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Games and Political Decisions

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Decision-Making in Committees

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 635))

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Abstract

This chapter provides an introduction to some fundamental aspects of decision-making in committees. ‘Committee’ is used to refer to a decision-making body that comprises a small number of members (as opposed to a referendum situation), and chooses from a set of well-defined policy alternatives (in contrast to the electorate in a general election which usually chooses between candidates or party platforms). Decisions are ultimately reached by putting alternatives to a vote according to some voting rule specifying which subsets of all committee members can pass a proposal. This notion of a committee differs from everyday language where the term also applies to expert panels with advisory function, or organizational subunits that make recommendations or submit proposals to some superordinate organization.

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Notes

  1. 1.

    This definition of a characteristic function was put forward by Von Neumann and Morgenstern (1944, pp. 238ff), and is also known as Von Neumann–Morgenstern characteristic function.

  2. 2.

    Instead of {1, 2}, v({1, 2}) etc. the notation {12}, v({12}) etc. is used throughout as there is no risk of confusion.

  3. 3.

    For \(u,w \in {\mathbb{R}}^{s}\), u ≥ w denotes u i ≥ w i for all i ∈ S, and u > w denotes u i > w i for all i ∈ S.

  4. 4.

    Besides the α- and β-characteristic functions dealt with here, characteristic functions can also be derived under the γ- and δ-assumptions in Hart and Kurz (1983, 1984).

  5. 5.

    S is β-effective for a payoff vector \(x \in {\mathbb{R}}^{s}\) if, for any strategy combination \({\sigma }_{N\setminus S} \in {\Sigma }_{N\setminus S}\), there exists a σ S ∈ Σ S such that u i (σ S , σ N ∖ S ) ≥ x i for all i ∈ S.

  6. 6.

    The convex hull of a set \(P \subset {\mathbb{R}}^{n}\) is the smallest convex set containing the points in P. It is defined by \({\it { conv}}(P) := \left \{{\sum \nolimits }_{i=1}^{n}{a}_{i}{p}_{i}\,\vert \,{p}_{i} \in P,\:{a}_{i} \in {\mathbb{R}}_{+},\:{\sum \nolimits }_{i=1}^{n}{a}_{i} = 1\right \}\). – If randomization is not possible, the comprehensive convex hull must be replaced by the comprehensive hull.

  7. 7.

    For each coalition structure, a partition function assigns a ‘worth’ to each coalition (for a discussion of the limitations of partition functions in the analysis of coalition formation see Ray and Vohra, 1997).

  8. 8.

    Following common practice, we include monotonicity in the definition of a simple game. See, for example, Von Neumann and Morgenstern (1944, pp. 420ff), or Shapley (1962). Note, however, that a simple game can also be defined more generally in terms of a set \(\mathcal{W}\) which satisfies conditions (a) and (b), but not necessarily condition (c) (see, e.g., Ordeshook, 1986, p. 324).

  9. 9.

    Taylor and Zwicker (1992, 1993) introduce a general method to decide whether a simple game is weighted, based on trades of players among coalitions.

  10. 10.

    Starting with Shapley (1962), the notation ×and + , instead of ∧ and ∨ , is often used in the literature for the special case that the player sets are disjoint. Thus, G 1 ×G 2 and G 1 + G 2 model bicameral, or more generally, multicameral decision structures.

  11. 11.

    For another example of a legislative system, consisting of a president, a senate, and a house, see Shapley and Shubik (1954).

  12. 12.

    See Braham and Steffen (2003) for an application of compound weighted voting games to insolvency law rules.

  13. 13.

    A utility function is of the Von Neumann–Morgenstern type if it satisfies the expected utility hypothesis. See, for example, Ordeshook (1986, pp. 37ff).

  14. 14.

    The setting consisting of the set of voters, the set of alternatives, and a voting rule has also been studied from a social choice perspective. See Peleg (2002) for an overview.

  15. 15.

    In voting situations, even taking pre-vote negotiations into account, it seems appropriate to ignore the possibility that a coalition can achieve outcomes x and y, but is not effective for outcome z (see Miller, 1982).

  16. 16.

    In Sect. 1.2.2, the NTU simple game will be defined to exclude the possibility of both free-riding and exploitation of non-winning players.

  17. 17.

    The example can be extended to an arbitrary number of players, see Example 1.3.1.

  18. 18.

    By contrast, in most purely economic applications, preferences are assumed to exhibit nonsatiation, i.e., more of some good (perhaps denominated in money) is always better.

  19. 19.

    The absence of value consensus is expressed in Arrow’s axiom of ‘unrestricted domain’.

  20. 20.

    The core of a game (N, v) in characteristic function form is defined by { Core}(N, v) : = { u ∈ I(v) | ∑ i ∈ S u i ≥ v(S){ for all }S ∈ 2N ∖ ∅} where I(v) is the set of imputations, i.e., vectors \(u \in {\mathbb{R}}^{n}\) such that (a) u i ≥ v({i}) for all i ∈ N, and (b) ∑ i = 1 n u i = v(N). Conditions (a) and (b) express individual rationality and efficiency, respectively.

  21. 21.

    In Arrow’s 1963[1951] model, individuals have binary preference relations over a finite set of alternatives, which are assumed to satisfy the usual consistency conditions such as reflexivity, transitivity, and completeness.

  22. 22.

    The theorem was proved by Black (1958) for a finite set of alternatives and extended by Arrow (1963[1951]) to arbitrary sets of alternatives.

  23. 23.

    A hyperplane is a plane in more than two dimensions.

  24. 24.

    It is not applicable when issues are purely distributive, or interpreted in terms of distribution by the decision-makers. These situations are more appropriately modeled by TU constant-sum games.

  25. 25.

    In the literature, the term ‘Banzhaf index’ is widespread. Yet, since L.S. Penrose proposed the measure well before Banzhaf, we think it more accurate to call it Penrose–Banzhaf index.

  26. 26.

    More formally, a power index is a family of functions \(\{{\mu {}^{n}\}}_{n=1,2,\ldots }\) because N and n are not fixed.

  27. 27.

    The Penrose–Banzhaf index, or measure, only sums to unity after ‘normalizing’ the number of swings for each player i with the total number of swings. The normalized Penrose–Banzhaf index here is \((\frac{3} {5}, \frac{1} {5}, \frac{1} {5})\).

  28. 28.

    Unlike the Shapley–Shubik and the Penrose–Banzhaf indices, the Deegan–Packel and the Public Good index both violate local monotonicity, i.e., for two players i and j with w i < w j , these indices do not always assign a higher value to player j. In the literature, local monotonicity has been discussed, along with other properties, as a major desideratum for a measure of voting power, and as a criterion for selecting among power indices (see Felsenthal and Machover, 1998, pp. 221ff, and Holler and Napel 2004a, 2004b).

  29. 29.

    The MLE could also be defined for arbitrary real values of p i . The restriction to the unit cube is due to the interpretation of the p i as probabilities which are naturally constrained by 0 ≤ p i ≤ 1.

  30. 30.

    The term ‘independence assumption’ may be misleading as voters’ decisions to vote ‘yes’ are independent under the homogeneity assumption as well.

  31. 31.

    Only in dictatorial games, probabilities can be assessed from the voting rule alone, without a behavioral assumption.

  32. 32.

    The probability of casting a decisive vote is greatest at p = 1 ∕ 2 and declines rapidly already for small deviations from p = 1 ∕ 2 (see Chamberlain and Rothschild, 1981). This observation translates into a substantial bias of the Penrose–Banzhaf index when applied to voting bodies that do not perfectly satisfy the ‘independence assumption’ (see Kaniovski, 2008).

  33. 33.

    The usage the ‘Principle of Insufficient Reason’ (or ‘Principle of Indifference’ in Keynes’ terminology) is discussed critically and in detail by Keynes (1921, Chaps. IV and VI) who also states conditions under which it is safely applicable.

  34. 34.

    Leech (1990) favors the Penrose–Banzhaf measure over Shapley–Shubik index arguing that the distributional assumption underpinning the latter is “unduly strong”.

  35. 35.

    The concept of ‘success’ was introduced in Penrose (1946) and Rae (1969), and elaborated further by Barry (1980a, 1980b). For an analysis of decisiveness and success also see Laruelle and Valenciano (2005).

  36. 36.

    To see that is a reasonable choice consider a game with a dictator (who faces a dichotomous choice): The equilibrium outcome is always the preferred alternative of the dictator, and he could only make a difference to that by choosing his dominated strategy. His power, however, is that he can impose his will on (the whole of) the other players.

  37. 37.

    Laruelle and Valenciano (2008b) can be considered as a contribution to the Nash programme of establishing connections between cooperative solution concepts and non-cooperative game theory. However, the fact that some bargaining protocol implements a given solution does not mean that all relevant protocols do.

  38. 38.

    As with the NTU game (1.4), it is also assumed here that players have Von Neumann–Morgenstern preferences over X. Thus the bargaining situation can be summarized by (U, d).

  39. 39.

    As is common practice in the literature, the terms ‘null player’ and ‘dummy player’ are used interchangeably here and refer to players whose marginal contribution is null with respect to every coalition. More accurately, a dummy player i is a null player if v({i}) = 0 (see Roth, 1988, p. 23).

  40. 40.

    Shapley’s 1953 original axiomatic characterization used another axiom, the carrier axiom, which bundles the efficiency axiom and the null player axiom into one.

  41. 41.

    Solution (1.15) can alternatively be derived from (ANO), (IIA), (IAT), the ⇒ -part of (NP ∗ ), and the following efficiency axiom: (EFF) For all \((B,\mathcal{W}) \in \mathcal{B}\times \mathfrak{W}\), there is no u ∈ U such that \(u > F(B,\mathcal{W})\). Then, φ must be an efficient anonymous function satisfying the ⇒ -part of (NP ∗ ).

  42. 42.

    Laruelle and Valenciano (2007) state their result also for the case that some weights are zero. Here, we simplify to the case of positive weights.

  43. 43.

    { Nash} i φ(B TU) is defined as the solution to the maximization problem \(\max {u}_{1}^{{\varphi }_{1}} \cdot \cdots \cdot {u}_{n}^{{\varphi }_{n}}\) s.t., u 1 + … + u n = 1. The first order conditions for the u i (\(i = 2,\ldots ,n\)) can be summarized to u i = u 1φ i ∕ φ1. Then, the constraint amounts to u 1 ∑ i = 1 nφ i = φ1. It follows that \({u}^{{_\ast}} = ({u}_{1},\ldots ,{u}_{n})\) with u i = φ i ∕ ∑ i = 1 nφ i for all i ∈ N is the solution to the above maximization problem.

  44. 44.

    Laruelle and Valenciano (2001) introduce an alternative version of the transfer axiom which states that the effect on any player’s power of eliminating a minimal winning coalition from \(\mathcal{W}\) is the same in any game in which this coalition is minimal winning.

  45. 45.

    Owen (1971) introduces a (n − 1)-dimensional space in order to describe ideological affinities between n voters. A legislative proposal is represented as a point which is chosen randomly from a uniform distribution over the space. The ordering of players is then taken in terms of increasing distances of players’ ideal points from that point. Shapley’ s (1977) modification, which is followed in Fig. 1.3, uses an ideological space of arbitrary dimension where dimensions can be interpreted as ‘pure’ issues. Then, a player ordering is induced by a random vector whose direction indicates the particular ideological ‘mixture’ of the (exogeneous) proposal faced by the committee.

  46. 46.

    The calculation of the modified power index for general weighted voting games in two dimensions is explained in Straffin (1994, pp. 1140ff).

  47. 47.

    For one-dimensional policy spaces, (minimum) connected winning coalitions were first suggested by Axelrod (1970) in his “conflict of interest” theory.

  48. 48.

    Discrete alternatives to the derivative – corresponding to an infinitesimally small preference change – are discussed in Napel and Widgrén (2004). Sensitivity may also be defined more directly as the reaction of x ∗ to a change in a given player’s actions rather than his preferences. However, as actions are usually thought to originate in preferences, the formulation of the approach in terms of preferences is more elementary.

  49. 49.

    The agenda-setter can also control the outcome under a sequential binary agenda, as described by a finite binary voting tree, if he has information about the majority preference relation.

  50. 50.

    Romer and Rosenthal (1978) analyzed the effect of agenda control for one-dimensional policy choices under closed rule.

  51. 51.

    In the closed rule game, players B and C could be characterized as spatially inferior (compared to A). The concept of inferior players, referring to players who are subject to credible ultimatum threats, is introduced by Napel and Widgrén (2001) for TU simple games, and by Napel and Widgrén (2002) and Widgrén ans Napel (2002) for (one-dimensional) spatial voting games. Napel and Widgrén propose power measures, called the Strict Power Index and the Strict Strategic Power Index, respectively, which regard inferior/spatially inferior players as powerless.

  52. 52.

    The yolk (McKelvey, 1986) is the smallest circle that intersects all median lines or median hyperplanes. Its radius can be seen as a measure of how close the game is to having a core outcome.

  53. 53.

    In fact, the core is always empty for superadditive constant-sum games; for a proof see Ordeshook (1986, pp. 350f).

  54. 54.

    For example, Littlechild and Thompson (1977) apply the Shapley value to the problem of allocating airport landing charges to different types of aircraft. Yet, the ‘fairness credentials’ of the Shapley value hinge on the axioms (see p. 48) which characterize it.

  55. 55.

    A game in characteristic function form exhibits decreasing returns to scale (McKelvey and Smith, 1974) if \(v(T)/\vert T\vert > v(S)/\vert S\vert \) for all \(T \subset S\), \(v(T),v(S) > 0\); for illustration see Example 1.1.2 with Table 1.3.

  56. 56.

    Let \({\mathcal{W}}_{-i}\) denote the set of winning coalitions to which player i does not belong. Then, it holds that \(\vert \mathcal{W}\vert = \vert {\mathcal{W}}_{i}\vert + \vert {\mathcal{W}}_{-i}\vert \) for any player i. Adding \(\vert {\mathcal{W}}_{i}\vert \) on both sides of this identity and rearranging yields \(\vert {\mathcal{W}}_{i}\vert -\vert {\mathcal{W}}_{-i}\vert = 2\vert {\mathcal{W}}_{i}\vert -\vert \mathcal{W}\vert \). The left hand side indicates the difference between the number of winning coalitions containing i, and the number not containing i, and is easily shown to be equivalent to the number of swings that player i has, i.e., to | {S ⊆ N : i is critical in S} | (for a proof see Dubey and Shapley, 1979, p. 102).

  57. 57.

    Under the consultation procedure, the Council can also amend the Commission’s proposal by unanimity. Moreover, the European Parliament needs to be consulted, but its opinion is non-binding. Generally, Article 155 grants the Commission the exclusive right to initiate legislation vis-à-vis the Council and the European Parliament, and according to Article 189a (2), it holds the right to modify a proposal at any point of procedure. Nevertheless, how much control the Commission can exercise over the proposal which ultimately comes to a vote in the Council probably varies with the EU’s different decision procedures (see Garrett and Tsebelis, 1996).

  58. 58.

    Indeed, a dummy player is always inferior, and in decisive games (cf. p. 23), the two concepts ‘inferior players’ and ‘dummy’ fully coincide (Napel and Widgrén, 2001, p. 213).

  59. 59.

    A stationary subgame perfect equilibrium is one in which (a) a proposer proposes the same distribution every time he is recognized, and (b) voting members vote only on the basis of the current proposal (and their expectations about future proposals which are time-invariant because of (a)), i.e., players’ actions are independent of the history of the game.

  60. 60.

    Holler and Schein (1979) provide a different model of ‘pie’ division by majority voting in which the shares of the ‘pie’ that players can secure for themselves are determined by their proposal rights and the sequence of their proposals.

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  1. School of Law, Business and Economics, University of Bayreuth, Universitätsstraße 30, 95447, Bayreuth, Germany

    Dr. Nicola F. Maaser

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Maaser, N.F. (2010). Games and Political Decisions. In: Decision-Making in Committees. Lecture Notes in Economics and Mathematical Systems, vol 635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04153-2_1

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