orders of criticality in voting games (and other situations) · dall’aglio, fragnelli, moretti...

Orders of Criticality in Voting Games(and other situations)

M. Dall’Aglio1 V. Fragnelli2 S. Moretti3

1LUISS Univ. Italy 2Univ. of Eastern Piedmont, Italy 3Univ. Paris-Dauphine, France

Game Theory and RationingLake Como School of Advanced Studies, September 7, 2018

Dall’Aglio, Fragnelli, Moretti Indices of Criticality Campione D’Italia 2018 1 / 25

Political Examples

In parliaments, not all winning coalitions are minimal

Example (Italy, 1983–1987)

Table: The majority in the Italian Chamber – IX Legislature.

Party seats

Democrazia Cristiana (DC) 225

Partito Socialista Italiano (PSI) 73

Partito Socialdemocratico Italiano (PSDI) 29

Partito Repubblicano Italiano (PRI) 23

Partito Liberale Italiano (PLI) 13

Majority quota q = 316

PSDI and PRI are not critical, but they can become critical if theyjoin forces.


More recent examples

Example (Israel,2013-2015)

Table: The majority in theKnesset – XIX Legislature.

Party seats

Likud-Beytenu 31

Yesh Atid 19

Jewish Home 12

Hatnua 6


One party (Hatnua) isnot critical (and cannotdo much about it).

Example (France, 2017-)

Table: The majority in the 15th NationalAssembly.

Party seats

La Republique en marche 314

Mouvement democrate 47


One party (MoDem) is notcritical (and cannot do much).

Goals

We describe the power of non-criticalplayers


Monotonic Simple Games

Given a set of players N = {1, 2, . . . , n} we consider simple games

v(S) ∈ {0, 1} for any S ⊆ N

with v(∅) = 0 and v(N) = 1

which are monotonic

v(S) ≤ v(T ) for S ⊆ T ⊆ N

Coalitions with value 1 are called winning coalitions.

Definition

Player i is critical for a coalition M if:

v(M)− v(M \ {i}) = 1


Main Definition

The following definition was given in D, Fragnelli and Moretti (2016):

Definition

Let k ≥ 0 be an integer, let M ⊆ N, with |M| ≥ k + 1, be a winningcoalition. We say that player i is critical of the order k + 1 for coalitionM, via coalition K ⊆ M \ {i}, with |K | = k iff

v(M \ K )− v(M \ (K ∪ {i})) = 1 (1)

and K is the set of minimal cardinality satisfying (1), i.e., when k ≥ 1,

v(M \ T ) = 0 or v(M \ (T ∪ {i}) = 1 (2)

for any T ⊂ M \ {i} with |T | < k

Notice that, when k = 0, K = ∅ and (2) does not operate. Thus, a playeris critical of order 1 if and only if it is critical in the usual sense.


An example

Example

Consider a simple game with N = {1, 2, 3, 4, 5, 6} players and the followingminimal winning coalitions (any superset of them is winning):

{1, 2, 3} {1, 2, 4} {1, 2, 5} {1, 3, 4} {1, 3, 5}

If M = {1, 2, 3, 4, 5} is the winning coalition:I Player 1 is critical of the first order, i.e. critical in the usual sense.I Players 2 and 3 are critical of the second order (K ∪ {i} = {2, 3})I Players 4 and 5 are critical of the third order (K ∪ {i} = {2, 4, 5} or

K ∪ {i} = {3, 4, 5}).

If N is the winning coalition, the order of criticality for the first 5players does not change, while player 6 is never critical: He does nothave any role in changing the status of a coalition.


Some general results

The following results are given in D., Fragnelli and Moretti (2016):

Proposition

Let M ⊆ N be a winning coalition, then the players in M may bepartitioned into those that are critical of some order and those who arenever critical.

Proposition

Let i ∈ M be a player critical of the order k + 1 for coalition M, viacoalition K ⊂ M; if a player j ∈ K leaves the coalition, then i is a playercritical of the order k for coalition M \ {j}, via coalition K \ {j}.


Indices of criticality

The Shapley-Shubik index for v is the average number of times in which aplayer is critical w.r.t. a coalition when players enter the coalition in arandom order.

φi (v) =1

n!

∑π∈Π

σi (π)

Π = class of all permutations of N

σi (π) =

{1 if player i is critical in Pπ ∪ {i}0 otherwise

Pπ = players that precede i in the order π.

We measure player i ’s power in being k order critical

SS index of criticality of order k

φi ,k(v) =1

n!

∑π∈Π

σi ,k(π) σi (π) =

{1 if i k-order critical in Pπ ∪ {i}0 otherwise


Computing the SS index of order kWe rewrite the index in a more familiar way:

φi ,k(v) =∑s 63i

|S |!(n − |S | − 1)!

n!dck(i , S ∪ {i})

dck(i ,M) =

{1 if i is critical of order i in M

0 otherwise

A whole distribution of indices to a player:

Example (continued)

Φ1(v) =

(17

30, 0, 0, 0, 0, 0

)Φ2(v) = Φ3(v) =

(3

20,

3

10, 0, 0, 0, 0

) Φ4(v) = Φ5(v) =

(1

15,

3

20,

1

5, 0, 0, 0

)Φ6(v) = (0, 0, 0, 0, 0, 0)


Some bounds

Proposition

1 ≤n∑

k=1

∑i∈N

φi ,k(v) ≤ n

These bounds are attained

Lower bound

v(S) =

{1 if S = N

0 otherwise

Φi =

(1

n, 0, . . . , 0

)∀i ∈ N

Upper bound

v(S) =

{0 if S = ∅1 otherwise

Φi =

(1

n, . . . ,

1

n

)∀i ∈ N


The Bahnzaf index of order k

The index counts the proportion of times that a certain player is criticalw.r.t. any coalition that includes her.

βi ,k(v) =∑s 63i

dck(i ,S ∪ {i})2n−1

Also in this case we have a distribution of indices:

Example (Continued)

B1(v) =

(5

8, 0, 0, 0, 0, 0

)B6(v) = (0, 0, 0, 0, 0, 0)

B2(v) = B3(v) =

(1

4,

3

16, 0, 0, 0, 0

)B4(v) = B5(v) =

(1

8,

3

16,

1

16, 0, 0, 0

)


The Bahnzaf index of order k

The index counts the proportion of times that a certain player is criticalw.r.t. any coalition that includes her.

βi ,k(v) =∑s 63i

dck(i ,S ∪ {i})2n−1

Also in this case we have a distribution of indices:

Example (Continued)

B1(v) =

(5

8, 0, 0, 0, 0, 0

)B6(v) = (0, 0, 0, 0, 0, 0)

B2(v) = B3(v) =

(1

4,

3

16, 0, 0, 0, 0

)B4(v) = B5(v) =

(1

8,

3

16,

1

16, 0, 0, 0

)Dall’Aglio, Fragnelli, Moretti Indices of Criticality Campione D’Italia 2018 11 / 25

Indices of non-criticality

Some players may be non-critical of any order for some or all winningcoalitions

We may define indices that measure this (lack of) power

Example (Continued)

Φ6,NC =23

60β6,NC =

5

16Φi ,NC = βi ,NC = 0 i = 1, 2, 3, 4, 5


Synthetic indices

We provide numeric indices that synthetise the whole distribution andallow the comparison of power among players.

Definition

The Collective Shapley-Shubik (CSS) power index through all orders ofcriticality for player i ∈ N

Φi (v) =

∑nh=1 φi ,h(v)h−1∑n

h=1 h−1

∈ [0, 1]

Similarly, The Collective Banzhaf power index through all orders ofcriticality for the same player

Bi (v) =

∑nh=1 βi ,h(v)h−1∑n

h=1 h−1

∈ [0, 1]


Example (Continued)

Φ1(v) =34

147Φ6(v) = 0

Φ2(v) = Φ3(v) =6

49

Φ4(v) = Φ5(v) =25

224and

B1(v) =25

98B6(v) = 0

B2(v) = B3(v) =55

392

B4(v) = B5(v) =115

1176

Clearly,1 �CSS 2 ∼CSS 3 �CSS 4 ∼CSS 5 �CSS 6 ,

and the same order applies through the Collective Bahnzaf index.


Weighted Majority Games

Players are parties in an assembly – each with a weight wi

The assembly has a quota q > 12

∑i∈N wi

A weighted majority situation [q;w1, . . . ,wn]

A weighted majority game

vm(S) =

{1 if

∑i∈S wi ≥ q

0 otherwise, S ⊆ N.

Proposition

In a voting game, if a coalition admits a critical player of order k , then thecoalition admits at least other k − 1 players critical of the same order.

Example

There can be more: In [51; 44, 3, 3, 3, 3, 3, 3], Players 2 through 7 arecritical of order 4 w.r.t. N


Monotonicity propertiesRankings in power and in size should agree within the same assembly andacross different assemblies, when the weights clearly show the advantageof a party over the others.

Definition (Turnovec, 1998)

Let [q;w1, . . . ,wn] be a w.m.s. A power index π = (π1, . . . , πn) is locallymonotone whenever wi > wj implies

πi (vm) ≥ πj(vm)

Consider [q;w1, . . . ,wn] and [q;w ′1, . . . ,w′n], with

∑i∈N wi =

∑i∈N w ′i

π is globally monotone if wi∗ > w ′i∗ and wj ≤ w ′j for all j 6= i∗ implies

πi∗(vm) ≥ πi∗(v ′m)

Theorem

CSS and CB are both locally and globally monotone


Graph Example

Consider a connected graph

If you want the graphconnected, even after theremoval of one or more edges:

Edge 1 is critical: if it is removed thegraph is disconnected

Edge 2 is safer: yet you have to makesure that it is not disconnected togetherwith edge 3.

Edge 4 is still safer: yet yet you have tomake sure that it is not disconnectedtogether with edges 2 and 3

Goal

Measure the safety of edges


Connectivity games

Let G = (V ,E ) be a connected, undirected graph.

Aziz et al. (2009) define the spanning connectivity game, SCG, withplayers E :

vg (S) =

{1 if ∃ spanning tree T = (V ,E ′) with E ′ ⊆ S

0 otherwise

The game is well defined forI Simple graphs = bridges in an attacked city,I Multigraphs (where two vertices may be joined by several arcs) =

soldiers on the bridges

Example

v(1, 2, 5, 6) = 1v(1, 2, 3, 4) = 0


Orders of criticality in SCG’sa winning coalition must contain a spanning tree;when the coalition is a itself spanning tree, it is a minimal winningcoalition: all its players are critical of the first order.

Example

1 is critical of the first order,

2, 3, 5 and 6 are critical of the secondorder. Either the pair {2, 3} or the pair{5, 6} disconnects the graph.

4 is critical of the third order – it needsa pair of carefully chosen edges in{2, 3, 5, 6} to disconnect the graph.

An important difference with weighted majority situations

Here a critical player of order k does not imply the presence of other k − 1players with the same order of criticality in the coalition.


A general algorithm

To compute the order of criticality of an edge e in a winning coalitionS , we need to find the minimal number of other edges in S that,together with e, disconnect, i.e. cut, the graph, leaving the twovertices of the edge, say s and t, on either side of the cut.

The problem can be therefore formulated as a classical minimum cutproblem in graph theory

The Algorithm

1 Remove e from S ;

2 Compute the minimum cut k from s tot (or viceversa), by means of theFord-Fulkerson algorithm;

3 The order of criticality is given by k + 1


the Ford-Fulkerson algorithm

1 First of all, transform each undirected edge as two directed edges inopposite direction with the same unit capacity;

2 Let k = 0

3 The cycle:

1 Find an augmenting path from s to t, otherwise exit, returning k;2 Compute the maximum flow f passing through this path;3 Now transform the graph into its residual by modifying the capacities

of the edges composing the augmenting path, and those in theopposite direction as follows:

F The capacity of each edge in the path is reduced by f ;F The corresponding edge in the opposite direction is augmented by f .

4 Let k := k + 1 and repeat


An example

Example

Consider S as the complete graphjoining 6 vertices. We wish tocompute the order of criticality forthe edge e


An example

Example


An example

Example

Since 4 cycles are allowed the edge iscritical of order 4 + 1 = 5


An example

Example

Since 4 cycles are allowed the edge iscritical of order 4 + 1 = 5Dall’Aglio, Fragnelli, Moretti Indices of Criticality Campione D’Italia 2018 24 / 25

Some references

Aziz H., Lachish 0., Paterson M., Savani R., Power Indices inSpanning Connectivity Games, 2009, Power indices in spanningconnectivity games. In International Conference on AlgorithmicApplications in Management (pp. 55-67). Springer Berlin Heidelberg.

Dall’Aglio M., Fragnelli V., Moretti S., 2016, Orders of Criticality inVoting Games, Opretaions Research and Decisions, Vol. 26 (2), 53–67.DOI: 10.5277/ord160204

Shapley L.S., Shubik M.(1954), A Method for Evaluating theDistribution of Power in a Committee System, American PoliticalScience Review 48 : 787-792

Turnovec F., Monotonicity and Power Indices, in T.J.Stewart and R.C.van den Honert (Eds.), Trends in Multicriteria Decision Making,Lecture Notes in Economics and Mathematical Systems, 465,Springer-Verlag, pp. 199–214.


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