Download Polyhedra and Nash equilibrium components: An

Polyhedra and Nash equilibrium components:
An elementary construction.
Dieter Balkenborgy
Dries Vermeulenz
February 5, 2012
Abstract
In this short note we characterize Nash equilibrium components in
terms of their topological properties. As is well known, every Nash
equilibrium component is a compact semi-algebraic set and can hence
be triangulated. It is thus homeomorphic to a connected union of
…nitely many simplices, i.e. a polyhedron. Conversely, we provide a
simple construction showing that every polyhedron is homeomorphic
to a Nash equilibrium component. Consequently, Nash equilibrium
components provide a very rich class of topological spaces including
all compact connected topological manifolds.
JEL Codes. C72, D44. Keywords. Strategic form games, Nash
equilibrium components, topology.
1
Introduction
In non-cooperative game theory the claim that a connected component of
Nash equilibria can have any conceivable shape seems to have the status of
a ”folk theorem”. Few researchers in the …eld doubt this claim, yet no proof
We are grateful to Fabrizio Germano and Rida Laraki for useful discussions. A discussion with Barry O’Neal many years back has now proved in‡uential.
y
Corresponding author; Department of Economics, University of Exeter, Rennes Drive,
Streatham Court, Exeter, EX, UK. Tel. 00441392263231, FAX 00441392263242, E-mail:
[email protected]
z
Department of Quantitative Economics, University Maastricht, P.O.n Box 616, 6200
MD Maastricht, The Netherlands. Email: [email protected]
1
seems to be available, and it is unclear what is meant by ”every conceivable
shape”. In this paper we provide the following “computational folk theorem”
on the topological structure of Nash equilibrium components. The ”only if”
result is, of course, well known.
Theorem 1 A connected topological space is homeomorphic to a Nash equilibrium component if and only if it is has a triangulation.
Thus, Nash equilibrium component have topologically no additional structure beyond what follows from the fact that they are compact semi-algebraic
sets (see Schanuel, Simon, and Zame (1991)) and hence have a triangulation
(see Lojasiewicz (1964) and Hironaka (1975)). The class of topological spaces
with a triangulation is very large. Classical combinatorial topology is primarily concerned with the topological classi…cation of such spaces, starting
with the observation that their Euler characteristic is a topological invariant.
The existence of a triangulation implies that the topological space is homeomorphic to the union of simplices in a simplicial complex, i.e. a polyhedron.
For the ‘if’direction we must construct an example of a game which has,
up to homeomorphism, a given polyhedron as a Nash equilibrium component. To do this, we notice …rst that every polyhedron is homeomorphic
to a standard simplicial set, i.e. a union of …nitely many faces of the standard simplex in some vector-space Rn of su¢ ciently high dimension. Our
construction starts with such a standard simplicial set.
Every line through the origin and a point on the standard simplex in Rn
intersects the boundary of the standard hypercube in the same Rn in exactly
one point which has a component with value one. This relation provides a
projection map from the standard simplex onto the standard hypercube such
that every face of the standard simplex is mapped onto a union of faces on
the hypercube. The hypercube can be viewed as the space of mixed strategy
combinations of a game with n players, where each player has two strategies.
Given a standard simplicial set we de…ne the payo¤s of this game as follows.
The payo¤ of all players at a pure strategy combination s, i.e. at a vertex
of the hypercube, is set equal to 1 if s is the image of a point in the given
standard simplicial set under the projection. Otherwise we set the payo¤ of
all players zero. Our given polyhedron is then shown to be homeomorphic to
the set of mixed strategy combinations of the game which yield payo¤ 1: It
is hence homeomorphic to a closed set of Nash equilibria. So far, so simple.
However, the simplicity is somewhat deceiving. While the closed set of Nash
2
equilibria so constructed is geometrically simple because it is a union of faces,
it is neither topologically nor algebraically trivial. It is topologically not trivial by construction. Algebraically it is the solution set to a highly degenerate
equation u = 1 intersected with the hypercube. This can have many nonisolated singularities who’s behavior under deformations, for instance, is not
well understood. Presumably for this reason we have so far not found a short
argument showing that the solution set is isolated, i.e. a component, of Nash
equilibria. Our proof relies on a rather lengthy argument using high-order
Taylor expansions made earlier by one of the authors (Balkenborg (1994))
and published as Proposition 4 in Balkenborg and Schlag (2007). The upside
of our construction is that the Nash equilibrium component we obtain is a
strict equilibrium sets in the sense of Balkenborg and Schlag (2007), so their
Theorem 6 yields the following.
Theorem 2 A topological space is homeomorphic to a Nash equilibrium component that is asymptotically stable under the replicator dynamics if and only
if it has a triangulation.
In general, the Nash equilibrium components we construct are clearly of
relevance to evolutionary game theory: They are the set of maximizers of
a potential in a potential game and hence asymptotically stable for a wide
class of evolutionary dynamics (see Monderer and Shapley (1996)).
In addition it is worthwhile to note the following elements of our construction.
–We construct “binary”games with many players and two strategies.
–The mapping from the grand simplex to the hypercube is piecewise linear and hence a semi-algebraic mapping. Thus we show the slightly stronger
result that every simplicial complex is homeomorphic to a Nash equilibrium
component via a semi-algebraic map. Notice that also the homeomorphisms
constructed in Hironaka (1975) are semi-algebraic,
– As, for instance, in Aumann (1974), one can think of a normal form
game as constructed in two steps. First, there is a map g : S ! X from
the set of strategy combinations to an abstract set of outcomes X. Secondly,
each player has preferences over the set of lotteries with prizes in X. If
these preferences satisfy the von Neumann Morgenstern axioms for decision
making under uncertainty then they can be represented by a von Neumann
Morgenstern expected utility functions Ui : X ! R. The composition yields
the utility functions ui = Ui g : S ! R as usually considered for normal
3
form games. The game is non-degenerate if the induced ordinal preferences
over the outcomes X are strict. In our construction we use only games which
have only two outcomes and are non-degenerate in this sense. As observed
and used by O’Neill (1987) (see also Roth and Malouf (1979)) games with
only two outcomes are of interest in particular to experimental economics
because the ordinal preferences over the two prizes uniquely determine the
preference over the lotteries with these prizes and hence the von Neuman
Morgenstern utility function up to a linear transformation.1 If the preferences
on the two outcomes are strict we can normalize expected utility such utility
is 1 on one outcome and 0 on the other. One special case is that of two
players with opposing preferences over the two outcomes. In that case of a
constant sum game the equilibrium does not have to be unique, but the set of
minimax strategies and hence of Nash equilibria is a convex polyhedron and
therefore topologically simple. We are studying here the polar case where
all player have identical and not opposing preferences. If the players could
communicate and coordinate on a correlated equilibrium, there would not be
a computational problem because the set of all correlated equilibria is again
convex. It is the lack of communication and the di¢ culty to coordinate which
generates the topological complexity of the Nash equilibrium component.
When we work with the utility functions ui : S ! R as the starting
point, the games we construct are, of course highly degenerate. For nondegenerate normal forms the number of Nash equilibria is …nite and the
interesting question concerning the structure of equilibrium sets is how many
Nash equilibria there can be at most. This question has been answered
in McKelvey and McLennan (1997). Early work on both non-generic and
degenerate games is found in Bubelis (1978). He gives, for instance, an
explicit example of a game where the set of Nash equilibria is di¤eomorphic
to a circle. He provides a construction showing that the set of totally mixed
Nash equilibria of any game with more than three players is di¤eomorphic
via an algebraic map to the set of totally mixed Nash equilibria of a three
player game.
Closest related to the question we address here is the work by Datta
(2003). She shows that every real algebraic variety is stable equivalent to the
set of all totally mixed equilibria of some game. Her beautiful result is di¤er1
For this reason O’Neill calculated the set of Nash equilibria in all 2 2 2 games
where the payo¤s are only zero or one, even if these are very degenerate games in the
traditional sense (personal communication).
4
ent from our very simple construction in a number of ways. Firstly, the Nash
equilibrium components we construct never contain any totally mixed equilibrium. Secondly, while we embed the triangulated space into a Euclidean
space of very high dimension, namely of the same dimension as the number
of vertices, Datta’s construction may increase the dimension of the real algebraic variety itself and does hence not provide a homeomorphism. Thirdly,
a priori real algebraic varieties are a strict subset of semi-algebraic sets. It is
currently not clear to us whether every polyhedron is homeomorphic to an
algebraic variety.
In the following sections we describe our construction in more detail. We
conclude with a few open questions.
2
Preliminaries
A …nite normal form game consists of a …nite set of players N = f1;
; ng,
and for each player i 2 N a …nite
pure
strategy
set
S
and
a
payo¤
function
i
Q
ui : S ! R on the set S := i2N Si of pure strategy pro…les. As indicated
above, one can think of the ui as obtained in two steps, …rst a map g : S ! X
from the set of pure strategy combinations to an abstract set of outcomes
and, secondly, a von Neumann / Morgenstern utility function Ui : X ! R
describing for each player his preferences over lotteries with prizes in X.
We denote the game by (N; u), where u = (ui )i2N is the vector of payo¤
functions. A mixed strategy i of player i is a vector ( i (si ))si 2SP
that assigns a
i
probability i (si ) 0 to each pure strategy si 2 Si such that si 2Si i (si ) =
1. We denote the set of mixed strategies of player i by i . The set of all
pro…les = ( i )i2N of mixed strategies is denoted by . The support of a
mixed strategy i is the set of all pure strategies si with i (si ) > 0. The
multilinear extension of the payo¤ function ui of player i to the set of all
strategy pro…les is given by the formula
XY
ui ( ) =
j (sj )ui (s):
s2S j2N
By ui ( j si ) we denote the payo¤ to player i when player i plays pure
strategy si 2 Si while his opponents adhere to the mixed strategy pro…le .
A strategy pro…le 2 is a Nash equilibrium when ui ( ) ui ( j si ) holds
for every player i and every pure strategy si of player i.
5
BINARY COORDINATION GAMES. A binary game is a …nite
normal form game (N; u) where each player has two pure strategies, for every
player i 2 N , and which has only two possible outcomes X = fa; bg. If the
ordinal preferences over prizes are strict, the von Neumann / Morgenstern
utility function representing the player’s preferences over lotteries with prizes
in X can be normalized such that it is 1 on one prize and zero on the other.
This implies for the normal form game that
ui (s) 2 f0; 1g
holds for all strategy pro…les s = (s1 ;
; sn ) 2 S. If the binary game is a
pure coordination game then all players will have the same ordinal preferences
over the outcomes a, b and hence have identical utility functions ui (s) = u (s)
over S. This is the class of games we employ in this paper.
For binary game we can identify a mixed strategy of a player with the
probability 0 xi 1 with which he plays his second strategy. The set of all
mixed strategy combinations is hence described by the standard hypercube.
3
Simplicial sets
A collection fv1 ; : : : ; vk g of points or vectors in Rn is called linearly independent if for all a1 ; : : : ; ak in R
a1 v1 +
+ ak v k = 0
implies a1 =
= ak = 0. A collection fv0 ; : : : ; vk g of points in Rn is called
geometrically (or a¢ ne) independent if the collection fv1 v0 ; : : : ; vk v0 g
is an linearly independent set in Rn . Note that a linearly independent set is
automatically geometrically independent. The convex hull
(
)
X
t0 v0 +
+ tk vk j ti 0 for all 1 i n and
ti = 1
i
of a geometrically independent set fv0 ; : : : ; vk g is called a simplex. We also
say that fv0 ; : : : ; vk g spans the simplex. For a geometrically independent
set fv0 ; : : : ; vk g we denote by [v0 ; : : : ; vk ] the simplex that is spanned by
fv0 ; : : : ; vk g.
A collection S = fS1 ; : : : ; Sp g of simplices in Rn is called a simplicial
complex if for every two simplices Si = [v0 ; : : : ; vk ] and Sj = [w0 ; : : : ; wm ]
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in S the intersection Si \ Sj is the simplex spanned by the geometrically
independent set
fv0 ; : : : ; vk g \ fw0 ; : : : ; wm g
and belongs to S. The set S =
p
[
Si is called the carrier of the simplicial
i=1
complex S and denoted by jSj. P
Rn is called a polyhedron if it is the
carrier of a simplicial complex. Clearly, a convex polyhedron is the convex
hull of …nitely many points and polyhedra are precisely the …nite unions of
convex polyhedra.
A triangulation of a topological space X is a homeomorphism f : jSj ! X
from the carrier jSj of a simplicial complex S onto X together with the
decomposition X = ff (S) j S 2 Sg.
Let ei denote the ith unit vector in Rn . The standard simplex N in RN
is the simplex spanned by all unit vectors ei in Rn . The face I of N for
a subset ? 6= I
N is the simplex spanned by the collection fei j i 2 Ig
of unit vectors in RN . A standard simplicial set is a union of faces of the
standard simplex N .
Lemma 3 Any polyhedron is homeomorphic to a standard simplicial set.
Proof. Suppose the polyhedron is the carrier of the simplicial complex S.
Let fv0 ; : : : ; vN g R be the set of all vertices which belong to some simplex
of the simplicial complex and let N be their total number. Assign to each
vertex vi , 1
i
n, the unit vector ei in RN . This assignment induces
for each face [vi0 ; : : : ; vik ] of S a linear homeomorphism onto a face of N .
Combining these homeomorphisms we obtain a well-de…ned mapping from
jSj onto a union of faces X in N . Every set U contained in a su¢ ciently
small neighborhood of a point x 2 jSj in RN is a relative neighborhood of x
in jSj if and only the intersection U with any of the …nite simplices Si of S
containing x is a relative neighborhood of x in Si . The same consideration
applied to points in X shows that the map constructed from jSj to X is a
homeomorphism.
Notice that the construction in the previous proof considerably ‘blows up’
dimensions. For instance, the union of the intervals [0; 1], [1; 2] and [2; 3] on
the real line would be mapped onto the union of line segments distributed
in four dimensional space. Notice also that the homeomorphism constructed
has a semialgebraic graph, but is not di¤erentiable.
7
The following triangulation theorem is a celebrated result in geometry
…rst proved by Lojasiewicz Lojasiewicz (1964) for semianalytic sets. A simpler construction for semialgebraic sets was given Hironaka (1975). For a
textbook treatment see Bochnak, Coste, and Roy (1998)
Theorem 4 Every compact semialgebraic set has a triangulation.
The set of Nash equilibria of a game is a compact semialgebraic set because it is given by …nitely many algebraic inequalities in the compact set of
all mixed strategy combinations. Hence this theorem applies. As a simple
application of the Tarski-Seidenberg theorem given in Schanuel, Simon, and
Zame (1991) shows the following.
Theorem 5 Every connected component of the set of all Nash equilibria of
a game is a compact semialgebraic set and has hence a triangulation.
Thus, from a topological point of view, the collection of all polyhedra
already captures all of the variety that we could reasonably expect for Nash
equilibrium components of normal form games.
4
Cubistic sets
For a set T
N , de…ne the characteristic vector eT 2 RN by
(
1 if i 2 T
eT i =
0 otherwise.
The standard hypercube K N in RN is the convex hull of all characteristic
vectors in RN . A subset F K N of the standard hypercube is a face if there
are sets Z and P in N such that
F = fx 2 K n j xi = 0; xj = 1 for all i 2 Z and j 2 P g:
If P 6= ?, we say that the face F is an upward face. Let K+N be the
union of all upward faces and for a subset I
N we let K+I = K+N \
fx 2 Rn j xi = 0 for i 2
= Ig. A subset C of K N is called a cubistic set if
[1] C is the union of faces of K N :
8
[2] C is n-convex in the sense that for all points (x0i ; x i ) 2 C, (x00i ; x i ) 2 C
and all scalars 0
1 the convex combination
(1
) (x0i ; x i ) +
(x00i ; x i )
is in C:2
An upward cubistic set is a cubistic set contained in K+N . In order to
connect the standard simplex with the “upper part” K+N of the standard
hypercube, it is useful to remind ourselves of two norms for vectors x =
(x1 ;
; xn ) 2 RN , namely of the norm
kxk =
and of the supremum norm
n
X
i=1
jxi j
kxk1 = max jxi j .
1 i n
Consider now a non-zero vector x 2 RN with non-negative components and
the ray Rx it generates. This ray intersects the standard simplex N in
the point x= kxk and the set K+N in the point x= kxk1 . This relationship
de…nes a homeomorphism between N and K+N . Moreover, each face I
for I N is mapped one-to-one onto K+I . We obtain the following result.
Theorem 6 Every standard simplicial set is homeomorphic to an upward
cubistic set.
Proof. Clearly, maps a standard simplicial set C onto a union D of
upward faces of the standard cube. To show that C is n-convex, take two
di¤erent points (x0i ; x i ), (x00i ; x i ) 2 D which di¤er only in the i’s component.
We can assume w.l.o.g. that x0i > x00i . Let I be the set of indices with nonzero components in (x0i ; x i ). Since (x0i ; x i ) = k(x0i ; x i )k 2 C, the simplex
I
is contained in C and contains both points (x0i ; x i ) = k(x0i ; x i )k and
(x00i ; x i ) = k(x00i ; x i )k and all their convex combinations. Because x00i < 1;
both (x0i ; x i ), (x00i ; x i ) have a component xj = 1 for some j 6= i. Thus
k(x0i ; x i )k1 = k(x00i ; x i )k1 = k((1
and therefore ((1
a point in C.
2
) x0i + x00i ; x i )k1 = 1
) x0i + x00i ; x i ) 2 K+N is for all 0
This extends the notion of biconvexity in Aumann and Hart (1986).
9
1 the image of
5
Nash equilibrium components
We now interpret N as the set of players in a game and interpret K N as
the set of mixed strategies in a binary game where each player has the pure
strategies 0 and 1. Fix a union of faces C of K N and let T be the set of
pure strategy combinations it contains. Let u : K N ! R be the multilinear
extension of the function u : f0; 1gn ! R de…ned by
u (s) =
1 for s 2 T
0 for s 2
=T
It is straightforward to check the following Lemma which describes cubistic
sets as maximizers of a potential.
Lemma 7 The equality of sets
C = x 2 K N j u (x) = 1
holds if and only if C is a cubistic set.
Now we suppose that C is a cubistic set and look at the game C where
each player i 2 N has the utility function ui = u. Then C is a strict equilibrium set of this game as de…ned in Balkenborg and Schlag (2007). This
means that if (x0i ; x i ) 2 C and if x00i is a best reply of player i against x i ,
then (x00i ; x i ) 2 C. Balkenborg and Schlag (2007) show that every strict
equilibrium set is a …nite union of Nash equilibrium components which consists of stable rest points and is an asymptotically stable set of the replicator
dynamics. As a corollary of their results we obtain
Theorem 8 Every cubistic set C is a …nite union of Nash equilibrium components of the binary game C . C is asymptotically stable and consists of
stable rest points for the replicator dynamics.
This concludes the proof of Theorem 2 and hence Theorem 1.
6
Summary and conclusions
Let us quickly summarize the …ndings. Every Nash equilibrium component
is a connected semi-algebraic set and hence has a triangulation. Conversely,
10
every polyhedron is homeomorphic to a standard simplicial set and hence
a cubistic set. Every cubistic set is a strict equilibrium set and hence, if
connected, a Nash equilibrium component of an associated binary game.
In our construction the number of strategies per player is as small as possible. However, it needs as many players as the simplicial complex has vertices.
Whether one can do a similar construction with few players (in particular,
with two or three players) and potentially many strategies is currently an
open problem. Bubelis (1978) provides a construction which identi…es the
set of completely mixed Nash equilibria for an n-player game (n 3) with a
set of Nash equilibria in a 3-player game. Unluckily the Nash equilibria we
construct are never totally mixed and so we cannot apply his construction.
The restriction to connected sets is not important for our argument. We
could just have worked throughout with …nite unions of Nash equilibrium
components instead. Our argument shows that every cubistic set (and hence,
up to homeomorphism, every polyhedron) is a …nite union of Nash equilibrium components of a binary game. However, typically the game constructed
will have additional Nash equilibria. This must be the case if the Euler characteristic of the set is not 1. This is so because the set we construct is
asymptotically stable and its index is hence equal to the Euler characteristic
(see Demichelis and Ritzberger (2003)) while the index of the set of all Nash
equilibria is always one (see Ritzberger (1994)). We do not know whether a
polyhedron of Euler characteristic one is always homeomorphic to the set of
all Nash equilibria in a suitably chosen game.
We considered here only the topological structure. One might more generally ask whether any compact connected semi-algebraic set is di¤eomorphic
via a semi-algebraic map to a Nash equilibrium component. One might ask
if this is true at least locally in the neighborhood of a point.
Daskalakis, Goldberg, and Papadimitriou (2006) showed that computing
a Nash equilibrium for a …nite strategic form game is PPAd complete (see
also McLennan (2011). The fact that from a topological perspective the set
of Nash equilibria may have any possible shape sheds light on this complexity
result. Again, in this context it might be interesting to see whether (and if
so, how!) our result can also be achieved for 2 player games.
11
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