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: robustness 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 I will study is a model of social communication which was introduced by Lewis (Reference Lewis1969). I will first review the results on the standard evolutionary dynamics of this game (Section 2). These results show that standard evolutionary dynamics is quite likely to lead to states of partial communication. 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 signaling game is based on three sets with n elements, where n is an arbitrary finite number: a set of world states $S=\{ \sigma _{1},\,\ldots,\,\sigma _{n}\} $ , a set of messages $M=\{ m_{1},\,\ldots,\, m_{n}\} $ and a set of possible acts $A=\{ \alpha _{1},\,\ldots,\,\alpha _{n}\} $ . For any i, act $\alpha _{i}$ is the right response to state $\sigma _{i}$ . It is the wrong response to any 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 outcome,Footnote ¹ then a simple signaling game $\Sigma _{n}$ may be defined as a triple $\left<I,\,\{ S_{i}\} _{i\in I},\,\{ u_{i}\} _{i\in I}\right>$ . $I=\{ 1,\, 2\} $ is the set of players: the sender, 1, and the receiver, 2. $S_{1}=\{ s_{k}\vert s_{k}\,\mathrm{is}\,\,\mathrm{a}\,\,\mathrm{function}\,\,\mathrm{from}\,\,S\,\mathrm{to}\,\,M\} $ is the set of sender strategies. $S_{2}=\{ a_{l}\vert a_{l}\,\mathrm{is}\,\,\mathrm{a}\,\,\mathrm{function}\,\,\mathrm{from}\,\,M\,\mathrm{to}\,\,A\} $ is the set of receiver strategies. And $u_{i}\thinspace{:}\thinspace S_{1}\times S_{2}\rightarrow \mathstrut{\Bbb R} $ are the payoff functions. Let $u_{i}=u$ and

Here, $\mathstrut{\Bbb P} $ is a probability distribution over S, $\circ $ is the operation of function composition and $u^{*}\thinspace{:}\thinspace S\times A\rightarrow \{ 0,\, 1\} $ such that $u(\sigma _{i},\,\alpha _{j}) =\delta _{ij}$ ( $\delta _{ij}$ being the Kronecker symbol: $\delta _{ij}=0$ if $i\neq j$ and $\delta _{ij}=1$ if $i=j$ ). The computation 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.

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 $m_{1}$ or sending message $m_{2}$ . The receiver, R, does not know which state has occurred (indicated by the dotted line). R chooses act $\alpha _{1}$ or act $\alpha _{2}$ .

Some combinations of sender strategies and receiver strategies allow perfect communication. They are called signaling systems in Lewis Reference Lewis1969. A strategy combination $(s_{k},\, r_{l}) $ is a signaling system if the composition $s_{k}\circ r_{l}$ maps $\sigma _{i}$ on $\alpha _{i}$ , for each i. Equivalently one may say that a signaling system guarantees the maximum payoff of 1 to both players regardless of the state of the world. A signaling system determines the meaning of 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 $n\geq 2$ , 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 Reference Skyrms1996, Reference Skyrms2000 and Huttegger Reference Huttegger2007). Let $\Sigma ^{r}_{n}$ be the two-player role conditioned game based on the signaling game $\Sigma _{n}$ . (See Cressman Reference Cressman2003 for details on role conditioned games.) That is, a strategy of $\Sigma ^{r}_{n}$ is a pair of strategies $(s,\, r) $ where s is a sender strategy and r is a receiver strategy of $\Sigma _{n}$ . It is assumed that each player of $\Sigma ^{r}_{n}$ is sender (receiver) with probability $1/ 2$ . This guarantees that the payoff matrix of $\Sigma ^{r}_{n}$ is symmetric. The payoff for each 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 $\Sigma _{n}$ . The $\phi (n) =n^{2n}$ strategies of $\Sigma ^{r}_{n}$ may be thought of as types of individuals in a population. If $\Delta ^{\phi (n) }$ denotes the simplex in $\mathstrut{\Bbb R} ^{\phi (n) }$ ,Footnote ² then the state of the population 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 $\mathbf{x}$ in terms of success relative to the current population average:

where $u(x_{i},\,\mathbf{x}) $ is the expected payoff for type i and $u(\mathbf{x},\,\mathbf{x}) $ is the average 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 $\mathbf{x}\in \Delta ^{k}$ is a rest point if $\dot{x}=0$ . That is, the population is at a rest point when its configuration does not change anymore. A point $\mathbf{x}\in \Delta ^{k}$ is stable if solutions starting near $\mathbf{x}$ stay nearby. It is asymptotically stable if there is a neighborhood U of $\mathbf{x}$ such that solutions starting at $\mathbf{y}\in U$ converge to $\mathbf{x}$ . $\mathbf{x}$ is unstable if it is not stable.

The replicator dynamics of signaling games has been studied in Huttegger Reference Huttegger2007. The main results are summarized in the following theorem. For a number of additional results, see Pawlowitsch Reference Pawlowitsch2006. Before stating Theorem 1, let me explain two concepts used in its statement. The interior of $\Delta ^{\phi (n) }$ is the part of $\Delta ^{\phi (n) }$ where all types have positive relative frequency. The boundary of $\Delta ^{\phi (n) }$ is the part of $\Delta ^{\phi (n) }$ where at least one type has zero relative frequency.

Suppose we are given a probability distribution over $\mathstrut{\Bbb R} ^{\phi (n) }$ which is absolutely continuous with respect to Lebesgue measure for $\mathstrut{\Bbb R} ^{\phi (n) }$ . Theorem 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, signaling 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 instance, does the evolution of perfect, or nearly perfect, communication systems get more likely if we add certain features to the replicator dynamics (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 communication 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 equations (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.Footnote 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 Reference Weibull1995, 144–148, and Hofbauer and Sigmund Reference Hofbauer and Sigmund1998, Section 8, for more information on payoff monotonic and adjustment dynamics).Footnote ⁴

3. Payoff Monotonic Dynamics.

Consider a dynamics of a simple signaling game $\Sigma ^{r}_{n}$ on the simplex $\Delta ^{\phi (n) }$ given by the system of differential equations

The functions $g_{i}\thinspace{:}\thinspace \Delta ^{\phi (n) }\rightarrow \mathstrut{\Bbb R} $ are assumed to be continuously differentiable. This guarantees the existence and uniqueness of solutions (see, e.g., Hirsch and Smale Reference Hirsch and Smale1974). Moreover, it is assumed that $\sum_{i}x_{i}g_{i}(\mathbf{x}) =0$ . This implies that the overall growth rate of the frequencies $x_{i}$ is constant. The frequency of one type can only increase if the frequency of other types decreases. As a consequence, $\Delta ^{\phi (n) }$ and its boundary faces are invariant.

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

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 assumption 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 even have $g_{i}(\mathbf{x}) -g_{j}(\mathbf{x}) =u(x_{i},\,\mathbf{x}) -u(x_{j},\,\mathbf{x}) $ . Other important examples of payoff monotonic dynamics include different kinds of imitation dynamics (see Hofbauer and Sigmund Reference Hofbauer and Sigmund1998 for more). There is a close relationship between payoff monotonic dynamics and the replicator equations (for a proof see, e.g., Weibull Reference Weibull1995, 147).

This does not imply, however, that the stability properties of $\mathbf{p}^{*}$ will be 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 $u(\mathbf{x},\,\mathbf{x}) $ is a global strict Liapunov function for the systems under consideration. This means that $u(\mathbf{x},\,\mathbf{x}) $ is strictly increasing along non-stationary solutions and constant on connected components of rest points. The significance of this result lies in the fact that $u(\mathbf{x},\,\mathbf{x}) $ is also a strict Liapunov function for the replicator dynamics of signaling games. (Indeed, it is even a potential for the replicator dynamics of signaling games [Huttegger Reference Huttegger2007]. For more information on Liapunov functions and potential functions see Hirsch and Smale Reference Hirsch and Smale1974.)

Proof. The payoff matrix A for $\Sigma ^{r}_{n}$ is symmetric. This is shown, for example, in Huttegger Reference Huttegger2007. The average payoff in the population is $u(\mathbf{x},\,\mathbf{x}) =\mathbf{x}\cdot A\mathbf{x}$ (where $\cdot $ denotes the dot-product). The symmetry of A yields

Inserting (2), we get

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

The last equality follows from the second constraint on payoff monotonic dynamics. Thus, $u(\mathbf{x},\,\mathbf{x}) $ is monotonically increasing along every nonstationary solution. If $\mathbf{x}$ is a stationary state, then $x_{i}g_{i}(\mathbf{x}) =0$ for all i. Hence $\dot{u}(\mathbf{x},\,\mathbf{x}) =0$ and u is constant. ▪

Theorem 3, together with the fact that $u(\mathbf{x},\,\mathbf{x}) $ is a potential for the replicator dynamics of $\Sigma ^{r}_{n}$ , allows us to draw some conclusions about the stability properties of rest points for $\Sigma ^{r}_{n}$ under payoff monotonic dynamics. Let us first consider interior rest points. If $\mathbf{p}^{*}$ is an interior rest point, then, by Theorem 1, every neighborhood contains almost exclusively solutions which tend away from $\mathbf{p}^{*}$ . Since $u(\mathbf{x},\,\mathbf{x}) $ is increasing along these 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 $u(\mathbf{x},\,\mathbf{x}) $ . Hence, s will attract nearby solutions under any payoff monotonic dynamics. That is to say, if a trajectory starting at $\mathbf{x}$ converges to a signaling system type s for the replicator dynamics of $\Sigma ^{r}_{n}$ , then the trajectory starting at $\mathbf{x}$ converges to s for any payoff monotonic dynamics (2) of $\Sigma ^{r}_{n}$ .

Thus we may conclude that, although trajectories of the replicator dynamics 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 $\Sigma ^{r}_{n}$ will be the same for any of these dynamics. The analysis becomes more difficult when we study rest points on the boundary which do not correspond to signaling systems. If $u(\mathbf{x},\,\mathbf{x}) $ is higher in the interior of the state space close to such rest points on the boundary, then they are unstable under any payoff monotonic dynamics. But, as is shown in Huttegger Reference Huttegger2007 and Pawlowitsch Reference Pawlowitsch2006, 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 $u(\mathbf{x},\,\mathbf{x}) $ is lower than on W. On the other hand, there exist points $\mathbf{p}$ on the boundary of W which are second order unstable. Hence, in every neighborhood of $\mathbf{p}$ there exist $\mathbf{x},\,\mathbf{y}$ such that $u(\mathbf{x},\,\mathbf{x}) < u(\mathbf{p},\,\mathbf{p}) $ and $u(\mathbf{y},\,\mathbf{y}) > u(\mathbf{p},\,\mathbf{p}) $ . Thus, it is possible for 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 (Reference Swinkels1993). See also Hofbauer and Sigmund Reference Hofbauer and Sigmund1998. Consider the requirement that a population moves toward a better reply relative to the current state. This means that $\mathbf{x}(t+h) \cdot A\mathbf{x}(t) > \mathbf{x}(t) \cdot A\mathbf{x}(t) $ , for h close to 0. A denotes the payoff matrix of some game. Adjustment dynamics are defined by taking the limit $h\rightarrow 0$ . Accordingly, a dynamics $\dot{x}=\mathbf{f}(\mathbf{x}) $ is an adjustment dynamics if and only if $\dot{x}\cdot A\mathbf{x}\geq 0$ and $\dot{x}\cdot A\mathbf{x}> 0$ whenever $\mathbf{x}$ is not a Nash equilibrium or a rest point of the replicator equation.

Every payoff monotonic dynamics is an adjustment dynamics. Moreover, best response dynamics and adaptive dynamics are also adjustment dynamics (see Hofbauer and Sigmund Reference Hofbauer and Sigmund1998 for details on those dynamics). The rationale of adaptive dynamics is that mutants use strategies close to the current state $\mathbf{x}$ such that the whole population is moving in the 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 $\mathbf{x}$ . On this interpretation, best response dynamics may be regarded as a boundedly rational dynamics. Payoff monotonic dynamics, best response dynamics and adaptive dynamics do not overlap. Thus, adjustment dynamics is a natural generalization of these three classes of dynamics.

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

Thus the average payoff is also a global strict Liapunov function for any adjustment dynamics of $\Sigma ^{r}_{n}$ . Since $\dot{x}\cdot A\mathbf{x}> 0$ for all $\mathbf{x}$ which are not rest 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 $\Sigma ^{r}_{n}$ as in the case of payoff monotonic dynamics.

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 generally, by adjustment dynamics. The evolution of simple communication 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.