Catastrophe theory and dynamic games

Quality and QuantiO', 1.4 (1980) 519-545 5 19 Elsevier Scientific Publishing Company, Amsterdam Printed in The Netherlands

CATASTROPHE THEORY AND DYNAMIC GAMES

NORMAN SCHOFIELD

Department of Economics, University of Essex, Wivenhoe Park, Colchester, Essex, Great Britain

1. Introduction

The notion of a dynamic system is fundamental in many of the natural sciences and in some social or behavioral sciences, such as ecology or economics. The qualitative theory of dynamic systems, sometimes called catastrophe theory, has recently been used to construct models of systems which can exhibit discontinuous behavior. Of more signifi- cance perhaps than the construction of simple models are the fundamental theoretical concepts of genericity and structural stability.

Although simple dynamic models, such as the prey predator model, are well behaved "almost always," more complex systems can display quite exotic behavior. Indeed simple difference equations, models of population growth, inter species competit ion and duopoly competition can display the form of indeterminacy known as chaos.

The concern of this paper is to use the conceptual notions of genericity and structural stability to provide some hints as to tile possible structural features of a political economy. The behavior of a political economic system is the result o f optimization by collections, or coalitions, o f actors in the society. The emphasis of the theory of dynamic games presented here is on local behavior rather than tile global notions o f game theory. In voting games, for example, it is now known that the set o f natural equilibria, the core, will generally he empty. Instead we consider the phenomena of cycling. For voting games the cycle set will be dense, and consequently we may regard such games as chaotic.

While the notion of natural equilibria is fundamental in economic and political theory, these results suggest the possibility that behavioral systems may in fact be highly unpredictable and indeterminate.

00334177/80/0000-0000/$02.25 �9 1980 Elsevier Scientific Publishing Company

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2. An Ecological Dynamic System

To introduce the notion of dynamic system consider the well known Vol t e r r a -Lo tka prey predator equations:

_ dp P' dt - (Aq - B) P

(1) dq q' = d-t = (C - Dp) q

Here p is the number of a predator species, q the number of its prey. In the absence of prey, the predators die of f at a rate proportional to the size of the predator population (B > 0). The predator per capita food supply we may take to be proportional to the prey population size (A < 0), and this gives us the predator growth equation. On the other hand, the prey is assumed to have a positive growth rate (C > 0) in the absence of predators, and to be depleted by the predators (D > 0).

Another way of looking at the differential equations (1)is to say that at each point (p, q) in an appropriate space W (in this example, R+ 2, the positive quadrant of two*dimensional euclidean space) the dynamical system X defined by eqns. (1) assigns a vector in what is called the tangent space at (p, q). Thus we can write:

X(p, q) = (p', q') = ((Aq - B ) p , (C - Dp) q) (2)

To integrate this equation, we suppose that the system is initially (at time t = 0) at a point (p(O), q(0)), say, and represent the points at any future time t, by a curve in W,

c(t) = (p(t), q(t)) .

For this curve to represent the integration of eqns. (1) we require that:

de(t) dt

- c ' ( t) = (p'( t) , q ' ( t ) ) = X(p ( t ) , q( t ) ) = X(c ( t ) ) (3)

Thus, if I = [0, T) is a half open interval in R+, and c: 1 -~ W is a curve such that c'(t) = X(c ( t ) ) we call c an integral curve of the dynamic system, or vector field, X.

An orbit through the point (p(0), q(0)) is simply the set of points that can be reached from (p(0), q(0)) by some integral curve. The phase portrait P(X) of X is the set of orbits o f X.

Fig. l(i) shows a typiCal phase portrait for the vector field of eqns. (1).

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C

Ci)

I

. . . . .

i

C

II ..Pl = ~- ~q

I B

q

( i i )

Fig. 1. (i) The phase portrait of a prey predator system with unlimited growth of prey. (ii). The phase portrait of a prey predator system with limited growth of prey.

The po in t (Pl, qt) = (C/D, B /A) is an equilibrium; in fact it is stable.

To give a de t in i t ion a s tabi l i ty in general , let x represent a po in t in the s ta te space, and x: I --> W represen t an integral curve o f a vec to r

field X on W, so tha t x'( t) = X(x(t)) . Call x an equi l ibr ium o f X when-

ever X(x) = 0. I f x0 is an equi l ib r ium o f X, then say it is s table i f f for

every n e i g h b o r h o o d U o f Xo in W there is a n e i g h b o r h o o d V o f Xo in U such tha t any integral curve x : 1 --> W, wi th x (0 ) in V, is def ined and in U for all t > 0 (see Hirsch and Smale, 1974). I f f u r t h e r m o r e Xo is a s table equi l ib r ium and for any integral curve s tar t ing in V, the l imit

o f x(t), as t -+ o~, is Xo, then call Xo asymptotically stable. Obvious ly while (Pt, ql) is a s table equi l ibr ium it is no t a s y m p t o t i -

cally s table.

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However consider eqns. (4) where the prey exhibit logistic growth, that is they are restrained by a possible resource limit:

p ' = (Aq - B ) p (4) q' = ( C - D p - Xq) q

For this dynamical system, call it X1, the equilibrium is given by:

(Pl, q t ) = D A '

Notice that this equilibrium is asymptotically stable, and near to the equilibrium of the system of eqns. (1).

The important point of comparison between Fig. l(i) and Fig. l(ii) is that they are qualitatively d i f f e r en t .

To be more explicit, if X~ and X2 are two vector fields on W, say they are e q u i v a l e n t iff there is a homeomorphism h of If onto itself such that 61 is an orbit in P ( X I ) iff h(q~m) is an orbit in P(X2) . A homeomorphism is a continuous map with continuous inverse. It should be clear that there is no homeomorphism taking Fig. l(i) to Fig. l(ii), which means that the dynamical systems of eqns. (1) and (4) are non- equivalent.

Now the phase portrait of the system X given by eqns. (1) is "un- usual" in the sense that any perturbation to eqns. (4), for no matter how small a parameter X, gives a qualitatively different dynamical system.

We can formalize this comment as follows. Let X(If) be a certain class of vector fields on If endowed with an appropriate topology. Say X in • is s t ruc tu ra l l y s tab le iff there is a neighborhood 0 of X in • such that for any Y in 0, X and Y are equivalent.

Now for X given by eqns. (1), there is as near to X as we like (that is to say for as small a X as we like) a dynamical system like the X1 of eqns. (4). Thus X is not structurally stable. As a matter of fact Fig. l(i) represents a conservative system, like a simple harmonic oscillator, where no energy is lost as "heat". Fig. 1 (ii) on the other hand repre.- sents a structurally stable system like a simple pendulum which eventually "comes to rest" (see Zeeman, 1968).

The next question to ask is whether almost all vector fields are structurally stable.

More generally say a property K of vector fields in • is gener i c if there is a dense set, defined as the countable intersection of open dense sets in • all of whose members satisfy K.

Structural stability has been shown to be generic for If two-dimensional and compact (Peixoto, 1962) but in general this is not so (Smale, 1966).

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In two dimensions, an equilibrium or critical point of a vector field may either be a sink (which "at t racts" integral curves), a source (which repels integral curves) or a saddle (which does both).

Peixoto obtained his result by showing that the set of structurally stable systems is characterized by the properties:

(i) there is a finite number of equilibria and closed orbits; (ii) orbits that are not closed start at sources or saddles, or wind

away from "repellor" closed orbits, and finish at saddles or sinks or wind towards "a t t rac tor" closed orbits; and

(iii) no orbit connects saddles.

(i)

Fig. 2. (i) A s t ructura l ly unstable vector field. (ii) The pe r tu rbed s t ructural ly stable vector field.

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To illustrate, in Fig. 2(i) is shown the phase portrait of a structurally unstable dynamic system. Orbit (a) is a repellor, orbit (b) is an attractor, point (c) is a sink, point (d) is a source. On the other hand orbit (g) connects the saddle points (e) and (f).

Any perturbation of Fig. 20) gives Fig. 2(ii) where the saddle con- nection is broken (see Hirsch and Smale, 1974 lbr this example).

To return to the phase portrait of Fig. l(i), since the number of closed orbits is not finite, this dynamical system is immediately seen by the Peixoto theorem to be structurally unstable.

The point however of the type of qualitative analysis briefly outlined above, is that it provides one way of studying dynamical systems without having to completely solve the equations, when this task may be extremely difficult.

Just to illustrate, Gilpin (1975) has studied predator prey systems of the form:

p ' = ( A q - B + Ep - F p 2) p

q' = (C - Dp + Gq H q 2 ) q (5)

where E, G are co-operation parameters, and F, H are competit ion parameters. Gilpin was interested in the possibility that predator co-operative efficiency could become sufficient to lead to prey, and thus predator, extinction. Since such a complex system may be difficult to completely analyze, it is none the less useful to characterize such systems by certain significant qual i ta t ive features and determine whether these persist under slight perturbation. The motivation of course is that features which disappear under perturbation are unlikely to be manifested by actual systems. Although the dynamical system associated with a two species prey-predator process is typically well behaved (May, 1972), this is not the case with multispecies systems.

Consider for example an n-species system of the form:

d)ci--= xiMi(Xl, Xn) for i = 1, ..., n (6) dt ....

where Mi: R~+ -~ R are smooth functions in (x~ . . . . . xn) representing a resource-scarce competitive situation (Smale, 1976a), and xi represents the number of the ith species.

For example, with n = 3 we may have:

dxl d t - x l ( A - B x l -- Cx2 - D x 3 ) and similarly for i = 2, 3 (7)

As May and Leonard (1975) have shown, this three-species dynamical process can exhibit interesting behavior. Suppose we normalize and

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let x~ = (1, O, O) represent tile state where species two and three are extinguished. From a typical initial state x = (Xl, x2, Xa) the evolution- ary path of development approaches the simplex:

A 3 = ( X ~ , X ~ , X ; )

swinging near to x~ then x~ then x~. In other words each species in turn will become dominant and in turn will be almost extinguished.

In a general n-species system (n > 4) of this form, the simplex A n = (x~ ... . , x,,) will be an attractor of the process. In particular with n > 5, as Smale (1976a) has observed, the flow on A n may have exotic properties. Since structural stability is not necessarily generic (Smale, 1966), the system may not be approximated by a structurally stable one, and may indeed be extremely complicated (Smale, 1967).

Since the system of eqn. (6) is in some way representative of competitive economic as well as ecological systems, these results raise the question whether general systems of this type are in any way capable of analysis. If small perturbations in exogeneous parameters may change the qualitative properties of the system, it is difficult to see how empiri- cal analysis of such a process can give definite clues about tile essential features of the system.

A somewhat similar problem occurs in discre te rather than dynamical processes. Instead of the one-species dynamical system:

d x / d t = x ( A - B x )

suppose:

xt+ 1 = x t ( A _ B x t) (8)

representing the transition for the species descriptor, x t , at time t to that at time (t + 1). Li and Yorke (1975) studied the general non-linear system,

x t+l = F ( x ~) (9)

A point x of such a process is per iod ic iff x = F r ( x ) , where F" is the rth iterate of F, and is asymptotically periodic if F r ( x ) - Fr ( p ) -* 0 as r -~ ~, for some periodic point p. They found that there was typically an uncountable subset S of points containing no asymptotically periodic points. This result implies that such discrete processes are non repeti- tive: Li and Yorke coined the phrase chao t i c to refer to such processes. For such a chaotic process statistical analysis of the time series of outcomes would not necessarily enable the investigator to infer the form of the transition function F. Indeed one would probably infer that the

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best fit was the autoregressive system,

xr+l =xt + e (10)

with most of the interesting features of the process residing in the resid- ual error term, e. For example public opinion may well change by a discrete transition rule like (9) even though statistical .analyses (Frey and Garbers, 1971) suggest that eqn. (10) is the best fit to the data. Later analyses have explored the bifurcation properties of discrete processes, by determining the parameter values (A, B) where multiple attractors appear (Oster and Guckenheimer, 1976; May, 1974, 1975 etc.). A further feature of such processes is that typically, for any initial point x0 and almost any other point x, there is some r such that x = F'(xo). In other words the discrete process, F, can wander almost anywhere. Such a process certainly deserves the term "chaotic".

An extension of system (9) to a two-player model,

(xt+l , y t + l ) = (El (yt), F2(xt)) = d~(X t, y t ) (11)

has been carried out by Rand (1978). The idea of this "Cournot duopo ly" model is that each player (1 and

2) has a utility function ui(x, y). For an initial value (x 1, y l ) player 1 modifies x ~ to x 2 to maximize u( , y~), assuming y fixed. Even though fixed points (given by (x, y) = qS(x, y)) occur, they will not be stable. Indeed the process is chaotic, since from any initial point the process can go arbitrarily close to almost any other point.

It is possible for example that any exchange process involving complex time lags could be chaotic in this way. For example consider the production of a commodity , such as copper or wheat. Production at time t + 1 is determined by decisions at time t and before over, say, capital investment in mines or acreage to be planted. Typically these decisions are made on the basis of demand at time t. Consump- tion at time (t + 1) may also be dependent on previous decisions based on earlier levels of supply. Cons.equently one obtains the wildly fluc- tuating price levels and demand/supply imbalances of the so called "pig cycle," observable almost always in the relevant commodities markets.

Although dynamical systems of the kind used to model inter- connected ecological or economic systems are well behaved in low dimensions, in higher dimensions, as Smale's result showed, quite complicated, even counter intuitive, behavior might occur. Even in simple discrete processes involving one or two variables, behavior will be chaotic in the sense that "almost anything" can occur. The next section will briefly review gradient and paretian dynamic processes, and then in

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the following section indicate why coalitional processes will be chaotic in the same way as the discrete processes.

3. Gradient Dynamic Systems

In this section we deal briefly with dynamical systems which essen-- tially involve the maximation (or minimization) of a single function. Such systems are common representations of physical processes, such as when a potential energy function is minimized. In this context cata-

( i )

xl x 2

P OO

P (X)

x 1

f u

x 3 R

Fig. 3.

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strophe theory provides some explanation of why small variations in parameters, or exogeneous variables, may produce sudden variations in internal variables.

Consider for example Fig. 3(i), which presents the graph of a smooth function u: W -+ R where the state space W = R.

Let X be the g r a d i e n t vector field X ( x ) = d u ( x ) / d x :

X has two stable equilibrium points, namely X l and x3, and an unstable point X2. The phase portrait of X is also shown. If the system starts at some point " to the left" of x~ , then it will "home in on" the stable equilibrium x~. Now suppose that u is parametized by some external variables and these change in such a way that x~ and x2 collapse to form a "degenerate" critical or inJTection point. If the system locally maximizes u, then only after degeneracy occurs can the observed state change to Xa. This change, when it occurs, will be "fast" in comparison with the "slow" parameter change. To give a possible example, Gribbin (1976) has discussed a possible hypothesis concerning ice ages. Two different stable polar ice states are supposed to exist, corresponding to different equilibria of a complex potential function. This function is in turn dependent on exogeneous variables (like the density of interstellar dust, and the behavior of our own sun). Small changes in these exogeneous variables may destroy the stability of a low ice state, and induce a rapid growth in ice coverage.

The point of Thom's work (1972, 1975) has been to characterize such possible rapid changes, or catastrophes. For example if u = - x 4 as in Fig. 3(iii), the perturbations of u are of the form:

u 1 = - x 4 + a x 2 + b x

The vector field of Fig. 3(ii) is structurally unstable, or catastrophic, since in any neighborhood of u, there exist functions whose gradient vector fields are like those of Fig. 3(i) or 3(iii) and are non equivalent.

Consider now the function u = - x 6. An unfolding of u, of the form:

U 1 = --X 6 + ax 4 + b x 3 + c x 2 + d x

is given in Fig. 4. The critical points {x~, x2, x3, x4, Xs} can collapse to form degeneracies in four different ways (xl to x z say) and these "catastrophes" are captured by the four-fold parametization of the unfolding.

Isnard and Zeeman (1976), for example, used the b u t t e r f l y catastrophe to illustrate the formation of compromise opinion (x3) between a hawk position (xs) and a dove position (x~) on a foreign policy issue.

To return to Fig. 3(i), let us call the set of points for which d u ( x ) /

dx = 0, the set I O ( u ) of i n f i n i t e s i m a l o p t i m a . The point x3 call a g loba l

P(X)

U

Fig. 4. The phase portrait of the gradient system for the funct ion u 1 b x 3 + c x 2 + d x .

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= x 6 + a x 4 +

optimum, GO(u) = {x3}, since u(x3) > u(x) for all x v~ x3. Let SO(u) be the set o f stable equilibria, or local opt ima, in this example SO(u) = {x~, x3}, and note that GO(u) c SO(u). Later we shall be using the difference between global, local and infinitesimal opt ima in the context o f game situations. For the momen t note that from the topological viewpoint the global op t ima are not very interesting, since optimiza- t ion processes may be unable to distinguish or reach such opt ima.

In the above examples we have considered a "de te rmin is t i c" dynamic process o f the form X(x) = du(x)/dx. For a state space h' o f d imension w, such a system would be o f tile form:

= du(x), where we write du(x) = ( 6u 6u ] X(x) (12) , \ ~ X l . . . . . ~ X w / x

One can however consider a "slightly inde te rmina te" gradient system const ructed in the following way.

At the point x in W let Hu(x) = (v: du(x) �9 v > 0} be the set o f vec- tors which have positive project ion on the vector du(x). Then Hu(x) is:

(i) convex: i f vl, v2 ~ Hu(x) then Xvl + (1 X)v2 E Hu(x) for all X c [0, 1],

(ii) con ic 'v E Hu(x) ~ Xv c Hu(x) for all X > 0. (iii) hal f open: Hu(x) is conta ined in (v: l.u > 0} for some tangent l. The set IO(u) can be seen to be equal to {x: Hu(x) = qS}, for W open.

If X is a vector field on W with the proper ty that X(x) c Hu(x) for x

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IO(u) and X(x) = 0 for x c IO(u) then we will call X a gradient field for u. Note that a vector field satisfying eqn. (12) is a deterministic

gradient field for u, and is characterized by a "law of motion". On the other hand a general gradient field is characterized not by such a law of motion but rather by sets of inequalities which determine a range with- in which change can occur. This seems more natural when discussing a dynamical framework for social behavior (Smale, 1976b). Note however that any two gradient systems for u will be equivalent, so this is a natural topological way of looking at such processes.

We now introduce the paretian gradient systems of Smale (1973). Let U(W) be the topological space of second differentiable functions

W -, R. Consider a society N each of whose members, i, has a utility function, ui ~ U(W), which represents this individual's preference. Thus individual i prefers a state x to a state y (and is presumed to try to bring about state x rather than state y) whenever ui(x) > ui(y). A collection of smooth utility functions for the society is called a smooth profile, written u, and regarded as a member of a topological space U(W) u. Ear- lier we introduced the idea of a cone Hu(x) at the point x, defined by the function u. When ui is a utility function for individual i, write Hug as Hi, and regard Hi as a conic field on W. We shall call Hi(x) the /-preference cone at x, and Hi the/-preference conic field.

Such a preference conic field has the following integrability property.

A conic field is called S-continuous iff whenever X(x) ~ H(x) at any x ~ W and any vector field X on W, then there is a neighborhood U of x in lg such that X(x') ~ H(x') for all x' in U.

What this property means is that locally there is a vector field Xi whose integral curves represent changes in the state (of the world) which i prefers i.e. if t' >. t then ui(c(t')) > ui(c(t)).

For the whole society now define the N-preference conic field by:

H N = O H i i E N

and observe that HN will be S-continuous. Let IO(N, u) = (x: Hu(x) = r and say a vector field X is gradient

for HN (or u) iff X(x) = 0 whenever x ~ IO(N, u), and X(x) c H ~ x ) whenever x r IO(N, u).

Such an N-gradient field represents changes which benefit all members of the society N. Thus if c is an integral curve of X, ui(c(t'))> ui(c(t)) for all in N, whenever t' > t.

In economics, game theory and social choice a state x is called a pareto optimum for a society N iff there is no x' in W such that:

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Ui(X' ) ~ lli(X ) all i in N and: u/(x ') > u/(x) some j in N

Since the set o f pareto opt ima is an analogue of tile set of global opt ima of a single funct ion , we may write GO(N, u) for the pareto op t imum set o f a society, and note that GO(N, u) c IO(N, u).

Call a point x a local pareto op t imum if for some ne ighborhood U of x, x is a pareto op t imum in U, and let LO(N, u) be the set o f local pareto opt ima. As in the single funct ion, or 1-gradient, case let x be a stable op t imum (in SO(N, u)) i f f any integral curve o f an N-gradient field for u stays near to x if it starts near to x. Obviously GO(N, u) c LO(N, u) c IO(N, u) (see Simon and Titus (1975) for more details). As it happens local opt ima and global op t ima need not be stable (Smale, 1974), even when the ut i l i ty funct ions are convex, a l though these opt ima coincide when t h e uti l i ty funct ions are strictly convex. Here a funct ion u: W ~ R is (strictly) convex iff u -1 [t, ,'~) is (strictly) convex for all t in R. When the ut i l i ty funct ions satisfy certain "differentiable convex i ty" condi t ions then SO(N, u) and IO(N, u) coincide.

To illustrate the difference between stable and infinitesimal opt ima consider Fig. 5 (due to Smale, 1973).

Here the state space is the two-dimensional square W 2 = [0, 1 ] 2. The ut i l i ty funct ion u j = - ( x + y) is convex, and u2 is non convex with a stable critical point at (1, 1). The stable opt ima set consists o f the two arcs GAB and CFE. The arc CDE is a set of unstable opt ima: the two points C, E are thresholds o f stabil i ty (Rand, 1976), or degenerate crit-

I I ~ , j a

G A Fig. 5. The infinitesimal optima set and phase portrait for a generalized Edgeworth box trade situation.

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ical points of u, rather like inflection points. If we consider points a, b we observe that Ul(b) > ul(a) so that a is not a global optimum. It is however a local optimum. If two individuals with these utilities found themselves at point a in this Edgeworth box trade game, there would be no integral or admissible curve along which they could trade to mutual satisfaction so as to reach a global optimum like b. On the other hand if they start from an unstable optimum like c, any slight perturbation away from c allows them to trade to a point like b. The reason for emphasizing the points of SO(N, u) rather than GO(N, u) is that stable optima could be final outcomes in such a system, when individuals are myopic or without knowledge of the nature of the "world" some distance from their present position. Thus integral curves are a realization of possible changes in an incrementally changing, or dynamic, system.

Any N-gradient field for u is characterized by the stability properties of the set IO(N, u). In Fig. 5 a phase portrait P(X) for some gradient field for u is shown. Any two u-gradient fields will have homeomorphic phase portraits, and thus be equivalent.

Similar to the 1-gradient case, we may regard U(W) u as a parametization of all N-gradient fields on W, and consider questions of structural stability of these fields. For example suppose in Fig. 5 we perturb u ; (Ul, u2) continuously so that at u = u ~ say, the component CDEF col- lapses to a degenerate critical point. This profile u ~ may be called catastrophic since near to u ~ will be profiles with phase portraits like that of Fig. 5, and also profiles whose optima are all stable. These different phase portraits are not homeomorphic, and thus u ~ is not structurally stable.

We can go further to determine the generic features of the infinitesimal optima set.

In our example, observe that for all points on IO({l, 2}, u), except for those on the arc GA, the direction gradients are semi-positively dependent (i.e. point in opposite directions), with Xldul(x) + X2du2(x) = 0 for some Xl, X2 both non negative, but not both zero.

More generally, for a society N of n players, at any point x, let d~(x) be the semi-positive span of the direction gradients {dui(x): i N}. Then d~(x) is a kind of generalized direction gradient for the society. If W is without boundary, then:

HN(X) = ~ iff 0 ~ dN(X)

Results from singularity theory (Golubitsky and Guillemin, 1973) can be used to determine the properties of the infinitesimal optima set, IO(N, u).

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Say that a property K which can be satisfied by some class of profiles in U(W) x is generic iff the class is defined by the countable intersection of open dense sets in U(W) x. If K is generic then {u ~ U(W)N: u satis- fies K} will be dense, and indeed open dense when W is compact.

Definitiotz ] (i) Let rank [dx(x)] be the number of linearly independent {dzli(X):

i ~ N } . (ii) Let S(N, u) be the set of points in W such that rank [dN(x)] <

rain(w, n), where w = dimension [W]. (iii) L(N, u) = W \ S(N, u). Since IO(N, u) is the set of points (other than on the boundary) such

that {du~(x): i ~ N} are semi-positively dependent, it is clear that IO(N, u) c S(N, u), when w > n.

Srnale's Proposition (19 73, 19 74) If w > n, then S(N, u) generically has the structure of an (n - 1)-

dimensional manifold with boundary various lower dimensional sub- manifolds. Under additional conditions IO(N, u) is an (n - 1)-dimensional submanifold.

Furthermore since S(N, u) is a manifold of dimension strictly less than dimension (W), L(N, u) will be generically dense in W. Thus so will be W \ IO(N, u).

To illustrate this proposition consider Fig. 5, the Edgeworth box dia- gram. Here n = 2 and IO(N, u) consists of two one-dimensional compo- nents.

Finally, suppose ul, u2 are profiles in U(W)'%, U(W) N2 respectively. The intersection, lO(Nl, ul)c~ IO(N> u2), will generically be of dimension at most nl + n2 w - 2. For example let n~ = n2 = w = 2. Then both IO(NI, u~) and IO(N> u2) will be 1-dimensional and will intersect in a point (of dimension 2 + 2 -- 2 - 2). We use this in the next section.

4. Dynamic Games

In this section we will at tempt to use the general orientation devel- oped in the previous section to present a way of looking at coalitional games. What we will be concerned about is an appropriate notion of equilibrium in co-operative situations. One way of starting is to introduce the notion of a social preference function a. Suppose the members of a society have preferences which are represented by a smooth "utility profile" u, and there is some rule adopted by the society which

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determines preferences for the society. We may write xo(u)y to mean that under the given rule, and with the profile u, the society prefers x to y. Thus we might write o: U(W) ~ ~ B(W) where B(W) stands for the set of binary relations on W.

One usually supposes that o(u) always be irreflexible (i.e. not(xo(u)x) for any x) and asymmetric (i.e. xo(u)y = not(yo(u)x)) . In this case o is called a strict preference function.

We are here concerned with simple voting games. A coalition M, or subset o f N, is called winning if:

ui(x) > ui(y) for all i ~ M ~ xo(u) y

The class of winning coalitions is wri t tenO. The rule, o, is called a simple game whenever xo(u)y iff x is preferred to y by all members of a winning coalition. For example simple majority rule is defined by taking c/) to be all coalitions with strictly more than n/2 players. We shall write om for this rule, and oq for the simple "q-game" defined by:

O q = { M c N : IMI > q}

It is well known that simple games cannot in general be well behaved. For example consider 02 when the preferences of the three individuals are represented by:

1 2 3

x y z y z x z x y

Here xa2(u)y since ~1, 3} both prefer x to y. But similarly y beats z and z beats x. This "cycle" z ~ y --, x --, z is a form of "irrationality" of the social rule o. On the other hand consider the "paretian rule," on, effectively discussed in the previous section. This rule cannot be cyclic. A second rationality property of a rule is the existence of a global core:

GO(N, u,o) = ix ~ W: there exists no y such that yo(u)x)

Since in general we assume the grand coalition, N, is winning it would be the case that GO(N, u, o) belongs to the global pareto set GO(N, u). While the paretian rule, on, is presumably well behaved in the sense that it is acyclic, most economic or political decision problems are not simply paretian, since individuals are generally able to act either singly, or in groups, without permission from the rest of society. A fundamental question for pluralistic conceptions of society is whether this interac- tion and cross-membership of different groups leads to natural equi-

535

( i )

R

p(x~2) - i Go ( [ 1 ,@u) 1

P ~x~) t ~(f2,31, ~) I P ( x~ ) �9 I Go({1.3t,~) H

( i i )

\ u3

~ " < \

X 1 X 4 X 5 X 2 X 6 X 3

P 0(23 ) ] ~

Fig. 6.

libria and social rationality. We seek to determine those situations under which cyclicity and the emptiness of the core occurs in voting games.

Consider the three-person majority rule voting game with a profile as in Fig. 6(i), due to Kramer and Klevorick (1974). Write GO({i, j}, u) for the optima set, and P(Xi/) for the phase portrait, for the three different pairwise {i,/} gradient systems.

It is clear that point x2 belongs to each of the three different coalition optima sets, and thus xz ~ GO({1 ,2 , 3}, u, oz). Note also that oz is acyclic for this profile. This type of example has led some scholars (Downs, 1957) to the conclusion that majority rule has some useful properties: here the equilibrium, x2, is a point rather than the larger pareto set GO({1, 2, 3}., u). However consider Fig. 6(ii) where the util-

536

ity function u3 has been slightly perturbed to produce a local mini- mum at xs, and a local maximum at x4. After this perturbation (1, 3} both prefer x4 to x2, so x2 can no longer be a global optimum. On the other hand (2, 3) both prefer x6 to x4, and (1, 2} both prefer x2 to x6. Thus the perturbation induced in the profile u destroyed both acyclicity (since there is now a cycle xz ~ x4 ~ x6 -+ x2) and the existence of the global core.

Indeed Rubinstein (1979) has shown that in any majority rule voting game on a policy space, if there is some u such that GO(N, u, ore) 4:0 then in any neighborhood V of u in U(W) ~, this time with the Kannai topology, there is some u' such that GO(N, u', ore) = 0. In other words the set of profiles such that GO(N, u, o~) 4:0 is nowhere dense.

Rubinstein's result indicates that in seeking equilibrium properties of voting games, we either must weaken the notion of the core, or restrict preferences in some way.

Observe that in Fig. 6(i) each utility function is convex. In one dimension if the utility functions are convex then the profile is single- peaked. The general property of single-peakedness has been shown to be sufficient for the acyclicity of major rule (Sen and Pattanaik, 1969, for example). However convexity of preference does not imply single- peakedness in dimensions greater than or equal to two (Kramer, 1973; Schofield, 1977a) and so convexity cannot be used in general to obtain rationality in spatial voting games.

Greenberg's Proposition (19 79) Suppose individual utility is convex, and let % be a q-game with n

players on a w-dimensional policy space, W.

(i) w < q ~ ~ GO(N,u,%)4:O, allu. n - q

(ii) W>~ q-- =~ there is some u such that GO(N, u, %) = 0 . n - q

To illustrate, suppose n is odd = 2k - 1, and consider majority rule, q = k. For (i) to be satisfied, we require:

k w < q -

n - q k - 1

Thus even if (n, q) = (3, 2), the core always exists only if w = 1. This of course is consistent with Fig. 6(i).

Instead of restricting attention to convex preferences, one may consider general preferences and a weaker core notion.

537

Definition 2 For a simple voting game o, with@ its class of winning coalitions:

(i) Let@ (x) = {M r HM(x) 4= (a}. (ii) Let Ho(x) = OHM(x), where the union is over@ (x). (iii) Define the infinitesimal core to be:

lO(N, u, o) = {x: Ho(x) = {?}

(iv) Say x ~ IO(N, u, o) is stable iff for any nbd. U of x in W there is a nbd. V o f x such that:

{y such that yo(u)z: z ~ V} c U.

Let SO(N, u, o) be the set of stable optima. (v) Say x in IO(N, u, o) is a local optima iff there is a nbd. U of x

such that for no y in U does yo(u)x. Let LO(N, u, o) be the set of local optima.

In Fig. 6(ii) the points {x4, xs, x2} are all infinitesimal optima, while (x4, x2} are both stable and local optima. Indeed in one dimension Kramer and Klevorick demonstrated that a stable optimum generically exists for simple majority rule. See also Salles and Wendell (1977).

Greenberg's proposition shows however that in two or more dimensions the existence of even the infinitesimal core is problematic.

Consider the case (n, q) = (4, 3). In two dimensions, since w < 3/1, the core is non-empty with convex preferences. Fig. 7 shows some of the various configurations which are possible. However, for (n, q) = (6, 4), 2 ~> 4/2, thus the core may be sometimes empty in even two dimensions. To determine what happens in this case, we proceed as follows.

Definition 3 (i) Say x is locally cyclic for o, iff, in any neighborhood U of x, there

is a finite sequence of points {Yl . . . . . Yr}, all in U, such that:

xo(u)y 1, Y 1 o(u)y2 ... . . yro(u)x

Write LC(N, u, o) for the set of locally cyclic points. (ii) Say x is infinitesimally cyclic for o, iff Ho(x) is not half open i.e.

there is no vector l such that l.v > 0 for all v ~ Ho(x). Write IC(N, u, o) for the infinitesimal cycle set.

By Schofield (1978a) it is known that IC(N, u, o) r LC(N, u, o). Thus if IC(N, u, o) vs ~ then o(u) will be cyclic.

Proposition 1 (Schofield, 1980a) In a q-game if dim(W) /> q then IO(N, u, Oq) is generically empty.

538

profi le u 1

[o (1) lo (2)

[ o (4 Io (3)

profi le u 2

io( 1 )

[o (4 1o(3)

Io(I )

prof i le u 3 0-3)

/ ' 0 (4 ) / ' 0 ( 3 )

Fig. 7. The non-empty core in the game (n, q) = (4, 3) wi th convex preferences in two dimensions. I O ( i ) stands for the bliss point of player i.

Indeed if dim(W) > q then IC(N, u, Oq) is generically dense. For example, we have seen with (n, q) = (4, 3) that 10(o4) exists in

two dimensions. By Proposition 1 in three dimensions the core lO(N, u, a) is almost always (i.e. generically) empty. Moreover IC(N, u, o) is generically non-empty. Indeed the cycle set in the whole of the pareto set IO({1,2 , 3, 4}, u). For that mat ter any majority rule game with n even is almost precisely like the game (n, q) = (4, 3).

Consider now the game (n, q) = (6, 4) in two dimensions. In Fig. 8(i), the core IO(N, u, o) is non-empty and identical to the bliss point I0(1 ) o f player 1. Moreover for sufficiently small perturbation u' of u, the core is still non-empty. By slight abuse of the terminology of section 2, say that the core, or infinitesimal optima set, in Fig. 8(i) is structurally stable. If we perturb the profile sufficiently however we destroy the core and create the cycle set IC(N, u", a) of Fig. 8(ii).

The situation for (n, q) -- (6, 4) in three dimensions is presented in

539

( i )

(ii)

I o ( 2 )

Io(3)

/o(6) ~ = lo(u'~

Io (5) Io (4) /o(2)

~ t o (3) Io (6

~ Io(4)

dul du 2 dG2

( i i i ) ~: du6 du 3 Y du 8 du 3

d d

du 4

Fig. 8. The game (n, q) = (6, 4) wi th convex preferences in two dimensions. (i) The structurally stable core IO(u, 04). (ii) An empty core and structurally stable cycle set (shaded). (iJi) Cyclic preferences at x and acyclic preferences at y.

Fig. 9. The core is given by the intersection of the opt ima sets 10((1, 2, 3}), 10((4, 5}), I0({6} ). By Smale's proposi t ion the first two intersect in an object o f d imension 3 + 2 - 3 - 2 = 0, a point. But two zero- dimensional objects do not generically intersect in three space: hence the core is generically empty . Appropriate , and small, per turbat ions of the profile o f Fig. 9 will result in a non-empty cycle set. One may regard profiles under which the core exists as "catastropll ic profiles," since per turbat ions result in quite different , chaotic, flows.

Proposition 2 (Schofield, 19 78b) For major i ty rule, ore, with n even:

540

Io(4)

4)

Io(5)

Fig. 9. The game (n, q) = (6, 4) with convex preferences, u, in three dimensions with a structurally unstable core,

IO(u, 04) = [0( {1,2, 3}, b/) n IO( {4, 5}, u) n IO( (6}, u).

(i) if dim(W) = 2 then there are structurally stable cores and cycle sets.

(ii) if dim(W) = 3 then the core is generically empty and the cycle set is generically n0n-empty.

(iii) if dim(W) > 3 then the cycle set is generically dense (and om chaotic).

The case with n odd is similar. If dim(W) > 2 then the core is generically empty and if dim(W) > 2 then the cycle set is generically dense (Plott, 1967; Mathews, 1978; Schofield, 1980b).

From the Greenberg results we know in the q-game that stability (core existence) only occurs when w < q/ (n - q).

In the range q / n - q < w < q, the core need not always exist, but structurally stable cores are possible. For w/> q the core is non generic and in particular w > q implies that chaos is generic. Here the fact that IC(N, u, o) is dense is taken to mean that any outcome can result. In other words, if we let Po(x) refer to the set o f points reachable by some particular winning coalition from x, and I'g(x) for the iterate of this, involving different coalitions, then in the chaotic state we find that in general W = l?r(x), for some r (see also McKelvey, 1976, 1979; Cohen, 1979).

Although it had been earlier argued (Tullock, 1967) that majority rule would be well behaved, these results seem to indicate that the con-

541

Fig. 10. The setsIC as n (odd) increases for the games (n, q) = (2k -. 1, k) with convex preferences in two dimensions.

verse is the case: tha t chaos occurs when the decision prob lem is o f high

dimension. In the pre-chaos d imension range it is possible that q-games are qui te

welt behaved. For example , consider Fig. 10, which represents the games (n, q) = (2k - 1, k) in two dimensions. Al though the core is empty , the cycle set, I C belongs to the pare to set l O ( o k , u) . As tl increases the cycle set "col lapses" . It seems to be the case that in the limit n -+~, that the cycle set vanishes and the core exists. This indeed was the pos-

sibility that Tul lock not iced . However this p h e n o m e n o n may not occur

in higher dimensions .

5. Equil ibria in Political and E c o n o m i c Systems

A problem as ye t unsolved is a full classification o f all coali t ional games in terms o f the topological s t ruc ture o f thei r infinitesimal opti-

ma and cycle sets. It is clear at least in vot ing games that in low dimensions a core

exists, while in high d imensions the inde te rminacy or chaos implied by

542

a dense cycle set is generic. There is thus a close parallel between the structure of voting games and the discrete ecological systems reviewed in the first section. In the middle dimension range the cycle set exists and would appear to have interesting properties. An examination of Fig. 8(ii) suggests, in the case of convex preferences at least, that the trajectories associated with coalition optimization leads eventually into the cycle set. Thus IC(N, u, o) suggests itself as a possible solution theory (Schofield, 1978c), whose stability properties may be examined (Matthews, 1977). While at a point in IO(N, u, o) no "coalitional force" acts, at a point in IC(N, u, o) the coalitionai forces are "balanced," in the sense that coalition preferences "point in all directions".

At a point outside IC(N, u, o), such as 3, in Fig. 8(ii), social preference is locally rational, in the sense that in a neighborhood of y social preference can be represented by a "social utility function". At the point y in fact player 6 is a kind of veto player or "local collegium" since his preference cone contains the social preference core H~(y). The existence of such a local collegium is sufficient to ensure some kind of local rationality near y. While it is possible that political decision struc- tures have built into them a form of local veto structure, it is certainly the case that general voting processes and also economic exchange processes need not in general be locally collegial. Consequently the following conjecture suggests itself: for any locally non-collegial decision process o, satisfying reasonable structural properties, there is some finite integer, w(o), such that d im(W)> w(o) implies that the infinitesimal cycle set is generically non-empty.

Economic systems are generally regarded as equilibratory, in the sense that for the process o there is some natural equilibrium set e(o), say a core or pareto set, with the property that, for almost all initial points, iteration of the process Fo leads close to the equilibrium (see for example, Melo, 1977, on paretian processes). If however an open cycle set exists in W then the process may hit this set and never ap- proach the equilibrium, even when the latter exists. Intuitively it would seem quite possible, in a real world involving nation states and multinational corporations all with competing interests and some ability to protect those interests, that the kind of instability implied by the existence of a cycle set can occur.

It is possible that general political economic processes are rather like Fig. 8: for certain parameter values and preferences the process is equilibratory. After sufficient perturbation of the system a degree of indeterminacy or irrationality is introduced. If new dimensions or complex interrelationships are introduced, then complete chaos occurs.

These results are of course at odds with the usual orientation of

543

economic theory towards a price equilibrium, where supply and demand are matched. Powerful results on the generic existence of equilibria in price economies have recently been obtained (see Debreu, 1976, for an excellent survey). In a certain sense a price equilibrium vector is a kind of social utility gradient, imposing rationality on the process. It seems to this writer however that real world processes lie "in be tween" chaotic voting processes and equilibratory (price) economic processes. The notions o f genericity and structural stability may possibly provide a common theoretical foundation on which to build a theory of political economy; a theory of both political and economic action.

Acknowledgement

This material is based upon work supported by the National Science Foundat ion under grant no. Soc. 77-21651.

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