- Email: [email protected]
The dynamics of locally adaptive parties under spatial voting
Journal of Economic Dynamics and Control 23 (1998) 171—189
The dynamics of locally adaptive parties under spatial voting J.H. Miller!,",*, P.F. Stadler",# ! Department of Social and Decision Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA " The Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, NM 87501, USA # Institut fu( r Theoretische Chemie, Universita( t Wien, Wa( hringerstr. 17, A-1090 Vienna, Austria Received 20 August 1996; accepted 13 January 1998
Abstract We explore the dynamics of a model of two-party competition under spatial voting. The parties are allowed to incrementally adapt their platforms by following the voting gradient imposed by the preferences of the electorate and the platform of the opposition. The emphasis in this model is on the dynamic system formed by these conditions, in particular, we examine the characteristics of the transient paths and the convergence points of the evolving platforms. We find that in a simple spatial model with probabilistic voting, regardless of the initial platforms of each party, platforms eventually converge to a unique, globally stable equilibrium matching the strength-weighted mean of the voters’ preferred positions. This result appears robust to many variations in voter preferences and party behavior. However, we do find some conditions under which other dynamic possibilities occur, including multiple equilibria and, perhaps, limit cycles. ( 1998 Elsevier Science B.V. All rights reserved. JEL classification: C61; C62; C63; D72 Keywords: Spatial voting; Local adaptation; Search; Dynamics
1. Introduction The importance of adaptive processes in economic theory has long been recognized (see, for example, Malthus, 1798). In this paper, we employ the tools
* Correspondence address. Department of Social and Decision Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. E-mail: [email protected] 0165-1889/98/$ — see front matter ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 0 4 - 9
172
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
of evolutionary game dynamics to better understand the dynamic behavior of an adaptive system in which agents are constrained to locally adapt to their world. We assume local adaptation for a variety of reasons. One justification is that agents are often constrained by external forces to only slowly alter their choices. For example, automobile manufactures may face extremely high retooling costs for extensive model changes, or political parties that rapidly shift their platform positions may be perceived as less trustworthy by voters than those that change only incrementally. Another factor favoring local adaptation is that agents are likely to have more, and better quality, information about choices that are ‘close’ to their current actions. Thus, an automobile manufacturer can more accurately estimate the consumer demand resulting from, say, offering a currently produced car in a new color, than from offering an entirely new type of car. We analyze incremental adaptation of complex actions in a model of two-party spatial voting, in which the parties compete for votes by making minor alterations to their respective platforms. A related application is that of firms competing for consumers by offering goods that embody multiple attributes (for example, a car). These types of scenarios represent an important class of phenomena that has been widely analyzed using more traditional tools, and thus they provide a nice benchmark from which to explore the impact of incremental adaptive dynamics. We examine a spatial voting model where candidates in a two-party system incrementally adjust their platforms to appeal to voters. In our model, a candidate is only allowed to make local adjustments to the previously endorsed platform. As mentioned above, such local adaptation could be justified on a variety of grounds, for example, incremental changes allow parties to maintain legitimacy and a coherent organization within the eyes of the electorate, special interests may demand only small changes, or polling information about closely related platforms may be relatively more available and accurate. We assume that parties follow the local voting gradient imposed on the system by the preferences of the voters and the position of the other candidate. This obviously implies that parties can accurately determine the optimal local response in any given situation. While models that degrade even this local optimization ability are certainly of interest, the case analyzed here represents an important first step in realizing such models (and, a significant departure from most current models that assume full knowledge of the entire space). We find that in a simple spatial model with probabilistic voting and local adaptation, regardless of the initial platforms of each party, platforms eventually converge to a unique, globally stable equilibrium matching the strengthweighted mean of the voters’ preferred positions. This result appears robust to many variations of our underlying assumptions about voter preferences and party behavior. However, if we allow less constrained voter utility functions, many dynamic possibilities occur, including multiple equilibria and, we suspect, limit cycles.
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 173
Our analysis focuses on an adaptive dynamic system, whereby global consequences emerge from locally adapting candidates. The inspiration for this work comes from the computational results discovered by Kollman et al. (1992). They found that parties following simple, locally adaptive rules rapidly converged toward common platforms of high aggregate utility. Moreover, they noted that such models appear to explain existing empirical regularities better than more traditional modeling approaches. While the ideas of probabilistic voting (for a general review see Coughlin, 1990) and locally restricted strategy searches in such models (for example, Coughlin and Nitzan, 1981; Samuelson, 1984) have been widely discussed, here we assume that parties do not start at an identical status quo platform, and that their ability to maximize voter support is limited to climbing the local voting gradient. Such a model serves as a necessary benchmark for spatial voting results when candidates’ abilities to optimize their platform positions are limited and where the dynamic behavior of platform modification is of interest. The emphasis in our analysis is on the dynamic system formed by the model, in particular, we examine the characteristics of the transient paths and the convergence points of the evolving platforms. Note that one important feature of the model is that incremental changes by each party may dramatically impact the payoff space facing the other party. Thus, the adapting parties are intimately coupled with one another: small movements by one party may alter the desired movements of the other party. The outcome of such a dance is the focus of our analysis.
2. The basic model Consider a model with I issues and » voters. We will use the following notation: platform position of party j on issue i yj i x voter v’s preferred position on issue i vi s voter v’s strength on issue i. We require for all i that s '0 for at least vi vi one v. (That is, at least one voter cares about each issue.) u (yj) voter v’s utility of party j’s platform: v
P(z)
I (1) u (yj)"! + s (yj!x )2. vi v vi i i/1 (This functional form will be generalized in Section 4.) probability of voting for a platform that has a difference z in utility against the opposing platform. If the two parties are distinguished only by their platforms, then P(z) will be symmetric about 0.5:
174
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
Prob[v votes for 1]"P(u (y1)!u (y2)), v v Prob[v votes for 2]"P(u (y2)!u (y1))"1!P(u (y1)!u (y2)). v v v v In the extreme case we have
G
0,
P(z)" 0.5, 1,
z(0, z"0,
(2)
z'0,
where a voter always votes for the party that offers the larger utility, and if both parties provide the same utility the voter is indifferent and flips a coin. To add both realism and mathematical tractability, we impose the following additional requirements on P(z). First, P(z) is assumed to be strictly monotonic, implying that the probability of choosing a party always increases as the difference in utility increases and that there is always a positive probability of voting for either party for any difference in utilities, z. Second, we require P(z) to be continuously differentiable. A simple class of functions that fit our requirements is the symmetric sigmoidal functions (such as the error integral). If P(z) is only monotonic, then when P@(z)"0 it is possible to have large areas in issue-space forming equilibria where incremental changes in the platforms do not cause any voters to change their expected votes. With strict monotonicity, platforms will only stabilize if there is an attractor present. Continuity is necessary for the results. In the extreme case of P(z) given in Eq. (2), the results derived below no longer hold due to the discontinuity implied by this assumption. In this latter case, Plott (1967) showed that in general parties can always locate a better platform, and hence no equilibrium configuration of the parties is possible. Given P(z), the expected number of votes for each party, E , is j V E (y1, y2)" + P(u (y1)!u (y2)), 1 v v v/1 (3) V E (y1, y2)" + P(u (y2)!u (y1))"»!E (y1, y2). 2 v v 1 v/1 We assume that a party’s utility, º , is given by the number of votes that it is j ahead of its rival: º (y1, y2)"E (y1, y2)!E (y1, y2)"2E (y1, y2)!», 1 1 2 1 (4) º (y1, y2)"E (y1, y2)!E (y1, y2)"»!2E (y1, y2)"!º (y1, y2). 2 2 1 1 1 2.1. Dynamics For the dynamics of platform adjustment we follow an approach used in the analysis of biological games (for an introduction to this area, see Hofbauer and Sigmund (1988), Friedman (1991) reviews economic applications of these
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 175
ideas). Hofbauer and Sigmund (1990), in a related application to continuous strategy spaces called ‘adaptive dynamics’, explore strategic stability around fixed points (unfortunately, our dynamics are not characterized by their results). The dynamics we explore here have not been previously analyzed. In our model both parties update their platforms according to the following dynamics: yR 1"+ 1º (y1, y2), y 1
yR 2"+ 2º (y1, y2). y 2
(5)
That is, each party alters its platform so as to locally improve its share of votes. The optimization is local rather than global since each party simply follows the gradient of its utility function. Since P(z) is continuously differentiable the vector field (+ 1º ,+ 2º ) is continuous everywhere, and therefore the existence and y 1 y 2 uniqueness of the trajectory for all initial conditions is guaranteed. (Note that the monotonicity of P is not necessary for existence and uniqueness of the trajectories.) More generally, we might assume yR k"a+ kº for k"1, 2 in y k Eq. (5). However, it is always possible to choose the time unit such that a"1. Therefore, the value of a does not influence the long-time dynamic behavior of the system. Although Eq. (5) seems to imply that parties are simply hill-climbing up a fixed ‘vote landscape’ imposed by the preferences of the electorate, in fact the ‘hill’ perceived by each party also depends on the platform position of the other party (as each voter makes a relative comparison between the two parties). Since both parties are moving on the same time-scale, their respective payoff landscapes are changing at the same speed. Thus, both parties confront a landscape that undulates with each move. We can rewrite the dynamics in a more explicit way. First, define p(z)"2P(z)!1. As a consequence of the differentiability, monotonicity and symmetry of P(z), p has the following properties: p(z)"!p(!z), p(0)"0, p@(z)'0, and p@(z)"p@(!z). The utility of the parties can be rewritten as º (y1, y2) " 2E (y1, y2)!» 1 1 " + [2P(u (y1)!u (y2))!1] v v v " + p(u (y1)!u (y2)), v v v º (y1, y2) " 2E (y1, y2)!» 2 2 " + [2P(u (y2)!u (y1))!1] v v v " + p(u (y2)!u (y1)). v v v
(6)
176
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
The dynamic equations now yield yR 1 " + 1º y 1 " + + 1p(u (y1)!u (y2)) v v y v " + p@(u (y1)!u (y2))+ 1u (y1), y v v v v yR 2 " + 2º y 2
(7)
" + + 2p(u (y2)!u (y1)) v v y v " + p@(u (y2)!u (y1))+ 2u (y2). y v v v v Using the voter’s utility function given by Eq. (1) we find L I (y !x )2"!2s (y !x ) [+ u (y)] "! + s vi vl l vl y v l vi Ly i l i/1
(8)
for the partial derivative in the direction of issue l. Combining Eqs. (1), (7) and (8) yields the system of differential equations
A A
B B
yR 1"!2+ p@ + s [(y2!x )2!(y1!x )2] s (y1!x ), vi i vi i vi vl l vl l i v
(9)
yR 2"!2+ p@ + s [(y1!x )2!(y2!x )2] s (y2!x ). i vi l vi vl vi i vl l v i The p@( ) ) terms in these equations weight the importance of a given voter for the changes that a party implements in its platform. If P(z) is sigmoidal, then these terms imply that the most important voters are the pivotal ones that currently receive similar utility from the two platforms. The multiplicative factor, s (yj!x ), implies that changes of position on issue l are influenced most by vl vl l those voters who both care a lot about issue l and who have preferred positions further away from the current platform position.
3. Analysis In this section we formally derive the properties of the dynamical system implied by the basic model discussed above. All results in this section are valid for an arbitrary number, I, of issues.
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 177
¸emma 1. ¸et x.*/ and x.!9 denote the extreme positions on issue i across the i i population of voters. ¹hen the box I B" < [x.*/, x.!9] i i i/1 is a compact, globally attracting, forward invariant1 set for the dynamical system given in Eq. (9). Proof. B is a direct product of closed finite intervals, and hence it is compact. If the position of a party on issue l lies outside the box, say, y 'x.!9, then yR (0. l l l If some voters have different positions on issue l, that is x.*/(x.!9, then yR can l l l be uniformly bounded from above by !d(0 for y 5x.!9, and hence y will l l l become smaller than x.!9 after finite time. (The actual time required depends on l the initial conditions.) An analogous argument holds for y (x.*/. In the l l degenerate case where all voters have the same position we have yR l "!a(y !x ), where a'0, and therefore y converges exponentially to x . h l l l l Lemma 1 simply shows that regardless of the initial platform positions, all platforms eventually end up in the box spanned by the extreme positions of the voters. In particular, no party will enter a region of issue space that is devoid of preferred voter positions. This lemma insures that the model harbors no absurd dynamical properties. We next show that, asymptotically, the platforms of the two parties must converge on one another. Using this result, we can simplify the dynamical system considerably, and in so doing show (in Lemma 3) that a unique equilibrium exists at the strength-weighted average of the voters’ preferred positions. ¸emma 2. For all possible choices of s and x , the difference between the vl vl platforms, y1!y2, vanishes for the dynamical equations given in Eq. (9) as tPR. ¹his convergence is exponential. Proof. Consider y1!y2 for an arbitrary issue l. Since p@(x)"p@(!x) we have l l
A
B A
B
p@ + s [(y1!x )2!(y2!x )2] "p@ + s [(y2!x )2!(y1!x )2] vi i vi i vi vi i vi i vi i i
1 The set B is forward invariant for a given dynamical system if each trajectory with initial condition in B remains in B for all future times.
178
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
and hence using Eq. (9) d (y1!y2)"yR 1!yR 2 l l l dt l
A
B
"!2+ p@ + s [(y2!x )2!(y1!x )2] s (y1!x !y2#x ) i vi l vl vi vl vi i vl l v i
A
B
"!(y1!y2) ) 2+ s p@ + s [(y2!x )2!(y1!x )2] . l l vl vi i vi i vi v i Since the sum is bounded away from 0 within the (finite) box B, say by d'0, we have Dy1!y2D4C exp(!d t)P0. h l l ¸emma 3. ¹here is a unique equilibrium +s x yL 1"yL 2" v vl vl"xN . l l l +s v vl ¹his equilibrium is globally stable within the manifold My1"y2N.
(11)
Proof. Lemma 2 implies that there is no equilibrium with y1Oy2. For y1"y2 the equations in Eq. (9) simplify considerably, since the arguments of the ‘response’ functions, p@( ) ), vanish and we are left with the non-degenerate set of linear equations yR j"!2p@(0)+ s (yj!x ). vl l vl l v Thus, the equations for the different issues decouple on the manifold My1"y2N. It is easily checked that yL 1 is a stable rest point within the manifold. The linearity of these equations implies that the rest point is unique and even globally stable within My1"y2N. h The above lemmas can be summarized by the following: ¹heorem. ¹he adaptive-platform, two-party voting model derived in Eq. (9) converges to a unique and globally stable equilibrium yL 1"yL 2"xN , where xN is the strength-weighted mean of the voters’ preferred positions as given by Eq. (11). Thus, two competing parties climbing their local voting gradients will converge to a globally stable equilibrium. While the convergence is guaranteed, the transient paths can be long and display unusual behavior. For example, Fig. 1 shows the transient paths, in the space of platform positions, for a case
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 179
Fig. 1. Transient paths for two parties are displayed in platform-position space. Three voters with random strengths, uniformly distributed on [0, 3), and random ideal positions, uniformly distributed on [0, 8), were initially generated. The starting platform positions were randomly generated uniformly on [0, 8), and, in this case, were located in the south-east of the diagram. P(z)"1/(1#e~20z). The strength-weighted equilibrium point is designated by the solid diamond to the north-east of the cycles.
with two parties, two issues, and three randomly generated voter utility functions. In this case, while the two platforms are getting closer to each other over time, they exhibit some unusual cycling behavior. Eventually, the distance between the platforms is small enough so that they converge on the predicted equilibrium point indicated by the solid diamond to the north-east of the top of the cycle. Finally, let us return to dynamics (7), and develop a simple result that will be needed later. Suppose we start with two identical platforms y1"y2. In this case, Eq. (7) reduces to yR j"+ p@(0)+ ju (yj)"p@(0)+¼(yj), y v v
(12)
that is, the restriction of Eq. (7) to the invariant manifold My1"y2N is a gradient system, and all trajectories converge to a (local) maximum of ¼(y)"+ u (y). v v Thus, parties that begin with identical platforms will converge to a local maximum of the additive social utility function, and therefore such political systems will, at least locally, maximize aggregate voter welfare. Note that for the
180
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
system discussed above, this insures that it ends up at a local maxima. With non-concave utility functions (which are investigated in Section 4 below) local maxima will also result when the stable equilibria have identical platforms, but as we will see, under non-concave utility it is possible for equilibria to exist with divergent platforms.
4. Generalizations of the model We modify the above model in a variety of directions. First, we analyze the situation where parties are restricted in how much they can change their positions on different issues. Next, we investigate the impact on our results of various modifications to the voters’ utility functions, including correlations among issues and non-quadratic (both concave and non-concave) utility functions. We also explore the case where parties adjust their platforms discretely, that is, instead of allowing parties to continuously adjust their platforms as above, parties are only allowed to alter their platforms at discrete points in time, for example, during an annual election. Finally, we analyze the situation where parties attempt to maximize their votes while simultaneously trying to stay close to a pre-defined “ideal” platform. 4.1. Limits on platform mobility Parties may be reluctant to change their positions on particular issues in an election, for example, special interests may demand that the party ‘moves slowly’ on a given issue. We model this behavior by yR 1"A1+ 1º (y1, y2), y 1 (5@) yR 2"A2+ 2º (y1, y2), y 2 where the Aj are I]I diagonal matrices with diagonal entries between 0 (the party does not change its position on a particular issue regardless of the demands of the voters) and 1 (the party changes its positions in a purely opportunistic way, as in our original model). It is easy to check that My1"y2N is no longer invariant if A1OA2. In this case the vote optimizing behavior of a party is distorted by its reluctance to change its positions on certain issues. In the following, we will assume Aj '0 for all issues (i) and both parties (j). ii ¸emma 4. ¹he voting model with dynamics (5@) has a unique rest point corresponding to the weighted mean voter, (xN , xN ), which is locally asymptotically stable. Proof. The fixed points of Eq. (5@) are given by 0"Aj+ jº (y1, y2). Multiplying y j by A~1 shows that the rest points must fulfill + jº (y1, y2)"0, hence Eq. (5@) has j y j
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 181
the same rest points as Eq. (5), that is, yL 1"yL 2"xN . It is simple to check stability since the Jacobian at (xN , xN ) is of the form J"!2p@(0)diag(A1, A2)]diag(s, s) where s is the vector with components s "+ V s . h i v/1 vi Note that we were unable to find a proof for global stability in this case as we did for model (5). The lack of other equilibria implies that any trajectory that does not converge to (xN , xN ) is periodic or chaotic. Nevertheless, we strongly suspect that the system is indeed globally stable. Thus, when parties are reluctant to adjust their positions on certain issues, the equilibrium point we found earlier still exists (and is a local attractor). 4.2. Preference correlations among issues Suppose that a voter’s preferred positions are correlated across the various issues. We can capture this case by generalizing Eq. (1) in a very natural way to general metric distances in the space of issues: u (yj)"!(yj!x )A (yj!x ), (1@) v v v v where x is the vector of voter v’s preferred positions, and A is a symmetric, v v positive-definite matrix that represents the covariance matrix across the issues for voter v. ¸emma 5. ¹he voting model (7), with voter utility function (1@), has a unique rest point that is (locally) asymptotically stable. ¹he manifold My1"y2N is invariant and (xN , xN ) is globally stable within My1"y2N. Proof. We know +u (y)"!2A (y!x ). For the evolution of Dy"y1!y2 we v v v find DyR "!2+ p@(u (y1)!u (y2)) ) A Dy"O(y1, y2)Dy, v v v v where O(y1, y2) is a symmetric positive-definite operator for all y1 and y2, hence My1"y2N is (at least locally) attracting. The dynamics on the invariant manifold My1"y2N are given by yR "!2p@(0)+ A (y!x )"!2p@(0)H(y!xN ), v v v with the (unique) equilibrium point, xN , defined by xN "H~1+ A x , v v v
182
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
where H"+ A is again a positive-definite matrix, and therefore invertible. v v Thus xN exists and is unique. Since !2p@(0)H is negative definite, xN is globally stable within the manifold My1"y2N. h Lemma 5 can be extended to a globally stable equilibrium point, if the Jacobi conjecture holds. The Jacobi conjecture concerns one of the old, open ODE problems (see, for example, Meister, 1982): ‘Let xR "f (x) be an ODE in Rn such that 0 is a rest point and the Jacobian is a stable matrix (all eigenvalues have a negative real part) for all x. Then 0 is globally stable’. If the Jacobi conjecture is true, then (xN , xN ) is globally stable under the dynamics (7) with voter utility functions (1@). Thus, we find that with correlated voter preferences, a unique equilibrium of the system exists as above (suitably adjusted for the correlated strengths). This equilibrium is definitely locally stable, and (conditional on the Jacobi conjecture holding) also globally stable. 4.3. Non-quadratic (but concave) utility functions In Eq. (1) we assumed that voters preferred the party platform that minimized the squared distance from each voter’s ideal platform. We can extend Eq. (1) to the non-quadratic (but concave) utility functions defined by u (y)"!+ / (y !x ), (1A) v vi i vi i where / (t)'0 for tO0, / (0)"0, and /@ (t)'0 for t'0 and /@ (t)(0 for vi vi vi vi t(0. ¸emma 6. ¹here is a unique and globally stable rest point (xN , xN ) for the dynamical system (7) with voter utility function (1@@) provided /A (t)'0. vi Proof. Here, the explicit dynamics are yR 1"!+ p@(u (y1)!u (y2))/@ (y1!x ), vl vl l l v v v yR 2"!+ p@(u (y2)!u (y1))/@ (y2!x ), vl vl l l v v k and thus the dynamics of the difference in the platforms is given by yR 1!yR 2"!+ p@(u (y1)!u (y2))[/@ (y1!x )!/@ (y2!x )]. vl l vl vl l v v vl l l v
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 183
If /A '0 the difference of the /@ terms is positive if and only if y1!y2 is vi vi l l positive; hence, the difference y1!y2 decreases and My1"y2N is globally stable. A fixed point must therefore fulfill + / (y !x )"U (y )"0. Since /A '0 we v vi i vi i i vi know U@'0, that is, U is strictly monotonically increasing. Therefore, there is i i at most one zero xN , which is defined by U (xN )"0, for all issues i. Since the box i i B is forward invariant, Brouwer’s fixed point theorem guarantees the existence of at least one rest point in B, and consequently (xN , xN ) is unique. h Therefore, as long as voter utility functions are strictly concave, the existence of a unique, globally stable equilibrium for the system (which is related to the equilibrium given in Eq. (11), but implicitly defined by U (xN )"0 above) is i i guaranteed under non-quadratic utility functions. Thus, the general tenor of the results found in the initial model hold under this much wider class of plausible voter utility functions. 4.4. Non-concave utility functions Our stability and uniqueness results concerning equilibrium required strictly concave utility functions (a typical assumption for similar dynamic game models). However, some non-concavity in these functions might be reasonable if, for example, once platforms are a sufficient distant from a voter’s preferred position, changes that move the platform even further away have little impact on the voter’s utility. While /A '0 everywhere on R is possible, /A (0 on all R leads to a contravi vi diction, since we require that / has a minimum at 0 and hence positive vi curvature in a neighborhood of 0. We can, however, construct a model where / has positive curvature only in a very small neighborhood of the preferred vi voter positions, and /A (0 everywhere else. Again My1"y2N is invariant. But vi now we can have more than one equilibrium. The eigenvalues of the Jacobian at an equilibrium fulfilling y1"y2 will determine its stability. We find that the Jacobian is diagonal with entries (13) !p@(0)+ /A (yL !x ) vi l vl v which can be both positive and negative, and thus unstable equilibria are possible. We illustrate the potential behavior of our model in the case where the concavity assumption on u is violated via the following example. Consider voter utility functions of the form
C
G
(yj!x )2 vi u (yj)"!c2+ s 1!exp ! i v vi c2 i
HD
.
(1@@@)
184
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
Note that the quadratic utility function (1) agrees with (1A@) up to the third order in (y!x ) (in Eq. (1) the utility diverges, while here it levels off ). We further v observe that the curvature of u v Lu (yj) 2(yj !x ) (yj!x )2 v "!2s 1! k vk exp ! i vk , vk c2 Lyj c2 k changes sign at
C
D G
H
c Dyj !x D" . k vk J2 Given the utility function in (1@@@), consider a simple example with only two issues and two voters. Let x "(!1,1) and x "(1,!1), and let each voter have 1 2 s "1 for both issues. The mean voter equilibrium of (7) is the origin (0,0). As vi a consequence of Eq. (13), the origin is a locally stable equilibrium (sink) if c'J2 and an unstable equilibrium (source) if c(J2. The symmetry of the problem and the forward invariance of the box [!1,1]][!1,1] (Lemma 1) imply that there cannot be a globally stable equilibrium. Furthermore, any equilibrium on the invariant manifold My1"y2N is unstable. We could not explicitly solve Eq. (7) with utility function (1@@@) for all equilibria, so we numerically integrated the dynamical system. The results are summarized in Fig. 2. The figure shows six pairs of platform paths induced by parties starting randomly about (0,0). The stable equilibria are of the form y1"(m, m),
y2"(m,!m),
y1"(m, m),
y2"(!m, m),
y1"(!m,!m),
y2"(m,!m),
y1"(!m,!m),
y2"(!m,m),
with m+0.955 for c"1 (solutions with y1 and y2 exchanged also exist). Which of these locally stable equilibria is actually reached depends on the initial conditions. Note that none of the equilibria are symmetric-the parties agree on one issue and dissent on the other. 4.5. Discrete time dynamics As elections are (usually) held at well-defined time intervals, the behavior of a discrete time model, instead of the continuous-time dynamics explored above, is of interest. In the discrete model, a party updates its current platform by taking a discrete step in the direction of its payoff gradient at fixed time intervals.
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 185
Fig. 2. The platform dynamics for two parties are shown in platform-position space. Two voters with strengths equal to one on all issues and ideal points of (!1, 1) and (1,!1) were used. The mean voter equilibrium of Eq. (7) is the origin (0, 0). Voters had utility functions given by (1A@) with c"1. Six different initial platform pairs were randomly generated about the origin (each pair has the same marker and line style). Similar markers end up at the same equilibrium points.
To simplify notation, define Wk(y1, y2)"+ kº (y1, y2) and W(y)"(W1(y1, y2), y k W2(y1, y2)). Thus, the system of ordinary differential equations in Eq. (5) can be written as yR k"Wk(y1, y2), or simply yR "W(y). Furthermore, let LW1 LW1 Ly1 Ly2 J(y)" (14) LW2 LW2 Ly1 Ly2 give the Jacobian matrix of this dynamical system at position y"(y1, y2). Each party is allowed to update its platform with a step in the direction of its payoff gradient:
A B
yk{"yk#qWk(y1, y2),
(15)
where q represents the mobility of each party in issue space between elections. Discrete- and continuous-time versions of a dynamical system are closely related. In fact, yL is a fixed point of Eq. (15) if and only if W(yL )"0; That is,
186
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
discrete and continuous-time models share the same equilibria. The stability of the fixed points is determined by the Jacobian matrices, J (y) and J(y), for the d discrete and continuous cases, respectively. For the discrete-time model we have
A
L(y1#qW1) L(y1#qW1) Ly1 Ly2
J (y)" d L(y2#qW2) L(y2#qW2) Ly1 Ly1
A
"
1#q
LW1 LW1 q Ly1 Ly2
LW2 q Ly1
1#q
LW2 Ly2
B
B
"I#qJ(y),
(16)
where I now denotes the identity matrix. Note that this correspondence is valid for all the modifications of Eq. (5) discussed above. A fixed point, yL , of a discrete-time dynamical system is (locally) asymptotically stable if and only if all eigenvalues of its Jacobian matrix, J (yL ), lie within the unit d circle. Suppose yL is stable for the continuous-time dynamics. Then all eigenvalues of J(yL ) have a negative real part. Therefore, there is a q '0 such that y is 0 (locally) asymptotically stable in the discrete-time model (15) for all choices of q in the interval (0, q ). On the other hand, complex dynamics, including 0 deterministic chaos, are commonly encountered in discrete-time models that allow large moves in each time step. In our original model, as well as for the modifications described in Lemmas 4—6, the equilibrium is (at least locally) stable, as the Jacobian, J(yL ), has only real, negative eigenvalues in all these cases. Denoting the spectral radius of J(yL ) by o, where the spectral radius is given by the largest absolute-valued eigenvalue, we find the yL is also stable in the discrete-time model as long as 1!qo'!1, that is, for 0(q(2/o. Thus, the mean voter equilibrium will be (locally) asymptotically stable under discrete-time dynamics provided that platform adaptation is slow and/or elections are frequent. 4.6. Policy-dependent party utilities Suppose that each party cares about achieving an ‘ideal’ platform, designated by qk. To capture this case, we assume that a party’s utility is a convex combination of both the number of expected votes and the distance, D(yk, qk), between the actual and ideal platform. Thus, we replace º (y1, y2) in Eq. (5) with k (1!m)º (y1, y2)!mD(yk, qk), where m parameterizes the relative importance of k the actual versus ideological platform. Under this new assumption, the voting
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 187
dynamics are governed by yR k"Wk(y1, y2)!m[Wk(y1, y2)#Ck(yk)],
(17)
with Ck(yk)"+ kD(yk,qk). y Restricting attention to the box B, we may assume that Ck and its partial derivatives with respect to yk are bounded, by some constant M. Without loss of j generality, we assume that Ck and m are scaled such that M"1 and m measures the size of the ideological payoff component. Thus, m[Wk(y1,y2)#Ck(yk)] forms a regular perturbation (in the sense of Hirsch and Smale, 1974 (Chapter 16)) of the vector field W(y) and the following proposition holds: Proposition (Hirsch and Smale, 1974). Suppose yL is a hyperbolic fixed point of yR "W(y). ¹hen there is m '0 and a neighborhood N(yL ) such that for all m in 0 the interval 04m4m there is a unique fixed point yL (m)3N(yL ). ¹he para0 meter m can be chosen such that yL (m) is hyperbolic and has the same index (that 0 is, the same number of positive and negative eigenvalues) as yL . In addition, yL (m) is a continuous function of the perturbation parameter m and yL (0)"yL , the fixed point of the unperturbed vector field. Introducing an ideological payoff component will, in general, break the symmetry of Eq. (5) and, hence, also the symmetry of the mean voter equilibrium. Thus, in general, we find yL 1(m)OyL 2(m) even for small values of m'0. Using a perturbation approach that was originally developed for selection-mutation equations in theoretical biology given in Stadler and Schuster (1992), it is possible to estimate the location of a perturbed mean-voter equilibrium: C1(yL 1)
A B
yL (m)"yL #mJ~1(yL )
C2(yL 2)
#O(m2).
(18)
Here we have incorporated the knowledge that W(yL )"0. Thus, the distance of the equilibrium positions of the two parties is proportional to the parameter m. In the other extreme, when m+1, and a party’s behavior is mostly determined by its ideology, we may consider the voter-dependent term W as a perturbation. Consequently, each party will settle down to an equilibrium close to qk, since in the limiting case where W"0 we have a gradient system in which each party converges to a minimum of Dk(yk, qk) independent of the other parties.
5. Conclusions By employing the tools of evolutionary game dynamics we analyze a twoparty spatial-voting system composed of locally adaptive parties. Our results indicate that in such a system, the two platforms eventually converge to
188
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
a globally stable equilibrium point located at the strength-weighted mean of the voters’ preferred positions. While the convergence is assured, the transient paths can be long and convoluted. The fundamental results are relatively invariant to a variety of modifications of the underlying model. Limited platform mobility, voter preferences correlated among issues, non-quadratic (but concave) utility functions, discrete adjustment dynamics, and ideological party preferences all have little influence on the underlying findings. However, the incorporation of non-concave voter utility functions can result in a variety of new behaviors, including multiple equilibria and, we suspect, limit cycles. The voting dynamics discussed here can also be extended to more than two parties. A recent study (Stadler, 1998) shows that the mean voter equilibrium can become unstable for three or more parties. Though the focus here has been on two-party spatial voting models, extensions to other phenomena are possible, for example, the underlying structure could be interpreted as an oligopolistic market in which firms locally alter sets of product attributes (platforms) to attract consumers (voters). Our model provides a necessary benchmark for the behavior of systems with agents that are limited in their abilities to directly solve for equilibria, and whose strategic choices are constrained by past actions. While we find that such systems display predictable behavior in the limit, we also note that the transient dynamics can be remarkable, and are worthy of further analysis.
Acknowledgements We thank Ken Kollman, Scott E. Page, and Ba¨rbel Stadler for many helpful comments on this research, and the Santa Fe Institute for general research support. Additional research support was received by Miller from the National Science Foundation (grants SBR-9411025 and SBR-9710014) and by Stadler from the German Max Planck Gesellschaft.
References Coughlin, P., Nitzan, S., 1981. Directional and local electoral equilibria with probabilistic voting. Journal of Economic Theory 24, 226—239. Coughlin, P.J., 1990. Candidate uncertainty and electoral equilibria. In: Enelow, J.M., Hinich, M.J., (Eds.), Advances in the Spatial Theory of Voting. Cambridge University Press, Cambridge, pp. 145—166. Friedman, D., 1991. Evolutionary games in economics. Econometrica 59(3), 637—666. Hirsch, M.W, Smale, S., 1974. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, Orlando FL, USA. Hofbauer, J., Sigmund, K., 1988. The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, UK.
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 189 Hofbauer, J., Sigmund, K., 1990. Adaptive dynamics and evolutionary stability. Appl. Math. Lett. 3(4), 75—79. Kollman, K., Miller, J.H., Page, S.E., 1992. Adaptive parties in spatial elections. American Political Science Review 86, 929—937. Malthus, T., 1798. An Essay on the Principle of Population. J.M. Dent (1933), London, UK. Meister, G.H., 1982. Jacobi problems in differential equations and algebraic geometry. Rocky Mountain J. Mathematics 12, 679—705. Plott, C.R., 1967. A notion of equilibrium and its possibility under majority rule. The American Economic Review 57, 787—806. Samuelson, L., 1984. Electoral equilibria with restricted strategies. Public Choice 43, 307—327. Stadler, B.M.R., Adaptive platform dynamics in spatial voting models. Ph.D. Thesis, University of Vienna, 1998. Stadler, P.F., Schuster, P., 1992. Mutation in autocatalytic networks — an analysis based on perturbation theory. J. Math. Biol. 30, 597—631.
The dynamics of locally adaptive parties under spatial voting J.H. Miller!,",*, P.F. Stadler",# ! Department of Social and Decision Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA " The Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, NM 87501, USA # Institut fu( r Theoretische Chemie, Universita( t Wien, Wa( hringerstr. 17, A-1090 Vienna, Austria Received 20 August 1996; accepted 13 January 1998
Abstract We explore the dynamics of a model of two-party competition under spatial voting. The parties are allowed to incrementally adapt their platforms by following the voting gradient imposed by the preferences of the electorate and the platform of the opposition. The emphasis in this model is on the dynamic system formed by these conditions, in particular, we examine the characteristics of the transient paths and the convergence points of the evolving platforms. We find that in a simple spatial model with probabilistic voting, regardless of the initial platforms of each party, platforms eventually converge to a unique, globally stable equilibrium matching the strength-weighted mean of the voters’ preferred positions. This result appears robust to many variations in voter preferences and party behavior. However, we do find some conditions under which other dynamic possibilities occur, including multiple equilibria and, perhaps, limit cycles. ( 1998 Elsevier Science B.V. All rights reserved. JEL classification: C61; C62; C63; D72 Keywords: Spatial voting; Local adaptation; Search; Dynamics
1. Introduction The importance of adaptive processes in economic theory has long been recognized (see, for example, Malthus, 1798). In this paper, we employ the tools
* Correspondence address. Department of Social and Decision Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. E-mail: [email protected] 0165-1889/98/$ — see front matter ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 0 4 - 9
172
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
of evolutionary game dynamics to better understand the dynamic behavior of an adaptive system in which agents are constrained to locally adapt to their world. We assume local adaptation for a variety of reasons. One justification is that agents are often constrained by external forces to only slowly alter their choices. For example, automobile manufactures may face extremely high retooling costs for extensive model changes, or political parties that rapidly shift their platform positions may be perceived as less trustworthy by voters than those that change only incrementally. Another factor favoring local adaptation is that agents are likely to have more, and better quality, information about choices that are ‘close’ to their current actions. Thus, an automobile manufacturer can more accurately estimate the consumer demand resulting from, say, offering a currently produced car in a new color, than from offering an entirely new type of car. We analyze incremental adaptation of complex actions in a model of two-party spatial voting, in which the parties compete for votes by making minor alterations to their respective platforms. A related application is that of firms competing for consumers by offering goods that embody multiple attributes (for example, a car). These types of scenarios represent an important class of phenomena that has been widely analyzed using more traditional tools, and thus they provide a nice benchmark from which to explore the impact of incremental adaptive dynamics. We examine a spatial voting model where candidates in a two-party system incrementally adjust their platforms to appeal to voters. In our model, a candidate is only allowed to make local adjustments to the previously endorsed platform. As mentioned above, such local adaptation could be justified on a variety of grounds, for example, incremental changes allow parties to maintain legitimacy and a coherent organization within the eyes of the electorate, special interests may demand only small changes, or polling information about closely related platforms may be relatively more available and accurate. We assume that parties follow the local voting gradient imposed on the system by the preferences of the voters and the position of the other candidate. This obviously implies that parties can accurately determine the optimal local response in any given situation. While models that degrade even this local optimization ability are certainly of interest, the case analyzed here represents an important first step in realizing such models (and, a significant departure from most current models that assume full knowledge of the entire space). We find that in a simple spatial model with probabilistic voting and local adaptation, regardless of the initial platforms of each party, platforms eventually converge to a unique, globally stable equilibrium matching the strengthweighted mean of the voters’ preferred positions. This result appears robust to many variations of our underlying assumptions about voter preferences and party behavior. However, if we allow less constrained voter utility functions, many dynamic possibilities occur, including multiple equilibria and, we suspect, limit cycles.
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 173
Our analysis focuses on an adaptive dynamic system, whereby global consequences emerge from locally adapting candidates. The inspiration for this work comes from the computational results discovered by Kollman et al. (1992). They found that parties following simple, locally adaptive rules rapidly converged toward common platforms of high aggregate utility. Moreover, they noted that such models appear to explain existing empirical regularities better than more traditional modeling approaches. While the ideas of probabilistic voting (for a general review see Coughlin, 1990) and locally restricted strategy searches in such models (for example, Coughlin and Nitzan, 1981; Samuelson, 1984) have been widely discussed, here we assume that parties do not start at an identical status quo platform, and that their ability to maximize voter support is limited to climbing the local voting gradient. Such a model serves as a necessary benchmark for spatial voting results when candidates’ abilities to optimize their platform positions are limited and where the dynamic behavior of platform modification is of interest. The emphasis in our analysis is on the dynamic system formed by the model, in particular, we examine the characteristics of the transient paths and the convergence points of the evolving platforms. Note that one important feature of the model is that incremental changes by each party may dramatically impact the payoff space facing the other party. Thus, the adapting parties are intimately coupled with one another: small movements by one party may alter the desired movements of the other party. The outcome of such a dance is the focus of our analysis.
2. The basic model Consider a model with I issues and » voters. We will use the following notation: platform position of party j on issue i yj i x voter v’s preferred position on issue i vi s voter v’s strength on issue i. We require for all i that s '0 for at least vi vi one v. (That is, at least one voter cares about each issue.) u (yj) voter v’s utility of party j’s platform: v
P(z)
I (1) u (yj)"! + s (yj!x )2. vi v vi i i/1 (This functional form will be generalized in Section 4.) probability of voting for a platform that has a difference z in utility against the opposing platform. If the two parties are distinguished only by their platforms, then P(z) will be symmetric about 0.5:
174
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
Prob[v votes for 1]"P(u (y1)!u (y2)), v v Prob[v votes for 2]"P(u (y2)!u (y1))"1!P(u (y1)!u (y2)). v v v v In the extreme case we have
G
0,
P(z)" 0.5, 1,
z(0, z"0,
(2)
z'0,
where a voter always votes for the party that offers the larger utility, and if both parties provide the same utility the voter is indifferent and flips a coin. To add both realism and mathematical tractability, we impose the following additional requirements on P(z). First, P(z) is assumed to be strictly monotonic, implying that the probability of choosing a party always increases as the difference in utility increases and that there is always a positive probability of voting for either party for any difference in utilities, z. Second, we require P(z) to be continuously differentiable. A simple class of functions that fit our requirements is the symmetric sigmoidal functions (such as the error integral). If P(z) is only monotonic, then when P@(z)"0 it is possible to have large areas in issue-space forming equilibria where incremental changes in the platforms do not cause any voters to change their expected votes. With strict monotonicity, platforms will only stabilize if there is an attractor present. Continuity is necessary for the results. In the extreme case of P(z) given in Eq. (2), the results derived below no longer hold due to the discontinuity implied by this assumption. In this latter case, Plott (1967) showed that in general parties can always locate a better platform, and hence no equilibrium configuration of the parties is possible. Given P(z), the expected number of votes for each party, E , is j V E (y1, y2)" + P(u (y1)!u (y2)), 1 v v v/1 (3) V E (y1, y2)" + P(u (y2)!u (y1))"»!E (y1, y2). 2 v v 1 v/1 We assume that a party’s utility, º , is given by the number of votes that it is j ahead of its rival: º (y1, y2)"E (y1, y2)!E (y1, y2)"2E (y1, y2)!», 1 1 2 1 (4) º (y1, y2)"E (y1, y2)!E (y1, y2)"»!2E (y1, y2)"!º (y1, y2). 2 2 1 1 1 2.1. Dynamics For the dynamics of platform adjustment we follow an approach used in the analysis of biological games (for an introduction to this area, see Hofbauer and Sigmund (1988), Friedman (1991) reviews economic applications of these
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 175
ideas). Hofbauer and Sigmund (1990), in a related application to continuous strategy spaces called ‘adaptive dynamics’, explore strategic stability around fixed points (unfortunately, our dynamics are not characterized by their results). The dynamics we explore here have not been previously analyzed. In our model both parties update their platforms according to the following dynamics: yR 1"+ 1º (y1, y2), y 1
yR 2"+ 2º (y1, y2). y 2
(5)
That is, each party alters its platform so as to locally improve its share of votes. The optimization is local rather than global since each party simply follows the gradient of its utility function. Since P(z) is continuously differentiable the vector field (+ 1º ,+ 2º ) is continuous everywhere, and therefore the existence and y 1 y 2 uniqueness of the trajectory for all initial conditions is guaranteed. (Note that the monotonicity of P is not necessary for existence and uniqueness of the trajectories.) More generally, we might assume yR k"a+ kº for k"1, 2 in y k Eq. (5). However, it is always possible to choose the time unit such that a"1. Therefore, the value of a does not influence the long-time dynamic behavior of the system. Although Eq. (5) seems to imply that parties are simply hill-climbing up a fixed ‘vote landscape’ imposed by the preferences of the electorate, in fact the ‘hill’ perceived by each party also depends on the platform position of the other party (as each voter makes a relative comparison between the two parties). Since both parties are moving on the same time-scale, their respective payoff landscapes are changing at the same speed. Thus, both parties confront a landscape that undulates with each move. We can rewrite the dynamics in a more explicit way. First, define p(z)"2P(z)!1. As a consequence of the differentiability, monotonicity and symmetry of P(z), p has the following properties: p(z)"!p(!z), p(0)"0, p@(z)'0, and p@(z)"p@(!z). The utility of the parties can be rewritten as º (y1, y2) " 2E (y1, y2)!» 1 1 " + [2P(u (y1)!u (y2))!1] v v v " + p(u (y1)!u (y2)), v v v º (y1, y2) " 2E (y1, y2)!» 2 2 " + [2P(u (y2)!u (y1))!1] v v v " + p(u (y2)!u (y1)). v v v
(6)
176
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
The dynamic equations now yield yR 1 " + 1º y 1 " + + 1p(u (y1)!u (y2)) v v y v " + p@(u (y1)!u (y2))+ 1u (y1), y v v v v yR 2 " + 2º y 2
(7)
" + + 2p(u (y2)!u (y1)) v v y v " + p@(u (y2)!u (y1))+ 2u (y2). y v v v v Using the voter’s utility function given by Eq. (1) we find L I (y !x )2"!2s (y !x ) [+ u (y)] "! + s vi vl l vl y v l vi Ly i l i/1
(8)
for the partial derivative in the direction of issue l. Combining Eqs. (1), (7) and (8) yields the system of differential equations
A A
B B
yR 1"!2+ p@ + s [(y2!x )2!(y1!x )2] s (y1!x ), vi i vi i vi vl l vl l i v
(9)
yR 2"!2+ p@ + s [(y1!x )2!(y2!x )2] s (y2!x ). i vi l vi vl vi i vl l v i The p@( ) ) terms in these equations weight the importance of a given voter for the changes that a party implements in its platform. If P(z) is sigmoidal, then these terms imply that the most important voters are the pivotal ones that currently receive similar utility from the two platforms. The multiplicative factor, s (yj!x ), implies that changes of position on issue l are influenced most by vl vl l those voters who both care a lot about issue l and who have preferred positions further away from the current platform position.
3. Analysis In this section we formally derive the properties of the dynamical system implied by the basic model discussed above. All results in this section are valid for an arbitrary number, I, of issues.
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 177
¸emma 1. ¸et x.*/ and x.!9 denote the extreme positions on issue i across the i i population of voters. ¹hen the box I B" < [x.*/, x.!9] i i i/1 is a compact, globally attracting, forward invariant1 set for the dynamical system given in Eq. (9). Proof. B is a direct product of closed finite intervals, and hence it is compact. If the position of a party on issue l lies outside the box, say, y 'x.!9, then yR (0. l l l If some voters have different positions on issue l, that is x.*/(x.!9, then yR can l l l be uniformly bounded from above by !d(0 for y 5x.!9, and hence y will l l l become smaller than x.!9 after finite time. (The actual time required depends on l the initial conditions.) An analogous argument holds for y (x.*/. In the l l degenerate case where all voters have the same position we have yR l "!a(y !x ), where a'0, and therefore y converges exponentially to x . h l l l l Lemma 1 simply shows that regardless of the initial platform positions, all platforms eventually end up in the box spanned by the extreme positions of the voters. In particular, no party will enter a region of issue space that is devoid of preferred voter positions. This lemma insures that the model harbors no absurd dynamical properties. We next show that, asymptotically, the platforms of the two parties must converge on one another. Using this result, we can simplify the dynamical system considerably, and in so doing show (in Lemma 3) that a unique equilibrium exists at the strength-weighted average of the voters’ preferred positions. ¸emma 2. For all possible choices of s and x , the difference between the vl vl platforms, y1!y2, vanishes for the dynamical equations given in Eq. (9) as tPR. ¹his convergence is exponential. Proof. Consider y1!y2 for an arbitrary issue l. Since p@(x)"p@(!x) we have l l
A
B A
B
p@ + s [(y1!x )2!(y2!x )2] "p@ + s [(y2!x )2!(y1!x )2] vi i vi i vi vi i vi i vi i i
1 The set B is forward invariant for a given dynamical system if each trajectory with initial condition in B remains in B for all future times.
178
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
and hence using Eq. (9) d (y1!y2)"yR 1!yR 2 l l l dt l
A
B
"!2+ p@ + s [(y2!x )2!(y1!x )2] s (y1!x !y2#x ) i vi l vl vi vl vi i vl l v i
A
B
"!(y1!y2) ) 2+ s p@ + s [(y2!x )2!(y1!x )2] . l l vl vi i vi i vi v i Since the sum is bounded away from 0 within the (finite) box B, say by d'0, we have Dy1!y2D4C exp(!d t)P0. h l l ¸emma 3. ¹here is a unique equilibrium +s x yL 1"yL 2" v vl vl"xN . l l l +s v vl ¹his equilibrium is globally stable within the manifold My1"y2N.
(11)
Proof. Lemma 2 implies that there is no equilibrium with y1Oy2. For y1"y2 the equations in Eq. (9) simplify considerably, since the arguments of the ‘response’ functions, p@( ) ), vanish and we are left with the non-degenerate set of linear equations yR j"!2p@(0)+ s (yj!x ). vl l vl l v Thus, the equations for the different issues decouple on the manifold My1"y2N. It is easily checked that yL 1 is a stable rest point within the manifold. The linearity of these equations implies that the rest point is unique and even globally stable within My1"y2N. h The above lemmas can be summarized by the following: ¹heorem. ¹he adaptive-platform, two-party voting model derived in Eq. (9) converges to a unique and globally stable equilibrium yL 1"yL 2"xN , where xN is the strength-weighted mean of the voters’ preferred positions as given by Eq. (11). Thus, two competing parties climbing their local voting gradients will converge to a globally stable equilibrium. While the convergence is guaranteed, the transient paths can be long and display unusual behavior. For example, Fig. 1 shows the transient paths, in the space of platform positions, for a case
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 179
Fig. 1. Transient paths for two parties are displayed in platform-position space. Three voters with random strengths, uniformly distributed on [0, 3), and random ideal positions, uniformly distributed on [0, 8), were initially generated. The starting platform positions were randomly generated uniformly on [0, 8), and, in this case, were located in the south-east of the diagram. P(z)"1/(1#e~20z). The strength-weighted equilibrium point is designated by the solid diamond to the north-east of the cycles.
with two parties, two issues, and three randomly generated voter utility functions. In this case, while the two platforms are getting closer to each other over time, they exhibit some unusual cycling behavior. Eventually, the distance between the platforms is small enough so that they converge on the predicted equilibrium point indicated by the solid diamond to the north-east of the top of the cycle. Finally, let us return to dynamics (7), and develop a simple result that will be needed later. Suppose we start with two identical platforms y1"y2. In this case, Eq. (7) reduces to yR j"+ p@(0)+ ju (yj)"p@(0)+¼(yj), y v v
(12)
that is, the restriction of Eq. (7) to the invariant manifold My1"y2N is a gradient system, and all trajectories converge to a (local) maximum of ¼(y)"+ u (y). v v Thus, parties that begin with identical platforms will converge to a local maximum of the additive social utility function, and therefore such political systems will, at least locally, maximize aggregate voter welfare. Note that for the
180
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
system discussed above, this insures that it ends up at a local maxima. With non-concave utility functions (which are investigated in Section 4 below) local maxima will also result when the stable equilibria have identical platforms, but as we will see, under non-concave utility it is possible for equilibria to exist with divergent platforms.
4. Generalizations of the model We modify the above model in a variety of directions. First, we analyze the situation where parties are restricted in how much they can change their positions on different issues. Next, we investigate the impact on our results of various modifications to the voters’ utility functions, including correlations among issues and non-quadratic (both concave and non-concave) utility functions. We also explore the case where parties adjust their platforms discretely, that is, instead of allowing parties to continuously adjust their platforms as above, parties are only allowed to alter their platforms at discrete points in time, for example, during an annual election. Finally, we analyze the situation where parties attempt to maximize their votes while simultaneously trying to stay close to a pre-defined “ideal” platform. 4.1. Limits on platform mobility Parties may be reluctant to change their positions on particular issues in an election, for example, special interests may demand that the party ‘moves slowly’ on a given issue. We model this behavior by yR 1"A1+ 1º (y1, y2), y 1 (5@) yR 2"A2+ 2º (y1, y2), y 2 where the Aj are I]I diagonal matrices with diagonal entries between 0 (the party does not change its position on a particular issue regardless of the demands of the voters) and 1 (the party changes its positions in a purely opportunistic way, as in our original model). It is easy to check that My1"y2N is no longer invariant if A1OA2. In this case the vote optimizing behavior of a party is distorted by its reluctance to change its positions on certain issues. In the following, we will assume Aj '0 for all issues (i) and both parties (j). ii ¸emma 4. ¹he voting model with dynamics (5@) has a unique rest point corresponding to the weighted mean voter, (xN , xN ), which is locally asymptotically stable. Proof. The fixed points of Eq. (5@) are given by 0"Aj+ jº (y1, y2). Multiplying y j by A~1 shows that the rest points must fulfill + jº (y1, y2)"0, hence Eq. (5@) has j y j
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 181
the same rest points as Eq. (5), that is, yL 1"yL 2"xN . It is simple to check stability since the Jacobian at (xN , xN ) is of the form J"!2p@(0)diag(A1, A2)]diag(s, s) where s is the vector with components s "+ V s . h i v/1 vi Note that we were unable to find a proof for global stability in this case as we did for model (5). The lack of other equilibria implies that any trajectory that does not converge to (xN , xN ) is periodic or chaotic. Nevertheless, we strongly suspect that the system is indeed globally stable. Thus, when parties are reluctant to adjust their positions on certain issues, the equilibrium point we found earlier still exists (and is a local attractor). 4.2. Preference correlations among issues Suppose that a voter’s preferred positions are correlated across the various issues. We can capture this case by generalizing Eq. (1) in a very natural way to general metric distances in the space of issues: u (yj)"!(yj!x )A (yj!x ), (1@) v v v v where x is the vector of voter v’s preferred positions, and A is a symmetric, v v positive-definite matrix that represents the covariance matrix across the issues for voter v. ¸emma 5. ¹he voting model (7), with voter utility function (1@), has a unique rest point that is (locally) asymptotically stable. ¹he manifold My1"y2N is invariant and (xN , xN ) is globally stable within My1"y2N. Proof. We know +u (y)"!2A (y!x ). For the evolution of Dy"y1!y2 we v v v find DyR "!2+ p@(u (y1)!u (y2)) ) A Dy"O(y1, y2)Dy, v v v v where O(y1, y2) is a symmetric positive-definite operator for all y1 and y2, hence My1"y2N is (at least locally) attracting. The dynamics on the invariant manifold My1"y2N are given by yR "!2p@(0)+ A (y!x )"!2p@(0)H(y!xN ), v v v with the (unique) equilibrium point, xN , defined by xN "H~1+ A x , v v v
182
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
where H"+ A is again a positive-definite matrix, and therefore invertible. v v Thus xN exists and is unique. Since !2p@(0)H is negative definite, xN is globally stable within the manifold My1"y2N. h Lemma 5 can be extended to a globally stable equilibrium point, if the Jacobi conjecture holds. The Jacobi conjecture concerns one of the old, open ODE problems (see, for example, Meister, 1982): ‘Let xR "f (x) be an ODE in Rn such that 0 is a rest point and the Jacobian is a stable matrix (all eigenvalues have a negative real part) for all x. Then 0 is globally stable’. If the Jacobi conjecture is true, then (xN , xN ) is globally stable under the dynamics (7) with voter utility functions (1@). Thus, we find that with correlated voter preferences, a unique equilibrium of the system exists as above (suitably adjusted for the correlated strengths). This equilibrium is definitely locally stable, and (conditional on the Jacobi conjecture holding) also globally stable. 4.3. Non-quadratic (but concave) utility functions In Eq. (1) we assumed that voters preferred the party platform that minimized the squared distance from each voter’s ideal platform. We can extend Eq. (1) to the non-quadratic (but concave) utility functions defined by u (y)"!+ / (y !x ), (1A) v vi i vi i where / (t)'0 for tO0, / (0)"0, and /@ (t)'0 for t'0 and /@ (t)(0 for vi vi vi vi t(0. ¸emma 6. ¹here is a unique and globally stable rest point (xN , xN ) for the dynamical system (7) with voter utility function (1@@) provided /A (t)'0. vi Proof. Here, the explicit dynamics are yR 1"!+ p@(u (y1)!u (y2))/@ (y1!x ), vl vl l l v v v yR 2"!+ p@(u (y2)!u (y1))/@ (y2!x ), vl vl l l v v k and thus the dynamics of the difference in the platforms is given by yR 1!yR 2"!+ p@(u (y1)!u (y2))[/@ (y1!x )!/@ (y2!x )]. vl l vl vl l v v vl l l v
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 183
If /A '0 the difference of the /@ terms is positive if and only if y1!y2 is vi vi l l positive; hence, the difference y1!y2 decreases and My1"y2N is globally stable. A fixed point must therefore fulfill + / (y !x )"U (y )"0. Since /A '0 we v vi i vi i i vi know U@'0, that is, U is strictly monotonically increasing. Therefore, there is i i at most one zero xN , which is defined by U (xN )"0, for all issues i. Since the box i i B is forward invariant, Brouwer’s fixed point theorem guarantees the existence of at least one rest point in B, and consequently (xN , xN ) is unique. h Therefore, as long as voter utility functions are strictly concave, the existence of a unique, globally stable equilibrium for the system (which is related to the equilibrium given in Eq. (11), but implicitly defined by U (xN )"0 above) is i i guaranteed under non-quadratic utility functions. Thus, the general tenor of the results found in the initial model hold under this much wider class of plausible voter utility functions. 4.4. Non-concave utility functions Our stability and uniqueness results concerning equilibrium required strictly concave utility functions (a typical assumption for similar dynamic game models). However, some non-concavity in these functions might be reasonable if, for example, once platforms are a sufficient distant from a voter’s preferred position, changes that move the platform even further away have little impact on the voter’s utility. While /A '0 everywhere on R is possible, /A (0 on all R leads to a contravi vi diction, since we require that / has a minimum at 0 and hence positive vi curvature in a neighborhood of 0. We can, however, construct a model where / has positive curvature only in a very small neighborhood of the preferred vi voter positions, and /A (0 everywhere else. Again My1"y2N is invariant. But vi now we can have more than one equilibrium. The eigenvalues of the Jacobian at an equilibrium fulfilling y1"y2 will determine its stability. We find that the Jacobian is diagonal with entries (13) !p@(0)+ /A (yL !x ) vi l vl v which can be both positive and negative, and thus unstable equilibria are possible. We illustrate the potential behavior of our model in the case where the concavity assumption on u is violated via the following example. Consider voter utility functions of the form
C
G
(yj!x )2 vi u (yj)"!c2+ s 1!exp ! i v vi c2 i
HD
.
(1@@@)
184
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
Note that the quadratic utility function (1) agrees with (1A@) up to the third order in (y!x ) (in Eq. (1) the utility diverges, while here it levels off ). We further v observe that the curvature of u v Lu (yj) 2(yj !x ) (yj!x )2 v "!2s 1! k vk exp ! i vk , vk c2 Lyj c2 k changes sign at
C
D G
H
c Dyj !x D" . k vk J2 Given the utility function in (1@@@), consider a simple example with only two issues and two voters. Let x "(!1,1) and x "(1,!1), and let each voter have 1 2 s "1 for both issues. The mean voter equilibrium of (7) is the origin (0,0). As vi a consequence of Eq. (13), the origin is a locally stable equilibrium (sink) if c'J2 and an unstable equilibrium (source) if c(J2. The symmetry of the problem and the forward invariance of the box [!1,1]][!1,1] (Lemma 1) imply that there cannot be a globally stable equilibrium. Furthermore, any equilibrium on the invariant manifold My1"y2N is unstable. We could not explicitly solve Eq. (7) with utility function (1@@@) for all equilibria, so we numerically integrated the dynamical system. The results are summarized in Fig. 2. The figure shows six pairs of platform paths induced by parties starting randomly about (0,0). The stable equilibria are of the form y1"(m, m),
y2"(m,!m),
y1"(m, m),
y2"(!m, m),
y1"(!m,!m),
y2"(m,!m),
y1"(!m,!m),
y2"(!m,m),
with m+0.955 for c"1 (solutions with y1 and y2 exchanged also exist). Which of these locally stable equilibria is actually reached depends on the initial conditions. Note that none of the equilibria are symmetric-the parties agree on one issue and dissent on the other. 4.5. Discrete time dynamics As elections are (usually) held at well-defined time intervals, the behavior of a discrete time model, instead of the continuous-time dynamics explored above, is of interest. In the discrete model, a party updates its current platform by taking a discrete step in the direction of its payoff gradient at fixed time intervals.
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 185
Fig. 2. The platform dynamics for two parties are shown in platform-position space. Two voters with strengths equal to one on all issues and ideal points of (!1, 1) and (1,!1) were used. The mean voter equilibrium of Eq. (7) is the origin (0, 0). Voters had utility functions given by (1A@) with c"1. Six different initial platform pairs were randomly generated about the origin (each pair has the same marker and line style). Similar markers end up at the same equilibrium points.
To simplify notation, define Wk(y1, y2)"+ kº (y1, y2) and W(y)"(W1(y1, y2), y k W2(y1, y2)). Thus, the system of ordinary differential equations in Eq. (5) can be written as yR k"Wk(y1, y2), or simply yR "W(y). Furthermore, let LW1 LW1 Ly1 Ly2 J(y)" (14) LW2 LW2 Ly1 Ly2 give the Jacobian matrix of this dynamical system at position y"(y1, y2). Each party is allowed to update its platform with a step in the direction of its payoff gradient:
A B
yk{"yk#qWk(y1, y2),
(15)
where q represents the mobility of each party in issue space between elections. Discrete- and continuous-time versions of a dynamical system are closely related. In fact, yL is a fixed point of Eq. (15) if and only if W(yL )"0; That is,
186
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
discrete and continuous-time models share the same equilibria. The stability of the fixed points is determined by the Jacobian matrices, J (y) and J(y), for the d discrete and continuous cases, respectively. For the discrete-time model we have
A
L(y1#qW1) L(y1#qW1) Ly1 Ly2
J (y)" d L(y2#qW2) L(y2#qW2) Ly1 Ly1
A
"
1#q
LW1 LW1 q Ly1 Ly2
LW2 q Ly1
1#q
LW2 Ly2
B
B
"I#qJ(y),
(16)
where I now denotes the identity matrix. Note that this correspondence is valid for all the modifications of Eq. (5) discussed above. A fixed point, yL , of a discrete-time dynamical system is (locally) asymptotically stable if and only if all eigenvalues of its Jacobian matrix, J (yL ), lie within the unit d circle. Suppose yL is stable for the continuous-time dynamics. Then all eigenvalues of J(yL ) have a negative real part. Therefore, there is a q '0 such that y is 0 (locally) asymptotically stable in the discrete-time model (15) for all choices of q in the interval (0, q ). On the other hand, complex dynamics, including 0 deterministic chaos, are commonly encountered in discrete-time models that allow large moves in each time step. In our original model, as well as for the modifications described in Lemmas 4—6, the equilibrium is (at least locally) stable, as the Jacobian, J(yL ), has only real, negative eigenvalues in all these cases. Denoting the spectral radius of J(yL ) by o, where the spectral radius is given by the largest absolute-valued eigenvalue, we find the yL is also stable in the discrete-time model as long as 1!qo'!1, that is, for 0(q(2/o. Thus, the mean voter equilibrium will be (locally) asymptotically stable under discrete-time dynamics provided that platform adaptation is slow and/or elections are frequent. 4.6. Policy-dependent party utilities Suppose that each party cares about achieving an ‘ideal’ platform, designated by qk. To capture this case, we assume that a party’s utility is a convex combination of both the number of expected votes and the distance, D(yk, qk), between the actual and ideal platform. Thus, we replace º (y1, y2) in Eq. (5) with k (1!m)º (y1, y2)!mD(yk, qk), where m parameterizes the relative importance of k the actual versus ideological platform. Under this new assumption, the voting
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 187
dynamics are governed by yR k"Wk(y1, y2)!m[Wk(y1, y2)#Ck(yk)],
(17)
with Ck(yk)"+ kD(yk,qk). y Restricting attention to the box B, we may assume that Ck and its partial derivatives with respect to yk are bounded, by some constant M. Without loss of j generality, we assume that Ck and m are scaled such that M"1 and m measures the size of the ideological payoff component. Thus, m[Wk(y1,y2)#Ck(yk)] forms a regular perturbation (in the sense of Hirsch and Smale, 1974 (Chapter 16)) of the vector field W(y) and the following proposition holds: Proposition (Hirsch and Smale, 1974). Suppose yL is a hyperbolic fixed point of yR "W(y). ¹hen there is m '0 and a neighborhood N(yL ) such that for all m in 0 the interval 04m4m there is a unique fixed point yL (m)3N(yL ). ¹he para0 meter m can be chosen such that yL (m) is hyperbolic and has the same index (that 0 is, the same number of positive and negative eigenvalues) as yL . In addition, yL (m) is a continuous function of the perturbation parameter m and yL (0)"yL , the fixed point of the unperturbed vector field. Introducing an ideological payoff component will, in general, break the symmetry of Eq. (5) and, hence, also the symmetry of the mean voter equilibrium. Thus, in general, we find yL 1(m)OyL 2(m) even for small values of m'0. Using a perturbation approach that was originally developed for selection-mutation equations in theoretical biology given in Stadler and Schuster (1992), it is possible to estimate the location of a perturbed mean-voter equilibrium: C1(yL 1)
A B
yL (m)"yL #mJ~1(yL )
C2(yL 2)
#O(m2).
(18)
Here we have incorporated the knowledge that W(yL )"0. Thus, the distance of the equilibrium positions of the two parties is proportional to the parameter m. In the other extreme, when m+1, and a party’s behavior is mostly determined by its ideology, we may consider the voter-dependent term W as a perturbation. Consequently, each party will settle down to an equilibrium close to qk, since in the limiting case where W"0 we have a gradient system in which each party converges to a minimum of Dk(yk, qk) independent of the other parties.
5. Conclusions By employing the tools of evolutionary game dynamics we analyze a twoparty spatial-voting system composed of locally adaptive parties. Our results indicate that in such a system, the two platforms eventually converge to
188
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189
a globally stable equilibrium point located at the strength-weighted mean of the voters’ preferred positions. While the convergence is assured, the transient paths can be long and convoluted. The fundamental results are relatively invariant to a variety of modifications of the underlying model. Limited platform mobility, voter preferences correlated among issues, non-quadratic (but concave) utility functions, discrete adjustment dynamics, and ideological party preferences all have little influence on the underlying findings. However, the incorporation of non-concave voter utility functions can result in a variety of new behaviors, including multiple equilibria and, we suspect, limit cycles. The voting dynamics discussed here can also be extended to more than two parties. A recent study (Stadler, 1998) shows that the mean voter equilibrium can become unstable for three or more parties. Though the focus here has been on two-party spatial voting models, extensions to other phenomena are possible, for example, the underlying structure could be interpreted as an oligopolistic market in which firms locally alter sets of product attributes (platforms) to attract consumers (voters). Our model provides a necessary benchmark for the behavior of systems with agents that are limited in their abilities to directly solve for equilibria, and whose strategic choices are constrained by past actions. While we find that such systems display predictable behavior in the limit, we also note that the transient dynamics can be remarkable, and are worthy of further analysis.
Acknowledgements We thank Ken Kollman, Scott E. Page, and Ba¨rbel Stadler for many helpful comments on this research, and the Santa Fe Institute for general research support. Additional research support was received by Miller from the National Science Foundation (grants SBR-9411025 and SBR-9710014) and by Stadler from the German Max Planck Gesellschaft.
References Coughlin, P., Nitzan, S., 1981. Directional and local electoral equilibria with probabilistic voting. Journal of Economic Theory 24, 226—239. Coughlin, P.J., 1990. Candidate uncertainty and electoral equilibria. In: Enelow, J.M., Hinich, M.J., (Eds.), Advances in the Spatial Theory of Voting. Cambridge University Press, Cambridge, pp. 145—166. Friedman, D., 1991. Evolutionary games in economics. Econometrica 59(3), 637—666. Hirsch, M.W, Smale, S., 1974. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, Orlando FL, USA. Hofbauer, J., Sigmund, K., 1988. The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, UK.
J.H. Miller, P.F. Stadler / Journal of Economic Dynamics and Control 23 (1998) 171–189 189 Hofbauer, J., Sigmund, K., 1990. Adaptive dynamics and evolutionary stability. Appl. Math. Lett. 3(4), 75—79. Kollman, K., Miller, J.H., Page, S.E., 1992. Adaptive parties in spatial elections. American Political Science Review 86, 929—937. Malthus, T., 1798. An Essay on the Principle of Population. J.M. Dent (1933), London, UK. Meister, G.H., 1982. Jacobi problems in differential equations and algebraic geometry. Rocky Mountain J. Mathematics 12, 679—705. Plott, C.R., 1967. A notion of equilibrium and its possibility under majority rule. The American Economic Review 57, 787—806. Samuelson, L., 1984. Electoral equilibria with restricted strategies. Public Choice 43, 307—327. Stadler, B.M.R., Adaptive platform dynamics in spatial voting models. Ph.D. Thesis, University of Vienna, 1998. Stadler, P.F., Schuster, P., 1992. Mutation in autocatalytic networks — an analysis based on perturbation theory. J. Math. Biol. 30, 597—631.