Abstract
The main purpose of the present paper is to disentangle the mix-up of the notions of success and satisfaction which is prevailing in the voting power literature. We demonstrate that both notions are conceptually distinct, and discuss their relationship and measurement. We show that satisfaction contains success as one component, and that both coincide under the canonical set-up of a simultaneous decision-making mechanism as it is predominant in the voting power literature. However, we provide two examples of sequential decision-making mechanisms in order to illustrate the difference between success and satisfaction. In the context of the discussion of both notions we also address their relationship to different types of luck.
JEL Classification: C79, D02, D71
We would like to thank Matthew Braham, Keith Dowding, William Gehrlein, Manfred Holler, Serguei Kaniovski, Dennis Leech, Moshé Machover, Peter Morriss, and Stefan Napel for comments and discussions. Considerable parts of the research contained in this paper were already developed between 2005 and 2006 when Frank Steffen was at Tilburg University under a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. He gratefully acknowledges this financial support.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
While Table 1 includes the most prominent sources, we do not claim completeness.
- 3.
We would like to note that Barry (1980) himself defines success as the probability to be successful as defined above. A discussion of this issue can be found in Dowding (1991, p. 65). Moreover, note that Barry’s (1980) additional requirement is always fulfilled under the canonical set-up as specified in Sect. 2 as under this set-up (1) abstention is not permissible and (2) all actors have to choose their action simultaneously. The issue of “abstention” and its implications for the relationship between success and satisfaction are addressed in Sect. 6.
- 4.
- 5.
This excludes the option of abstention as a tertium quid, which can have a considerable impact on the power distribution among the actors (see, for instance, Felsenthal and Machover 1997, 1998, 2001, and Braham and Steffen 2002). In Sect. 6 we will briefly address the possibility of abstention in the context of the analysis of the present paper.
- 6.
Note that this implies that a secret but sequential decision-making procedure is permissible as well.
- 7.
For a detailed description of the “action-based approach” we refer to van den Brink and Steffen (2008).
- 8.
So, \(p(\cdot,\Gamma )\) is a probability distribution over \(\mathcal{A}^{N}\), while \(\tilde{p}(\cdot,a,\Gamma )\) is a conditional probability distribution over K(a).
- 9.
Note that it would be possible to “code” the actions, inclinations and outcomes by 0 and 1, in which case the correspondences ∼ defined above could be simply written as equalities. We chose not to do that in this paper to make clear the distinction between actions and inclinations, which is essential for the difference between success and satisfaction (and luck) later on.
- 10.
This can be illustrated by the following example which we owe to Matthew Braham. Assume that you have the desire to become rich, but you are not doing anything to achieve this. Instead, you are lying on the beach enjoying the sun and the fresh air. However, suddenly one of the seagulls circling over your head drops a valuable diamond which just falls into your lap. Thus, your desire has been fulfilled even you have not attempted anything to achieve this, if we assume that you were not aware of the fact that it might happen that a seagull drops a valuable diamond during the time you are lying on that beach.
- 11.
- 12.
Note that Barry (1980) does not use the notion of “power” in this context, but refers to decisiveness. However, by this notion he means what is usually called “power” in the voting power literature. For a discussion of this issue see, for instance, Dowding (1991, pp. 63–68, 1996, pp. 52–54) or Felsenthal and Machover (1998, p. 41).
- 13.
As van den Brink and Steffen (2008) point out it is important to draw attention to the interpretation of the ceteris paribus condition in this context. Its common interpretation is that the actions of all other actors remain constant. That is, if i alters its action the only effect that can result out of this is a change in the collective outcome (then we say that i has a swing and we ascribe power to i). While this “all other things being equal” interpretation is appropriate for simultaneous DMMs, it no longer applies for our more general case of a sequential DMM, which may allow certain actors to exclude other actors from the decision-making as a result of their choices. If we have an action profile and we alter i’s choice of action it can happen that the decision-making process requires either the exclusion of actions of other actors from the domain of the decision rule and, hence, from the action profile, or the inclusion of actions by other actors in the domain of the decision rule and, therefore, in the action profile. If such information would be ignored, we can end up with an inappropriate power ascription. In order to avoid this problem we have to go back to the idea behind the literal “all other things being equal” interpretation of the ceteris paribus clause. The basic idea of the ceteris paribus clause is a comparison between two possible worlds: the world as it is (our initial action profile and its associated collective outcome) and the world as it would be if an action were changed (the resulting action profile and its associated collective outcome if i’s choice of action were altered). In contrast to the standard interpretation of the ceteris paribus clause our analysis does not necessarily require that all other components of the action profile remain constant after we altered i’s choice of action; it requires that the action profiles after the initial change by one actor are consistent with the DMM. This interpretation of the ceteris paribus clause is underlying Definitions 4.1 and 4.2.
- 14.
Note that van den Brink and Steffen (2008) demonstrate that it is not necessary to specify the value of ε for binary DMMs.
- 15.
Note that, as pointed out by Dowding (1991, p. 64) with respect to one of these notions, both notions must be carefully distinguished from what he call’s personal identity luck, i.e., “the luck of being the particular person one happens to be” which is discussed by egalitarians (see, for instance, Roemer 1986 or Cohen 1989).
- 16.
For a critical discussion of the probability requirement in Barry’s (1980) luck definition see Dowding (1991, p. 65; 1996, 52f). Moreover, we would like to point out an inconsistency in Barry’s (1980) analysis. His definition of decisiveness being: “the difference between his success [making use of the IOC] and his luck. …it represents the difference that it makes to his success if he tries.” Hence, whenever an actor tries according to this definition it is decisive, i.e., if i is a dummy actor and i chooses an action, i.e., i tries to get what it wants, i would be decisive. However, this is not what is usually meant by the notion of decisiveness (see Footnote 11) and Barry (1980) himself later in his essay writes that decisiveness means to be “critical”, i.e., to have a swing. Now one might argue, that Barry (1980) has meant this and that a dummy actor by definition cannot “try”, but this would mean that the notion of a “try” presupposes the ability to be successful with the “try”. However, this contradicts also the very basic meaning of a “try” as being just an attempt in order to achieve something, whether one has the ability to do so or not. Taking this criticism into account one could re-define Barry’s (1980) definition of luck to be: “getting what one wants if one does not try or if one tries without being critical with respect to the action profile in question”. Note that the second part of this definition is the definition of action luck as contained in Definition 4.8.
- 17.
Dworkin (1981) illustrates the difference between both types of luck by an example of bad luck: “If someone develops cancer in course of a normal life, and there is no particular decision to which we can point as a gamble risking the disease, then we will say that he has suffered brute bad luck. But if he smoked cigarettes heavily then we may prefer to say that he took an unsuccessful gamble”, i.e., he has suffered bad option luck.
- 18.
In fact, it is also possible to allow for any ε ∈ [0, 1] but in that case we also need to redefine action luck taking account of weak and strong swings. Since this paper focusses on the distinction between success and satisfaction, we will not do that.
- 19.
Note that the terminology successor-predecessor is opposite to the one as used in van den Brink and Steffen (2012). In van den Brink and Steffen (2012) both notions are used to refer to the positions of actors in a hierarchy, i.e., if actor i directly dominates an actor j, we say that i is a predecessor of j, and that j is a successor of i. In the present paper we make use of the same terminology to refer to actors in a sequential DMM, i.e., if an actor i chooses its action after actor j has made its choice of action, we say that actor i is a successor of j, and that j is a predecessor of i.
- 20.
Note that \(\Gamma _{1}\) and \(\Gamma _{2}\) in Examples 5.3 and 5.4, respectively, could be regarded as examples for DMMs in hierarchical organizations, where the structure of the hierarchy is a “line” in case of \(\Gamma _{1}\) and a “star” in case of \(\Gamma _{2}\) (see van den Brink and Steffen 2008, 2012).
References
Banzhaf, J. F. (1965). Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Review, 19, 317–343.
Barry, B. (1980). Is it better to be powerful or lucky? Political Studies, 28, 183–194, 338–352.
Braham, M. (2008). Social power and social causation: towards a formal synthesis. In M. Braham, & F. Steffen (Eds.), Power, freedom, and voting (pp. 1–21). Berlin, Heidelberg: Springer.
Braham, M., & Steffen, F. (2002). Voting power in games with abstentions. In M. J. Holler, H. Kliemt, D. Schmidtchen, & M. E. Streit (Eds.) Power and fairness (pp. 333–348). Tübingen: Mohr-Siebeck.
Brams, S. J., & Lake, M. (1978). Power and satisfaction in a representative democracy. In P. C. Ordeshook (Ed.), Game theory and political science (pp. 529–562). New York: New York University Press.
van den Brink, R., Rusinowska, A., & Steffen, F. (2011). Measuring power and satisfaction in societies with opinion leaders: dictator and new opinion leader properties. Homo Oeconomicus, 28, 161–185.
van den Brink, R., Rusinowska, A., & Steffen, F. (2013). Measuring power and satisfaction in societies with opinion leaders: an axiomatization. Social Choice and Welfare, 41, 671–683.
van den Brink, R., & Steffen, F. (2008). Positional power in hierarchies. In M. Braham, & F. Steffen (Eds.), Power, freedom, and voting (pp. 57–81). Berlin, Heidelberg: Springer.
van den Brink, R., & Steffen, F. (2012). Axiomatizations of a positional power score and measure for hierarchies. Public Choice, 151, 757–787.
Cohen, G. A. (1989). On the currency of egalitarian justice. Ethics, 99, 906–944.
CollinsDictionary (2009) Collins English Dictionary, 10th edn. Glasgow: HarperCollins.
Dowding, K. (1991). Rational choice and political power. Aldershot: Edward Elgar.
Dowding, K. (1996). Power. Buckingham: Open University Press.
Dubey, P., & Shapley, L. S. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research, 4, 99–131.
Dworkin, R. (1981). What is equality? Part 2: equality of resources. Philosophy of Public Affairs, 10, 283–345.
Felsenthal, D., & Machover, M. (1997). Ternary voting games. International Journal of Game Theory, 26, 335–351.
Felsenthal, D., & Machover, M. (1998). The Measurement of voting power: theory and practice, problems and paradoxes. London: Edward Elgar.
Felsenthal, D., & Machover, M. (2001). Models and reality: the curious case of the absent abstention. In M. J. Holler, & G. Owen (Eds.), Power indices and coalition formation (pp. 87–103). Dordrecht: Kluwer.
Grabisch, M., & Rusinowska, A. (2010). A model of influence in a social network. Theory and Decision, 69, 69–96.
Holler, M. J. (1982). Forming coalitions and measuring voting power. Political Studies, 30, 262–271.
Hosli, M. O., & Machover, M. (2004). The nice treaty and voting rules in the council: a reply to Moberg (2002). Journal of Common Market Studies, 42, 497–521.
Laruelle, A., Martinez, R., & Valenciano, F. (2006). Success versus decisiveness: conceptual discussion and case study. Journal of Theoretical Politics, 18, 185–205.
Laruelle, A., & Valenciano, F. (2005). Assessing success and decisiveness in voting situations. Social Choice and Welfare, 24, 171–197.
Laruelle, A., & Valenciano, F. (2008). Voting and collective decision-making. Cambridge: Cambridge University Press.
Morriss, P. (1987/2002). Power: a philosophical analysis. Manchester: Manchester University Press.
Nevison, C. H. (1979). Structural power and satisfaction in simple games. In S. J. Brams, et al. (Eds.), Applied game theory (pp. 39–57). Würzburg: Physica.
Nevison, C. H. (1982). Structural satisfaction in simple games. Mathematical Social Sciences, 3, 397–401.
Penrose, L. S. (1946). The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109, 53–57.
Pinker, S., et al. (Eds.). (2011) The American heritage dictionary of the English language, 5th edn. Boston: Houghton Mifflin Harcourt.
Rae, D. (1969). Decision rules and individual values in constitutional choice. American Political Science Review, 63, 40–56.
Roemer, J. (1986). Equality of resources implies equality of welfare. Quarterly Journal of Economics, 101, 751–784.
Rusinowska, A., & de Swart, H. (2006). Generalizing and modifying the Hoede-Bakker index. In H. de Swart et al. (Eds.), Theory and applications of relational structures as knowledge instruments. Springer’s Lecture Notes in Artificial Intelligence, LNAI 4342 (pp. 60–88). Heidelberg: Springer.
Straffin, P. D. (1977). Majority rule and general decision rules. Theory and Decision, 8, 351–360.
Straffin, P. D. (1978). Probability models for power indices. In P. C. Ordeshook (Ed.), Game theory and political science (pp. 477–510). New York: New York University Press.
Straffin, P. D., Davis, M. D., & Brams, S.J. (1982). Power and satisfaction in an ideologically divided voting body. In M. J. Holler (Ed.) Power, voting and voting power (pp. 239–255). Würzburg: Physica.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
van den Brink, R., Steffen, F. (2014). On the Measurement of Success and Satisfaction. In: Fara, R., Leech, D., Salles, M. (eds) Voting Power and Procedures. Studies in Choice and Welfare. Springer, Cham. https://doi.org/10.1007/978-3-319-05158-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-05158-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05157-4
Online ISBN: 978-3-319-05158-1
eBook Packages: Business and EconomicsEconomics and Finance (R0)