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Stable outcomes in spatial voting games
Mathematical Social Sciences 19 (1990) 269-279 North-Holland
STABLE
OUTCOMES
IN SPATIAL
269
VOTING
GAMES*
Guillermo OWEN Department of Mathematics, Naval Postgraduate School, Monterey, CA 93943, U.S.A. and School of Social Sciences, University of California, Irvine, CA 92717, U.S.A. Communicated by S. Barbera Received 24 September 1987 Revised 10 July 1989
The traditional representation of outcomes for an n-person game as vectors in n-space was superseded in the 1960s when it became clear that, for political and related applications, a lowerdimensional space was generally sufficient for the purpose. Thus, instead of vectors in utility space, vectors in policy space came to be used. Utility to the various decision-makers (usually voters or parliamentarians) could then be defined in terms of distance from given ideal points. A well-known result states that, when the underlying policy space is one-dimensional, a nonempty core (one or more undominated points) will exist. In two or more dimensions, it is known that, for decisivegames, where a bare majority is sufficient to carry motions, the core will be non-empty only under very special circumstances (degenerate cases). Stability is then sought in the form of near-core outcomes. Among the best known of the near-core outcomes are (a) the Copeland winner, defined as that outcome which defeats the greatest proportion of alternatives; (b) the yolk center, defined as that alternative which comes closest to satisfying all possible winning coalitions of voters; and (c) the minimal response (finagle) point, defined as the point which comes closest to defeating any other alternative. We show that each of these near-core outcomes has a well-defined analogue for games with side payments (classical von Neumann-Morgenstern games). In particular, the Copeland winner’s analogue is the Shapley value and the yolk center’s analogue is the nucleolus. The side-payment analogue of the finagle point has not been previously studied; a straightforward adaptation, known also as the finagle point, is however possible. Key words: Utility space; ideal point; policy space: spatial games.
1. Spatial games In their original development of n-person games, von Neumann and Morgenstern (1944) represented all outcomes as payoff vectors, i.e. points in n-dimensional utility space. This representation in utility space continued even after new models for nperson games, i.e. games without transferable utility (see Aumann, 1967) were developed. * Research supported by the National Science Foundation, 01654896/90/$3.50
0 1990-Elsevier
under Grant 8503676.
Science Publishers B.V. (North-Holland)
270
G. Owen
/ Spatial
wring
games
In the 196Os, however, a new representation of outcomes, in terms of policy In this model, outcomes of the games are policies, which are assumed to lie in some low-dimensional Euclidean space: utilities to the players are usually defined in terms of distance from their most preferred, or ideal, points. Such a representation is most useful when the outcomes correspond to public goods (so that several individuals might obtain comparable benefits from the same outcome) and so have been used mainly in connection with applications to political science. We refer the reader to Black (1958), Plott (1967), Owen (1971), Ordeshook (1971), and Riker and Ordeshook (1973) for some of the earlier developments; Enelow and Hinich (1984) give a good overview of the theory as it has developed. In what follows, a simple n-person game is a set N= { 1,2, . . . , n} of players together with a collection %Vof subsets of N, the winning coalitions. All other subsets of N are called losing coalitions. It is assumed that (i) NE % (ii) 06 GV,and (iii) if SE %, and SC T, then TE W. The game will be called proper if, for any S and TE W, Sfl T# 0. It will be called decisive if, for any S, either S or N-SE %V. We shall deal exclusively with proper, decisive simple games. Some of our results are valid for other simple games, but it should not be assumed that this is so in general. An m-dimensional spatial simple n-person game is a simple game, together with a collection of n points P’, i= 1, . . . , n, in m-dimensional Euclidean space R”‘. Point P’ will be said to be voter i’s ideal point. The convex hull of the points P’ is the Pareto-optimal set. The points of space R”’ are called positions or policies. Given two policies, x and y, we define a partition of N: space, was introduced.
Q(xY) = {i\d(x,P’)
= {i\d(xP’)>d(xP’)l,
I(x,y) = {i\d(x,P’)=d(y,P’)), where d represents the usual (Euclidean) distance between two points. The interpretation is that each voter prefers points which are closer to his ideal point; hence, Q(x, y) is the set of voters which prefer x to y; Z(x, y) is the set of voters indifferent between x and y. We say x dominates y if Q(x, y) E w. The core is the set of undominated policies.
2. Non-emptiness
of the core
It is not difficult to see (Black, 1958) that, when m = 1, so that the voters can, in fact, be ordered along a line, there will be a point in the core obtained as follows.
G. Owen / Spatial voting games
271
Fig. 1.
If the linear ordering is strict (no two ideal points coincide) there will be a voter such that the set of all voters to either side of him forms a losing coalition. For decisive proper games, he will be unique - the median or swing voter. If the linear ordering is not strict, so that several voters’ ideal points coincide, the swing voter need not be unique, but there will be a unique swing position. This position can easily be seen to lie in the core (see Fig. 1). For m = 2, and a fortiori for greater m, the core is usually empty, i.e. each policy is dominated by some other. We see this in Fig. 2, where points A, B, and C represent the ideal points for voters 1, 2, and 3, respectively. (In this three-voter game, we assume each coalition of two or more voters wins.) To see that the core is empty, we consider point x, which does not lie on line AB. It is easy to see that y, the foot of the perpendicular to AB through x, is closer to both A and B than is x. Thus, Q(y, x) = { 1,2}, and we conclude that y dominates x. But y does not lie on AC, and so y is dominated by z - the foot of the perpendicular to AC through y. In turn, z is dominated by w, etc. Since no point can lie on all three lines AB, AC, and BC simultaneously (unless AB, AC, and BC are concurrent) we conclude that the core is empty. In fact, it is possible for a two-dimensional array of voters to have a non-empty core. Fig. 3 illustrates this, for a five-voter game (in which all coalitions of three or more voters win). The point here is that one of the ideal points - Ps in this case is very centrally located at the intersection of the two line segments PIP3 and P2P4. The effect of this is that every line that passes through P’ is a median line: the set of ideal points on each side of the line represents a losing coalition. Conversely, all median lines pass through P’.
272
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Fig. 2.
[The above analysis suggests that core points exist only in degenerate cases, i.e. two or more coincident points, three or more collinear points, etc. In fact, that is the case when each voter has only one vote. In general, weighted voting games, however, if one voter has many votes, a core point may exist even under non-
Fig. 3.
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213
degeneracy. The obvious example occurs when one voter (a dictator) has more than half the votes. In such a case the dictator’s position is clearly a core point.] In general, in m-dimensional space we define a median hyperplane as a hyperplane such that all points to either side of it form a losing coalition. For a decisive game, this can only happen if at least one of the ideal points lies on the hyperplane. No two median hyperplanes can be parallel, but there is one median hyperplane in each direction in space. Now, the necessary and sufficient condition for a nonempty core is that all median hyperplanes pass through the same point. That point (the intersection of all median hyperplanes) is necessarily the ideal point of at least one voter. (This principle, in a slightly different form, was first enunciated by Plott, 1967.) In fact, this point is not only undominated; it actually dominates all other points, so that it is a very strong ‘solution’ to the game. (In game theory generally, the fact that a point is undominated does not mean that it dominates all other points.) To see why this is so, suppose that a point x does not lie on a medium hyperplane, H - as is the case with x and the hyperplane (line) AB in Fig. 1. In this case, the pointy, at the foot of the normal line (perpendicular) to H through x, must dominate x, for Q(y,x) contains at least all j whose ideal points lie on H, as well as those that lie on the opposite side of H from x. Since H is a median hyperplane, that set is a winning coalition. The consequence of this is that, if x does not lie on all median hyperplanes, it is dominated by at least one pointy. Conversely, if x does lie on all such hyperplanes, then it will dominate all other points. For, if y is any other point, the hyperplane through x normal to the line xy is a median hyperplane, and so x dominates y.
3. Near-core solution concepts As was discussed above, the core is very frequently empty for higher-dimensional games. This deprives us of one of the most appealing solution concepts. We look, then, for points which come ‘as close as possible’ to being in the core. Three approaches seem reasonable. 3.1. The Copeland winner The core consists of points which are undominated by any other. Absent such a point, we might look for a point which is dominated by ‘as few’ points as possible. This idea was originally advanced by Copeland (1951). In the context of Euclidean space, we take this ‘as few’ to mean a set with minimal m-dimensional Lebesgue measure (i.e. area or volume). Fig. 4 shows the same configuration as Fig. 2. Three circles are drawn, passing through x, and centered at the ideal points A, B, and C, respectively. Now, x is dominated by all points in the interior of the three cigar-shaped areas which are bounded
274
G. Owen / Spatial voting games
Fig. 4.
by these circles. It dominates all other points, except for a set of area 0, namely points on the three circular arcs. It can be shown (Grofman, Glazer et al., 1987) that the total shaded area is then: H(x) = (CAB~(A,X)~+
(ABC~(B,X)~+
@CA d(C,x)‘-2T,
where the angles are measured in radians, and T is the area of the triangle. This being so, it will follow without much difficulty that H(x) is minimized by choosing x to be the weighted arithmetic mean, with weights equal to the three angles, of points A, B, and C, i.e. x = ((CAB) A + ((ABC) B+ ((B&l) C 180” In this last expression, the angles are measured in degrees; if in radians, the denominator should be changed to R. The x obtained in this manner is the Copeland winner. For more than three voters, the Copeland winner can still be obtained as a weighted mean of the several ideal points, Pi, of the voters. Calculation of the weights is not, however, a trivial matter; the reader is referred to Grofman et al. (1987) and to Owen and Shapley (1989) for a discussion of this problem. 3.2. The yolk As discussed above, the core point, if it exists, must lie at the intersection of all the median hyperplanes. When there is no such point, we might look for the point
G. Owen / Spatial voting games
275
Fig. 5.
which is closest to all such planes, in the sense of minimizing its distance from them, i.e. a point x which gives r = min rnF d(x, K), X where d(x,K) is the (perpendicular) distance from x to hyperplane K, and the maximum is taken over all median hyperplanes. Such an x is the yolk center, and the ball of radius r, centered at x (the smallest ball which touches all median hyperplanes) is known as the yolk. This concept was first introduced by McKelvey (1986); its principal properties have been developed in Feld and Grofman (1985) and Feld et al. (1987). In Fig. 5 the same configuration as Fig. 2 is used once again. The yolk here is the circle inscribed in the triangle ABC; its center - the incenter of the triangle - can be found, as is well known from elementary Euclidean geometry, at the intersection of the three angle bisectors. For more complex configurations, locating the yolk is more complicated. We note that there is an infinite number of hyperplanes, one in each direction in space. However, only a finite number of hyperplanes need be considered in searching for x, namely those which bound the convex hull of the ideal points for some minimal winning coalition. Since distance from a plane is a linear function of position, it follows that the yolk can be found by linear programming. Readers are referred to McKelvey (1983) and Ferejohn et al. (1984) for details. 3.3. The minimal response point We mentioned
above that the core point (if it exists) will dominate
all other
G. Owen / Sparial voring games
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points. In its absence, we might look for the point which ‘comes closest’ to dominating all others. Let x be a policy, and suppose y (an alternative policy) dominates x (or at least, suppose x does not dominate y). If the core is empty, y can certainly be dominated, by a third alternative, say, called z. If there is such a z close to x, then we can say that x ‘comes close’ to dominating y. If there is no such z near x, then x is far from dominating y. We look, then, for the z closest to x (in the sense of Euclidean distance) which dominates y. In fact, there will usually be no closest z, since domination is (topologically) an open relationship. Nevertheless, an infimal distance will exist. Let, then s(x, y) = inf {d(x, z) \z dominates y]. We may think of x as the position chosen by an incumbent politician, whereas y is that chosen by a challenger seeking to unseat him (or her). Then s(x, y) is the ‘change of position’ necessary for the incumbent to repel this challenger. Of course, the challenger is also able to choose (alternatively, the incumbent’s opposition is free to choose a challenger), and so we might expect that, given x, y will be chosen so as to maximize this s(x,y). Thus, let
a(x) = max s(x, y) Y-
be the finagle radius of position x. The minimal response point is that x which minimizes the finagle radius Q(X), while the minimal value, Q, of a(x) is the minimal response distance. This point and distance, under the names finagle point and finagle radius, respectively, were introduced in Wuffle et al. (1987). Their properties have not yet been studied in detail. In Fig. 6 three mutually tangent circles have been drawn, centered at points A, B, and C, respectively. [The three points of tangency are, coincidentally, the three points of tangency of the inscribed circle (yolk) described above.] A small, fourth circle, J, has been drawn tangent to these three circles. The center, x, and radius, Q, of J, are the minimal response point and the minimal response distance, respectively. For other configurations, calculation of the minimal response point is considerably more complicated: it reduces to a non-linear programming problem.
4. Analogues of the near-core solutions for side-payment games In our discussion above, three near-core concepts for spatial games were introduced. It is interesting to see how these could be modified to suit other types of games. In particular, we shall see how they can be modified to side-payment games. In so doing, we shall find two of the better known solution concepts.
G. Owen / Spatial voting games
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Fig. 6.
4.1. The Shapley value The Shapfey value was introduced by Shapley (1953) as a method of obtaining an ‘expectation’ for game-type economic situations. Later, Shapley and Shubik (1954) showed that this same value could be used as a measure of power in abstract voting situations. As is generally known, the Shapley value, as an index of voting power, can be obtained by considering, for each possible permutation of the set N of voters, the pivot or median voter (as defined above). Voter i’s power is then the number of permutations in which i appears as voter, divided by n!, the number of all permutations. This can be restated by saying that i’s voting power is the probability that i will be a pivot in a randomly chosen permutation, given that all permutations are given equal probability 1/n ! . In spatial games, it does not seem reasonable to assign equal probabilities to all permutations of the players; rather, the spatial configuration might be used to change probabilities. Thus, Owen (1971) suggests ordering by distance from a randomly chosen point of the space; Shapley (1977) suggests ordering voters along a randomly chosen direction of space. Both of these indices reduce to the ordinary power index in case of symmetry, namely when the n ideal points are the vertices of a regular (n - 1) -simplex in n-space. More precisely, Shapley (1977) considers elements of the dual space, which might be called issue space. Each issue is a linear function f on policy space. Then, voters can be ranked according to the value f(Pi) of the functional f at voter i’s ideal point. The modlfed power index assigns to voter i the probability of being the
G. Owen
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pivotal voter for a randomly chosen issue (assuming a uniform distribution of issues from the unit sphere in issue space). Now, it turns out (see Grofman et al., 1987, and Owen and Shapley, 1989) that if each ideal point is assigned weight w;, where Wi is voter i’s modified index, then, in two-dimensional space, the Copeland winner is given by: x = w,P’ + w,P2+ a**+ w,P”. We conclude, then, that the side-payments the Shapley value.
analogue of the Copeland winner is
4.2. The nucleolus The nucleolus, which was first introduced by Schmeidler (1969), is based on the concept of excess. For a game with characteristic function u, the excess of coalition S at imputation (payoff vector) x is defined to be u(S) -x(S), where x(S) is the sum of payoffs to members of S. The nucleolus is then that imputation which minimizes the maximum excess to all coalitions of players. If there are two or more such imputations, that which minimizes the second largest excess is then chosen, and so on. It is easy to see that, if a game has a non-empty core, then the nucleolus will always belong to the core. In a simple game, the excess is 1 -x(S) for winning coalitions S, and thus measures ‘how far’ imputation x is from satisfying coalition S (which presumably would be satisfied if it could get the full amount 1). In this sense, then, the nucleolus is the point which comes closest to satisfying the least satisfied winning coalition. (This least satisfied coalition is necessarily a minimal winning coalition.) In spatial games, the yolk center is the point which minimizes the maximum distance from all median hyperplanes. However, as mentioned above, it suffices to minimize the maximum distance from those median hyperplanes which are bounding planes for the convex hulls of minimal winning coalitions. Now, if a point x lies in the convex hull of the ideal points of a winning coalition S, then S is ‘satisfied’ with x in the sense that there is no point which all members of S prefer to x. Thus, the yolk center can also be considered the point which comes closest to satisfying the least satisfied winning coalition. We conclude from this that the side-payment analogue of the yolk center is the nucleolus. 4.3. The finagle point No side-payments analogue to the minimal response point appears in the literature - at least none that is known to this author. It is not, however, difficult to see that the definition of minimal response point can be directly transposed to sidepayments games; all that is needed is a specification of the metric d(x,z) which appears in the definition. In Owen (1987) this concept is developed, using the L, met-
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ric in Euclidean n-space. This finagle point, which (for simple games) is relatively easy to calculate, has not as yet been studied in any detail; we hope that some interesting properties may be found for it.
References R.J. Aumann, A survey of games without side payments, in: M. Shubik, ed., Essays in ,Mathematical Economics (Princeton University Press, 1967), pp. 3-27. D. Black, The Theory of Committees and Elections (Cambridge University Press, 1958). A.H. Copeland, A reasonable social welfare function, Unpublished manuscript, 1951. J.E. Enelow and M.J. Hinich, The Spatial Theory of Voting (Cambridge University Press, 1984). S.L. Feld and B.N. Grofman, Necessary and sufficient conditions for a majority winner in n-dimensional spatial voting games, Amer. J. Polit. Sci. (1987). S.L. Feld, B.N. Grofman and N. Miller, Centripetal forces in spatial voting: on the size of the yolk, Unpublished manuscript, 1985. J.A. Ferejohn, M. Fiorina and E.W. Packel, Non-equilibrium solutions for legislative systems, Behav. Sci. (1980) 140-148. J.A. Ferejohn. R.D. McKelvey and E.W. Packel, Limiting distributions for continuous state .Markov models, Social Choice and Welfare (1984) 45-67. B.N. Grofman, A. Glazer, N. Noviello and G. Owen, Stability and centrality of legislative choice in the spatial context, Amer. Polit. Sci. Rev. (1987) 539-553. G.H. Kramer, Sophisticated voting over multi-dimensional choice spaces, J. Math. Sociology (1973) 165-180. R.D. McKelvey, Covering, dominance, and institution-free properties of social choice, Amer. J. Polit. Sci. (1986) 283-314. P.C. Ordeshook, Pareto-optimality in electoral competition, Amer. Polit. Sci. Rev. (1971) 1141-l 145. G. Owen, Political games, Naval Res. Logistics Quart. (1971) 345-354. G. Owen, The finagle point for characteristic function games, Unpublished manuscript, 1987. G. Owen and L.S. Shapley, Optimal location of candidates in ideological space, Internat. J. Game Theory (1989). C.R. Plott, A notion of equilibrium and its possibility under majority rule, Amer. Econ. Rev. (1967) 787-806. W. Riker and P.C. Ordeshook, An Introduction to Positive Political Theory (Prentice-Hall, Englewood Cliffs, 1973). D. Schmeidler, The nucleolus of an n-person game, SIAM J. Appl. Math. (1969) 1163-1170. L.S. Shapley, A value for n-person games, in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games Ii, Annals of Mathematics Study 24 (Princeton University Press, 1953). pp. 307-317. L.S. Shapley, A comparison of power indices and a non-symmetric generalization, P-5872, The RAND Corporation, Santa Monica, CA, 1977. L.S. Shapley and M. Shubik, A method for evaluating the distribution of power in a committee system, Amer. Polit. Sci. Rev. (1954) 787-792. J. von Neumann and 0. Morgenstern, The Theory of Games and Economic Behavior (Princeton University Press, 1944). A. Wuffle, S. Feld, G. Owen and B. Grofman, Finagle’s law and the finagle point, a new solution concept for two-candidate competition in spatial games voting without a core, Amer. J. Polit. Sci. (1987).
STABLE
OUTCOMES
IN SPATIAL
269
VOTING
GAMES*
Guillermo OWEN Department of Mathematics, Naval Postgraduate School, Monterey, CA 93943, U.S.A. and School of Social Sciences, University of California, Irvine, CA 92717, U.S.A. Communicated by S. Barbera Received 24 September 1987 Revised 10 July 1989
The traditional representation of outcomes for an n-person game as vectors in n-space was superseded in the 1960s when it became clear that, for political and related applications, a lowerdimensional space was generally sufficient for the purpose. Thus, instead of vectors in utility space, vectors in policy space came to be used. Utility to the various decision-makers (usually voters or parliamentarians) could then be defined in terms of distance from given ideal points. A well-known result states that, when the underlying policy space is one-dimensional, a nonempty core (one or more undominated points) will exist. In two or more dimensions, it is known that, for decisivegames, where a bare majority is sufficient to carry motions, the core will be non-empty only under very special circumstances (degenerate cases). Stability is then sought in the form of near-core outcomes. Among the best known of the near-core outcomes are (a) the Copeland winner, defined as that outcome which defeats the greatest proportion of alternatives; (b) the yolk center, defined as that alternative which comes closest to satisfying all possible winning coalitions of voters; and (c) the minimal response (finagle) point, defined as the point which comes closest to defeating any other alternative. We show that each of these near-core outcomes has a well-defined analogue for games with side payments (classical von Neumann-Morgenstern games). In particular, the Copeland winner’s analogue is the Shapley value and the yolk center’s analogue is the nucleolus. The side-payment analogue of the finagle point has not been previously studied; a straightforward adaptation, known also as the finagle point, is however possible. Key words: Utility space; ideal point; policy space: spatial games.
1. Spatial games In their original development of n-person games, von Neumann and Morgenstern (1944) represented all outcomes as payoff vectors, i.e. points in n-dimensional utility space. This representation in utility space continued even after new models for nperson games, i.e. games without transferable utility (see Aumann, 1967) were developed. * Research supported by the National Science Foundation, 01654896/90/$3.50
0 1990-Elsevier
under Grant 8503676.
Science Publishers B.V. (North-Holland)
270
G. Owen
/ Spatial
wring
games
In the 196Os, however, a new representation of outcomes, in terms of policy In this model, outcomes of the games are policies, which are assumed to lie in some low-dimensional Euclidean space: utilities to the players are usually defined in terms of distance from their most preferred, or ideal, points. Such a representation is most useful when the outcomes correspond to public goods (so that several individuals might obtain comparable benefits from the same outcome) and so have been used mainly in connection with applications to political science. We refer the reader to Black (1958), Plott (1967), Owen (1971), Ordeshook (1971), and Riker and Ordeshook (1973) for some of the earlier developments; Enelow and Hinich (1984) give a good overview of the theory as it has developed. In what follows, a simple n-person game is a set N= { 1,2, . . . , n} of players together with a collection %Vof subsets of N, the winning coalitions. All other subsets of N are called losing coalitions. It is assumed that (i) NE % (ii) 06 GV,and (iii) if SE %, and SC T, then TE W. The game will be called proper if, for any S and TE W, Sfl T# 0. It will be called decisive if, for any S, either S or N-SE %V. We shall deal exclusively with proper, decisive simple games. Some of our results are valid for other simple games, but it should not be assumed that this is so in general. An m-dimensional spatial simple n-person game is a simple game, together with a collection of n points P’, i= 1, . . . , n, in m-dimensional Euclidean space R”‘. Point P’ will be said to be voter i’s ideal point. The convex hull of the points P’ is the Pareto-optimal set. The points of space R”’ are called positions or policies. Given two policies, x and y, we define a partition of N: space, was introduced.
Q(xY) = {i\d(x,P’)
= {i\d(xP’)>d(xP’)l,
I(x,y) = {i\d(x,P’)=d(y,P’)), where d represents the usual (Euclidean) distance between two points. The interpretation is that each voter prefers points which are closer to his ideal point; hence, Q(x, y) is the set of voters which prefer x to y; Z(x, y) is the set of voters indifferent between x and y. We say x dominates y if Q(x, y) E w. The core is the set of undominated policies.
2. Non-emptiness
of the core
It is not difficult to see (Black, 1958) that, when m = 1, so that the voters can, in fact, be ordered along a line, there will be a point in the core obtained as follows.
G. Owen / Spatial voting games
271
Fig. 1.
If the linear ordering is strict (no two ideal points coincide) there will be a voter such that the set of all voters to either side of him forms a losing coalition. For decisive proper games, he will be unique - the median or swing voter. If the linear ordering is not strict, so that several voters’ ideal points coincide, the swing voter need not be unique, but there will be a unique swing position. This position can easily be seen to lie in the core (see Fig. 1). For m = 2, and a fortiori for greater m, the core is usually empty, i.e. each policy is dominated by some other. We see this in Fig. 2, where points A, B, and C represent the ideal points for voters 1, 2, and 3, respectively. (In this three-voter game, we assume each coalition of two or more voters wins.) To see that the core is empty, we consider point x, which does not lie on line AB. It is easy to see that y, the foot of the perpendicular to AB through x, is closer to both A and B than is x. Thus, Q(y, x) = { 1,2}, and we conclude that y dominates x. But y does not lie on AC, and so y is dominated by z - the foot of the perpendicular to AC through y. In turn, z is dominated by w, etc. Since no point can lie on all three lines AB, AC, and BC simultaneously (unless AB, AC, and BC are concurrent) we conclude that the core is empty. In fact, it is possible for a two-dimensional array of voters to have a non-empty core. Fig. 3 illustrates this, for a five-voter game (in which all coalitions of three or more voters win). The point here is that one of the ideal points - Ps in this case is very centrally located at the intersection of the two line segments PIP3 and P2P4. The effect of this is that every line that passes through P’ is a median line: the set of ideal points on each side of the line represents a losing coalition. Conversely, all median lines pass through P’.
272
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Fig. 2.
[The above analysis suggests that core points exist only in degenerate cases, i.e. two or more coincident points, three or more collinear points, etc. In fact, that is the case when each voter has only one vote. In general, weighted voting games, however, if one voter has many votes, a core point may exist even under non-
Fig. 3.
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degeneracy. The obvious example occurs when one voter (a dictator) has more than half the votes. In such a case the dictator’s position is clearly a core point.] In general, in m-dimensional space we define a median hyperplane as a hyperplane such that all points to either side of it form a losing coalition. For a decisive game, this can only happen if at least one of the ideal points lies on the hyperplane. No two median hyperplanes can be parallel, but there is one median hyperplane in each direction in space. Now, the necessary and sufficient condition for a nonempty core is that all median hyperplanes pass through the same point. That point (the intersection of all median hyperplanes) is necessarily the ideal point of at least one voter. (This principle, in a slightly different form, was first enunciated by Plott, 1967.) In fact, this point is not only undominated; it actually dominates all other points, so that it is a very strong ‘solution’ to the game. (In game theory generally, the fact that a point is undominated does not mean that it dominates all other points.) To see why this is so, suppose that a point x does not lie on a medium hyperplane, H - as is the case with x and the hyperplane (line) AB in Fig. 1. In this case, the pointy, at the foot of the normal line (perpendicular) to H through x, must dominate x, for Q(y,x) contains at least all j whose ideal points lie on H, as well as those that lie on the opposite side of H from x. Since H is a median hyperplane, that set is a winning coalition. The consequence of this is that, if x does not lie on all median hyperplanes, it is dominated by at least one pointy. Conversely, if x does lie on all such hyperplanes, then it will dominate all other points. For, if y is any other point, the hyperplane through x normal to the line xy is a median hyperplane, and so x dominates y.
3. Near-core solution concepts As was discussed above, the core is very frequently empty for higher-dimensional games. This deprives us of one of the most appealing solution concepts. We look, then, for points which come ‘as close as possible’ to being in the core. Three approaches seem reasonable. 3.1. The Copeland winner The core consists of points which are undominated by any other. Absent such a point, we might look for a point which is dominated by ‘as few’ points as possible. This idea was originally advanced by Copeland (1951). In the context of Euclidean space, we take this ‘as few’ to mean a set with minimal m-dimensional Lebesgue measure (i.e. area or volume). Fig. 4 shows the same configuration as Fig. 2. Three circles are drawn, passing through x, and centered at the ideal points A, B, and C, respectively. Now, x is dominated by all points in the interior of the three cigar-shaped areas which are bounded
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Fig. 4.
by these circles. It dominates all other points, except for a set of area 0, namely points on the three circular arcs. It can be shown (Grofman, Glazer et al., 1987) that the total shaded area is then: H(x) = (CAB~(A,X)~+
(ABC~(B,X)~+
@CA d(C,x)‘-2T,
where the angles are measured in radians, and T is the area of the triangle. This being so, it will follow without much difficulty that H(x) is minimized by choosing x to be the weighted arithmetic mean, with weights equal to the three angles, of points A, B, and C, i.e. x = ((CAB) A + ((ABC) B+ ((B&l) C 180” In this last expression, the angles are measured in degrees; if in radians, the denominator should be changed to R. The x obtained in this manner is the Copeland winner. For more than three voters, the Copeland winner can still be obtained as a weighted mean of the several ideal points, Pi, of the voters. Calculation of the weights is not, however, a trivial matter; the reader is referred to Grofman et al. (1987) and to Owen and Shapley (1989) for a discussion of this problem. 3.2. The yolk As discussed above, the core point, if it exists, must lie at the intersection of all the median hyperplanes. When there is no such point, we might look for the point
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Fig. 5.
which is closest to all such planes, in the sense of minimizing its distance from them, i.e. a point x which gives r = min rnF d(x, K), X where d(x,K) is the (perpendicular) distance from x to hyperplane K, and the maximum is taken over all median hyperplanes. Such an x is the yolk center, and the ball of radius r, centered at x (the smallest ball which touches all median hyperplanes) is known as the yolk. This concept was first introduced by McKelvey (1986); its principal properties have been developed in Feld and Grofman (1985) and Feld et al. (1987). In Fig. 5 the same configuration as Fig. 2 is used once again. The yolk here is the circle inscribed in the triangle ABC; its center - the incenter of the triangle - can be found, as is well known from elementary Euclidean geometry, at the intersection of the three angle bisectors. For more complex configurations, locating the yolk is more complicated. We note that there is an infinite number of hyperplanes, one in each direction in space. However, only a finite number of hyperplanes need be considered in searching for x, namely those which bound the convex hull of the ideal points for some minimal winning coalition. Since distance from a plane is a linear function of position, it follows that the yolk can be found by linear programming. Readers are referred to McKelvey (1983) and Ferejohn et al. (1984) for details. 3.3. The minimal response point We mentioned
above that the core point (if it exists) will dominate
all other
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points. In its absence, we might look for the point which ‘comes closest’ to dominating all others. Let x be a policy, and suppose y (an alternative policy) dominates x (or at least, suppose x does not dominate y). If the core is empty, y can certainly be dominated, by a third alternative, say, called z. If there is such a z close to x, then we can say that x ‘comes close’ to dominating y. If there is no such z near x, then x is far from dominating y. We look, then, for the z closest to x (in the sense of Euclidean distance) which dominates y. In fact, there will usually be no closest z, since domination is (topologically) an open relationship. Nevertheless, an infimal distance will exist. Let, then s(x, y) = inf {d(x, z) \z dominates y]. We may think of x as the position chosen by an incumbent politician, whereas y is that chosen by a challenger seeking to unseat him (or her). Then s(x, y) is the ‘change of position’ necessary for the incumbent to repel this challenger. Of course, the challenger is also able to choose (alternatively, the incumbent’s opposition is free to choose a challenger), and so we might expect that, given x, y will be chosen so as to maximize this s(x,y). Thus, let
a(x) = max s(x, y) Y-
be the finagle radius of position x. The minimal response point is that x which minimizes the finagle radius Q(X), while the minimal value, Q, of a(x) is the minimal response distance. This point and distance, under the names finagle point and finagle radius, respectively, were introduced in Wuffle et al. (1987). Their properties have not yet been studied in detail. In Fig. 6 three mutually tangent circles have been drawn, centered at points A, B, and C, respectively. [The three points of tangency are, coincidentally, the three points of tangency of the inscribed circle (yolk) described above.] A small, fourth circle, J, has been drawn tangent to these three circles. The center, x, and radius, Q, of J, are the minimal response point and the minimal response distance, respectively. For other configurations, calculation of the minimal response point is considerably more complicated: it reduces to a non-linear programming problem.
4. Analogues of the near-core solutions for side-payment games In our discussion above, three near-core concepts for spatial games were introduced. It is interesting to see how these could be modified to suit other types of games. In particular, we shall see how they can be modified to side-payment games. In so doing, we shall find two of the better known solution concepts.
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Fig. 6.
4.1. The Shapley value The Shapfey value was introduced by Shapley (1953) as a method of obtaining an ‘expectation’ for game-type economic situations. Later, Shapley and Shubik (1954) showed that this same value could be used as a measure of power in abstract voting situations. As is generally known, the Shapley value, as an index of voting power, can be obtained by considering, for each possible permutation of the set N of voters, the pivot or median voter (as defined above). Voter i’s power is then the number of permutations in which i appears as voter, divided by n!, the number of all permutations. This can be restated by saying that i’s voting power is the probability that i will be a pivot in a randomly chosen permutation, given that all permutations are given equal probability 1/n ! . In spatial games, it does not seem reasonable to assign equal probabilities to all permutations of the players; rather, the spatial configuration might be used to change probabilities. Thus, Owen (1971) suggests ordering by distance from a randomly chosen point of the space; Shapley (1977) suggests ordering voters along a randomly chosen direction of space. Both of these indices reduce to the ordinary power index in case of symmetry, namely when the n ideal points are the vertices of a regular (n - 1) -simplex in n-space. More precisely, Shapley (1977) considers elements of the dual space, which might be called issue space. Each issue is a linear function f on policy space. Then, voters can be ranked according to the value f(Pi) of the functional f at voter i’s ideal point. The modlfed power index assigns to voter i the probability of being the
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pivotal voter for a randomly chosen issue (assuming a uniform distribution of issues from the unit sphere in issue space). Now, it turns out (see Grofman et al., 1987, and Owen and Shapley, 1989) that if each ideal point is assigned weight w;, where Wi is voter i’s modified index, then, in two-dimensional space, the Copeland winner is given by: x = w,P’ + w,P2+ a**+ w,P”. We conclude, then, that the side-payments the Shapley value.
analogue of the Copeland winner is
4.2. The nucleolus The nucleolus, which was first introduced by Schmeidler (1969), is based on the concept of excess. For a game with characteristic function u, the excess of coalition S at imputation (payoff vector) x is defined to be u(S) -x(S), where x(S) is the sum of payoffs to members of S. The nucleolus is then that imputation which minimizes the maximum excess to all coalitions of players. If there are two or more such imputations, that which minimizes the second largest excess is then chosen, and so on. It is easy to see that, if a game has a non-empty core, then the nucleolus will always belong to the core. In a simple game, the excess is 1 -x(S) for winning coalitions S, and thus measures ‘how far’ imputation x is from satisfying coalition S (which presumably would be satisfied if it could get the full amount 1). In this sense, then, the nucleolus is the point which comes closest to satisfying the least satisfied winning coalition. (This least satisfied coalition is necessarily a minimal winning coalition.) In spatial games, the yolk center is the point which minimizes the maximum distance from all median hyperplanes. However, as mentioned above, it suffices to minimize the maximum distance from those median hyperplanes which are bounding planes for the convex hulls of minimal winning coalitions. Now, if a point x lies in the convex hull of the ideal points of a winning coalition S, then S is ‘satisfied’ with x in the sense that there is no point which all members of S prefer to x. Thus, the yolk center can also be considered the point which comes closest to satisfying the least satisfied winning coalition. We conclude from this that the side-payment analogue of the yolk center is the nucleolus. 4.3. The finagle point No side-payments analogue to the minimal response point appears in the literature - at least none that is known to this author. It is not, however, difficult to see that the definition of minimal response point can be directly transposed to sidepayments games; all that is needed is a specification of the metric d(x,z) which appears in the definition. In Owen (1987) this concept is developed, using the L, met-
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ric in Euclidean n-space. This finagle point, which (for simple games) is relatively easy to calculate, has not as yet been studied in any detail; we hope that some interesting properties may be found for it.
References R.J. Aumann, A survey of games without side payments, in: M. Shubik, ed., Essays in ,Mathematical Economics (Princeton University Press, 1967), pp. 3-27. D. Black, The Theory of Committees and Elections (Cambridge University Press, 1958). A.H. Copeland, A reasonable social welfare function, Unpublished manuscript, 1951. J.E. Enelow and M.J. Hinich, The Spatial Theory of Voting (Cambridge University Press, 1984). S.L. Feld and B.N. Grofman, Necessary and sufficient conditions for a majority winner in n-dimensional spatial voting games, Amer. J. Polit. Sci. (1987). S.L. Feld, B.N. Grofman and N. Miller, Centripetal forces in spatial voting: on the size of the yolk, Unpublished manuscript, 1985. J.A. Ferejohn, M. Fiorina and E.W. Packel, Non-equilibrium solutions for legislative systems, Behav. Sci. (1980) 140-148. J.A. Ferejohn. R.D. McKelvey and E.W. Packel, Limiting distributions for continuous state .Markov models, Social Choice and Welfare (1984) 45-67. B.N. Grofman, A. Glazer, N. Noviello and G. Owen, Stability and centrality of legislative choice in the spatial context, Amer. Polit. Sci. Rev. (1987) 539-553. G.H. Kramer, Sophisticated voting over multi-dimensional choice spaces, J. Math. Sociology (1973) 165-180. R.D. McKelvey, Covering, dominance, and institution-free properties of social choice, Amer. J. Polit. Sci. (1986) 283-314. P.C. Ordeshook, Pareto-optimality in electoral competition, Amer. Polit. Sci. Rev. (1971) 1141-l 145. G. Owen, Political games, Naval Res. Logistics Quart. (1971) 345-354. G. Owen, The finagle point for characteristic function games, Unpublished manuscript, 1987. G. Owen and L.S. Shapley, Optimal location of candidates in ideological space, Internat. J. Game Theory (1989). C.R. Plott, A notion of equilibrium and its possibility under majority rule, Amer. Econ. Rev. (1967) 787-806. W. Riker and P.C. Ordeshook, An Introduction to Positive Political Theory (Prentice-Hall, Englewood Cliffs, 1973). D. Schmeidler, The nucleolus of an n-person game, SIAM J. Appl. Math. (1969) 1163-1170. L.S. Shapley, A value for n-person games, in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games Ii, Annals of Mathematics Study 24 (Princeton University Press, 1953). pp. 307-317. L.S. Shapley, A comparison of power indices and a non-symmetric generalization, P-5872, The RAND Corporation, Santa Monica, CA, 1977. L.S. Shapley and M. Shubik, A method for evaluating the distribution of power in a committee system, Amer. Polit. Sci. Rev. (1954) 787-792. J. von Neumann and 0. Morgenstern, The Theory of Games and Economic Behavior (Princeton University Press, 1944). A. Wuffle, S. Feld, G. Owen and B. Grofman, Finagle’s law and the finagle point, a new solution concept for two-candidate competition in spatial games voting without a core, Amer. J. Polit. Sci. (1987).