POWER MEASUREMENT AS SENSITIVITY ANALYSIS A UNIFIED APPROACH

POWER MEASUREMENT AS SENSITIVITY

ANALYSIS

A UNIFIED APPROACH

Stefan Napel and Mika Widgren

ABSTRACT

This paper proposes a unified framework that integrates the traditional index-based approach and the competing non-cooperative approach to poweranalysis. It rests on a quantifiable notion of ex post power as the (counter-factual) sensitivity of the expected or observed outcome to individual players.Thus, it formalizes players marginal impact on outcomes in both cooperativeand non-cooperative games, for both strategic interaction as well as purelyrandom behavior. By taking expectations with respect to preferences, actions,and procedures, one obtains meaningful measures of ex ante power. Estab-lished power indices turn out to be special cases.

KEY WORDS . decision procedures . equilibrium analysis . power indices .spatial voting

1. Introduction

Scientists who study power in political and economic institutions seemdivided into two disjoint methodological camps. The first one uses non-cooperative game theory to analyze the impact of explicit decision pro-cedures and given preferences over a well-defined usually Euclidean policy space.1 The second one stands in the tradition of cooperative game

Journal of Theoretical Politics 16(4): 517538 Copyright & 2004 Sage PublicationsDOI: 10.1177/0951629804046152 London, Thousand Oaks, CA and New Delhi

Our research has been supported by the Center for Interdisciplinary Research (ZiF), Bielefeld,the Center for Economic Studies (CES), Munich, the Yrjo Jahnsson Foundation, and theGerman Academic Exchange Service (DAAD). The paper has benefitted from discussions atthe 2002 San Sebastian Workshop on Modelling the European Decision-making, the 2002LSE Workshop on Voting Power Analysis, the 2002 Spring Meeting of Young Economists,the Fifth Spanish Game Theory Meeting, the Third Meeting of Public Economic Theory,ICM2002 Game Theory and Applications Satellite Conference and the constructive commentsof three anonymous referees,M. Braham, I. Dittmann,M. J. Holler, S. Seifert and F. Valenciano.

1. See, for example, Steunenberg (1994), Tsebelis (1994, 1996), Crombez (1996, 1997) andMoser (1996, 1997).

at UNIV AUTO DE CIUDAD JUAREZ on May 27, 2015jtp.sagepub.comDownloaded from

theory with much more abstractly defined voting bodies: the consideredagents have no particular preferences and form winning coalitions whichimplement unspecified policies. Individual chances of being part of and influ-encing a winning coalition are then measured by a power index.2

Proponents of either approach have recently intensified their debate, whichwas sparked by the critique by Garrett and Tsebelis (1996), in the context ofdecision-making in the European Union (EU).3 The non-cooperative campsverdict is that power indices exclude variables that ought to be in a politicalanalysis (preferences, institutions and strategies) and include variables thatought to be left out (computational formulas and hidden assumptions)(Garrett and Tsebelis, 1999a: 337). The cooperative camp has responded byclarifying the assumptions underlying its power formulas and giving somereasons for notmaking institutions and strategies corresponding to decisionprocedures and rational preference-driven agents more explicit.4 There alsohave been some attempts to include actors preferences in the cooperativeapproach.5

Several authors have concluded that it is time to develop a unified frame-work for measuring decision power (cf. Steunenberg et al., 1999; Felsenthaland Machover, 2001a).6 On the one hand, such a framework should allowfor predictions and ex post analysis of decisions based on knowledge ofprocedures and preferences. On the other hand, it must be open to ex anteand even completely a priori7 analysis of power when detailed information

518 JOURNAL OF THEORETICAL POLITICS 16(4)

2. See, for example, Brams and Affuso (1985a, b), Widgren (1994), Hosli (1993), Laruelle andWidgren (1998), Baldwin et al. (2000, 2001), Felsenthal and Machover (2001b) and Leech (2002)for recent applications of traditional power indices. Felsenthal and Machover (1998) and Nurmi(1998) contain a more general discussion regarding index-based analysis of power.3. Cf. the contributions to the symposium in Journal of Theoretical Politics 11(3, 1999),

together with Tsebelis and Garrett (1997), Dowding (2000), Garrett and Tsebelis (2001) andFelsenthal and Machover (2001a).4. See, in particular, Holler andWidgren (1999), Berg and Lane (1999), Felsenthal andMach-

over (2001a), and Braham and Holler (2003).5. See, for example, Straffin (1977, 1988), Widgren (1995), Kirman and Widgren (1995) and

Hosli (2002).6. Gul (1989) and Hart and Mas-Colell (1996) already give non-cooperative foundations for

the Shapley value and thus indirectly the ShapleyShubik index.7. There has been controversy at several workshops on power analysis about the correct usage

of the terms a priori and a posteriori. We use the term ex ante to mean before a decision issingled out or taken and the term ex post to mean for a particular expected or observed deci-sion. We reserve a priori to the complete ignorance about any aspect of decision-making otherthan somewhat arbitrarily voting weights and quota. Note that this definition makes mean-ingful ex ante or ex post analysis of real institutions for which more information than weightsand quota e.g. the strategic resources awarded to players by a particular agenda-setting ormulti-stage voting procedure is known and relevant necessarily a posteriori.


may either not be available or should be ignored for normative reasons.Unfortunately, the first attempt to provide such a framework, by Steunen-berg et al. (1999), is problematic.8 It confounds power and the success thatmay, but need not, result from it. This paper replies to the critique by Garrettand Tsebelis (1996) by proposing an alternative framework.In particular, we generalize the concept of a players marginal impact or

marginal contribution to a collective decision in order to establish a common(ex post) primitive of power for cooperative and non-cooperative analysis.Its evaluation amounts to the comparison of an actual outcome with acounterfactual shadow outcome which alternatively could have been broughtabout by the considered player. That is we look at the sensitivity of a givenoutcome to the considered players behavior. In our view, sensitivity analysisof outcomes goes a long way towards a reconciliation of equilibrium-based non-cooperative measurement and winning coalition-based traditionalpower indices. One can transparently measure ex ante power as a playersexpected ex post power. This is in line with the probabilistic interpreta-tion of traditional power indices (cf. Owen [1972, 1995] and Straffin [1977,1978, 1988]; Laruelle and Valenciano [2002] provide an up-to-date discussionand extensions). Expectation is to be taken with respect to an appropriateprobability measure on the power-relevant states of the world.The proposed framework is flexible and allows for different degrees of a

priori-ness, concerning players either purely random or preference-basedactions as well as details about decision procedures. Only inclusion of thelatter allows us to capture the power implications of resources other thanpure voting weight players procedural and strategic resources. Traditional(ex ante) power indices, such as the Penrose index or ShapleyShubik index,are obtained as special cases.The index approach to power analysis has evolved significantly in the last

50 years. It has reached a point where its integration into a framework thatalso allows for explicit decision procedures and preference-driven strategicbehavior seems a natural step. We first give a short overview of the indexapproach in Section 2, which takes up some arguments from the funda-mental critique of index-based studies by Garrett and Tsebelis. The creativeresponse by Steunenberg et al. to the latter is sketched and briefly discussed inSection 3. The main Section 4 then lays out our unified framework. Section 5concludes.

NAPEL AND WIDGREN: POWER MEASUREMENT AS SENSITIVITY 519

8. Partly in response to Widgren and Napel (2002) and an earlier version of this paper,Steunenberg and Schmidtchen (2002) have suggested modifications of their framework thatpromise to significantly improve it.


2. The Traditional Power Index Approach

The traditional object of studies of decision power has been a weighted votinggame characterized by a set of players, N f1; . . . ; ng, a voting weight foreach player, wi " 0 i 2 N, and a minimal quota of weights, k > 0, that isneeded for the passage of a legislative proposal. Subsets of players, S % N,are called coalitions. If a coalition S meets the quota, i.e.

Pi2S wi " k, it is

a winning coalition. The formation of a winning coalition is assumed to bedesirable to its members. More generally, a winning coalition need not bedetermined by voting weights. One can conveniently describe an abstractdecision body v by directly stating either the set Wv of all its winning co-alitions or its subset of minimal winning coalitions,Mv.9 The latter containsonly those winning coalitions which are turned into a losing coalition if oneof its members leaves the coalition. An equivalent representation is obtainedby taking v to be a mapping from the set of all possible coalitions, }N , tof0; 1g, where vS 1 (0) indicates that S is winning (losing). Function v isusually referred to as a simple game. The difference vS & vS& fig isknown as player i s marginal contribution to coalition S.The most direct approach to measuring players power is to state a map-

ping ! called an index from the space of simple games to Rn togetherwith a verbal story of why !iv indicates player i s power in the consideredclass of decision bodies. Despite the plausibility of some stories, their verbalform easily disguises incoherence or even inconsistency. The axiomatic orproperty-based approach, in contrast, explicitly states a set of mathematicalproperties fA1; . . . ;Akg that an index is supposed to have together withan (ideally unique) index ! which actually satisfies them. The requirementsAj are usually referred to as axioms. A prominent example for the axiomaticapproach is the ShapleyShubik index " (cf. Shapley, 1953; Shapley andShubik, 1954). Though this may not be immediately obvious, four propertiesA1A4 imply that "iv must be player i s weighted marginal contribution toall coalitions S, where weights are proportional to the number of playerorderings ( jl; . . . ; i; . . . jn such that S f j1; . . . ; ig.10 Axioms can give aclear reason of why " and no other mapping is used in particular, if a con-vincing story for them is provided. A drawback of the axiomatic approach is,


9. Typically, one requires that the empty set is losing, the grand coalition N is winning, andany set containing a winning coalition is also winning.10. The ShapleyShubik index is characterized by the requirements that (Al ) a (dummy)

player who makes no marginal contribution in v has index value 0; (A2 ) that the labelling ofthe players does not matter; (A3 ) that players index values add up to 1; and (A4 ) that in thecomposition u _ v of two simple games u and v, having the union of Wv and Wv as its setof winning coalitions Wu _ v, each players power equals the sum of his power in u and hispower in v minus his power in the game u ^ v obtained by intersecting Wv and Wu.


however, that axioms clarify the tool with which one measures11 but not whatis measured based on which (behavioral and institutional) assumptionsabout players and the decision body.The probabilistic approach to the construction of power indices entails

explicit assumptions about agents behavior together with an explicit defini-tion of what is measured. Agent behavior is specified as a probability distri-bution P for players acceptance rates, denoting the probabilities of a yesvote by individual players. A given players ex ante power is then taken tobe his probability of casting a decisive vote, i.e. to pass a proposal thatwould not have passed had he voted no instead of yes. Thus power isinferred from the hypothetical consequences of an agents behavior. Theobject of analysis with this approach is no longer described only by the setof winning coalitions or v but also by an explicit model of (average) beha-vior.12 For example, the widely applied Penrose index (Penrose, 1946) also known as the non-normalized Banzhaf index is based on the distribu-tion assumption that each player independently votes yes with probability1/2 (on an unspecified proposal). The corresponding joint distribution ofacceptance rates then defines the index # where #iv turns out to be,again, player i s weighted marginal contribution to all coalitions in v,where weight is this time equal to 1=2n&1 for every coalition S % N.13Just as the direct approach and the axiomatic approach require stories to

justify the index ! or fA1; . . . ;Akg, respectively, the assumption of a particu-lar distribution P of acceptance rates has to be motivated. This pointstowards drawbacks of the probabilistic index approach. First, the describedbehavior is usually not connected to any information on the agents prefer-ences or decision procedures.14 Second, decisions by individual players areassumed to be stochastically independent. This will, in practice, only rarelybe the case since it is incompatible with negotiated coalition formation andvoting based on stable player preferences. The imposition of stochastic ordeterministic restrictions for coalitions containing particular players orsub-coalitions can alleviate some of these shortcomings of traditional indices(see van den Brink, 2001, or Napel and Widgren, 2001). Still, the application


11. Even this cannot be taken for granted. Axioms can be too general or mathematicallycomplex to give much insight.12. Acceptance rates and assumptions on their stochastic relation among different players can

be interpreted as an implicit way of taking preferences into account in power index models (seeStraffin, 1988, for discussion).13. This means that #iv is the ratio of the number of swings that player i does have to the

number of swings that i could have. One can alternatively derive # from the assumption thatplayers acceptance rates are independently distributed on 0; 1) with mean 1/2.14. 1f such information is not available, the principle of insufficient reason seems a valid argu-

ment for the assumptions behind the Penrose index.


of traditional power indices has been severely criticized. Concerning decision-making in the European Union, Garrett and Tsebelis (1999a, b, 2001) havetaken a particularly critical stance, pointing out indices ignorance of decisionprocedures and player strategies.15

We agree with Garrett and Tsebelis that institutions and strategies have tobe taken into account by political analysis. Nevertheless, both the normativeor constitutional ex ante (possibly even a priori) analysis of political institu-tions and the positive or practical political analysis of actual and expecteddecisions are valuable. It is legitimate to ask: Which voting weights in theEU Council of Ministers would be equitable? For an answer, countriesspecial interests and their potentially unstable preferences in different policydimensions should not matter. Hence, they are best concealed behind aveil of ignorance as accomplished by the Penrose index. When multi-level decision bodies are designed, the objective of minimizing the probabilityfor the referendum paradox (an upper-level decision taken against a majorityat a lower level) provides a similar case for a priori analysis. In contrast,evaluation of the medium-run expected influence on EU policy from a par-ticular countrys point of view benefits if available preference informationis taken into account.16 Garrett and Tsebeliss goal of understanding . . .policy changes on specific issues . . . and negotiations about treaty revisions(Garrett and Tsebelis, 1999b: 332) or, in general, understanding decision-making in the EU (Garrett and Tsebelis, 2001: 105) seems impossible toachieve by looking only at the voting resources of different EU membersand the relevant qualified majority rule. The traditional power indexapproach is not a suitable framework in which to discuss these positive ques-tions related to power.We disagree with Garrett and Tsebelis (2001: 100) call for a moratorium

on the proliferation of index-based studies. As already pointed out by others,it is a matter of taste whether one deems the pursuit of positive or normativeanalysis more worthwhile. We believe in both and agree with Garrett andTsebelis that the strategic implications, which are hard to separate fromplayers preferences, of particular institutional arrangements in the EU andelsewhere have received too little attention so far. We think it desirable tohave a general unified framework which allows for positive and normativeanalysis and actual political and constitutional investigations.


15. See the replies of Berg and Lane (1999), Holler and Widgren (1999), Steunenberg et al.(1999) and Felsenthal and Machover (2001a).16. The medium run can, of course, be short-lived: A social democratic Council member can

quickly turn into a right-wing conservative through elections. For example, news about the firstnational or another foreign outbreak of mad-cow disease have quickly changed voters and agovernments views on agricultural, health or trade policy.


3. The Strategic Power Index of Steunenberg et al.

Replying to the critique by Garrett and Tsebelis, Steunenberg et al. (1999)have proposed a framework originally believed to reconcile traditionalpower index analysis and analysis of non-cooperative games, which explicitlydescribe agents choices in a political procedure and (their beliefs about)agents preferences. They consider a spatial voting model with n playersand an m-dimensional outcome space. In our notation, let N f1; . . . ; ngbe the set of players and X % Rm be the outcome or policy space. ! denotesthe procedure or game form describing the decision-making process andq 2 X describes the status quo before the start of decision-making. Playersare assumed to have Euclidean preferences with $i 2 X i 2 N as player i sideal point. A particular combination of all players ideal points and thestatus quo point define a state of the world %. Assuming that it exists andis unique, let x*% denote the equilibrium outcome of the game based on !and %.17 Steunenberg et al. are aware of Barrys (1980a, b) distinctionbetween power and luck and explicitly strive to isolate the ability of aplayer to make a difference in the outcome (p. 362). They note that[h]aving a preference that lies close to the equilibrium outcome of a particu-lar game does not necessarily mean that this player is also powerful (p. 345). Therefore, they suggest considering not one particular state of theworld % but many.In particular, one can consider each $i and the status quo q to be realiza-

tions of random variables ~$$i and ~qq, respectively. If P denotes the joint distri-bution of random vector ~%% : ~qq; ~$$1; . . . ; ~$$n and jj + jj the Euclidean norm,then

"!i : Zjj ~$$i & x* ~%%jj dP 1

gives the expected distance between the equilibrium outcome for decisionprocedure ! and player i s ideal outcome. Steunenberg et al. (1999: 348)all other things being equal consider a player . . . more powerful thananother player if the expected distance between the equilibrium outcomeand its ideal point is smaller than the expected distance for the otherplayer. Interested not only in a ranking of players but also a cardinalmeasure of their power, they proceed by considering a dummy player d either already one of the players or added to N whose preferences varyover the same range as the preferences of actual players. This leads to theirdefinition of the strategic power index (StPI) as


17. Non-uniqueness may be accounted for by either equilibrium selection or, in ex anteanalysis, explicit assumptions about different equilibrias probability.


#!i : "!d &"!i"!d

The remainder of Steunenberg et al.s paper is then dedicated to thedetailed investigation of particular game forms ! which model the consulta-tion and cooperation procedures of EU decision-making. They derive thesubgame perfect equilibrium of the respective policy game for any state ofthe world %, and aggregate the distance between players ideal outcomeand the equilibrium outcome assuming independent uniform distributionsover a one-dimensional state space X for the ideal points and the statusquo. The uniformity assumption is not innocuous (see Garrett and Tsebelis,2001: 101) but Steunenberg et al.s numerical calculations could quite easilybe redone with more complex distribution assumptions. The important ques-tion is: Does #!i measure what it is claimed to, i.e. the ability of a player tomake a difference in the outcome?The answer is: only under very special circumstances. In particular, (1)

turns out to define #!i to be player i s expected success. Just like actualdistance measures success (a function of luck and power), so does averagedistance measure average success. Unless one regards average success asthe defining characteristic of power (which neither Steunenberg et al. normany others do), taking expectations will only by coincidence achievewhat Steunenberg et al. aim at, namely to level out the effect of luck ora particular preference configuration on the outcome of a game (p. 362).This point is discussed in considerable detail in Napel and Widgren (2002:9ff). There, various examples illustrate that the StPI is a good measure ofexpected success but, in general, it fails to capture power:18 #!i may alsobecome negative. Only for particular distribution assumptions is lucklevelled out by taking averages. Similarly, only under special conditions which eliminate all strategic aspects from the StPI does a link betweenthe StPI and Penrose index discovered by Felsenthal and Machover (2001a)exist.


18. A non-technical example refers to a group of boys with a leader who makes proposals ofwhat to do in the afternoon (play football, watch a movie, etc.) which have to be accepted bysimple majority. Boys preferences (mappings from the weather conditions, pocket money,etc.) are assumed to be identically but independently distributed. The agenda setter enjoyssmaller average distance to the equilibrium outcome than the others; amongst the latter, expecteddistance is the same. Then, the little brother of the groups leader is allowed to participate in thegroups afternoon activity albeit without any say in selecting the daily programme. He does notalways agree with his elder brothers most desired outcome but does so more often than theothers (the brothers ideal points are positively correlated). Then, mean distance between thegroups equilibrium activity and its new members most desired recreation is smaller andhence his StPI power value is larger than that of the established members who actually havea vote on the outcome.


These points seem to have been taken. In particular, Steunenberg andSchmidtchen (2002) have proposed the incorporation of a distinct dummyplayer (meaning a reference player without decision rights) for each indi-vidual player. This solves some of the problems but not all.

4. An Alternative Approach

Steunenberg et al.s original framework is suited to studying success both expost and, by taking expectations, ex ante. The chief reason why the StPI doesnot measure power is its reliance on information only about the outcome ofstrategic interaction. Power refers to the ability to make a difference to some-thing (which is implicitly regarded as subjectively valuable to someone):a players power derives from and needs to be measured by recurring to his available strategies in the considered decision procedure or game form.Power means potential and thus refers to consequences of both actual andhypothetical actions.In our view, the key to isolating a players power in the context of collective

decision-making is his marginal impact or his marginal contribution to theoutcome x*. As mentioned in Section 2, this concept is well established inthe context of simple games and also general cooperative games, where itmeasures the implication of some player i joining a coalition S. The funda-mental idea of comparing a given outcome with one or several other out-comes, taking the considered player i s behavior to be variable, amountsto analysis of the sensitivity of the outcome with respect to player is actions.This can be generalized to a non-cooperative setting which explicitly describesa decision procedure.

4.1. An Example

For illustration, consider the player set N [ fbg, the rather restricted policyspace X f0; 1g embedded in R, and status quo ~qq 0. Let the decision pro-cedure ! be such that, first, bureaucrat b sets the agenda, i.e. either proposes1 or ends the game and thereby confirms the status quo. Formally, he choosesan action ab from Ab f1; qg. If a proposal is made, then all players i 2 Nsimultaneously vote either yes, denoted by ai 1, or no (ai 0). Thismakes Ai f0; 1g their respective set of actions. The proposal is acceptedif the weighted number of yes votes meets a fixed quota k, where the voteby player i is weighted by wi " 0. Otherwise, the status quo prevails.Formally, the function

xa xab; a1; . . . ; an 1 ab 1 ^P

i2N aiwi " k0 otherwise

!



maps each action profile a ab; a1; . . . ; an to an outcome. Traditionalpower index analysis for players i 2 N can easily be mimicked with this set-ting.19 Take

D0i a : xab; a1; . . . ; ai; . . . ; an & xab; a1; . . . ; 0; . . . ; an 2as player i s marginal contribution for action profile a corresponding tovS & vS nfig for coalition S f j jaj 1g and make probabilisticassumptions over the set of all action profiles. The latter replaces the prob-ability distribution over the set of all coalitions which is usually consideredvia assumptions on acceptance rates. Assume, for example, that bureaucratb always chooses ab 1 and let P denote the joint distribution over actionprofiles a 2Qi2fbg[N Ai. Then,

!!i :ZD0i a dPa 3

corresponds exactly to the traditional probabilistic measures obtained viaOwens multilinear extension (1972, 1995) and Straffins power polynomial(1977, 1978, 1988). For example, the uniform distribution over yes- andno-vote configurations

Pa 1=2n ab 1

0 ab 0!

makes !! (a rescaled version of) the Penrose index. For

Pa "P

i2N ai & 1#!"n&Pi2N ai#!

n!Pn

k1 1=kab 1

0 ab 0

8

players i 2 N have random spatial and procedural preferences with uni-formly distributed ideal points ~$$i taking values in X and the procedural com-ponent that for given policy outcome x 2 X they prefer to have votedtruthfully.21 Let the bureaucrat have only a procedural preference, namelyone for putting up a proposal if and only if it is accepted.Now consider a particular realization of ideal points $ $1; . . . ; $n. In

the unique equilibrium of this game, first, the bureaucrat chooses ab 1,i.e. he proposes 1, if and only if the set Y : fij$i 1g meets the quota, i.e.P

i2Y wi " k. Second, every voter votes truthfully according to his preferencein case that a proposal is made. Hence, the function

x*$ x*$1; . . . ; $n 1P

i2N $iwi " k0 otherwise

!maps all preference profiles (as determined by the vector of players idealpoints) to a unique equilibrium outcome. Assumptions about the distribu-tion P 0 of random vector ~$$ ~$$1; . . . ; ~$$n can be stated such that P 0 inducesthe same distribution P over action profiles a in equilibrium which has beendirectly assumed above. For example, P 0$1; . . . ; $n , 1=2n implies equili-brium behavior which makes !!i in (3) the Penrose index.

4.2. Measuring Ex Post Power

We propose extending the previous analysis from the simple coalition frame-work typical of ex ante power measurement and the very basic examplevoting game just considered to a more general setting. Player i s marginalcontribution is a measure for the sensitivity of the outcome to player i sbehavior. We consider it the best available indicator of player i s potentialor ability to make a difference for a given (expected or observed) collectivedecision, i.e. his ex post power. If this is of normative interest or for thelack of precise data, one can calculate the ex ante power based on the expost power as the latters expected value. This equates ex ante power ofplayer i with the expected sensitivity of the outcome to i s behavior. Expecta-tions can be taken with respect to several different aspects which affect expost power such as actions, preferences, or the procedure. This allows forthe (re-)foundation of ex ante measures (including a priori indices) on awell-specified notion of ex post power.


21. See Hansson (1996) on the importance of procedural preferences in the context of collec-tive decision-making. Our procedural preference assumption can, for example, be motivated byregarding each player as the representative of a constituency to which he wants do demonstratehis active pursuit of its interests.


There are several to us, at this stage, all promising directions in whichthe notion of power as marginal impact or power as sensitivity can bemade precise. The uniting theme is the identification of the potential to influ-ence an outcome of group decision-making by looking at the sensitivity ofthat outcome with respect to the considered set of agents. Influence canequally refer to the impact of a random yes or no decision, as assumedby the traditional probabilistic index approach and to the impact of astrategic yes or no vote based on explicit preferences.Crucially, impact is always relative to a what-if scenario or what we

would like to call the shadow outcome. The shadow outcome is the groupsdecision which would have resulted if the player i whose power is underconsideration had chosen (were to choose) differently than he actually did(is expected to), e.g. if he had stayed out of coalition S when he ex postbelongs to it or had ideal point 0 instead of 1. In simple games, the differencebetween shadow outcome and actual outcome, i.e. the sensitivity of the out-come to i s behavior for a given action profile or coalition, is either 0 or 1.A richer decision framework allows for more finely graded ex post power.It also requires a choice between several candidates for the shadow outcomeand, possibly, the subjective evaluation of differences.A natural way to proceed is to measure player i s ex post power as the

difference in outcome for given actions of all players, e.g.a* a*b; a*1; . . . ; a*i ; . . . ; a*n also denoted by a*i ; a*& i, and the case inwhich for whatever reasons player i chooses a different action, i.e. fora 0i; a*& i a*b; a*1; . . . ; a 0i; . . . ; a*n with a*i 6 a 0i. For preference-based actionswith a unique equilibrium a*, this defines a players ex post power as thehypothetical impact of a tremble in the spirit of Seltens (1975) perfectnessconcept, i.e. of irrational behavior or imperfect implementation of his pre-ferred action. More generally, a tremble can refer to just any deviationfrom reference behavior or reference preferences.If Ai consists of more than two elements, different degrees of irrationality

or if preferences are left out of the picture potential deviations fromthe observed action profile can be considered. In particular, one may con-fine attention to the impact of a local action tremble. If Ai X f0; &; 2&; . . . ; 1g for some S > 0 that uniformly divides 0; 1),22 a suitabledefinition of player i s marginal contribution is

D10i a* :

xa*i ; a*& i & xa*i & &; a*& i&

a*i & & " 00 otherwise

8

If & 1, this corresponds exactly to D0i + and the marginal contributiondefined in the traditional power index framework (see p. 526). As playerschoice set approaches the unit interval, i.e. & ! 0, one obtains

D1i a* : lim&!0xa*i ; a*& i & xa*i & &; a*& i

& @xa

@ai

$$$$aa*

5

for a*i 2 0; 1.Both (4) and (5) measure player i s power in a given situation, described by

the ex post action vector a*, as the (marginal) change of outcome whichwould be caused by a small (or marginal) change of i s action. It is, however,not necessary to take only small trembles into account; one may plausibly use

D100i a* : max

ai 2Xxa*i ; a*& i & xai; a*& i) 6

to define an alternative measure of ex post power.23 D1i ;D10i , and D

100i are all

based on the question:

. If a player acted differently, would this alter the outcome of collectivedecision-making and, if yes, by how much?

Making different assumptions about which possible differences in a playersbehavior are relevant, they give a different cardinal answer to this question.Players preferences may enter (4)(6) to define x*$1; . . . ; $n , xa* as

the reference point for action trembles, i.e. the point at which the (general-ized) derivative of outcome function x+ with respect to player i s action isevaluated. If x*$1; . . . ; $n is the unique equilibrium outcome, any actiondeviation resulting in a distinct outcome is irrational. A meaningful alterna-tive to studying the potential damage or good that a players irrationalitycould cause is to consider instead the effect of variations in his preferenceswhile maintaining rationality. This refers to the following two criteria forex post power as sensitivity for given preferences:

. If a player wanted to, could he alter the outcome of collective decision-making?


23. The total range Dmaxi : maxa & i 2X n & 1 maxai 2X xai; a& i &minai 2X xai; a& i) of playeri s possible impact on outcome lacks any ex post character. It seems a reasonable a priorimeasure which requires no distribution assumptions on ~aa* or ~$$. However, it is a rather coarseconcept and would only discriminate between dummy and non-dummy players in the contextof simple games. Dmax

0i a*& i : maxai 2X xai; a*& i &minai 2X xai; a*& i holds an interesting

intermediate ground. Another promising measure of ex post power is (the inverse of ) the mini-mal tremble size by which a player i would affect a given outcome (Matthew Braham, personalcommunication).


. Would the change of outcome in magnitude (and direction) match theconsidered change in preference?

Precise answers to these questions can be given by replacing xa* byx*$1; . . . ; $n in the previous definitions. As in the case of hypotheticalaction changes, one may consider either any conceivable change of prefer-ences relative to some reference point or restrict attention to slight variations.The latter requires some metric on preferences, which is, however, naturallygiven for Euclidean preferences.For illustration, consider players N f1; 2; 3g with Euclidean preferences

on policy space X 0; 1), described by individual ideal points $i 2 X, andsimple majority voting on proposals made by the players. Let $ j denotethe j th smallest of players ideal points, i.e. $1 - $2 - $3. Depending onthe precise assumptions on the order of making proposals and voting,many equilibrium profiles of player strategies exist. However, they yieldthe median voters ideal point, $2, as the unique equilibrium outcomex*$1; $2; $3. One can then investigate player 1s power for given $2 and$3, where, without loss of generality, we assume $2 - $3. The mapping

x*$1; $2; $3 $2 $1 < $2

$1 $2 - $1 - $3$3 $3:$1

8

To illustrate this, let $ $1; . . . ; $n be the collection of n players idealpoints in Rm (an m. n matrix having as columns the $i-vectors representingindividual players ideal points).25 In a policy space X % Rm, the opportu-nities even for only marginal changes of preference are manifold. A givenideal point $i can locally be shifted to $i ' where ' is an arbitrary vectorin Rm with small norm. It will depend which tremble directions are particu-larly meaningful in applications.26 Multiples of the vector 1; 1; . . . ; 1 2 Rmseem reasonable if the m policy dimensions are independent of each other.In any case,

D2i $ : limt!0

x*$i t'; $& i & x*$i; $& it

@'x*$i; $& i@$i

7

defines a suitable measure of player i s ex post power provided that this limitexists. This is simply the directional derivative of the equilibrium outcome indirection '. Alternatively, measures for the multidimensional case can bebased on the gradient of x*$i; $& i (holding $& i constant). In case of idealpoints in a discrete policy space, a preference-based measure D2

0i $ can be


Figure 1. Equilibrium Outcome of Simple Majority Voting as $1 is Varied

25. At the level of national elections, m 2 is for most countries a sufficiently high dimen-sion (Norman Schofield, personal communication).26. For example, one may be interested in the expected effect of a general swing towards eco-

nomically and/or socially more liberal or conservative positions across parties. One may alsoconsider not a particular direction ' but rather an entire neighborhood of $i (e.g. taking thesupremum of (7) for all possible directions ').


defined by replacing the derivative in (7) with a difference quotient in analogyto (4) and (5). Interestingly, if x*$ is the Copeland winner, denoting thepolicy which is majority-preferred to the highest proportion of other alterna-tives in X and having a number of other desirable properties, then theprevious power measure for any direction ', since x*$ is a convex combi-nation of players ideal points corresponds to the OwenShapley spatialpower index (see Grofman et al., 1987; Owen and Shapley, 1989).

4.3. Calculating Ex Ante Power

Mappings D1i ;D2i , and their discrete versions measure ex post power as the

difference between distinct shadow outcomes and the expected or observed(equilibrium) collective decision. We do not want to discuss at this pointwhich is the most relevant shadow outcome and hence measure. All ofthem clearly distinguish power from the luck of a satisfying group decision;they do not require any averaging out of luck. Taking expectations merelyserves the purpose of obtaining ex ante conclusions if these are of interest.Having selected a meaningful measure of ex post power, it is straight-

forward to define a meaningful ex ante measure. It has to be based on explicitinformational assumptions concerning players preferences or if one doesnot want to assume preference-driven behavior actions. Denoting by ~%%the random state of the world as given either by preferences (and statusquo) or players actions, and by P its distribution,

!!i :ZDi ~%% dP 8

is the ex ante power index based on ex post measure Di+ and decision pro-cedure or game form !.27

Traditional power indices, such as the Penrose or ShapleyShubik index,consider the particularly simple decision procedure in which playersi 2 N f1; : . . . ; ng choose an action ai 2 Ai f0; 1g and the outcome ofdecision-making, xa, is 1 if set Y : fi j$i 1g is a winning coalition, i.e.if vY 1, and 0 otherwise. They use D0i a (or D1i a with & 1). TheStPI proposed by Steunenberg et al., too, is a linear transform of (8), albeitusing the unreasonable a posteriori power measure Di ~%% jj ~$$i & x* ~%%jj.Let us illustrate our sensitivity analysis approach to measuring power

more explicitly. As an example assume a simple procedural spatial voting


27. We have omitted a sub- or superscript ! in the definition of ex post measures for a concisenotation. The procedure is, however, the central determinant of outcome functions x+ or x*+.If several different game forms ! 2 G are to be considered ex ante, one has to take expectationover !!i with the appropriate probability measure on G.


game where a fixed agenda setter makes a take it or leave it offer to a groupof five voters and needs four sequentially cast votes to pass it.28 The policyspace is X 0; 1), voters ideal points are $1; . . . ; $5, that of the agenda-setter is ( and the status quo is 0. The unique equilibrium outcome of thisgame is

x*$ x*$1; . . . ; $5 2$2 0 - $2 < 12 (( 12 ( - $2 - 1

(

where $2 is the second-smallest of the $i. For agenda-setter j, we get the expost power

D2j (; $ :0 0 < $2 < 12 (1 12 ( < $2 < 1

(and for voter i

D2i (; $ : 2 0 < $2

POWER MEASUREMENT AS SENSITIVITY ANALYSIS A UNIFIED APPROACH

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