Philosophy of Science, 74 (December 2007) pp. 839–847. 0031-8248/2007/7405-0024$10.00
Copyright 2007 by the Philosophy of Science Association. All rights reserved.

839

Robustness in Signaling Games

Simon M. Huttegger†‡

The spontaneous emergence of signaling has already been studied in terms of standard
evolutionary dynamics of signaling games. Standard evolutionary dynamics is given
by the replicator equations. Thus, it is not clear whether the results for standard
evolutionary dynamics depend crucially on the functional form of the replicator equa-
tions. In this paper I show that the basic results for the replicator dynamics of signaling
games carry over to a number of other evolutionary dynamics.

1. Introduction. Various kinds of social behavior have been explained by
evolutionary game theoretic models. Such models usually remain agnostic
about many details of the phenomenon under consideration, for example,
the mechanisms that might produce a specific behavior. Not specifying
details leaves evolutionary game theoretic models open to a number of
criticisms. In particular, one might argue that including some of these
details might result in a dynamic which is considerably different from the
dynamic of the less detailed model.

Appealing to robustness may sometimes help to escape such criticisms.
A result of some evolutionary game theoretic model, like the emergence
of a certain social behavior, is robust relative to particular changes if it
continues to hold in models which resemble the original one except in
those changes. If some result is robust in this sense, then certain details
of the original model don’t matter.

At one level, this paper may be regarded as an exercise in providing
mathematically sound arguments for a specific kind of robustness: ro-
bustness with respect to qualitative changes in the evolutionary dynamics.
Such changes generate different classes of dynamical systems which share
certain features. At another level, this paper may be seen as a contribution
to research on the evolution of simple communication systems. The game

†To contact the author, please write to: Konrad Lorenz Institute for Evolution and
Cognition Research, Adolf Lorenz Gasse 2, A-3422 Altenberg, Austria; e-mail: simon
.huttegger@kli.ac.at.

‡This research was supported by the Konrad Lorenz Institute for Evolution and Cog-
nition Research.



840 SIMON M. HUTTEGGER

I will study is a model of social communication which was introduced by
Lewis (1969). I will first review the results on the standard evolutionary
dynamics of this game (Section 2). These results show that standard evo-
lutionary dynamics is quite likely to lead to states of partial communi-
cation. But it does not always lead to states of perfect communication.
In Sections 3 and 4 I will show that the results for the replicator dynamics
basically carry over to some general classes of evolutionary dynamics.

2. Signaling Games and Standard Evolutionary Dynamics. A Lewis sig-
naling game is based on three sets with n elements, where n is an arbitrary
finite number: a set of world states , a set of messagesS p {j , . . . , j }1 n

and a set of possible acts . For anyM p {m , . . . , m } A p {a , . . . , a }1 n 1 n
i, act is the right response to state . It is the wrong response to anya ji i
other state. Moreover, it is assumed that there are two players. A sender
observes the state of the world and may choose one of n messages from
the set M. A receiver, who is, for whatever reasons, incapable of observing
the state of the world, may choose an act after she has received the sender’s
signal. If we assume that the players get the same payoff for each out-
come,1 then a simple signaling game may be defined as a tripleSn

. is the set of players: the sender, 1, and theAI, {S } , {u } S I p {1, 2}i i!I i i!I
receiver, 2. is the set of senderS p {s Fs is a function from S to M}1 k k
strategies. is the set of receiverS p {a Fa is a function from M to A}2 l l
strategies. And are the payoff functions. Let andu : S # S r ! u p ui 1 2 i

n

u(s , r ) p "(j ) 7 u*(j , (r s )(j )).! !k l j j l k j
jp1

Here, is a probability distribution over S, is the operation of function" !
composition and such that ( being theu* : S # A r {0, 1} u(j , a ) p d di j ij ij
Kronecker symbol: if and if ). The computationd p 0 i ( j d p 1 i p jij ij
of the players’ payoffs implements the assumption of complete common
interest between the players. If some state of the world obtains, they both
get a positive payoff just in case the right act is chosen by the receiver.
Figure 1 shows an extensive form representation of a simple signaling
game.

Some combinations of sender strategies and receiver strategies allow
perfect communication. They are called signaling systems in Lewis 1969.
A strategy combination is a signaling system if the composition(s , r )k l

maps on , for each i. Equivalently one may say that a signalings r j a!k l i i
system guarantees the maximum payoff of 1 to both players regardless
of the state of the world. A signaling system determines the meaning of

1. This assumption expresses complete common interest between the players.



ROBUSTNESS IN SIGNALING GAMES 841

Figure 1. Extensive form representation of a simple signaling game. There are two
states, two acts and two messages. Nature, N, decides which state occurs. The sender,
S, chooses between sending message or sending message . The receiver, R, doesm m1 2
not know which state has occurred (indicated by the dotted line). R chooses act ora1
act .a2

signals. That is to say, in a signaling system players use the signals in
such a way as to allow information to be transmitted. Since, for ,n ! 2
there is always more than one signaling system, meaning is conventional.

Signaling games have already been studied in terms of evolutionary
game theory by looking at the replicator dynamics (see Skyrms 1996, 2000
and Huttegger 2007). Let be the two-player role conditioned gamerSn
based on the signaling game . (See Cressman 2003 for details on roleSn
conditioned games.) That is, a strategy of is a pair of strategiesrS (s, r)n
where s is a sender strategy and r is a receiver strategy of . It is assumedSn
that each player of is sender (receiver) with probability . This guar-rS 1/2n
antees that the payoff matrix of is symmetric. The payoff for eachrSn
player, and each outcome, of the role conditioned game may then be
obtained by computing expected values. A signaling system type s is a
pair of strategies which constitute a signaling system of . TheS f(n) pn

strategies of may be thought of as types of individuals in a pop-2n rn Sn
ulation. If denotes the simplex in ,2 then the state of the popu-f(n) f(n)D !
lation may be described by the proportion of those types. The replicator
dynamics determines the growth rate of each type i given the current
population state in terms of success relative to the current populationx
average:

ẋ p x (u(x , x) ! u(x, x)), i p 1, . . . , k, (1)i i i

where is the expected payoff for type i and is the averageu(x , x) u(x, x)i

2. The simplex in is the -dimensional manifold given byk k! k ! 1 D p {x p
.k(x , . . . , x ) ! ! : ! x p 1)}1 k ii



842 SIMON M. HUTTEGGER

payoff in the population. (1) may be thought of as a model for cultural
evolution or as a model for biological evolution. Before we proceed, recall
the following concepts from the theory of dynamical systems. A point

is a rest point if . That is, the population is at a rest pointk ˙x ! D x p 0
when its configuration does not change anymore. A point is stablekx ! D
if solutions starting near stay nearby. It is asymptotically stable if therex
is a neighborhood U of such that solutions starting at convergex y ! U
to . is unstable if it is not stable.x x

The replicator dynamics of signaling games has been studied in Hut-
tegger 2007. The main results are summarized in the following theorem.
For a number of additional results, see Pawlowitsch 2006. Before stating
Theorem 1, let me explain two concepts used in its statement. The interior
of is the part of where all types have positive relative frequency.f(n) f(n)D D
The boundary of is the part of where at least one type has zerof(n) f(n)D D
relative frequency.

Theorem 1. Let be a symmetrized simple signaling game. Then therSn
following statements are true:
1. Denote the set of points in the interior of which do notf(n)D

converge to the boundary of by S. Then S has Lebesguef(n)D
measure zero.

2. A state is asymptotically stable if and only if is af(n)p* ! D p*
signaling system type.

3. Denote by W the set of solutions which do not converge to a
signaling system. Then W has Lebesgue measure zero if and only
if and .n p 2 "(j ) p "(j )1 2

Suppose we are given a probability distribution over which is ab-f(n)!
solutely continuous with respect to Lebesgue measure for . Theoremf(n)!
1 tells us that the replicator dynamics will with probability 1 carry the
population to some state where not all types are present. Thus, some
degree of coherence for communication is achieved almost surely. But,
although signaling systems are the only asymptotically stable states, there
is a positive probability of not reaching them. This is expressed by the
third part of Theorem 1. If at least one of the conditions of this statement
fails to hold, then there exist connected components of rest points on the
boundary which attract a set of positive measure from the interior of state
space. It can be shown that these connected sets of rest points are not
attractors. This means that there exists no neighborhood U such that all
states in U converge to the connected set of rest points. Some states on
the boundary of these connected components are unstable. Hence, sig-
naling systems are the only states which are stable relative to selection
and relative to neutral drift.

These results leave open a number of interesting questions. For in-



ROBUSTNESS IN SIGNALING GAMES 843

stance, does the evolution of perfect, or nearly perfect, communication
systems get more likely if we add certain features to the replicator dy-
namics (such as mutation or correlated encounters between individuals)?
If we assume a reasonably high number of signals n, the evolutionary
dynamics might spend a very long time in states of partial communi-
cation which are far from optimal (even if we take into account neutral
drift). Numerical simulations suggest that these states are not observed
under replicator-mutation dynamics.

The robustness of the results stated in Theorem 1 is a related issue. Do
these results depend on the specific functional form of the replicator equa-
tions (1)? To answer this question, we will first look at a quite large class
of evolutionary dynamics called payoff monotonic, which contains the
replicator dynamics.3 We can get still more general results when we study
adjustment dynamics. This is a class of games which contains all payoff
monotonic dynamics. (See Weibull 1995, 144–148, and Hofbauer and
Sigmund 1998, Section 8, for more information on payoff monotonic and
adjustment dynamics).4

3. Payoff Monotonic Dynamics. Consider a dynamics of a simple sig-
naling game on the simplex given by the system of differentialr f(n)S Dn
equations

ẋ p x g (x). (2)i i i

The functions are assumed to be continuously differentiable.f(n)g : D r !i
This guarantees the existence and uniqueness of solutions (see, e.g., Hirsch
and Smale 1974). Moreover, it is assumed that . This implies! x g (x) p 0i ii
that the overall growth rate of the frequencies is constant. The frequencyxi
of one type can only increase if the frequency of other types decreases.
As a consequence, and its boundary faces are invariant.f(n)D

A game dynamics (2) is said to be payoff monotonic if and only if

g (x) 1 g (x) "# u(x , x) 1 u(x , x). (3)i j i j
Thus, a payoff monotonic dynamics is characterized by the property that
the proportion of types with a higher payoff grows at a higher rate than
the proportion of types with a lower payoff. This is a reasonable as-
sumption for any dynamics for which the relative payoffs are assumed to
influence the evolution of types.

The replicator dynamics is clearly payoff monotonic. In this case we

3. Similar classes of dynamics have already been studied for a signaling mini-game in
Skyrms 2000.

4. Another kind of robustness concerns structural stability, i.e., small perturbations of
the differential equations in function space. See D’Arms et al. 1998 and Skyrms 2000.



844 SIMON M. HUTTEGGER

even have . Other important examples ofg (x) ! g (x) p u(x , x) ! u(x , x)i j i j
payoff monotonic dynamics include different kinds of imitation dynamics
(see Hofbauer and Sigmund 1998 for more). There is a close relationship
between payoff monotonic dynamics and the replicator equations (for a
proof see, e.g., Weibull 1995, 147).

Theorem 2. is a rest point for (1) if and only if is a rest pointp* p*
of a payoff monotonic dynamics (2).

This does not imply, however, that the stability properties of will bep*
the same under any payoff monotonic dynamics. The next proposition
implies that, for simple signaling games, some stability results for rest
points of the replicator dynamics indeed carry over to payoff monotonic
dynamics. It shows that the average payoff is a global strict Lia-u(x, x)
punov function for the systems under consideration. This means that

is strictly increasing along non-stationary solutions and constantu(x, x)
on connected components of rest points. The significance of this result
lies in the fact that is also a strict Liapunov function for the rep-u(x, x)
licator dynamics of signaling games. (Indeed, it is even a potential for the
replicator dynamics of signaling games [Huttegger 2007]. For more in-
formation on Liapunov functions and potential functions see Hirsch and
Smale 1974.)

Theorem 3. is monotonically increasing along every nonsta-u(x, x)
tionary solution and is constant on every connected set of stationary
states for any payoff monotonic dynamics (2) of .rSn

Proof. The payoff matrix A for is symmetric. This is shown, forrSn
example, in Huttegger 2007. The average payoff in the population
is (where denotes the dot-product). The symmetryu(x, x) p x 7 Ax 7
of A yields

˙ ˙ ˙˙ ˙u(x, x) p x 7 Ax " x 7 Ax p 2x 7 Ax p 2 x u(x , x).! i i
i

Inserting (2), we get

u̇(x, x) p 2 x g (x)u(x , x).! i i i
i

Suppose the solution starting at is not stationary. Then, since thex
stationary states for payoff monotonic dynamics coincide with the
stationary states for the replicator dynamics, there exists a j such
that for all k with a strict inequality holding for atu(x , x) $ u(x , x)j k
least one k. Hence

u̇(x, x) p 2 x g (x)u(x , x) 1 2u(x , x) x g (x) p 0.! !i i i j i i
i i



ROBUSTNESS IN SIGNALING GAMES 845

The last equality follows from the second constraint on payoff mono-
tonic dynamics. Thus, is monotonically increasing along everyu(x, x)
nonstationary solution. If is a stationary state, then forx x g (x) p 0i i
all i. Hence and u is constant. "u̇(x, x) p 0

Theorem 3, together with the fact that is a potential for the rep-u(x, x)
licator dynamics of , allows us to draw some conclusions about therSn
stability properties of rest points for under payoff monotonic dynamics.rSn
Let us first consider interior rest points. If is an interior rest point,p*
then, by Theorem 1, every neighborhood contains almost exclusively so-
lutions which tend away from . Since is increasing along thesep* u(x, x)
non-stationary solutions, the same holds for any payoff monotonic
dynamics.

Moreover, it is quite obvious that signaling system types s continue to
be asymptotically stable for payoff monotonic dynamics. Since s attracts
all nearby solutions, s locally maximizes . Hence, s will attractu(x, x)
nearby solutions under any payoff monotonic dynamics. That is to say,
if a trajectory starting at converges to a signaling system type s for thex
replicator dynamics of , then the trajectory starting at converges torS xn
s for any payoff monotonic dynamics (2) of .rSn

Thus we may conclude that, although trajectories of the replicator dy-
namics and trajectories of some payoff monotonic dynamics will in general
be different, the qualitative behavior of trajectories close to interior rest
points and signaling systems of will be the same for any of theserSn
dynamics. The analysis becomes more difficult when we study rest points
on the boundary which do not correspond to signaling systems. If

is higher in the interior of the state space close to such rest pointsu(x, x)
on the boundary, then they are unstable under any payoff monotonic
dynamics. But, as is shown in Huttegger 2007 and Pawlowitsch 2006,
there exist connected components of rest points which attract a set of
points with positive measure from the interior. Thus, close to such a
connected component W of rest points is lower than on W. Onu(x, x)
the other hand, there exist points on the boundary of W which arep
second order unstable. Hence, in every neighborhood of there existp

such that and . Thus, it is possible forx, y u(x, x) ! u(p, p) u(y, y) 1 u(p, p)
some payoff monotonic dynamics to be such that although orbits tend
toward W due to increasing average payoff, they also turn outward toward
boundary points, where average payoff is increasing away from W.

4. Adjustment Dynamics. The results presented in the preceding section
can be improved by studying another class of dynamics called adaptive
dynamics. Adaptive dynamics were introduced by Swinkels (1993). See
also Hofbauer and Sigmund 1998. Consider the requirement that a pop-



846 SIMON M. HUTTEGGER

ulation moves toward a better reply relative to the current state. This
means that , for h close to 0. A denotes thex(t " h) 7 Ax(t) 1 x(t) 7 Ax(t)
payoff matrix of some game. Adjustment dynamics are defined by taking
the limit . Accordingly, a dynamics is an adjustment dy-˙h r 0 x p f(x)
namics if and only if and whenever is not a Nash˙ ˙x 7 Ax ! 0 x 7 Ax 1 0 x
equilibrium or a rest point of the replicator equation.

Every payoff monotonic dynamics is an adjustment dynamics. More-
over, best response dynamics and adaptive dynamics are also adjustment
dynamics (see Hofbauer and Sigmund 1998 for details on those dynamics).
The rationale of adaptive dynamics is that mutants use strategies close
to the current state such that the whole population is moving in thex
most promising direction. Best response dynamics may also be interpreted
in terms of a large population model. A small fraction of individuals in
a large population revises strategies from time to time by choosing a best
reply to the current mean population strategy . On this interpretation,x
best response dynamics may be regarded as a boundedly rational dynam-
ics. Payoff monotonic dynamics, best response dynamics and adaptive
dynamics do not overlap. Thus, adjustment dynamics is a natural gen-
eralization of these three classes of dynamics.

The analogue to Theorem 3 for adjustment dynamics follows easily
from the above definition of adjustment dynamics.

Theorem 4. is monotonically increasing along every nonsta-u(x, x)
tionary solution and is constant on every connected set of stationary
states for any adjustment dynamics of .rSn

Proof. If A denotes the payoff matrix of , then the symmetry of ArSn
implies that . Thus˙ ˙ ˙ ˙ ˙x 7 Ax p x 7 Ax u(x, x) p x 7 Ax " x 7 Ax p

. By definition, the last term is greater than 0 for nonsta-˙2x 7 Ax
tionary solutions, and it is 0 if and only if is a rest point. "x

Thus the average payoff is also a global strict Liapunov function for any
adjustment dynamics of . Since for all which are not restr ˙S x 7 Ax 1 0 xn
points of the replicator equation and since Nash equilibria are rest points
of the replicator dynamics, adjustment dynamics do not have more rest
points than the corresponding replicator dynamics. This allows us to draw
the same conclusions concerning the stability of rest points for adjustment
dynamics of as in the case of payoff monotonic dynamics.rSn

5. Conclusion. Often it is difficult to judge whether one model is more
realistic than another one. Robustness of a result across a variety of
models—each of them being plausible—may be used as a substitute. In
this sense, the emergence of states of partial or perfect communication is
a robust result relative to changes described by payoff monotonic or, more



ROBUSTNESS IN SIGNALING GAMES 847

generally, by adjustment dynamics. The evolution of simple communi-
cation systems in these classes of dynamics is at least as likely as it is in
the replicator dynamics. This, on the other hand, implies that our results
do not show that there exist adjustment dynamics which improve on the
replicator dynamics, that is, in which the evolution of signaling systems
is more likely than in the replicator dynamics. Results in this direction
might be achieved only by studying more specific dynamics.

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