power in voting games: axiomatic and probabilistic approaches filepower in voting games: axiomatic...
TRANSCRIPT
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Power in voting games: axiomatic and probabilisticapproaches
Josep Freixasa
aDepartment of MathematicsTechnical University of Catalonia
Summer School, Campione d’ItaliaGame Theory and Voting Systems
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Outline
1 Power indices, several interpretations
2 Decisiveness as a payoff
3 Power as a measure of influence
4 Success, luckiness and inclusiveness
5 Probabilistic model ((N,W ), p)Probability distributionBarry’s equation:Dubey and Shapley’s equation
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Notion(s) of power
What do you want to measure?
Depending on your answer we should look for a suitable tool.
An open definition of “power”...
Roughly speaking...
a power index is a numerical measure that estimates the a priori. . . of each voter in a simple game.
What could the . . . be?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Notion(s) of power
What do you want to measure?
Depending on your answer we should look for a suitable tool.
An open definition of “power”...
Roughly speaking...
a power index is a numerical measure that estimates the a priori. . . of each voter in a simple game.
What could the . . . be?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Notion(s) of power
What do you want to measure?
Depending on your answer we should look for a suitable tool.
An open definition of “power”...
Roughly speaking...
a power index is a numerical measure that estimates the a priori. . . of each voter in a simple game.
What could the . . . be?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Notion(s) of power
What do you want to measure?
Depending on your answer we should look for a suitable tool.
An open definition of “power”...
Roughly speaking...
a power index is a numerical measure that estimates the a priori. . . of each voter in a simple game.
What could the . . . be?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff:
divide a cake (P-power)
Decisiveness as influence: a priori influence capacity to makedecisions in a committee (I-power)
Success: expectation to achieve the desired result
Inclusiveness: expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff: divide a cake (P-power)
Decisiveness as influence: a priori influence capacity to makedecisions in a committee (I-power)
Success: expectation to achieve the desired result
Inclusiveness: expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff: divide a cake (P-power)
Decisiveness as influence:
a priori influence capacity to makedecisions in a committee (I-power)
Success: expectation to achieve the desired result
Inclusiveness: expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff: divide a cake (P-power)
Decisiveness as influence: a priori influence capacity to makedecisions in a committee (I-power)
Success: expectation to achieve the desired result
Inclusiveness: expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff: divide a cake (P-power)
Decisiveness as influence: a priori influence capacity to makedecisions in a committee (I-power)
Success:
expectation to achieve the desired result
Inclusiveness: expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff: divide a cake (P-power)
Decisiveness as influence: a priori influence capacity to makedecisions in a committee (I-power)
Success: expectation to achieve the desired result
Inclusiveness: expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff: divide a cake (P-power)
Decisiveness as influence: a priori influence capacity to makedecisions in a committee (I-power)
Success: expectation to achieve the desired result
Inclusiveness:
expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff: divide a cake (P-power)
Decisiveness as influence: a priori influence capacity to makedecisions in a committee (I-power)
Success: expectation to achieve the desired result
Inclusiveness: expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
What do you want to measure?
Several options:
Decisiveness as a payoff: divide a cake (P-power)
Decisiveness as influence: a priori influence capacity to makedecisions in a committee (I-power)
Success: expectation to achieve the desired result
Inclusiveness: expectation to be part in the winning side
Luckiness, etc.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Measuring decisiveness as a payoff: P-power
The property of efficiency (∑
i∈N ψi = 1) for P-power seemsinescapable. Thus, it is a requirement rather than an axiom.
Axiomatic approach: Properties for a power index ψ : SN → Rn:
Efficiency,
If i is a null voter then ψi [W ] = 0,
If i and j are equivalent voters then ψi [W ] = ψj [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 many indices exist with these properties?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Measuring decisiveness as a payoff: P-power
The property of efficiency (∑
i∈N ψi = 1) for P-power seemsinescapable. Thus, it is a requirement rather than an axiom.
Axiomatic approach: Properties for a power index ψ : SN → Rn:
Efficiency,
If i is a null voter then ψi [W ] = 0,
If i and j are equivalent voters then ψi [W ] = ψj [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 many indices exist with these properties?Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
...only one!
the Shapley–Shubik index which is the restriction to simple gamesof the Shapley value for cooperative games.Formulation for simple games, Wa are the winning coalitions
SSa[W ] =∑
S :S /∈W ,S∪a∈W
s!(n − s − 1)!
n!(|S | = s)
Probabilistic approach: derivation of the model from a bargainingmodel
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
...only one!
the Shapley–Shubik index which is the restriction to simple gamesof the Shapley value for cooperative games.Formulation for simple games, Wa are the winning coalitions
SSa[W ] =∑
S :S /∈W ,S∪a∈W
s!(n − s − 1)!
n!(|S | = s)
Probabilistic approach: derivation of the model from a bargainingmodel
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Shapley value bargaining model, 1953
y
1
y
2
ys
y
s + 1‖a
y
s + 2
yn
v(S)
v(S ∪ a)
Comment: Rationality of players + superadditivity stimulatethe formation of N and to divide revenues or costs accordingto the Shapley value.
Question: Is the probabilistic model provided a convincingargument for cooperative games?
yesand for the “restriction” to simple games? ...maybe “not”
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Shapley value bargaining model, 1953
y
1
y
2
ys
y
s + 1‖a
y
s + 2
yn
v(S)
v(S ∪ a)
Comment: Rationality of players + superadditivity stimulatethe formation of N and to divide revenues or costs accordingto the Shapley value.
Question: Is the probabilistic model provided a convincingargument for cooperative games? yes
and for the “restriction” to simple games? ...maybe “not”
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Shapley value bargaining model, 1953
y
1
y
2
ys
y
s + 1‖a
y
s + 2
yn
v(S)
v(S ∪ a)
Comment: Rationality of players + superadditivity stimulatethe formation of N and to divide revenues or costs accordingto the Shapley value.
Question: Is the probabilistic model provided a convincingargument for cooperative games? yesand for the “restriction” to simple games?
...maybe “not”
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Shapley value bargaining model, 1953
y
1
y
2
ys
y
s + 1‖a
y
s + 2
yn
v(S)
v(S ∪ a)
Comment: Rationality of players + superadditivity stimulatethe formation of N and to divide revenues or costs accordingto the Shapley value.
Question: Is the probabilistic model provided a convincingargument for cooperative games? yesand for the “restriction” to simple games? ...maybe “not”
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Standard formula for both S-value and SS-index
y
1
y
2
ys
y
s + 1‖a
y
s + 2
yn
s! (n − s − 1)!
Well-known formula by taking common factors:
The Shapley value φ is given by
φa(v) =∑
S∈2N\a
s!(n − s − 1)!
n![v(S ∪ a)− v(S)]
for any a ∈ N, where s = |S |.Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
The S&S bargaining model for the S&S index, 1954
1 Assume that all orderings of voters are equally probable.
2 Assume that everybody votes “yes” in his/her turn.
3 A player is “pivotal” if the coalition of his/her predecessors inthe queue is losing and his/her addition to it does the newcoalition winning.
The S&S index is the probability of being pivotal under deabove scheme, or equivalently
it is the expected value of the marginal contributions underthis scheme.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
y
1
y
2
ys
y
s + 1‖a
y
s + 2
yn
LOSING
WINNING
Question: Is this the most natural probabilistic scheme for theindex?
...Should it be possible for a voter to vote “no”?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
y
1
y
2
ys
y
s + 1‖a
y
s + 2
yn
LOSING
WINNING
Question: Is this the most natural probabilistic scheme for theindex? ...Should it be possible for a voter to vote “no”?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Example Consider the 1958 EU voting system:
[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 country is 4times pivotal in the third position
B
a big country is 24 times pivotal in the fourth position.
Thus the
power of a big country is28
120=
7
30, and the power of a small
country is:1
2
(1− 3
7
30
)=
3
20.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Example Consider the 1958 EU voting system:
[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 country is 4times pivotal in the third position
B
a big country is 24 times pivotal in the fourth position. Thus the
power of a big country is28
120=
7
30, and the power of a small
country is:1
2
(1− 3
7
30
)=
3
20.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probabilistic approach
The Shapley-Shubik index relies for its justification on theaxiomatic derivation of the Shapley value, not on any modelof voting protocol, bargaining or coalition formation.
In particular the queue formation procedure of voting ismerely a heuristic device for calculating the values of the SS.
It is not intended as a justification of the Shapley-Shubikindex, and is certainly not to be taken seriously as adescription of how voting actually takes place.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Alternative bargaining model:
A voter can vote either “in favor” or “against” the proposalsubmitted to vote.
y
1
n2
ns
y
s + 1
·s + 2
·n
WINNING
y
1
n2
ns
n
s + 1
·s + 2
·n
LOSING
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Some other indices of P-power indices
the Banzhaf, normalized, index
7→ Banzhaf index (I-power)
the Johnston index
the Deegan-Packel index
the Holler index
they do not come from the cooperative context
the nucleolus
it comes from the cooperative context
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Some other indices of P-power indices
the Banzhaf, normalized, index 7→ Banzhaf index (I-power)
the Johnston index
the Deegan-Packel index
the Holler index
they do not come from the cooperative context
the nucleolus
it comes from the cooperative context
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Some other indices of P-power indices
the Banzhaf, normalized, index 7→ Banzhaf index (I-power)
the Johnston index
the Deegan-Packel index
the Holler index
they do not come from the cooperative context
the nucleolus
it comes from the cooperative context
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Reasonable requirements for P-power
Minimum requirements:
E Efficiency
N+ET Null and equal treatment properties
S Sensitivity
Weak sensitivity: if v(S ∪ i) ≥ v(S ∪ j) for allS ⊆ N \ i , j, then ψi (v) ≥ ψj(v)Strong sensitivity: if, moreover, v(S ∪ i) > v(S ∪ j) forsome S ⊆ N \ i , j, then ψi (v) > ψj(v).
Only ... SS, Bz, Jh fulfill them.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Some other reasonable requirements for P-power
D Let u and w two simple games, and Ci (u) ⊂ Ci (w), thenψi (u) ≤ ψi (w).
[8; 5, 3, 1, 1, 1] and [8; 4, 4, 1, 1, 1] (donation)
AV Let u and w be derived from u by adding a veto player i , thenfor all j , k ∈ N the power of the players in N should remainproportional
ψj(u)/ψk(u) = ψj(w)/ψk(w)
No known index satisfies all these axioms: E+N+ET+S+D+AV(conjecture)No index satisfies: E+N+ET+T+AV (trivial impossibility theorem)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Some other reasonable requirements for P-power
D Let u and w two simple games, and Ci (u) ⊂ Ci (w), thenψi (u) ≤ ψi (w). [8; 5, 3, 1, 1, 1] and [8; 4, 4, 1, 1, 1] (donation)
AV Let u and w be derived from u by adding a veto player i , thenfor all j , k ∈ N the power of the players in N should remainproportional
ψj(u)/ψk(u) = ψj(w)/ψk(w)
No known index satisfies all these axioms: E+N+ET+S+D+AV(conjecture)No index satisfies: E+N+ET+T+AV (trivial impossibility theorem)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Some other reasonable requirements for P-power
D Let u and w two simple games, and Ci (u) ⊂ Ci (w), thenψi (u) ≤ ψi (w). [8; 5, 3, 1, 1, 1] and [8; 4, 4, 1, 1, 1] (donation)
AV Let u and w be derived from u by adding a veto player i , thenfor all j , k ∈ N the power of the players in N should remainproportional
ψj(u)/ψk(u) = ψj(w)/ψk(w)
No known index satisfies all these axioms: E+N+ET+S+D+AV(conjecture)
No index satisfies: E+N+ET+T+AV (trivial impossibility theorem)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Some other reasonable requirements for P-power
D Let u and w two simple games, and Ci (u) ⊂ Ci (w), thenψi (u) ≤ ψi (w). [8; 5, 3, 1, 1, 1] and [8; 4, 4, 1, 1, 1] (donation)
AV Let u and w be derived from u by adding a veto player i , thenfor all j , k ∈ N the power of the players in N should remainproportional
ψj(u)/ψk(u) = ψj(w)/ψk(w)
No known index satisfies all these axioms: E+N+ET+S+D+AV(conjecture)No index satisfies: E+N+ET+T+AV (trivial impossibility theorem)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Measuring...decisiveness as influence
The Banzhaf index is simply the probability of being crucial in thegame.
Bza[W ] =|Ca[W ]|
2n−1= Probability for a of being crucial
where Ca[W ] = S ⊆ N : S ∈W , S \ a /∈W .
Only a winning coalition will be formed.
All winning coalitions have equal probability of being formed.
All crucial voters receive equal shares.
Efficiency is not a requirement.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Measuring...decisiveness as influence
The Banzhaf index is simply the probability of being crucial in thegame.
Bza[W ] =|Ca[W ]|
2n−1= Probability for a of being crucial
where Ca[W ] = S ⊆ N : S ∈W , S \ a /∈W .
Only a winning coalition will be formed.
All winning coalitions have equal probability of being formed.
All crucial voters receive equal shares.
Efficiency is not a requirement.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Consider the 1958 EU voting system:
[12; 4, 4, 4, 2, 2, 1] ≡ [6; 2, 2, 2, 1, 1, 0])
CB [W ] = 3B, 3B + S , 2B + 2S
models that represent 1, 2 and 2 coalitions respectively.
CS [W ] = 2B + 2S
model that represents 3 coalitions.
Bz [W ] =
(5
16,
5
16,
5
16,
3
16,
3
16, 0
)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Consider the 1958 EU voting system:
[12; 4, 4, 4, 2, 2, 1] ≡ [6; 2, 2, 2, 1, 1, 0])
CB [W ] = 3B, 3B + S , 2B + 2S
models that represent 1, 2 and 2 coalitions respectively.
CS [W ] = 2B + 2S
model that represents 3 coalitions.
Bz [W ] =
(5
16,
5
16,
5
16,
3
16,
3
16, 0
)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
To emulate the SS index, some scholars provide axiomatizationsfor the Banzhaf (value) index.
Owen (1978)
Lehrer (1988) (Banzhaf value)
Feltkamp (1995)
Barua et al. (2005)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Measuring success, luckiness and inclusiveness
Success:
Raei [W ] =|S : i ∈ S ∈W |
2n+|S : i /∈ S /∈W |
2n
Luckiness:
Luci [W ] =|S : i ∈ S ∈W ,S \ i ∈W |
2n+|S : i /∈ S /∈W ,S ∪ i /∈W |
2n
Inclusiveness:
KBi [W ] =|Wi ||W |
where Wi = S ∈W : i ∈ S
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Measuring success, luckiness and inclusiveness
Success:
Raei [W ] =|S : i ∈ S ∈W |
2n+|S : i /∈ S /∈W |
2n
Luckiness:
Luci [W ] =|S : i ∈ S ∈W , S \ i ∈W |
2n+|S : i /∈ S /∈W ,S ∪ i /∈W |
2n
Inclusiveness:
KBi [W ] =|Wi ||W |
where Wi = S ∈W : i ∈ S
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Measuring success, luckiness and inclusiveness
Success:
Raei [W ] =|S : i ∈ S ∈W |
2n+|S : i /∈ S /∈W |
2n
Luckiness:
Luci [W ] =|S : i ∈ S ∈W , S \ i ∈W |
2n+|S : i /∈ S /∈W ,S ∪ i /∈W |
2n
Inclusiveness:
KBi [W ] =|Wi ||W |
where Wi = S ∈W : i ∈ S
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Wm = 1, 2, 1, 3, 2, 3
Rae1[W ]→ 1, 2, 1, 3, 1, 2, 3, 2, 3, ∅
Luc1[W ]→ ∅, 1, 2, 3
Rae1[W ] =6
8=
3
4, Luc1[W ] =
2
8=
1
4, Bz1[W ] =
1
2
Observe...
Rae1[W ] = Bz1[W ] + Luc1[W ]
but also
Rae1[W ] =1
2+
1
2Bz1[W ].
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Wm = 1, 2, 1, 3, 2, 3
Rae1[W ]→ 1, 2, 1, 3, 1, 2, 3, 2, 3, ∅
Luc1[W ]→ ∅, 1, 2, 3
Rae1[W ] =6
8=
3
4, Luc1[W ] =
2
8=
1
4, Bz1[W ] =
1
2
Observe...Rae1[W ] = Bz1[W ] + Luc1[W ]
but also
Rae1[W ] =1
2+
1
2Bz1[W ].
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Barry’s equation and Dubey and Shapley’s equation:
Raei [W ] = Bzi [W ] + Luci [W ] for all i ∈ N
Success = Decisiveness + Luckiness
Raei [W ] =1
2+
1
2Bzi [W ] for all i ∈ N
linear relationship between success and decisiveness
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Barry’s equation and Dubey and Shapley’s equation:
Raei [W ] = Bzi [W ] + Luci [W ] for all i ∈ N
Success = Decisiveness + Luckiness
Raei [W ] =1
2+
1
2Bzi [W ] for all i ∈ N
linear relationship between success and decisiveness
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probability distributionBarry’s equation:Dubey and Shapley’s equation
Model: ((N,W ), p)
Assume that a probability distribution over vote configurations p(exogenous information) enters as a second input besides thesimple game W . Of course,
0 ≤ p(S) ≤ 1 for all S ⊆ N, and∑S⊆N
p(S) = 1.
In a voting situation thus described by the pair (W , p) the ease ofpassing proposals or probability of acceptance is given by
α(W , p) = Prob (acceptance) =∑
S :S∈Wp(S)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probability distributionBarry’s equation:Dubey and Shapley’s equation
Model: ((N,W ), p)
Assume that a probability distribution over vote configurations p(exogenous information) enters as a second input besides thesimple game W . Of course,
0 ≤ p(S) ≤ 1 for all S ⊆ N, and∑S⊆N
p(S) = 1.
In a voting situation thus described by the pair (W , p) the ease ofpassing proposals or probability of acceptance is given by
α(W , p) = Prob (acceptance) =∑
S :S∈Wp(S)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probability distributionBarry’s equation:Dubey and Shapley’s equation
Success and Decisiveness in the model ((N,W ), p))
Ωi (W , p) = Prob (i is successful) =∑
S :i∈S∈Wp(S) +
∑S :i /∈S /∈W
p(S).
Φi (W , p) = Prob (i is decisive) =∑
S:i∈S∈WS\i/∈W
p(S) +∑
S :i /∈S /∈WS∪i∈W
p(S).
Λi (W , p) = Prob (i is lucky) =∑S:i∈S
S\i∈W
p(S) +∑S:i /∈S
S∪i/∈W
p(S)
Barry’s equation is still true: success = decisiveness + luckiness
Ωi (W , p) = Φi (W , p) + Λi (W , p)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probability distributionBarry’s equation:Dubey and Shapley’s equation
Success and Decisiveness in the model ((N,W ), p))
Ωi (W , p) = Prob (i is successful) =∑
S :i∈S∈Wp(S) +
∑S :i /∈S /∈W
p(S).
Φi (W , p) = Prob (i is decisive) =∑
S:i∈S∈WS\i/∈W
p(S) +∑
S :i /∈S /∈WS∪i∈W
p(S).
Λi (W , p) = Prob (i is lucky) =∑S:i∈S
S\i∈W
p(S) +∑S:i /∈S
S∪i/∈W
p(S)
Barry’s equation is still true: success = decisiveness + luckiness
Ωi (W , p) = Φi (W , p) + Λi (W , p)
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probability distributionBarry’s equation:Dubey and Shapley’s equation
Dubey and Shapley’s equation:
Ωi (W , p) 6= 1
2+
1
2Φi (W , p)
does not extend in the more general context ((N,W ), p)
Only for p =
(1
2,
1
2, . . . ,
1
2
)the equality
Ωi (W , p) =1
2+
1
2Φi (W , p)
holds.
For p = (p, . . . , p) the two indices rank voters in the sameway, but
For p 6= (p, . . . , p) is is always possible to find W such thatthe rankings do not coincide.
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probability distributionBarry’s equation:Dubey and Shapley’s equation
To be or not to be?
What is more important, to be decisive or to be successful?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probability distributionBarry’s equation:Dubey and Shapley’s equation
Questions?
Josep Freixas Power in voting games
Power indices, several interpretationsDecisiveness as a payoff
Power as a measure of influenceSuccess, luckiness and inclusiveness
Probabilistic model ((N,W ), p)
Probability distributionBarry’s equation:Dubey and Shapley’s equation
THANKS FOR YOUR
ATTENTION
Josep Freixas Power in voting games