Abstract We study the structure of unstable power mechanisms. A powermechanism is modeled by an interaction form, the solution of which is called a set-tlement. By stability, we mean the existence of some settlement for any preferenceprofile. Configurations that produce instability are called cycles. We introduce a sta-bility index that measures the difficulty of emergence of cycles. Structural propertiessuch as exactness, superadditivity, subadditivity and maximality provide indicationsabout the type of instability that may affect the mechanism. We apply our analysis tostrategic game forms in the context of Nash-like solutions or core-like solutions. Inparticular, we establish an upper bound on the stability index of maximal interactionforms.
Keywords Interaction form · Effectivity function · Stability index ·Nash equilibrium · Strong equilibrium · Solvability · Acyclicity · Nakamura number ·Collusion
JEL Classification C70 · D71
1 Introduction
Stability is an essential requirement for political systems; however, it is knownthat most political institutions are unstable. In this paper, we study the structure of
J. M. Abdou (B)Centre d’Economie de la Sorbonne (CES) CNRS, UMR 8174,Université Panthéon-Sorbonne-Paris I, Paris, Francee-mail: [email protected]
J. M. AbdouÉcole d’Économie de Paris-Paris School of Economics (EEP-PSE), Paris, France
123
390 J. M. Abdou
instability of power systems. A system enters into instability when it is submitted tocontradictory forces that prevent any outcome from being established in such a waythat commonly accepted institutions work normally. However, no matter how harm-ful it may be deemed for the viability of collectivities, instability is not necessarily acourse of chaotic actions and events that obey no law: the main thesis of this paper isthat there are patterns of instability.
Many countries present the property of being politically split over two main issues.History and geography are accountable for this bipolarity. The main issues can be ofsocioeconomic type, or of ethnic or religious type. Almost all Western countries aredivided between left and right, conservative and liberal, democrats and republicans.Many Middle-Eastern societies are split into pro-Western and anti-Western coalitions.Bipolarity of opinions does not necessarily translate into instability. In most Westerncountries (“democracies”), where governance is based on a written constitution andelections, rules of government are immune to that crude type of instability. Lawmak-ers strive to define constitutions that avoid instability generated by any bipolar split.In contrast, some countries in the Middle-East did experience recently this type ofinstability. Some pro-Western coalition formally constituted the ruling power; never-theless, it could not force any significant outcome. The anti-Western coalition itselfcould oppose any outcome but could force none: political analysts express this situationby the vocable “deadlock”, “impasse” or “stalemate”.
Some countries, though immune to bipolar instability, could experience moresophisticated types of instability. Many parties with distinct political agendas existsimultaneously. Legal institutions work correctly and choose some ruling coalitionwith some program. The ruling coalition includes two or more parties who agree onmost issues. But the exercise of power becomes impossible when there is a disagree-ment within the coalition over the implementation of some important issue. Some partyin the opposition may propose an alliance to some component of the ruling coalition.As a result, the ruling coalition will eventually be overthrown, and new elections willbe held. This scenario may repeat itself. Lawmakers have designed institutions thatare immune against bipolar stalemates, but the political structure is more complex anda different type of instability may occur. It is important from the point of view of polit-ical science to distinguish between the types of instability and analyze the contexts oftheir emergence.
In this paper, using game theoretic tools, we wish to shed some light on the structureof instability.
The present study extends the theory and the results of Abdou (2010). In that study,the scope was limited to coalitional power distributions. Here, we intend to includestrategic interactions ruled by standard solutions, e.g., Nash equilibrium or strong Nashequilibrium. For that purpose, we need a framework that goes beyond coalitions. Themodel that we adopt is similar to that of effectivity structures introduced by Abdouand Keiding (2003, 2009). Here, the general concept is that of interaction form andthe solution is called a settlement.
Note that we consider power mechanisms with abstract outcomes rather than inter-actions with specified preferences. This approach allows for the study of systems assuch (institutions), where the profile of actual agents is drawn from an arbitrary popu-lation. Therefore this article can be viewed as a study in political or social engineering.
123
The structure of unstable power mechanisms 391
In our approach, there is no loss in focusing on the power structure rather than thestrategic form. As it turns out, an interaction form can be associated with any strategicform and an equilibrium concept, in such a way that given any preference profile, anoutcome is a settlement of the interaction form if, and only if, it is an equilibriumoutcome of the strategic game form. However, other interaction forms exist that arenot derived from any strategic game form. An interaction form is said to be stable ifany preference profile gives rise to some settlement.
Now, assume that the interaction form is not stable, as it is indeed often the case, inpolitical life. As one of the advantages of the model adopted in this paper, a comparisonbetween different power systems is possible within the same framework. Therefore,at least in some cases, the model suggests why some systems are deemed more stablethan others. This question is most relevant to political institutions, such as constitutionsor protocols of government formation.
The general idea in studying unstable systems is to obtain some typology of insta-bility. An instability type would determine the general features along which instabilityis likely to emerge and, consequently, the lines along which the society is likely to split.This idea applied to local effectivity functions has led to the definition of a stabilityindex (Abdou 2010). We extend that definition to interaction forms.
Our stability index is an integer associated with the mechanism itself. Looselyspeaking, it measures the difficulty of provoking a deadlock. If this number is low,for instance two, then a simple split in the society with strong opposition power oneach side can lead, at polarized preferences, to a stalemate. If the index is high, thenunless agents possess some intricate preference profile, a settlement can be reached.The stability index marks a threshold on the level of sophistication that a society mustnot exceed if it wishes to use safely the mechanism. The stability index is relatedclosely to the Nakamura number for simple games (Nakamura 1979), the differencebeing that the Nakamura number is defined on the winning coalition structure only,whereas the stability index depends on the whole interaction form. In Abdou (2010),it is shown that, in case of instability, the index of a maximal local effectivity functionis either 2 or 3. Here, even for maximal interaction forms, the index can take any valuebetween 2 and the cardinality of the alternative set.
To refine our knowledge of the structure of instability, one has to go beyond thestability index. The difficulty lies in the fact that interaction forms, as compared toeffectivity functions, are more complex objects. This is no surprise since in the interac-tion form model, we capture the power structure that underlies behavior in the contextof equilibrium (Nash, strong Nash, etc.), while in effectivity functions, the powerstructure is related to the core. In this framework, we cannot satisfy ourselves withclassical properties such as maximality, superadditivity and subadditivity, even thoughthey are involved in the lower part of the interaction form, the part that one can identifyas the “effectivity function”. Some machinery is needed for our purpose. However,our effort is rewarded, since this construction allows us to identify an important prop-erty that we call exactness: the latter describes whether the joint action (interactionarrays) of some coalition structure is stronger or not than the independent actions ofthe coalitions that compose that structure. If r is an integer, the absence of r -exactnessimplies that some active coalition structure of size r can oppose more power thanthat of the coalitions where they are independent. Strategically, this situation offers
123
392 J. M. Abdou
an opportunity for collusion and, for maximal interactive forms, signals the existenceof some cycle. If the interaction form is not r -exact for some degree r , then the upperbound on the index is r + 2.
The paper is organized as follows. In Sect. 2, we define the interaction form as themain object of our model and recall some families of interaction forms that relate ourmodel to more classical concepts of game theory. In Sect. 3, we define, motivate andinterpret the stability index: this is one of the main contributions of the paper. Sec-tion 4 is the most technical part of the paper; it is devoted to the structure of instability.This is where we define the notion of exactness, the second main contribution of thepaper. Special attention is paid to the case of bipolarity. In Sect. 5, we apply our theoryto strategic game forms with standard equilibrium notions and give an example ofcalculation of the stability index. In Sect. 6, we conclude.
2 Interaction forms as a model of power
A model that could serve as a framework of a power instability theory must fulfilltwo basic requirements. First, it must describe power and not the background stra-tegic mechanism that supposedly underlies that power; that is, the model must beintrinsic. This condition allows for comparison of different power distributions. Sec-ond, the model has to be sufficiently general so as to encompass a large variety ofinteraction contexts. Thus, we cannot be content with a simple power model such assimple games or coalitional games. The effectivity functions model itself seems to bequite restrictive for our purpose. Although effectivity functions play an important rolein implementation theory (Moulin and Peleg 1982; Moulin 1983; Peleg 1984, 2004;Abdou and Keiding 1991; Peleg and Peters 2009; Vannucci 2002, 2008; Peleg andWinter 2002), the power that they convey when they are associated to game formsis essentially related to the core solution. Only in a few particular cases, effectiv-ity functions proved to be relevant to dealing with solvability problems. This wasthe case for Nash solvability of two-player game forms, as well as that of rectangulargame forms (Gurvich 1975, 1978, 1989; Abdou 1998, 2000). However, characterizingstrong Nash solvability, even for two-player game forms, necessitates the introductionof more complex effectivity structures (Abdou 1995; Abdou and Keiding 2003, 2009).For all these reasons, our basic model in this study will be that of the interaction form.An interaction form can be viewed as the power structure adapted to the solvabilityproblem of game forms in the context of some (pure) Nash-like equilibrium concept.
In an interaction form, we take into account (1) the dependence of the interactionpower on the actual state 1 (the local aspect) and (2) the whole range of joint oppositionpower that may be activated in all available scenarios that lead to that state (the inter-active aspect). Technically, interaction forms are to game forms and equilibria whateffectivity functions are to game forms and the core. But, interaction forms can bestudied abstractly without any reference to game forms. This was first done in Abdou(1995) and later on in Abdou (2000) and Abdou and Keiding (2003, 2009).
1 The idea of introducing a power description that is conditional on the state goes back to Rosenthal (1972).
123
The structure of unstable power mechanisms 393
The model that we adopt in this paper is identical to that of Abdou and Keiding(2009), the only difference being that in the present article we restrict ourselves tofinite alternative sets. The notion of effectivity structures introduced in Abdou andKeiding (2003) is similar to our interaction form; but in a theoretical study, the presentmodel has some advantages: (1) it allows for the representation of various equilibriumconcepts within the same interaction form, whereas in Abdou and Keiding (2003) wehave to define an effectivity structure for each equilibrium concept; (2) it allows foroperations like projections that faithfully reflect the change in the underlying activecoalition structure, while in the framework of Abdou and Keiding (2003), projectionsdo not exist. This unification, a by-product of our model, is in the same spirit as theone obtained in Greenberg’s theory of social situations (Greenberg 1990), even thoughour purpose remains different. On one hand, a general theory of interactions is notwithin the scope of our paper; on the other hand, the model that we adopt applies togame forms rather than games because our aim is to study properties of power systems(institutions) and not properties of specific interactions.
2.1 Notations
Throughout this paper, we shall consider a finite set N = {1, . . . , n} the elements ofwhich are called players, and a finite set A = {a1, . . . , ap} the elements of which arecalled alternatives or states. We make use of the following notational conventions: Forany set X , we denote by P(X) the set of all subsets of X and by P0(X) = P(X)\{∅}the set of all non-empty subsets of X . Q(X) [resp. L(X)] will denote the set of all pre-orders (resp. linear orders) on X : that is, all binary relations on X , which are transitiveand complete (resp. transitive, complete and antisymmetric). If R ∈ Q(X), we denoteby R◦ (resp. R∼) the strict binary relation (resp. the equivalence relation) induced byR on X . Elements of P0(N ) are called coalitions. If S ∈ P0(N ), then N\S is denotedby Sc. Similarly, if B ∈ P(A), A\B is denoted by Bc. A preference profile (over A)is a map from N to Q(A), so that a preference profile is an element of Q(A)N . Forevery preference profile RN ∈ Q(A)N and S ∈ P0(N ), we have
P(a, S, RN ) = {b ∈ A | b R◦
i a, ∀i ∈ S}
(so that P(a, S, RN ) consists of all the outcomes considered to be strictly better thana by all members of the coalition S), and Pc(a, S, RN ) = A\P(a, S, RN ).
2.2 The basic model
Definition 2.1 An interaction array on (N , A) is a mapping ϕ : P0(N ) → P(A).
Let � ≡ �(N , A) be the set of all interaction arrays. We endow � with the partialorder ≤ where ϕ ≤ ϕ′ if, and only if, ϕ(S) ⊂ ϕ′(S) for all S ∈ P0(N ). By activecoalition structure (ACS hereafter) M, we mean any subset of P0(N ). By federationF, we mean any subset of active coalition structures. The support of ϕ denoted by [ϕ] isthe active coalition structure formed by all coalitions S ∈ P0(N ) such that ϕ(S) �= ∅.
123
394 J. M. Abdou
We denote by �0 ≡ �0(N , A) the subset of interaction arrays with non-empty sup-port. The range of ϕ is the set ρ(ϕ) := ∪S∈P0(N )ϕ(S). More generally for any ACSM, the range of ϕ in M is the set ρM(ϕ) := ∪S∈Mϕ(S).
Definition 2.2 An interaction form over (N , A) is a mapping E from P0(A) to subsetsof �0 satisfying the following conditions:
An interaction form E is said to be standard if it satisfies the following properties:
(iii) For all U, V ∈ P0(A): E[U ∪ V ] = E[U ] ∩ E[V ],(iv) For all U ∈ P0(A) : U ⊂ ρ(ϕ) ⇒ ϕ ∈ E[U ].
If E and E ′ are two interaction forms, we shall write E � E ′ if E[U ] ⊂ E ′[U ] forall U ∈ P0(A).
We may think of an interaction array in E[U ] as a description of an available moveof the agents given any state in U . To interpret the statement ϕ ∈ E[{a}], one hasto assume that a may occur in different scenarios that are not explicit in the model;any scenario leading to state a may arouse some coalition S that has the power todrive the outcome into ϕ(S). In an interaction array, all such potential “moves” aredescribed. If, for instance, the support of ϕ is the set {S, T }, then for some situationswith outcome a, coalition S has the power to reach ϕ(S) and for some situations withthe same outcome, coalition T has the power to reach ϕ(T ). The interaction array doesnot sharply define in which scenarios S can reach ϕ(S) and in which scenarios T canreach ϕ(T ). Within each coalition, action is coordinated (as in any coalitional game),but there is no coordination between S and T . Thus, an interaction array represents ajoint potential action. Note that, in this model, an interaction array is defined for thewhole set of coalitions. Nevertheless, ϕ(S) = ∅ means that coalition S is inhibited ordeactivated and therefore that the power represented by ϕ holds without the partici-pation of S. Therefore, the support of ϕ is in fact the active coalition structure behindϕ. Whether coalitions have a real incentive to make their move depends on the actualpreferences. This is why we introduce the following:
Definition 2.3 Let RN ∈ Q(A)N . An alternative a is dominated at RN if there existssome U ∈ P0(A), U � a, and some ϕ ∈ E(U ) such that ϕ(S) ⊂ P(a, S, RN ) for allS ∈ P0(N ). The alternative a is a settlement at RN if it is not dominated at RN . Theset of all settlements at RN will be denoted by Stl(E, RN ).
It follows that, even if there is no coordination of actions between coalitions,some collusion may exist between them (see Subsect. 4.5). For instance, if for ϕ ∈E[{a}], ϕ(S) and ϕ(T ) are non-empty and if, given the preference profile, both coali-tions wish to oppose a, they can do it: the implicit interpretation is that S will opposethe outcome in some scenarios and T will oppose it in some others. This is why acannot survive. The possibility of representing collusion is that which gives our modelthe ability to cope with situations that, if represented in the strategic form, wouldinvolve solutions that we classically call “equilibrium”(e.g., Nash, or strong Nash).
123
The structure of unstable power mechanisms 395
In contrast, collusion does not appear in effectivity functions, where the solution isusually termed “core”. To discard any misinterpretation of the model, we show that,in an interaction array, the joint action is disjunctive. To clarify this point, we give auseful definition. For any ϕ : P0(N ) → P(A) we have: ϕ∗(S) = ϕ(S)c, and for anymapping E from P0(A) to subsets of � we define:
E∗[U ] = {ϕ ∈ �|ϕ∗ /∈ E[U ]} (1)
Definition 2.4 An interaction coform over (N , A) is a mapping E from P0(A) tosubsets of � such that E∗ is an interaction form.
All our theory could have been formulated using interaction coforms instead ofinteraction forms. The two approaches are mathematically equivalent. Interactionforms highlight the opposition power (that is the β-version of power, in which thejoint action is disjunctive). They describe the power to upset some state while coformsdescribe the conservation power that forces that state (that is the α-version of power,in which the joint action is conjunctive). Those are the two faces of the same powermechanism, but since we focus on destabilizing power we adopt the β-version.
The model of interaction forms is a unifying model that imbeds the main featuresof standard strategic models, as well as those of standard coalitional models. The lastsection of this paper is devoted to interaction forms derived from strategic game forms.The following subsection shows how they generalize (local) effectivity functions.
2.3 Effectivity functions as interaction forms
Local effectivity functions can be viewed as particular interaction forms. They areformally defined as follows:
Definition 2.5 A local effectivity function on (N , A) is a family E ≡ (E[U ], U ∈P0(A)) where for any U ∈ P0(A), E[U ] : P(N ) → P(P0(A)) and such that thefollowing conditions are satisfied:
(i) E[U ](∅) = ∅,(ii) B ∈ E[U ](S), B ⊂ B ′ ⇒ B ′ ∈ E[U ](S),
(iii) U ⊂ V ⇒ E[V ](S) ⊂ E[U ](S).
The core of E at RN denoted by C(E, RN ) is the set of outcomes a ∈ A, such that thereis no S ∈ P0(N ), U ∈ P0(A) such that P(a, S, RN ) ∈ E[U ](S). The “canonical”interaction form associated with E[·] is defined by:
E[U ] = {ϕ ∈ �|∃S ∈ P0(N ) : ϕ(S) ∈ E[U ](S)} (2)
Clearly, one has Stl(E, ·) = C(E[·], ·). A local effectivity function E[·] is called aneffectivity function if E[U ][S] does not depend on U . Local effectivity functions havebeen the object of Abdou (2010).
123
396 J. M. Abdou
3 Stability and stability index
We start by introducing a graded notion of stability. For any r ∈ N∗, let �r denote the
set of all partitions of A with at least r elements (classes). If π ∈ �r and a ∈ A, wedenote by π(a) the element of the partition that contains a. Let Q•(π) be the set ofall R ∈ Q(A), such that a R∼b for all a and b with π(a) = π(b). There is a naturalbijection between Q•(π) and Q(π). Since |A| = p one has: �r = �p for all r ≥ p.
Definition 3.1 Let E be an interactive form and let r ∈ N∗, E is r-stable if
Stl(E, RN ) �= ∅ for all RN ∈ Q•(π)N and all π ∈ �r . E is stable if Stl(E, RN ) �= ∅for all RN ∈ Q(A)N .
It is easy to see that: r -stability of E implies its r − 1 stability (r ≥ 2); E is stableif, and only if, E is p-stable; and any interaction form E is 1-stable.
Let E be an interaction form. Cycles appear naturally in the study of interactionforms stability (see Keiding (1985) for cycles in effectivity functions and Abdou andKeiding (2003) for cycles in more general effectivity structures). The following def-inition is adapted from Abdou and Keiding (2009), where it has been defined forinteraction forms on continuous domains. For any ϕ ∈ � and i ∈ N , we have:ρi (ϕ) = ∪S�iϕ(S) (the range of player i in ϕ).
Definition 3.2 An E-dominance configuration is any r -tuple F = ((U1, ϕ1), . . . ,
(Ur , ϕr )) where: (U1, . . . , Ur ) is a partition of A and ϕk ∈ E[Uk] (k = 1, . . . , r).The partition (U1, . . . , Ur ) will be called the basis of the dominance configurance.The natural number r is its order (or length).An E-dominance configuration ((U1, ϕ1), . . . , (Ur , ϕr )) is a cycle in E if it satisfiesthe following:
(Property (C)) : For any i ∈ N and ∅ �= J ⊂ {1, . . . , r} there exists k ∈ J suchthat for all l ∈ J : Ul ∩ ρi (ϕk) = ∅.
The interaction form E is said to be acyclic if it has no cycles.
The following result is a particular case of Theorem 4.4 of Abdou and Keiding(2009):
Theorem 3.3 E is stable if, and only if, E is acyclic.
The existence of a cycle is equivalent to instability. One of the most attractivefeatures of the present model is that it allows for a unified expression of instabilityconditions across models of different types. Whether the background is strategic ornot, any failure in stability can be viewed as the effect of some generalized Condor-cet cycle (a Condorcet cycle is a situation where, given three alternatives, a, b, c, amajority prefers a to b, a majority prefers b to c and a majority prefers c to a so thatthere is no Condorcet winner). Moreover, as argued in the introduction, instability thatmay occur in a political system is not a chaotic or unanalyzable matter. In a countrygoverned by perennial institutions, unstable situations often present the same features.Governments are toppled almost in the same way. Institutions are generally paralyzedalong the same opposition lines. Recurrent instability clearly indicates some flaw inthe current institutions or the governance system. Since this flaw has been identi-fied as the existence of some cycle, studying the structure of instability amounts to
123
The structure of unstable power mechanisms 397
characterizing the nature of cycles. For that purpose, some typology of cycles may beneeded.
3.1 Stability index
In this section, we propose a measure attached to the stability of interaction forms.That such a measure is needed in political theory and indeed in effective governanceis largely motivated by the evidence that most institutions often present instabilitysymptoms and many current states are politically unstable.
Proposition and definition 3.4 The following numbers are equal:
(i) The minimal order of some cycle in E (with the convention that this number is+∞ if no cycle exists),
(i i) The smallest integer s for which E is s-unstable (with the convention that thisnumber is +∞ if no such integer exists).
This number is called the stability index of E and shall be denoted as σ(E).
Remark 3.5 In the context of local effectivity functions, a stability index has beendefined in Abdou (2010). This index is based on the notion of cycles introduced indefinition 3.1 of that paper. Since we associate to any local effectivity function E[·] acanonical interaction form E (Subsect. 2.3), one has to check whether the definitionof Abdou (2010) is consistent with the one given here. Indeed, one can verify thatany cycle in the sense of Abdou (2010) gives rise to a cycle of the same order in thesense of definition 3.2, and conversely. It follows that the stability index defined hereextends the stability index defined in that paper.
In what follows, we prove Proposition 3.4. It is instructive to start by some opera-tion that has to do with merging alternatives.Let f : A → A′ be a map. If ϕ′ ∈ �
(N , A′) we denote f −1 ◦ ϕ′, the element ϕ
of �(N , A) defined by ϕ(S) = f −1((
ϕ′) (S))
for all S ∈ P0(N ). For any interac-tion form E on (N , A) we define the interaction form E f on
(N , A′) as follows. For
U ′ ∈ P0(
A′) :
E f [U ′] =
{ϕ′ ∈ �
(N , A′) | f −1 ◦ ϕ′ ∈ E
[f −1 (
U ′)]}
Therefore, we have the following characterization of the index that generalizes a sim-ilar result obtained for local effectivity functions (Abdou 2010, Theorem 4.3).
Proposition 3.6 A cycle of order r exists in E if, and only if, there exists a surjectivemapping f : A → {1, . . . , r} such that E f is unstable.
Proof Assume that there exists a surjective mapping f : A → {1, . . . , r} such thatE f .
((U ′
1, ϕ′1
), . . . ,
(U ′
s, ϕ′s
))is a dominance configuration (resp. a cycle) in E f if and
only if((
f −1(U ′1), f −1 ◦ ϕ′
1
), . . . ,
(f −1
(U ′
s
), f −1 ◦ ϕ′
s
))is a dominance configu-
ration (resp. a cycle) in E . Assume that E f is unstable, then by Theorem 3.3, E f hassome cycle of order s ≤ r ; moreover, it is easy to see that we can choose s = r (justdivide the basis if necessary). It follows that E has a cycle of order r .
123
398 J. M. Abdou
Conversely, let ((U1, ϕ1), . . . , (Ur , ϕr )) be a cycle of E based on the partition(U1, . . . , Ur ). Let A′ be some set with r elements A′ := {u1, . . . , ur } and let f : A →A′ be defined by f (a) = uk for a ∈ Uk . Define ϕ′ ∈ �
(N , A′) by putting ϕ′
k(S) :=f (ϕk(S))(S ∈ P0(N )). For any S ∈ P0(N ) and k, l ∈ {1, . . . , r}, one has Uk∩ϕl(S) =∅ if and only if {uk} ∩ f (ϕl(S)) = ∅. It follows that
({(u1}, ϕ′1), . . . ,
({ur }, ϕ′r
))is a
cycle of E f based on the partition ({u1}, . . . , {ur }). Therefore, E f is not stable. ��Proof of Proposition 3.4 1 Let r ≤ p be a positive integer such that E is not r -stable.There exists π ∈ �r and RN ∈ Q•(π)N such that the settlement set of E at RN
is empty. Let f : A → π be the quotient map. Since RN ∈ Q•(π)N , there existsR′
N ∈ Q(π)N such that E f has an empty settlement set at R′N . By Proposition 3.6,
there exists a cycle of order r in E . Conversely, let r be the order of some cycle in E ,then by Proposition 3.6, there exists f : A → {1, . . . , r}, a surjective map such thatE f is not stable. Let π ≡ A/ f ∈ �s be its quotient set. To any R′ ∈ Q({1, . . . , r}),one can associate R ∈ Q•(π) by putting a R b if, and only if, π(a) R′ π(b). Therefore,E has an empty settlement set at some profile in Q•(π). ��.
We readily have some straightforward properties of the stability index:
(P1) The stability index takes integer values between 2 and |A|. Any cycle has anorder greater or equal to 2. Indeed, if r = 1 in definition 3.2, then by condition(i) one has U1 = A and by condition (ii) ϕ(S) = 0 for all S ∈ P0(N ). Thiscontradicts definition 2.2. Since the index is the cardinality of some partitionof A, one has 2 ≤ σ(E) ≤ |A|.
(P2) If E and E ′ are two interaction forms such that E ′ � E , then σ(E) ≤ σ(E ′).
If in a mechanism the opposition power of some player or coalition increases,then the stability index decreases.
(P3) For any map f : A → A′, σ (E) ≤ σ(E f
).
3.1.1 Relation to literature
As mentioned earlier, the stability index has first been defined in Abdou (2010) in thecontext of local effectivity functions. As far as I know, there has been no other attemptto define a notion (index, degree, number...), which gives hints about the structure ofpotential instability that may occur in mechanisms. Clearly, many stability conceptsexist for games and markets (strategic stability, dynamical stability, evolutionary sta-bility) and indices based on local topological features can be defined as a propertyof the equilibrium. But, there is little more than a mere vocabulary coincidence withour concept. Our index makes sense when there is no equilibrium. For cooperativegames, applied to politics, many power indices (e.g., Shapley, Banzhaf) exist in theliterature (see Straffin (1994)), but to the best of my knowledge, they have not beeninterpreted as relevant measures of the stability of power distribution. In my opinion,some indications on stability can be extracted from the study of power indices. Anotherlimitation of power indices comes from the range of the model where they are defined:indeed, most of the definitions apply only to maximal simple games. On the otherhand, one can find in literature studies concerning the degree of manipulability. Theyare mainly concerned with measuring to what extent a nonstrategy-proof social choice
123
The structure of unstable power mechanisms 399
function is manipulable, or else what are the losses and gains from manipulating sucha function (see Campbell and Kerry (2009) for a recent article using this approach).Applied to our context, this idea would lead to assessing the likelihood (or probability)of instability that occurs in the exercise of a mechanism, while our approach aims at thestructure (shape and number) of potential configurations that generate instability. Theonly exception is the Nakamura number (Nakamura 1979), which provides a notionthat can be compared to our index.
The Nakamura number is defined only for simple games (a simple game W is anysubset of non-empty coalitions). Since the operation (or the action) of a simple gameon an alternative set A can be viewed as a particular effectivity function, and indeedas an interaction form, we can compare our index to the Nakamura number. Let ν bethe Nakamura number of the simple game W and let σ be the stability index of theresulting interaction form. ν is classically viewed as a criterion for stability. What ourinterpretation adds to this viewpoint is that this number can also be viewed as an indexof stability. Precisely, for Nakamura, a simple game W acting on A is stable if, andonly if, ν > |A|. For us, even if ν ≤ |A| the number ν is meaningful. In fact, one has:σ = ν if ν ≤ |A| and σ = +∞ if ν > |A|. From this relation, one sees readily that,in particular, stability is obtained if and only if ν(W ) > |A| (see Abdou (2010) formore details). Example 5.10 highlights the role that the Nakamura number may playin the determination of the stability index of game forms that are not actions of simplegames. In view of this remark, we can state:
(P4) The index can take any value between 2 and A. Let p = |A| and let ν be suchthat 2 ≤ ν ≤ p. Let N = {1, . . . , ν} be some set of players. The simple gameW = {N \ {1}, . . . , N \ {ν}} has Nakamura number ν. Therefore, the stabilityindex of the action of W on A is precisely ν.
3.2 What does the stability index capture?
The stability index marks a threshold on the sophistication level of a society thatwishes to use safely some given mechanism. In the absence of a stable mechanism,the social engineer has to recommend institutions with a rather high stability index.Assume that an interaction form is unstable with a stability index σ , then merging somesocial states (or alternatives) results in a decrease of the number of alternatives and atransformation of the interaction form in a way that respects power distribution. Thisis the interpretation of the transformation E → E f . This transformation may occur,for instance, when the agents do not distinguish any more between two previouslydifferent alternatives. If the number of the new alternatives is less than σ , then the newinteraction form will be stable. When σ = 2 (see special Subsect. 4.7 on bipolarity),alternatives can be partitioned into two aggregates, or two major issues, over which thesociety can be opposed, and the power of agents or institutions allowed by the rules issuch that either issue can be opposed and neither can be forced. Efficient institutionsare designed to avoid this situation. Most modern political systems, though immuneto bipolarity, may suffer higher order cycles.
We consider that an interaction form with a higher index as more desirable than aninteraction form with a lower index. We base our opinion on two arguments of quite
123
400 J. M. Abdou
different nature. On the one hand, if an interaction form has a higher index, then to pro-duce the cycle the society has to exhibit sophisticated preferences that are compatiblewith that cycle. Thus, the index of a power mechanism measures the degree of polit-ical sophistication that a society must possess in order that the mechanism enters adeadlock.
On the other hand, the formation of a cycle of lower order is less costly (hencemore likely) than that of a cycle of higher order. In any cycle, one may distinguishtwo factors: the cooperation factor (coalitions must coordinate their strategies) and thecollusion factor (many coalitions may be needed to contribute to upset the same out-come, because multiple scenarios with the same outcome are potentially available).The cooperation factor may be absent in some cases (e.g., Nash equilibrium), butplays the main role in other cases (e.g., core or local core). Clearly, one can considerthat the difficulty to form a cycle increases with the number of coalitions involvedin that cycle (for instance, in the core). The collusion factor is present in all interac-tion arrays that have a support with two elements or more. To fix ideas, assume thatM = {S, T }, and the cycle (ϕ1, ϕ2, ϕ3) has a basis (U1, U2, U3). Given any scenariowith outcome in U1, then either (S, ϕ1(S)) or (T, ϕ1(T )) would have to form in orderto oppose that scenario. The disjunctive nature of this operation requires that onlyone coalition has to form and to propose the same subset of alternatives against thisscenario. This suggests that the cost of formation of an interaction array is comparableto that of a coalition. The order of the cycle is the relevant number and we take it asan a priori measure of the difficulty of cycle formation.
We illustrate our interpretation by a political example, which formed the back-ground in motivating this research. Compare two societies that have been endowedwith the same institutions, with index 3 for instance. The first society is split by anethnic or religious strife, while the second society lives with many social issues con-sidered as relevant. In the first society, ethnic or religious conflict cannot create adeadlock because those types of stakes are mostly bipolar and that the index is 3,while it is possible, in the second case, to observe instability, though the occurrenceof the latter is tempered by the fact that conditions of cyclicity are somehow difficultto crystallize. Therefore, if the society seeks stability and if a stable mechanism is notavailable (after all, a dictatorship is a stable mechanism), the more sophisticated is thesociety, the higher index must be recommended.
It is worth noting that the machinery deployed in the following section is not neededif we restrict ourselves to local effectivity functions. In particular, this reflects the factthat the structure of instability for M-equilibrium of a game form is much morecomplex than that of the (local) core.
4 The structure of instability
In this section, we shall attempt to determine or at least provide some bounds onthe stability index. For that purpose, we need to devise some operations on interac-tion forms. Precisely, we shall derive some appropriate substructures from the basicstructure. The simplest substructure is the one that contains the local and the globaleffectivity functions E1 and E0, respectively. This will be the object of Subsect. 4.3.
123
The structure of unstable power mechanisms 401
Higher order substructures may also be extracted (Subsect. 4.4) and the notion ofexactness is defined (Subsect. 4.5) for each. The tools that we provide are not merelytechnical. Each of the notions and operations that we define below (projection, fed-eration, exactness) have a theoretical role and possess an intuitive interpretation. Westart by the projection operation.
4.1 Projection
An interaction array is defined on P0(N ) (definition 2.1), so that the model allows apriori the surge of any coalition. However, it may be the case that institutionally (bylaw or structural constraints) some coalitions are not allowed to form: only coalitionsin some M ⊂ P0(N ) can actually be active. For instance, in a legislature (Senate,House of representatives) only some coalitions are practically possible. The definitionscan be adapted to take into account these institutional limitations. An alternative a isM-dominated at the preference profile RN if there exists some U ∈ P0(A), U � aand ϕ ∈ E(U ) such that [ϕ] ⊂ M and ϕ(S) ⊂ P(a, S, RN ) for all S ∈ M. Thealternative a is an M-settlement at RN if it is not M-dominated at RN . The set of allM-settlements at RN will be denoted as: Stl(M)(E, RN ). To cope with this situationwithout changing our model, we would like to define an interaction form that reflectsthe activity of the active coalition structure M. This is precisely the role of the follow-ing operation. For ϕ ∈ �, the projection of ϕ on M, denoted by ϕM is defined by:
ϕM(S) :={
ϕ(S) if S ∈ M∅ if S /∈ M
The projection of E on M is the interaction form E(M) defined by:
E(M)[U ] := {ϕ ∈ � | ϕM ∈ E[U ]} (3)
In particular, E(∅)[U ] = ∅. The following properties express that projection representsfaithfully the transformation of power induced by a restriction of the ACS.
(1) For any U ∈ P0(A), and for any M1,M2s ⊂ P0(N ), one has: E(M1)(M2)
(2) For any RN ∈ Q(A)N , one has : Stl(M)(E, RN ) = Stl(E(M), RN ).
As a general remark, one can see that allowing that some ϕ(S) be empty is not aninnocuous convention; it has to be interpreted as meaning that the intervention of S isnot needed in the interaction array ϕ.
4.2 Federation
Now, consider the case of a legislative body composed of two chambers. If a, a con-fidence motion for instance, is disrupted by an active coalition structure, say M, inChamber 1, then a is discarded. If a passes Chamber 1 unopposed, then it has to be
123
402 J. M. Abdou
presented to Chamber 2, where the active structure is T . We assume that M and Texert their sovereignty independently. Then to analyze the whole governance structure,we need to introduce the federation {M, T }.
We shall call federation any set F of active coalition structures. Federations will bedenoted by symbols F,M,P . . . Examples of federations are the Congress in the USA(Senate and House of representatives, where some proposal is submitted successivelyto both legislatures) and the Parliament in France (Assemblée Nationale and Sénat),or generally any institution with two or more levels of independent legislatures. In afederation, any component is an active coalition structure. It is assumed that each com-ponent acts independently of other components. One can note a difference betweenthe activity of an ACS and that of a federation. For an ACS, to rule out some state a, ajoint action of the components’ coalitions may be needed: in general, no coalition ofits own is assumed to have the power to rule out that state. On the federation level, ifthe state is ruled out by some active structure (e.g., Chamber 1), then it is discarded.
The notions of F-settlement and F-stability are defined in consequence: an alterna-tive a is F-dominated at the preference profile RN if there exists some M ∈ F, U ∈P0(A), U � a and ϕ ∈ E(M)[U ] and ϕ(S) ⊂ P(a, S, RN ) for all S ∈ M. Thealternative a is an F -settlement at RN if it is not F-dominated at RN . The set ofF-settlements at RN will be denoted by StlF(E, RN ). The second operation that weneed is the following:
The restriction of E to the federation F is defined by:
EF[U ] :=⋃
M∈FE(M)[U ] (4)
It is clear from the definition that: if F1 and F2 are two federations, then EF1∪F2[U ] =
Example 4.1 (a) Let S = P0(N ). S is the active coalition structure correspondingto the situation where all coalitions have some joint power. A settlement for Sis similar to a strong Nash equilibrium outcome (see Subsect. 5).
(b) Let M = N := {{i}| i ∈ N }. N is the active coalition structure correspondingto the situation where only individuals have some joint power. A settlement forN is similar to a Nash equilibrium outcome (Subsect. 5). In the case of inter-action forms associated with strategic game forms, EG
β (N ) is the projection of
EGβ (P0(N )) on N .
(c) Let F ≡ M1 = {{S} | S ∈ P0(N )}. M1 is the federation where every activecoalition structure is a single coalition. This is a context where every coalitionhas an independent (as opposed to joint) power. A settlement in this case issimilar to an element of the local core (Subsect. 5).
4.3 Induced effectivity: E0, E1, E1, E0
In Subsect. 2.3 we have shown how interaction forms generalize local effectivityfunctions; now we show how they induce local effectivity functions.
123
The structure of unstable power mechanisms 403
If S ∈ P(N ) and B ∈ P0(A), then δS,B denotes the element of � such that:
δS,B(T ) ={∅ if T �= S
B if T = S
Let E be an interaction form and let M1 := {{S} | S ∈ P0(N )} be the federationcomposed of all ACS that are singletons. We denote by E1 the restriction of E to M1.For S ∈ P0(N ) and U ∈ P0(A), we define E1[·] as follows:
E1[U ](S) = {B ∈ P0(A) | δS,B ∈ E[U ]} (5)
E1 is the local effectivity function induced by E . Moreover, for any U ∈ P0(A), onehas:
Let E0 := E1[A]. We denote by E0 the interaction form defined by E0[U ] := E0 forall U ∈ P0(A). Let E0(S) = E1[A](S). E0 is the (global) effectivity function inducedby E . One has:
E0 = {ϕ ∈ �|∃S ∈ P0(N ) : ϕ(S) ∈ E0(S)} (7)
Let C(E1, RN ) (resp. C(E0, RN )) be the core of E1 (resp. E0) at RN (Subsect. 2.3).One has Stl(E1, RN ) = C(E1, RN ) and Stl(E0, RN ) = C(E0, RN ). From now on, weshall use E1 or E1 (resp. E0 or E0) indifferently when we want to refer to the local(resp. global) effectivity function extracted from E .
Definition 4.2 An interaction form E is said to be maximal (resp. regular, superaddi-tive, subadditive), if E0 satisfies that property (see Abdou (2010) for definitions).
4.4 Higher order derived structures: Er , r ≥ 2
To study the whole structure of instability, one may need higher level of “effectivity”than those described by E0 and E1. Thus, we are led to the definition of a gradedfamily of derived structures. Let A1 = M1 ≡ {{S} | S ∈ P0(N )}. For r ≥ 2, let Ar
be the set of active coalition structures M such that |M| = r, N /∈ M and for allS, T ∈ M such that S �= T we have S ∪ T = N . When r ≥ 2,M ∈ Ar if, and onlyif, M∗ := {T | T c ∈ M} is composed of r non-empty disjoint subsets of N . We haveMr := ∪k=r
k=1Ak . An element of M ≡ Mn will be called an admissible ACS.Let E be an interaction form. Let Er ≡ EMr
To obtain some indications on the existence of cycles in some derived substructure Er ,we define the notion of exactness as a generalization of the notion bearing the samename introduced in Abdou (1995). In what follows, if ϕ ∈ �,M an ACS, we haveρM(ϕ) := ∪S∈Mϕ(S). For any active coalition structure M, we define the followingsets:
�∗(M) := {ϕ ∈ � | ρM(ϕ) �= A} (9)
D∗(M) := {ϕ ∈ �∗(M) | S, T ∈ M, S �= T ⇒ ϕ(S) ∩ ϕ(T ) = ∅} (10)
We have the inclusion: E0(M) ∩ �∗(M) ⊂ Eξ (M).Eξ (M) will be called the exact power held by M (this is the union of all the localpower to reach, at some state, interaction arrays with range excluding that state. E0(M)
represents the sum of the global power that coalitions in M hold independently andseparately. We are now in a position to provide a definition that gives a new insightinto the structure of instability.
Definition 4.3 Let M be an ACS. E is M-exact if one has:
Eξ (M) ∩ D∗(M) = E0(M) ∩ D∗(M) (14)
Let r ∈ N, 1 ≤ r ≤ n. E is r -exact if it is M-exact for all M ∈ Mr . In particular, Eis said to be exact if it is 1-exact and fully exact if it is n-exact.
If M = {S}, then M-exactness expresses that the (global) effectivity power of Scoincides with the exact power. 1-exactness has been discussed in Abdou (2010) Sub-sect. 4.4. When M contains more than one coalition, then one can ask what is the addi-tional exact power that the ACS M derives from the interaction compared to the sumof the global power that coalitions hold separately? If ϕ ∈ Eξ (M) and ϕ /∈ E0(M),then for some a /∈ ρM(ϕ), we have ϕ ∈ E(M)[a] and ϕ /∈ E0(M)[A]. M-exactnessamounts to say that restricted to D∗(M), there is no such additional power. Therefore,when restricted to D∗(M), the joint power of the ACS M can be uncoupled, that isto say, distributed between the coalitions that compose the ACS, which may use itindependently. Thus, r -exactness expresses that the power of an admissible ACS ofsize r is, in a sense, additive. If the interaction power of the coalitions of M is strongerthan their independent power, we say that there is an opportunity of collusion at a.Indeed, if a is proposed and coalitions of M prefer to reject a, then it may be the casethat ϕ cannot be in their power because no coalition can do it independently since∀S ∈ M, ϕ(S) /∈ E0(S), whereas they can very well do it jointly since ϕ ∈ E(M[a]:that is, in any scenario where a is proposed, some coalition S ∈ M can counter it byϕ(S). Collusion is not cooperation. Collusion (in politics, diplomacy or war) expresses
123
The structure of unstable power mechanisms 405
precisely a situation where two or more distinct forces, though not formally cooper-ating, have an objective interest to target the same state (in our interpretation, theircommon goal is to upset that state). In the following example 2-exactness fails thoughthe interaction form does not present any local dependence.
Example 4.4 Let |A| = p ≥ 3. For any U ∈ P0(A):
E[U ] := {ϕ | ∃S, T �= N , S ∪ T = N , |ϕ(S) ∪ ϕ(T )| ≥ p − 1} ∪ {ϕ | ϕ(N ) �= ∅}
It is clear that E does not depend on U . Moreover for any S ∈ P0(N )a ∈ A, S �= N :E1[a](S) = E0(S) = {B | |B| ≥ p − 1}, E0(N ) = E1[a](N ) = P0(A), and for anyS, T �= ∅, N with S ∪ T = N :
E0({S, T }) = {ϕ | |ϕ(S)| ≥ p − 1 or |ϕ(T )| ≥ p − 1}E0({S, T }) ∩ D∗({S, T }) = {ϕ | |ϕ(S)| = p − 1, ϕ(T ) = ∅}
∪ {ϕ | |ϕ(T )| = p − 1, ϕ(S) = ∅}Eξ
({S, Sc}) = {
ϕ | |ϕ(S) ∪ ϕ(Sc) | = p − 1
}
Eξ
({S, Sc}) ∩ D∗({S, T }) = {
ϕ | |ϕ(S) ∩ ϕ(T ) = ∅, ϕ(S) ∪ ϕ(Sc) | = p − 1
}
E is not 2-exact: indeed E0 ({S, Sc}) ∩ D∗({S, Sc}) �= Eξ ({S, Sc}) ∩ D∗ ({S, Sc}).There is some additional power in Eξ ({S, Sc}) that exceeds the union of the separateeffectivity power of S and Sc as represented by E0(S) and E0 (Sc), respectively. Sincethere is no dependence on U, E is 1-exact.
Example 4.5 Let |A| = p ≥ 2. For any a ∈ A, U ∈ P0(A), let wa : P0(N ) →{1, . . . , p}, E[a] = {ϕ | ∃S ∈ P0(N ) : |ϕ(S)| ≥ wa(S)} and E(U ) = ∩a∈U E[a].Let w(S) = maxa∈A wa(S), then one obtains:
E1[a](S) = {B | |B| ≥ wa(S)}, E0(S) = {B | |B| ≥ w(S)},Eξ ({S} = {ϕ | ϕ(S) �= A and |ϕ(S)| ≥ mina /∈ϕ(S)wa(S)}
Then it is easy to see that E is 1-exact if, and only if, ∀a ∈ A,∀S ∈ P0(N ) :wa(S) = w(S) (E is independent of U ). Assume that this condition is satisfied. Forany ACS M and ϕ ∈ �, with ρM(ϕ) �= A, one has ϕ ∈ Eξ (M) if, and only if,∃S ∈ M : |ϕ(S)| ≥ w(S). It follows that : Eξ (M) = E0M). Since this equality istrue is for any ACS M then, in particular, E is fully exact.
We have the remarkable result:
Lemma 4.6 Assume that E0(N ) = P0(A) and let r ∈ N, 1 ≤ r ≤ n. If E is r-exactthen for all RN ∈ Q(A)N : C (E0, RN ) = Stl(E1, RN ) = · · · = Stl (Er , RN ).
Proof For any interaction form E and any profile RN we have Stl (Er , RN ) ⊂C (E0, RN ). Assume that E is r -exact and E0(N ) = P0(A). Let a ∈ A be domi-nated in Er at RN . We have ϕ(S) = P(a, S, RN )(S ∈ P0(N )). Then, a /∈ ρM(ϕ) and
123
406 J. M. Abdou
ϕ ∈ E(M)({a}) for some M ∈ Mr so that ϕ ∈ Eξ (M). If for some S, T ∈ M, S �=T, ϕ(S) ∩ ϕ(T ) �= ∅, then ϕ(N ) = ϕ(S) ∩ ϕ(T ) ∈ E0(N ) and it follows that a isdominated in E0. If for all S, T ∈ M, S �= T, ϕ(S) ∩ ϕ(T ) = ∅, then, by r -exactnessϕ ∈ E0(M) ∩ D∗(M). It follows that there exists S ∈ M such that ϕ(S) ∈ E0(S) sothat a is dominated in E0. ��
4.6 Calculating the stability index
We shall prove that full exactness is a necessary condition of stability of maximalinteraction forms. More precisely, the absence of r -exactness signals the presence ofsome cycle of order ≤ r + 2. We start by the following basic result:
Lemma 4.7 Let r ∈ N, 1 ≤ r ≤ n. Assume that E is maximal. If E is not r-exact then:
(i) Er has a cycle of order ≤ r + 2,(ii) If further E is superadditive, then there exists some RN ∈ L(A)N such that
Stl(Er , RN ) = ∅ and |C(E0, RN )| = 1.
Proof If E is not r - exact, then there exists, an active coalition structure M ∈ Mr , a ∈A, ϕ ∈ D(M) such that ϕ ∈ E(M)[a], a /∈ ρM(ϕ), and for all S ∈ M, ϕ(S) /∈E0(S). Without loss of generality, we can assume M ∈ Ar , so that M = {S1, . . . , Sr }.Let Tk = Sc
k (k = 1, . . . , r), Bk = ϕ(Sk), T0 = N \ ∪r1Tk , and B0 = A \ ∪r
1 Bk . SinceE0 is maximal, we have that Bc
k ∈ E0(Tk)(k = 1, . . . , r). Consider the r + 2-tuple((U0, ϕ0), . . . , (Ur+1, ϕr+1)) defined as follows: U0 := B0 \{a}, ϕ0 := δN ,{a}, Uk :=Bk, ϕk := δTk ,Bc
a cycle of order r + 2. If B0 \ {a} = ∅, we can remove index 0 and thus have a cycleof order r + 1.
(ii) We construct a profile RN = (Ri )i∈N with the following properties:
(i ∈ Tk, k �= 0) : A \ B0 ∪ Bk Ri {a} Ri B0 \ {a} Ri Bk
(i ∈ T0) : A \ B0 Ri {a} Ri B0 \ {a}An alternative b ∈ Bk where k ∈ {1, . . . , r} is dominated in E0, since Bc
k ∈ E0(Tk) andBc
k ⊂ P(b, Tk, RN ). An alternative b ∈ B0\{a} is dominated in E0, since by maximal-ity of E0, {a} ∈ E0(N ) and {a} ⊂ P(b, N , RN ). It follows that C (E0, RN ) ⊂ {a}.For k = 1, . . . , r, Sk = ∪l �=k Tl , P(a, Sk, RN ) = ∩l �=k P(a, Tl , RN ) = ∩l �=k A \(B0 ∪ Bl) = Bk = ϕ(Sk). So that a is not a settlement in Er . Since Stl(Er , RN ) ⊂C (E0, RN ), it follows that Stl(Er , RN ) = ∅. If furthermore E0 is superadditive, onecan prove that C (E0, RN ) = {a}. This verification is left to the reader. ��We thus have an easy characterization of r -exactness when E is maximal and super-additive:
Proposition 4.8 Let r ∈ N, 1 ≤ r ≤ n. Let E be maximal and superadditive. Then Eis r-exact if, and only if, Stl (Er , RN ) = C (E0, RN ) for all RN ∈ L(A)N (or Q(A)N ).
Proof Since E0 is maximal, then in particular E0(N ) = P0(A). If E is r -exact, inview of lemma 4.6, Stl (Er , RN ) = C (E0, RN ) for all RN ∈ Q(A)N . The conversefollows from lemma 4.7. ��
123
The structure of unstable power mechanisms 407
The following provides necessary and sufficient conditions for stability of Er formaximal interaction forms:
Proposition 4.9 Let r ∈ N, 1 ≤ r ≤ n. Assume that E is maximal. Er is stable if, andonly if, E0 is stable and E is r-exact. Moreover, in this case Stl (Er , RN ) = C (E0, RN )
for all RN ∈ Q(A)N .
Proof Assume that E0 is maximal. If Er is stable then in view of lemma 4.7 (ii) E isr-exact. Moreover, since C(E0, RN ) ⊃ Stl(Er , RN ) for all RN ∈ Q(A)N , E0 is stable.Conversely, if E0 is stable then, in particular E0 is superadditive and if in addition Eis r -exact then Stl (Er , RN ) = C (E0, RN ) (Proposition 4.8). ��Now, we deduce necessary conditions for stability of any maximal interaction form E .
Theorem 4.10 Assume that E is maximal. If E is stable, then: E is superadditive andsubadditive, and E is fully exact.
Proof Full exactness follows from lemma 4.7(i). Stability and maximality of E0 implysuperadditivity and subadditivity. ��
As a second consequence of lemma 4.7, we obtain the following partial localizationof the index in case of unstable E :
Theorem 4.11 Assume that E is maximal :
(i) If E is not regular, then σ(E) = 2,(ii) If E is not subadditive or not superadditive, then σ(E) ≤ 3,
(iii) If E is not r-exact, then σ(E) ≤ r + 2.
Proof In view of Abdou (2010) Theorem 4.9 (i) (ii), non regularity of E implies theequality σ(E0) = 2. Since 2 ≤ σ(E) ≤ σ(E0) = 2 ≤, it follows that σ(E) = 2. By thesame reference, if E fails to be superadditive or subadditive, then σ(E) ≤ σ(E0) ≤ 3.This proves (i) and (ii). (iii) is a consequence of Lemma 4.7 (i). ��
4.7 Bipolarity: when the index is 2
Intuitively, a conflictual situation is bipolar when there are two main forces that opposeany settlement (σ = 2). In this short subsection, we show the following: whatever isthe solution concept that underlies the power mechanism, index 2 is the symptom ofthe existence of two disjoint coalitions that can veto any settlement. It is not obviousfrom the definition of a cycle that a cycle of order 2 marks the existence of two disjointcoalitions with incompatible effectivity power. This will be done under the conditionof monotonicity that we now define.
For any μ : P0(N ) → P0(N ) and ϕ ∈ �, we associate ϕμ ∈ � defined byϕμ(S) = ∪{ϕ(T ) : μ(T ) = S}. μ is said to be an inclusion map if for all T ∈P0(N ) : T ⊂ μ(T ).
Definition 4.12 E is said to be monotonic if for any inclusion map μ and any U ∈P0(A): if ϕ ∈ E[U ] then ϕμ ∈ E[U ].
123
408 J. M. Abdou
Lemma 4.13 Let E[·] a local effectivity function and let E be the interaction formcanonically associated with E (Subsect. 2.3). Then E is monotonic in the sense ofthe definition 4.12 if, and only if, E[·] is monotonic w.r.t. players (that is wheneverU ∈ P0(A) and S ⊂ T then E[U ](S) ⊂ E[U ](T )).
Proof Assume that E is monotonic. Let S ⊂ T and let μ be defined by μ(S′) = S′
if S′ �= S and μ(S) = T . Then μ is an inclusion map. If B ∈ E[U ](S) then ϕ ≡δS,B ∈ E[U ]. By the monotonicity of E , it follows that ϕμ = δT,B ∈ E[U ] so thatB ∈ E[U ](T ). Conversely, assume that E[·] is monotonic w.r.t. players and let μ
be any inclusion map. If ϕ ∈ E[U ], then for some S ∈ P0(N ), ϕ(S) ∈ E[U ](S) orequivalently ϕ′ ≡ δS,ϕ(S) ∈ E[U ]. Let T = μ(S). Then S ⊂ T and ϕ′μ (
S′) = ∅for all S′ �= T . It follows that ϕ′μ = δT,ϕ(S) and ϕ(S) ∈ E[U ](T ). We conclude thatϕ′μ ∈ E[U ]. Now since ϕ′μ ≤ ϕμ, it follows that ϕμ ∈ E[U ]. ��
Let E be monotonic, let μ be an inclusion map, and let M,M′ any ACS such thatμ(M) ⊂ M′. If ϕ ∈ E(M)[U ], then ϕμ ∈ E
(M′) [U ]. For any strategic game form
G, EGβ is monotonic.
Theorem 4.14 Let E be monotonic.
(i) The index of E is 2 if, and only if, the index of its derived local effectivity functionE1 is 2.
(ii) If in addition E is standard (definition 2.2), then the index of E is 2 if, and onlyif, the index of its derived (global) effectivity function E0 is 2, that is if, and onlyif, E is not regular.
Proof Let (U1, ϕ1), (U2, ϕ2) be a cycle where wlog, we assume that (U1, U2) is apartition of A. Put M := [ϕ1] ∪ [ϕ2], T0 := {
i ∈ N | ρi (ϕ1) ∪ ρi (ϕ2) = ∅}and let
Ts be the set of i ∈ N \ T0 such that Us ∩ (ρi (ϕ1) ∪ ρi (ϕ2)
) = ∅(s = 1, 2). Byproperty (ii) of definition 3.2, Ts �= ∅(s = 1, 2), T1 ∩ T2 = ∅ and T0 ∪ T1 ∪ T2 = N .For any i ∈ N , let Hi := {S | S � i}. We have i ∈ T0 if, and only if, Hi ∩ M = ∅. Ifi ∈ T1 and S ∈ Hi then ϕ1(S) ∪ ϕ2(S) ⊂ U2. Similarly, if j ∈ T2 and S ∈ H j , thenϕ1(S) ∪ ϕ2(S) ⊂ U1. It follows that for any (i, j) ∈ T1 × T2,Hi ∩ H j contains onlycoalitions S such that ϕ1(S))∪ϕ2(S) = ∅ or put otherwise Hi ∩H j ∩M = ∅. That isequivalent to say: for any S ∈ M either S ⊂ T1 or S ⊂ T2. By definition of T1 and T2,if S ⊂ T1, then ϕ2(S)∩ (U1 ∪U2) = ∅. Since U1 ∪U2 = A, it follows that ϕ2(S) = ∅.Similarly, if S ⊂ T2, then ϕ1(S) = ∅. Let μ : P0(N ) → P0(N ) defined by μ(S) = Tk
if S ∈ M, S ⊂ Tk(k = 1, 2) and μ(S) = N if S /∈ M. Then μ is an inclusion map.Since E is monotonic: ϕ
μk ∈ E[Uk]. Moreover, if S �= Tk, ϕ
μk (S) = ∅. It follows that
ϕμk (Tk) ∈ E1[Uk](Tk)(k = 1, 2). Since ϕ
μ1 (T1) ⊂ U2 and ϕ
μ2 (T2) ⊂ U1, we have that((
U1, ϕμ1
),(U2, ϕ
μ2
))is a 2-cycle for E1. This proves (i). Now, U2 ∈ E1[U1](T1) and
U1 ∈ E1[U2](T2). If we assume in addition that E is locally effective, and satisfiesthe sheaf property, then U1 ∈ E0(S2) and U2 ∈ E0(S1), so that E is not regular. Thisproves (ii). ��
123
The structure of unstable power mechanisms 409
5 Instability structure of strategic game forms
Starting with a strategic game form and an equilibrium concept, we derive a descrip-tion of the underlying power distribution, thus defining some interaction form. Thisderivation of the power imbedded in a strategic form follows the same pattern as thederivation of the α- and the β-effectivity functions (Moulin and Peleg 1982) and morerecently the derivation of the more general effectivity structures (Abdou 1995; Abdouand Keiding 2003, 2009).
Let G = 〈X1, . . . , Xn, A, g〉 be a strategic game form. The set of players is N ={1, . . . , n}, Xi is the strategy set of players i, g : ∏
i∈N Xi → A is the outcome func-tion, assumed to be surjective. For each preference profile RN ∈ Q(A)N , the gameform G induces a game (X1, . . . , Xn; Q1, . . . , Qn) with the same strategy spaces andwhere Qi is the preorder on X N defined by: xN Qi yN if, and only if, g(xN ) Ri g(yN )
for xN , yN ∈ X N . We denote this game by G(RN ).
5.1 M- equilibrium
Let M be an active coalition structure. A strategy array xN ∈ X N is an M-equilibriumof the game G(RN ) if there is no coalition S ∈ M and yS ∈ X S such that g (yS, xSc )
R◦i g(xN ) for all i ∈ S.An alternative a is an M-equilibrium outcome of G at RN if there exists some
equilibrium xN ∈ X N of G(RN ) such that g(xN ) = a. Denote by E O(M) (G, RN )
the set of all M-equilibrium outcomes of (G, RN ). In particular, when M = N ≡{{1}, . . . , {n}}, anM-equilibrium is a Nash equilibrium. Similarly, whenM = P0(N ),an M-equilibrium is a strong Nash equilibrium. The set of Nash (resp. strong Nash)equilibrium outcomes will be denoted by N E O(G, RN ) (resp. SE O(G, RN )). Thegame form G is said to be solvable in M-equilibrium or M-solvable, if for eachpreference profile RN ∈ Q(A)N , the game G(RN ) has an M-equilibrium. G is saidto be r -M-solvable if E O(M) (G, RN ) �= ∅ for all RN ∈ Q•(π)N and all π ∈ �r
(see 3.1). G is said to be M-solvable if G is r -M-solvable for all r ≥ 1. Define thestability index of G relatively to the M-equilibrium solution as the minimal integerr for which G is not r - solvable (with the convention that the index is equal to +∞if such integer does not exist). We denote by σ(M)(G) this index. In particular, thestability index relative to Nash equilibrium (resp. strong Nash equilibrium) will bedenoted by σN (G) (resp. σS(G)).
5.2 Cores
There are other solution concepts for game forms, that cannot be formulated directlyas an M-equilibrium. This is the case for all the family of solutions usually termed“core”. Let F be a federation. For any preference profile RN ∈ Q(A)N , the F-coreof G at RN is defined by CF (G, RN ) = ∩M∈FE O(M) (G, RN ). In particular, theM1- core is the exact core, the M2-core is the biexact core (see Abdou (2000) fordirect definitions). Finally, the β-core Cβ (G, RN ) ≡ C0 (G, RN ) is defined as theset of alternatives a such that there is no coalition S ∈ P0(N ) with the following
123
410 J. M. Abdou
property: for any zN ∈ X N , there exists yS ∈ X S such that g (yS, zSc ) R◦i a for all
i ∈ S. G is said to be r -F-stable if CF (G, RN ) �= ∅ for all RN ∈ Q•(π)N and allπ ∈ �r (see 3.1). G is said to be F-stable if G is r -F-stable for all r ≥ 1. Definethe stability index of G relatively to the F-core solution as the minimal integer r forwhich G is not r - solvable (with the convention that the index is equal to +∞ if suchinteger does not exist). It will be denoted by σF(G). In particular, the stability indexrelative to the β-core, the exact core, the biexact core will be denoted, respectively, asσ0(G), σ1(G), σ2(G).
Definition 5.1 The interaction form associated with G is the mapping EG defined asfollows: For U ∈ P0(A):
EG[U ]={ϕ∈�0(N , A) | ∀yN ∈g−1(U ), ∃S ∈P0(N ), ∃xS ∈ X S : g(xS, ySc )∈ϕ(S)
}
(15)
If M is an ACS define EG(M) by projection (see Subsect. 4.1) and if F is a federationdefine EG
Fby restriction (see Subsect. 4.2). Moreover, EG
1 = EM1, and EG
0 is defined
by EG0 [U ] = E1[A] for any U ∈ P0(A).
Lemma 5.2 Let G be a game form and let RN ∈ Q(A)N .
(i) For any ACS M, the set of M-equilibrium outcomes of G at RN coincides withthe settlement set of EG(M) at RN . Therefore, G is r-M-solvable if, and onlyif, EG(M) is r-stable.
(ii) For any federation F, the F-core of G at RN coincides with the settlement setof EG
Fat RN . Therefore, G is r-F-solvable if, and only if, EG
Fis r-stable.
(iii) The β-core of G at RN coincides with the settlement set of EG0 at RN . Therefore,
G is r-β stable if, and only if, EG0 is r-stable.
According to this lemma, the solvability problem for any solution (M-equilibrium, F-core or β-core) can be formulated as a stability problem of the associated interactionform. In view of proposition and definition 3.4, the stability index relative to each ofthe solutions is equal to the stability index of the corresponding interaction form.
Special attention must be paid to the lower effectivity structures of G. Describ-ing stability requires properties of the classical effectivity function EG
β . The latter isdefined by:
EGβ (S) = {B ∈ P0(A) | ∀yN ∈ X N , ∃xS ∈ X S : g(xS, ySc )∈ B} (16)
Actually, EGβ ≡ E0 is the (global) effectivity function derived from EG
0 (see Sub-sect. 4.3). The α-effectivity function is defined by:
EGα (S) =
{B ∈ P0(A) | Bc /∈ EG
β (S)}
(17)
It is easy to see that EGβ is maximal, and that EG
α is superadditive (hence regular).
Moreover, EGβ is monotonic (Definition 4.12). G is said to be tight if EG
α = EGβ . G is
123
The structure of unstable power mechanisms 411
tight if, and only if, EGα is maximal or equivalently if EG
β regular. A property (subad-ditivity, r -exactness, etc.) is said to be fulfilled by G if the property bearing the samename is fulfilled by EG .
Regarding the β-core and the exact core, we refer the reader to the classificationobtained in Abdou (2010), Proposition 4.18. The general results of Sect. 4 allow forfurther classifications. Here is a precise description of the situation concerning thebiexact core:
Proposition 5.3 Let G be a game form. Then:
(i) σ2(G) = 2 if, and only if, G is not tight.(ii) σ2(G) = 3 if G is tight, but G is either not subadditive or not 1-exact.
(iii) σ2(G) ∈ {3, 4} if G is tight, but G is not 2-exact.(iv) σ2(G) = +∞ if, and only if, G is tight, subadditive and 2-exact.
Proof In view of Theorem 4.9 σ2(G) = +∞ if, and only if, EG2 is biexact and EG
0stable. But E0 is stable if, and only if, G is tight and subadditive. This proves (iv). Statet-ement (i) is straightforward. Moreover, if σ2(G) < +∞, then by Theorem 4.11(iii),σ2(G) ≤ 4. This proves statements (i), (ii) and (iii). ��
In the rest of this section, we shall establish some facts about the stability indexrelated to equilibrium concepts. It is very interesting that bipolarity reveals an intrinsicproperty of game forms, in the sense that it does not depend on the underlying solution.
Proposition 5.4 Let G be a strategic game form. If σ(M)(G) = 2 or σF(G) = 2 forsome active coalition structure M or for some federation F, then σ0(G) = 2, that isthe game form is not tight.
Proof If σ(M)(G) = 2 or σF(G) = 2, then σ(EG) = 2. Since EG is a standardmonotonic interaction form, the result follows from Theorem 4.14. ��
For strong Nash solvability, we have the following:
Theorem 5.5 If G is strongly solvable, then G is tight, subadditive and fully exact.Moreover, for any preference, the β-core, the exact core and the biexact core coincide.
Proof EG is maximal so that (i) is a straightforward consequence of Theorem 4.9. ��Remark 5.6 In Abdou (2000), Example 2.3, a two-player game form is given such
that Stl(EG
1 , RN) �= Stl
(EG
0 , RN) ≡ C
(EG
β , RN
). In Example 2.4, a two-player
game form is given such that Stl(EG
2 , RN) �= Stl
(EG
1 , RN), and in example 2.5 a
three-player game form is given where Stl(EG
2 , RN)
is strictly larger than the cor-responding strong outcome set denoted as SE O (G, RN ). It would be interesting toexhibit an example of an n-player game form (where necessarily n ≥ 3) such that G isstrongly solvable and SE O (G, RN ) ≡ Stl
(EG, RN
) �= Stl(EG
n , RN)
for some RN .By Theorem 5.5, this is equivalent to finding G such that G is strongly solvable and
SE O (G, RN ) �= C(
EGβ , RN
)for some RN .
123
412 J. M. Abdou
In case of strong Nash instability, we can determine upper bounds for the strong Nashindex.
Proposition 5.7 Let G be a game form. Then:
(i) σS(G) = 2 if, and only if G is not tight,(ii) σS(G) = 3 if G is tight and G is not subadditive,
(iii) σS(G) ≤ r + 2 if G is not r-exact (1 ≤ r ≤ n).
Proof (i) is a straightforward consequence of Theorem 4.11(i) and Proposition 5.4.The rest is consequence of Theorem 4.11(ii) and (iii). ��Now, we give a precise localization of the index for some classes of games that canbe obtained when we combine our results with well-known results in literature.
Proposition 5.8 Let G be a two-player game form.
(i) The Nash stability index σN (G) = 2 if, and only if, G is not tight. If G is tightσN (G) = +∞. In any case σN (G) = σ0(G).
(ii) The strong Nash stability index σS(G) = 2 if, and only if, G is not tight.σS(G) = 3 if G is tight and not 1-exact. σS(G) ∈ {3, 4} if G is tight and not2-exact. σS(G) = +∞ if, and only if, G is tight and biexact.
Proof For the Nash case, the result follows from the fact that for two-player gameforms, Nash solvability is equivalent to tightness [see Gurvich (1975, 1989) or Abdou(1995)]. For the strong Nash case, this is a corollary of Theorem 5.3(ii) since, whenn = 2, then the biexact core is equal to the set of strong Nash outcomes, and when Gis tight then G is subadditive. ��
We recall that a game form G = 〈X1, . . . , Xn, A, g〉 is said to be rectangularif for any a ∈ A, g−1(a) = ∏n
i=1 Yi , for some Yi ⊂ Xi , (i = 1, . . . , n) (Gurvich1979; Abdou 1998, 2000). The strong Nash index for rectangular game forms can bedetermined with good precision.
Proposition 5.9 Let G be a rectangular game form. The strong Nash index σS(G) =+∞ if, and only if G is 1-exact and this is the case if, and only if, G is essentiallya one-player game form. σS(G) = 2 if, and only if, G is not tight. σS(G) = 3 in allother cases.
Proof In view of Theorem 4.7 of Abdou (2000) any 1-exact rectangular game formG is essentially a one-player game form. It follows that a rectangular game form isstrongly solvable if, and only if, it is 1-exact and in this case the strong Nash indexis +∞. If G is not strongly solvable, G is not 1-exact, then by theorem 5.7 (iii) theindex is less or equal to 1 + 2 = 3, and in fact is equal 3 if G is tight. ��
We end this section by an example that shows how the stability index and Nakamuranumber can be strongly related in a framework that is different from simple games.
Example 5.10 Let X1 = X2 = · · · A = {1, . . . , p}. We define an n-player game formG = 〈X1, . . . , Xn, A, g〉 by g(x1, . . . , xn) = min(x1, . . . , xn). Let M be any ACS
123
The structure of unstable power mechanisms 413
and let M = {{S}|S ∈ M}. Let E(M) (resp. EM) be the interaction form associatedwith the M equilibrium solution (resp. the M- exact core). Let E1[·] be the localeffectivity function (5) associated with G. Let EM[·] be the local effectivity functiondefined by EM[a](S) = E1[a](S) if S ∈ M and EM[a](S) = ∅ if S /∈ M. Clearly,EM and EM[·] are (for any game form) equivalent objects. In our game form G, onecan compute easily the interaction structure. For any a ∈ A one has:
The local effectivity function is given by E1[a](N ) = P0(A) and:
E1[a](S) = {B ∈ P0(N )|∃b ∈ A : b ≤ a} (S �= N )
The effectivity function is given by: E0(S) = {B ∈ P0(A)|B � 1} if S �= N andE0(N ) = P0(A). It follows that for this game form and for any ACS M, one hasthe remarkable equality E(M) = EM. Thus, for any preference profile the set ofM-equilibrium outcomes is equal to the M-exact core, so that M- solvability of Gamounts to stability of the local effectivity function EM[·]. In the context of localeffectivity functions, an equivalent definition of cycles is given in definition 3.1 ofAbdou (2010) (see remark 3.5). Let νM be the Nakamura number of M. We recallthat νM is defined as the minimal cardinality of a subfamily S ⊂ M that verifiesthe property: ∩S∈S S = ∅ (with the convention νM = +∞ if there exists no such asubfamily). We thus have a good general statement about stability and index of G.
Proposition 5.11 Let M be any ACS, let M be the associated federation and let ν bethe Nakamura number of M.
(i) G is β-core stable, that is σ0(G) = +∞.(ii) If N /∈ M or p ≤ ν, then σM(G) = σ(M)(G) = +∞.
(iii) If N ∈ M and p ≥ ν + 1, then σM(G) = σ(M)(G) = ν + 1.
Proof (i) Assume that C(E0, RN ) = ∅ for some RN ∈ L(A)N . In particular,P(1, S, RN ) ∈ E0(S) for some S ∈ P0(N ). It follows that S = N , that is 1is Pareto dominated. One can construct by induction a sequence a0, . . . , at+1,such that a0 = 1, ak is Pareto dominated by ak+1 for k = 0, . . . , t − 1 andat+1 not Pareto dominated. It follows that for some S ∈ P0(N ), S �= N onehas P(at+1, S, RN ) ∈ E0(S). Therefore, 1 ∈ P(at+1, S, RN ), which is a con-tradiction.
(ii) Let ((C1, B1, S1), (C2, B2, S2) · · · (Cr , Br , Sr )) be a cycle of the local effectiv-ity function EM[·]. Without loss of generality, we can assume that (Ck, k =1, . . . , r) is a partition of A. Let k0 be such that 1 ∈ Ck . Since 1 /∈ Bk0 andBk0 ∈ EM[1] one has Sk0 = N . This proves that the existence of a cycle in EMimplies that N ∈ M. Assume that p ≤ ν. Since r ≤ p, one has r ≤ ν, so that∩r
k=1Sk = ∩rk=1,k �=k0
Sk �= ∅. It follows from the definition of a cycle that there
exists some k ∈ {1, . . . , r} such that Bk ∩ (∪nk=1Ck
) = ∅. Since ∪nk=1Ck = A,
and Bk �= ∅, this is a contradiction. We conclude that the existence of a cyclein E implies that p ≥ ν + 1.
123
414 J. M. Abdou
(iii) Let S1, . . . , Sν be a subfamily of M with empty intersection and minimal cardi-nality. If p ≥ ν +1, then using this family, and adding Sν+1 = N , we are goingto exhibit a cycle of length ν + 1, namely ((C1, B1, S1), (C2, B2, S2), . . . ,
(Cν+1, Bν+1, Sν+1)), where: Bk := {k}, Ck := {k + 1} for k = 1, . . . , ν;Bν+1 := {ν + 1}, Cν+1 := {1, . . . , p} \ {2, . . . , ν + 1}. By construction,(C1, . . . , Cν+1) is a partition of A and Bk ∈ EM(Sk)(k = 1, . . . , ν + 1).We need to prove that for any ∅ �= I ⊂ {1, . . . , ν + 1}, such that ∩k∈I Sk �= ∅there exists with the property : C ∩(∪k∈I Bk) = ∅. Take such a subset I . Thenthere exists some k0 ≤ ν, k0 /∈ I . Choose k0 /∈ I minimal with this property. Ifk0 = 1 take = max I one has C ∩ (∪k∈I Bk) = ∅. If k0 �= 1 take = k0 − 1,then again C ∩ (∪k∈I Bk) = ∅. We conclude that we have a cycle of length lessor equal to ν + 1. Moreover, by the proof of (ii) there is no cycle of length lessor equal to ν. It follows that the stability index is ν + 1.
��Thus G is not exactly stable: σ1(G) = 3. Similarly σ2(G) = 3. G is not strongly
solvable: the strong equilibrium index σS(G) = 3. If we take M = {N } ∪ {N \{1}, . . . , {N \ {n}}, then σ(M)(�) = n + 1 if |A| ≥ n + 1 and σ(M)(G) = +∞ if|A| ≤ n. But, G is Nash solvable. The latter fact could be deduced from the well-knownresult that G is dominance-solvable [see Moulin (1983) problem 16].
6 Concluding remarks
The model of interaction form of a set of agents N over a set of alternatives A encom-passes aspects of power distributions underlying both cooperative and strategic models.Interactive forms defined on the same sets of agents and alternatives can be comparedwith each other. In particular, they can be compared with respect to their stability,a main issue in political science and social choice. Stability is known to be equiv-alent to the absence of cycles. The minimal order of some cycle in an interactionform is called the stability index. In order to avoid deadlocks in political decisions,the stability index must be high. Loosely speaking, it must overcome the degree ofsophistication of the society. A graded notion of exactness has been introduced. Exact-ness is related to the intuitive notion of collusion. A failure of exactness of some orderfor a maximal interaction form introduces an upper bound on the stability index. Asapplication, we can determine in many cases the stability index of strategic gameforms in the context of classical solution concepts. However, since computing cyclesin an effectivity function and even the Nakamura number is an NP complete prob-lem [see Boros and Gurvich (2000); Mizutani et al. (1993); Takamiya and Tanaka(2006)], computing cycles, or checking instability in a general interaction form is NP-hard.2 Moreover, many questions about the stability index are still open. Structuralproperties—including exactness—proved to be efficient criteria only when the inter-action form was maximal. Since interaction forms related to Nash equilibrium are notmaximal, the present tools do not provide valuable information about the Nash index.
2 I am indebted to an anonymous referee for pointing out this complexity issue.
123
The structure of unstable power mechanisms 415
Moreover, a finer analysis of the structure of instability requires the definition of atypology of cycles that goes beyond the notion of index.
References
Abdou, J.: Nash and strongly consistent two-player game forms. Int J Game Theory 24, 345–356 (1995)Abdou, J.: Rectangularity and tightness: a normal form characterization of perfect information extensive
game forms. Math Oper Res 23, 553–567 (1998)Abdou, J.: Exact stability and its applications to strong solvability. Math Soc Sci 39, 263–275 (2000)Abdou, J.: A stability index for local effectivity functions. Math Soc Sci 59, 306–313 (2010)Abdou, J., Keiding, H.: Effectivity functions in social choice. Dordrecht: Kluwer Academic Publish-
ers (1991)Abdou, J., Keiding, H.: On necessary and sufficient conditions for solvability of game forms. Math Social
Sci 46, 243–260 (2003)Abdou, J., Keiding, H.: Interaction sheaves on continuous domains. J Math Econ 45, 708–719 (2009)Boros, E., Gurvich, V.: Stable effectivity functions and perfect graphs. Math Soc Sci 39, 175–194 (2000)Campbell, D.E., Kerry, J.S.: Gains from manipulating social choice rules. Econ Theory 40, 349–371 (2009)Greenberg, J.: The theory of social situations: an alternative game-theoretic approach. Cambridge: Cam-
bridge University Press (1990)Gurvich, V.A.: Solvability in pure strategies. Zh Vychisl Mat i Mat Fiz 15, 358–371 (1975); English transl.
in USSR Comput Math and Math Phys 15Gurvich, V.A.: Application of Boolean functions and contact schemes in game theory. Candidate’s disser-
tation Moskow, Fiz-Tekn. Inst., Russian (1978)Gurvich, V.A.: Equilibrium in pure strategies. Soviet Math Dokl 38, 597–602 (1989)Keiding, H.: Necessary and sufficient conditions for stability of effectivity functions. Int J Game The-
ory 14, 93–101 (1985)Mizutani, M., Hiraide, Y., Nishino, H.: Computational complexity to verify the unstability of effectivity
function. Int J Game Theory 22, 225–239 (1993)Moulin, H.: The strategy of social choice. Advanced textbook in Economics. Amsterdam: North-
Holland (1983)Moulin, H., Peleg, B.: Cores of effectivity functions and implementation theory. J Math Econ 10, 115–
162 (1982)Nakamura, K.: The vetoers in a simple game with ordinal preferences. Int J Game Theory 8, 55–61 (1979)Peleg, B.: Game theoretic analysis of voting in committees. Gambridge: Cambridge University Press (1984)Peleg, B.: Effectivity functions, game forms, games, and rights. Soc Choice Welfare 15, 67–80 (1998)Peleg, B.: Representation of effectivity functions by acceptable game forms: a complete characteriza-
tion. Math Soc Sci 47, 275–287 (2004)Peleg, B., Peters, H.: Nash consistent representations of effectivity functions through lottery models. Games
Econ Behav 65, 503–515 (2009)Peleg, B., Winter, E.: Constitutional implementation. Rev Econ Design 7, 187–204 (2002)Rosenthal, R.W.: Cooperative games in effectiveness form. J Econ Theory 5, 88–101 (1972)Straffin, P.D.: Power and stability in politics. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game
Theory, vol 2, pp. 1128–1151. Amsterdam: North-Holland (1994)Takamiya, K., Tanaka, A.: Computational complexity in the design of voting rules. Discussion Paper No.
653, The Institute of Social and Economic Research Osaka University (2006)Vannucci, S.: Effectivity functions and stable governance structures. Ann Oper Res 109, 99–127 (2002)Vannucci, S.: A coalitional game-theoretic model of stable government forms with umpires. Rev Econ
