting sstem: [12; 4, 4, 4, 2, 2, 1] [6; 2, 2, 2, 1, 1, 0] B B B S S (we ignore the null voter because receives 0) a big countr is 4 times pivotal in the third position a big countr is 24 times pivotal in the fourth position. Thus the power of a big countr is = 7, and the power of a small 30 countr is: 1 ( ) = B

30 Probabilistic approach The Shaple-Shubik index relies for its justification on the axiomatic derivation of the Shaple value, not on an model of voting protocol, bargaining or coalition formation. In particular the queue formation procedure of voting is merel a heuristic device for calculating the values of the SS. It is not intended as a justification of the Shaple-Shubik index, and is certainl not to be taken seriousl as a description of how voting actuall takes place.

31 Alternative bargaining model: A voter can vote either in favor or against the proposal submitted to vote. 1 n 2 n s s + 1 s + 2 n WINNING 1 n 2 n s n s + 1 s + 2 n LOSING

32 Some other indices of P-power indices the Banzhaf, normalized, index

33 Some other indices of P-power indices the Banzhaf, normalized, index Banzhaf index (I-power)

34 Some other indices of P-power indices the Banzhaf, normalized, index Banzhaf index (I-power) the Johnston index the Deegan-Packel index the Holler index the do not come from the cooperative context the nucleolus it comes from the cooperative context

35 Reasonable requirements for P-power Minimum requirements: E Efficienc N+ET Null and equal treatment properties S Sensitivit Weak sensitivit: if v(s {i}) v(s {j}) for all S N \ {i, j}, then ψ i (v) ψ j (v) Strong sensitivit: if, moreover, v(s {i}) > v(s {j}) for some S N \ {i, j}, then ψ i (v) > ψ j (v). Onl... SS, Bz, Jh fulfill them.

36 Some other reasonable requirements for P-power D Let u and w two simple games, and C i (u) C i (w), then ψ i (u) ψ i (w).

37 Some other reasonable requirements for P-power D Let u and w two simple games, and C i (u) C i (w), then ψ i (u) ψ i (w). [8; 5, 3, 1, 1, 1] and [8; 4, 4, 1, 1, 1] (donation) AV Let u and w be derived from u b adding a veto plaer i, then for all j, k N the power of the plaers in N should remain proportional ψ j (u)/ψ k (u) = ψ j (w)/ψ k (w)

38 Some other reasonable requirements for P-power D Let u and w two simple games, and C i (u) C i (w), then ψ i (u) ψ i (w). [8; 5, 3, 1, 1, 1] and [8; 4, 4, 1, 1, 1] (donation) AV Let u and w be derived from u b adding a veto plaer i, then for all j, k N the power of the plaers in N should remain proportional ψ j (u)/ψ k (u) = ψ j (w)/ψ k (w) No known index satisfies all these axioms: E+N+ET+S+D+AV (conjecture)

39 Some other reasonable requirements for P-power D Let u and w two simple games, and C i (u) C i (w), then ψ i (u) ψ i (w). [8; 5, 3, 1, 1, 1] and [8; 4, 4, 1, 1, 1] (donation) AV Let u and w be derived from u b adding a veto plaer i, then for all j, k N the power of the plaers in N should remain proportional ψ j (u)/ψ k (u) = ψ j (w)/ψ k (w) No known index satisfies all these axioms: E+N+ET+S+D+AV (conjecture) No index satisfies: E+N+ET+T+AV (trivial impossibilit theorem)

40 Measuring...decisiveness as influence The Banzhaf index is simpl the probabilit of being crucial in the game. Bz a [W ] = C a[w ] 2 n 1 = Probabilit for a of being crucial where C a [W ] = {S N : S W, S \ {a} / W }.

41 Measuring...decisiveness as influence The Banzhaf index is simpl the probabilit of being crucial in the game. Bz a [W ] = C a[w ] 2 n 1 = Probabilit for a of being crucial where C a [W ] = {S N : S W, S \ {a} / W }. Onl a winning coalition will be formed. All winning coalitions have equal probabilit of being formed. All crucial voters receive equal shares. Efficienc is not a requirement.

42 Consider the 1958 EU voting sstem: [12; 4, 4, 4, 2, 2, 1] [6; 2, 2, 2, 1, 1, 0]) C B [W ] = {3B, 3B + S, 2B + 2S} models that represent 1, 2 and 2 coalitions respectivel.

43 Consider the 1958 EU voting sstem: [12; 4, 4, 4, 2, 2, 1] [6; 2, 2, 2, 1, 1, 0]) C B [W ] = {3B, 3B + S, 2B + 2S} models that represent 1, 2 and 2 coalitions respectivel. C S [W ] = {2B + 2S} model that represents 3 coalitions. ( 5 Bz[W ] = 16, 5 16, 5 16, 3 16, 3 ) 16, 0

44 To emulate the SS index, some scholars provide axiomatizations for the Banzhaf (value) index. Owen (1978) Lehrer (1988) (Banzhaf value) Feltkamp (1995) Barua et al. (2005)

45 Measuring success, luckiness and inclusiveness Success: Rae i [W ] = S : i S W S : i / S / W 2 n + 2 n

46 Measuring success, luckiness and inclusiveness Success: Rae i [W ] = S : i S W 2 n + Luckiness: S : i / S / W 2 n Luc i [W ] = S : i S W, S \ i W S : i / S / W, S i / W 2 n + 2 n

47 Measuring success, luckiness and inclusiveness Success: Rae i [W ] = S : i S W 2 n + Luckiness: S : i / S / W 2 n Luc i [W ] = S : i S W, S \ i W S : i / S / W, S i / W 2 n + 2 n Inclusiveness: KB i [W ] = W i W where W i = {S W : i S}

48 W m = {{1, 2}, {1, 3}, {2, 3}} Rae 1 [W ] {1, 2}, {1, 3}, {1, 2, 3}, {2}, {3}, { } Luc 1 [W ] { }, {1, 2, 3} Observe... Rae 1 [W ] = 6 8 = 3 4, Luc 1[W ] = 2 8 = 1 4, Bz 1[W ] = 1 2

49 W m = {{1, 2}, {1, 3}, {2, 3}} Rae 1 [W ] {1, 2}, {1, 3}, {1, 2, 3}, {2}, {3}, { } Luc 1 [W ] { }, {1, 2, 3} Rae 1 [W ] = 6 8 = 3 4, Luc 1[W ] = 2 8 = 1 4, Bz 1[W ] = 1 2 Observe... but also Rae 1 [W ] = Bz 1 [W ] + Luc 1 [W ] Rae 1 [W ] = Bz 1[W ].

50 Barr s equation and Dube and Shaple s equation: Rae i [W ] = Bz i [W ] + Luc i [W ] for all i N Success = Decisiveness + Luckiness

51 Barr s equation and Dube and Shaple s equation: Rae i [W ] = Bz i [W ] + Luc i [W ] for all i N Success = Decisiveness + Luckiness Rae i [W ] = Bz i[w ] for all i N linear relationship between success and decisiveness

52 Probabilit distribution Barr s equation: Dube and Shaple s equation Model: ((N, W ), p)

53 Probabilit distribution Barr s equation: Dube and Shaple s equation Model: ((N, W ), p) Assume that a probabilit distribution over vote configurations p (exogenous information) enters as a second input besides the simple game W. Of course, 0 p(s) 1 for all S N, and S N p(s) = 1. In a voting situation thus described b the pair (W, p) the ease of passing proposals or probabilit of acceptance is given b α(w, p) = Prob (acceptance) = p(s) S:S W

54 Probabilit distribution Barr s equation: Dube and Shaple s equation Success and Decisiveness in the model ((N, W ), p)) Ω i (W, p) = Prob (i is successful) = Φ i (W, p) = Prob (i is decisive) = Λ i (W, p) = Prob (i is luck) = S:i S W S:i S W S\{i}/ W S:i S S\{i} W p(s) + p(s) + p(s) + S:i / S / W S:i / S / W S {i} W S:i / S S {i}/ W p(s). p(s). p(s)

55 Probabilit distribution Barr s equation: Dube and Shaple s equation Success and Decisiveness in the model ((N, W ), p)) Ω i (W, p) = Prob (i is successful) = Φ i (W, p) = Prob (i is decisive) = Λ i (W, p) = Prob (i is luck) = S:i S W S:i S W S\{i}/ W S:i S S\{i} W p(s) + p(s) + p(s) + S:i / S / W S:i / S / W S {i} W S:i / S S {i}/ W p(s). p(s). p(s) Barr s equation is still true: success = decisiveness + luckiness Ω i (W, p) = Φ i (W, p) + Λ i (W, p)

56 Probabilit distribution Barr s equation: Dube and Shaple s equation Dube and Shaple s equation: Ω i (W, p) Φ i(w, p) does not extend in the more general context ((N, W ), p) Onl for p = ( 1 2, 1 2,..., 1 ) the equalit 2 Ω i (W, p) = Φ i(w, p) holds. For p = (p,..., p) the two indices rank voters in the same wa, but For p (p,..., p) is is alwas possible to find W such that the rankings do not coincide.

57 Probabilit distribution Barr s equation: Dube and Shaple s equation To be or not to be? What is more important, to be decisive or to be successful?

58 Probabilit distribution Barr s equation: Dube and Shaple s equation Questions?

59 Probabilit distribution Barr s equation: Dube and Shaple s equation THANKS FOR YOUR ATTENTION