Power in voting games: axiomatic and probabilistic approaches filePower in voting games: axiomatic and probabilistic approaches Josep Freixasa aDepartment of Mathematics Technical

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




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





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?





Notion(s) of power




Roughly speaking...







Notion(s) of power




Roughly speaking...







Notion(s) of power




Roughly speaking...








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.






Several options:

Decisiveness as a payoff: divide a cake (P-power)




Luckiness, etc.






Several options:


Decisiveness as influence:

a priori influence capacity to makedecisions in a committee (I-power)



Luckiness, etc.






Several options:





Luckiness, etc.






Several options:



Success:

expectation to achieve the desired result


Luckiness, etc.






Several options:





Luckiness, etc.






Several options:




Inclusiveness:

expectation to be part in the winning side

Luckiness, etc.






Several options:





Luckiness, etc.






Several options:





Luckiness, etc.





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?





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




...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





...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





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”






y

1

y

2

ys

y

s + 1‖a

y

s + 2

yn

v(S)

v(S ∪ a)


Question: Is the probabilistic model provided a convincingargument for cooperative games? yes

and for the “restriction” to simple games? ...maybe “not”






y

1

y

2

ys

y

s + 1‖a

y

s + 2

yn

v(S)

v(S ∪ a)


Question: Is the probabilistic model provided a convincingargument for cooperative games? yesand for the “restriction” to simple games?

...maybe “not”






y

1

y

2

ys

y

s + 1‖a

y

s + 2

yn

v(S)

v(S ∪ a)


Question: Is the probabilistic model provided a convincingargument for cooperative games? yesand for the “restriction” to simple games? ...maybe “not”





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




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.





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”?





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”?





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.





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.





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.





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





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






the Banzhaf, normalized, index 7→ Banzhaf index (I-power)

the Johnston index


the Holler index


the nucleolus







the Banzhaf, normalized, index 7→ Banzhaf index (I-power)

the Johnston index


the Holler index


the nucleolus






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.





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)






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)












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)














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.





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.





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

)





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

)





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)





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






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 |







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 |






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 ].





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 ].





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





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





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)






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)






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)






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)






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.






To be or not to be?

What is more important, to be decisive or to be successful?






Questions?






THANKS FOR YOUR

ATTENTION


Documents

Power in voting games: axiomatic and probabilistic approaches filePower in voting games: axiomatic and probabilistic approaches Josep Freixasa aDepartment of Mathematics Technical