1. Signaling Games and Evolution

Common interest signaling games were introduced by David Lewis (Reference Lewis1969) as a game theoretic framework that treats communicative conventions as solutions to coordination problems. In recent years, this has informed a growing body of work on the evolution of communication, incorporating signaling games into an evolutionary game theoretic approach to modeling the evolution of communication and cooperation in humans (Skyrms Reference Skyrms1996, Reference Skyrms2010).

As the basis for such work, Lewis signaling games are attractive in their intuitive simplicity and clear outcomes. They are coordination games between world-observing senders and action-making receivers using costless signals, in contrast to games where interests may differ and where costly signals are typically invoked. In the standard two-player, two-state, two-option Lewis signaling game (the ‘$2\times 2\times 2$ game’), the first agent (signaler) observes that the world is in one of two possible states and broadcasts one of two possible signals that are observed by the second agent (receiver) who performs one of two possible actions. If the act chosen by the receiver matches the underlying state of the world, both agents receive a greater payoff than otherwise.

Most important, though, the game theoretic results are unequivocal. There exist two Nash equilibria that are, in Lewis’s words, signaling systems, where senders condition (otherwise arbitrary) signaling behavior on the state of the world, and receivers act on those signals so as to secure the mutual payoff. The two systems only differ on which signal is associated with each state of the world.Footnote ¹ Huttegger (Reference Huttegger2007) and Pawlowitsch (Reference Pawlowitsch2008) have shown that under certain conditions a signaling system is guaranteed to emerge under the replicator dynamics, a standard model of evolution further discussed in section 4.

Of course the degree to which Lewis’s approach makes sense is to some extent determined by the level of confidence we have in the interpretation and application of such idealized models to the more complex target systems (such as animal and human communication). Thus, one obvious worry is that by introducing more realistic features into the model, one may break or significantly dilute previous findings on the evolution of signaling.

Not surprisingly, then, recent work on Lewis signaling games has investigated the many ways in which such de-idealizations could occur and tested the robustness of signaling in the face of them. Some deviations from the standard Lewis signaling game include more and varied states of the world, the possibility of observational error or signal error, noisy signals, the introduction of a partial conflict of interest between senders and receivers, the reception of more than one signal, and so on. Many such concerns are dealt with favorably by Skyrms (Reference Skyrms2010) and in work by others. For example, Bruner et al. (Reference Bruner, O’Connor, Rubin and Huttegger2014) generalize beyond the $2\times 2\times 2$ case, while Godfrey-Smith and Martinez (Reference Godfrey-Smith and Martínez2013) and Martinez and Godfrey-Smith (Reference Martínez and Godfrey-Smith2016) mix signaling games of common interest and conflict of interest. One result (particularly important for our purposes) is that signaling systems are not guaranteed in the simple $2\times 2\times 2$ case when ‘nature is biased’. In other words, when the probabilities of the world being in one state or the other are not equal, a pooling equilibrium in which no communication occurs between sender and receiver is evolutionarily significant (Huttegger Reference Huttegger2007; Pawlowitsch Reference Pawlowitsch2008).

2. Symmetry Breaking

Our article will focus on the idealization that senders and receivers are equally responsive in strategic settings. Senders and receivers (in the evolutionary treatment of such games) are two populations of highly abstract and constrained agency roles: all that signalers do upon observing the state of the world is send a signal, and the receivers choose to act as though the world is in one of the two possible sender-observable states. Of those two roles, it is the restriction on receivers that appears to be particularly unrealistic.

Imagine, for example, a forager sighting a prey animal at a location inaccessible to her, but close enough to be acquired by an allied conspecific (who cannot observe the animal). In this case, it is easy for the first forager to slip into the signaling role and execute it, whistling or gesturing to her counterpart. To play the receiver role, however, the second forager has to actually reorient his attention (to some degree) and attempt to engage in appropriate behavior for the world-state the first has observed (e.g., prey is to the east or to the west).

The Lewis signaling model by design is constrained such that the receiver’s actions are limited to just those acts associated with the sender’s observed world-states. It is of course sensible to begin inquiry with as simple of a model as possible and consider a limited range of responses to stimuli. However, our point is that it is more plausible to make these idealizations for signalers than for receivers. Signals are (by stipulation) cheap and easy to send, yet the actions available to the receiver are less plausibly interpreted as intrinsically cheap and free of opportunity cost.

In addition, the informational states drawn on by sender and receiver are also likely to be very different. Any real-life sender’s observation of a world state will likely inform her motivations (‘we should catch that animal’) to dictate a fairly clear course of action (‘try to direct the other agent’s behavior’). But all the receiver gets is a whistle, gesture, or other signal that (by stipulation) has no preestablished meaning. The experience of observing a strategically relevant state of the world will typically be richer and more detailed than that of observing a strategically relevant artificial signal.

All this leads to two concerns. First, asymmetries in the strategic situations are likely to exist between senders and receivers. Receivers are likely to have locally reasonable options available to them other than those relevant to signaler-observed states of the world, and their responsiveness to the strategic situation is therefore less satisfactorily modeled by the strictly symmetric payoff structures of standard signaling games. Call this the structural responsiveness concern.

Second, given the likely differences in informational states, goal directness, workload, and opportunity cost between sender and receiver roles, we can expect the mechanisms (cognitive and otherwise) that instantiate them to differ as well, quantitatively and qualitatively. This implies that we should not expect the update responsiveness between sender and receiver to be equal either. Yet the working evolutionary assumption is that senders and receivers update their strategies in an identical manner, modeled using either learning dynamics or replicator dynamics. Call this the evolutionary responsiveness concern.

3. Hedgehog Strategies and Update Asymmetry

The first of these concerns might sound like an argument for abandoning coordination games and moving toward ‘conflict of interest’ or ‘partial conflict of interest’ models. However, the issue is more specific than this.

The structural responsiveness concern provides parallel motivation to one of Sterelny’s (Reference Sterelny2012) worries about Skyrms’s (Reference Skyrms2010) use of the Lewis model. Sterelny asks whether the availability of ‘third options’ on the part of the receiver might undermine the evolution of signaling even when these third options are less valuable than the payoff for successful coordination. As part of a discussion of animal threat responses, he labels this a ‘hedgehog’ strategy, for it provides the agent with an action that pays off modestly regardless of the state of the world. To make this concrete, hedgehogs often roll into a ball in response to predators. This is a stark contrast to the more sophisticated behavior of vervets, who have specific responses to specific threats. Yet the optimal response a vervet takes to one threat—climb a tree when confronted by a leopard—may lead to total disaster when used in response to another threat, such as an eagle. Hedgehogs avoid such outcomes by ‘hedging’ unconditionally so as to secure a modest payoff. Translated to signaling games, such a gambit may, in many cases, be more attractive than attempting to respond optimally to a signal.Footnote ²

Sterelny’s ‘hedgehog’ strategy compliments the structural responsiveness concern: receivers (especially) might have other options of value that will stand in competition to those assumed in the standard signaling game. Something like these hedgehog strategies is a plausible departure from the idealization of the baseline Lewis signaling game and better captures the demandingness of the receiver role. The question is whether (as Sterelny suspects) including hedgehog strategies might undermine the evolutionary robustness of signaling systems.

Our second concern pertaining to evolutionary responsiveness parallels a well-known evolutionary hypothesis: the so-called Red Queen effect. In competitive relationships such as predator-prey or parasite-host, the Red Queen hypothesis states that species will be constantly adapting and evolving in response to one another just to “stay in the same place” (van Valen Reference van Valen1973). This should also be the case in competitive signaling situations—such as predator-prey signaling systems or courtship displays among conspecifics. Signalers and receivers come to not just update their strategies but do so at faster or slower rates depending on the nature of the strategic encounter they are entwined in.Footnote ³

It might seem that independent update rates should play no role in Lewis signaling games (along with games of common interest more generally). However, any realistic interpretation of the Lewis signaling game makes it plausible to consider asymmetry in evolutionary responsiveness as likely, if not the norm. First, as argued, the precise cognitive mechanisms and procedures employed by senders and receivers are likely to be different. Different systems (or systems used differently) will admit to different degrees of plasticity and evolvability and will have a different set of crosscutting tasks and utilities exerting distinct demands and pressures. Quick and easy signaling responses will have pathways of update and adaptation different from the (typically) more complex set of systems that appropriate receiver responses require. We should not assume that the evolution of sender and receiver strategies always proceeds at the same pace.

Finally, there is at least some evidence of a basic asymmetry between sender and receiver roles in the literature on great ape communication. For example, Hobaiter and Byrne (Reference Hobaiter and Byrne2014) stress the great sophistication and flexibility on the receiver side of Chimpanzee gestural communication, while Seyfarth and Cheney (Reference Seyfarth and Cheney2003) discuss how greater inferential sophistication on the receiver side is a feature of many primate communication systems. While these findings do not directly support the structural and evolutionary responsiveness concerns, they show that real-life sender and receiver strategies (in our near biological cousins at least) exhibit important differences, suggesting cognitive asymmetries compatible with those concerns.

In summary then, there is reason to consider two structural modifications to the Lewis signaling game as especially salient to the issue of responsiveness: the addition of ‘hedgehog’ strategies for receivers and differing rates of change in sender and receiver strategies.

4. The Model

The evolutionary model we use as the basis for our analysis is the pure strategy $2\times 2\times 2$ Lewis signaling game, with the two-population discrete-time replicator dynamics. Exact components of the model include two states of the world (L and R), a world-observing signaler with two possible signals (V ₁ and V ₂), and a signal-observing receiver with two possible actions (A _L and A _R).

If the receiver’s action matches the state of the world, then both signaler and receiver get a fixed positive success payoff; otherwise, their payoff is zero. Signalers and receivers both have four pure strategies available to them (see table 1).

Table 1. Signaler and Receiver Strategies in the Standard $2\times 2\times 2$ Common Interest Signaling Game

	Strategy Description
S1	Signal V 1 if L and signal V 2 if R
S2	Signal V 2 if L and signal V 1 if R
S3	Signal V 1 always
S4	Signal V 2 always
S5	Act AL if V 1 and act AR if V 2
S6	Act AR if V 1 and act AL if V 2
S7	Act AL always
S8	Act AR always

For the evolutionary model, the proportions of the different strategies within sender and receiver populations are initially randomly generated. The fitness of each strategy at a time period t is determined by the composition of the opposing population and the payoff associated with each strategy pairing. The proportion of each strategy at play in the next time period $t+1$ is determined by the standard discrete-time replicator dynamics. For the sender population this is

$\begin{array}{}& X_i(t+1)=X_i\left(t\right)\frac{F_i^S}{F^S},\end{array}$

where X _i is the proportion of senders using the ith sender strategy at a given point in time, $F_i^S$ is the fitness of that strategy, and F^S is the average sender strategy fitness. Likewise, for receivers

$\begin{array}{}& Y_j(t+1)=Y_j\left(t\right)\frac{F_j^R}{F^R},\end{array}$

where Y _j is the proportion of receivers using the jth receiver strategy, $F_j^R$ is the fitness of that strategy, and F^R is the average receiver strategy fitness. This is repeated until the populations settle into an evolutionarily stable arrangement. The update process is deterministic with no randomization or mutation.

5. Modifications and Results

We introduce two novel modifications to this model. First, we add a ‘hedgehog’ action A _H for the receiver. Second, we allow the rate of generational change of senders and receivers to vary relative to one another. In addition, the bias of nature is varied, and we investigate the effects these three departures from the Lewis $2\times 2\times 2$ game have on the evolutionary significance of signaling systems.

5.1. Hedgehog Strategy

Turning to our first modification, the receiver now has three possible actions upon observing the signal: A_L, A_R, and A_H. As before, a success payoff of 1 is received by both players in the case that the receiver plays A_L while the world is in state L or the receiver plays A_R while the world is in state R. A payoff of zero is received if A_L or A_R is played otherwise. A payoff of H is received unconditionally if the receiver plays A_H, where the value of H is between 0 and 1. The sender has four familiar pure strategies, whereas the receiver now has five (for simplicity we omit conditional strategies involving A_H).

To adapt the earlier forager story, we can imagine the sender and receiver as an egalitarian hunting party, and the game as a situation in which the sender remotely observes the location of a valuable prey animal (left or right) and calls out to the receiver. The receiver is initially unable to observe the prey but can choose to go left or go right (catching the prey if he goes in the matching direction) or alternatively to abandon the hunt in order to obtain a less valuable resource he does not need help from the sender to acquire (the hedgehog strategy). Varying the prior probability of the world is equivalent to it being in a situation in which it is systematically more likely that the prey is to the left or the right.

In the simple unbiased $2\times 2\times 2$ signaling game, one of the two signaling equilibria is guaranteed to be reached under the replicator dynamics (Huttegger Reference Huttegger2007). In our notation, these equilibria are S1-S1 and S2-S2. Increasing the bias of the world (i.e., making L more probable than R or vice versa) will undermine this, with an increasing proportion of populations instead collapsing to a pooling equilibrium. This will occur when there are initially few conditional signaling strategies (such as S1) in the sender population. In such situations, receivers do best to simply perform the act that is most appropriate for the more likely state of the world. The incentive for senders to adopt a signaling system then disappears, and the community is locked into a pooling equilibrium.

Not surprisingly, we find a similar effect with the hedgehog strategy as values of H, the payoff for A_H, become significant. The hedgehog strategy provides the receiver with an additional unilateral response and is able to attract some proportion of initial populations away from the signaling equilibria when H is in excess of 0.5 (i.e., the average payoff for ‘guessing’). This result, for an unbiased world, is illustrated in figure 1.Footnote ⁴

Figure 1. Effect of varying hedgehog payoff H on the number of simulations reaching signaling equilibria.

We observe a more surprising result when the bias and H are varied in combination. Figure 2 shows the results of varying bias for different values of H. As expected, when H is set to zero signaling is guaranteed when nature is unbiased and is less likely to emerge as bias increases. As before, when the hedgehog strategy pays significant dividends ($H>0.5$) signaling is less likely when nature is unbiased. For these high values of H, however, increasing nature’s bias does not immediately reduce the likelihood of signaling. In fact we observe a ‘plateau’ followed by an increase in signaling behavior, peaking at the point where bias is equal to the value of H, that is, at $P\left(L\right)=H$ and at $P\left(L\right)=1-H$. Within these bounds, all populations reach either a signaling equilibrium or a Hedgehog equilibrium (in which all receivers have adopted the hedgehog strategy), but for values of H outside of this interval traditional pooling instead occurs when the population fails to reach a signaling convention.

Figure 2. Effect of hedgehog strategy and bias of nature on number of simulations reaching signaling equilibria.

5.2. Generational Asymmetry

We now turn to our second modification of the Lewis signaling framework in which we introduce a generational asymmetry. We introduced a ‘slow-down factor’ Z to the replicator dynamics in order to control the rate at which sender and receiver populations change over time. Composition of the sender and receiver populations is now governed by the following equations:

$\begin{array}{}& \begin{array}{ccc}{X}_i(t+1)& =& (1-Z_S)X_i\left(t\right)\frac{F_i^S}{F^S}+X_i\left(t\right)Z_S,\\ Y_j(t+1)& =& (1-Z_R)Y_j\left(t\right)\frac{F_j^R}{F^R}+Y_j\left(t\right)Z_R\mathrm{.}\end{array}\end{array}$

Note that when both $Z_R=0$ and $Z_S=0$ the above equations are simply the standard replicator dynamics. However, the rate of change is reduced as these values are increased; for example, setting $Z_S=.5$ halves the rate of change for sender strategies. Setting Z_R to 1 (while holding Z_S fixed at 0) means the composition of the receiver population remains fixed, and only the sender population is allowed to evolve.

Introducing this generational asymmetry between senders and receivers has the effect of making signaling more likely when sender strategies evolve faster than receiver strategies. This is illustrated in figure 3, where, for various levels of bias, senders (receivers) are slowed down to half and one-tenth speeds while receivers (senders) are left unaltered.

Figure 3. Effect of generational asymmetry and bias of nature on number of simulations reaching signaling equilibria.

Slowing the evolution of the sender population leads to more pooling because, as before, receivers facing a sender population consisting of few conditional signalers will do best to simply perform that act that is best suited for the more likely state of the world. As figure 3 illustrates, this effect is more pronounced for high levels of bias. Slowing the evolution of the receiver population has the opposite effect. Senders now have time to adopt the best separating strategy given the initial mix of receiver strategies. Once this occurs, the receiver population slowly responds, and the end result is a robust signaling system. By a similar logic, it is easy to see that a faster-evolving sender population also mitigates against the effect of receiver hedgehog strategies.Footnote ⁵

6. Discussion

We have explored a few well-motivated departures from the highly idealized and simple Lewis signaling game typically considered in the literature. As shown in section 4, breaking the symmetry between senders and receivers often significantly reduces the likelihood that a separating equilibrium emerges. For one, providing receivers with a safe third option that allows them to secure a decent payoff regardless of the state of the world significantly reduces the size of the basin of attraction of the separating equilibrium. Likewise, separating is a remote possibility when receivers outpace senders in the race to adapt.

However, the situation is less bleak when senders evolve at a faster pace than receivers.Footnote ⁶ Interestingly, many scholars in the animal communications literature have noted that such a response asymmetry holds between sender and receiver when the interests of the parties partially conflict. For instance, Owren, Rendall, and Ryan (Reference Owren, Rendall and Ryan2010) note that senders can easily adapt their signaling behavior while receivers for the most part have responses to the stimuli produced by senders that are more difficult to change. This has lead some to the conclusion that signaling primarily involves the manipulation of receivers by senders.

Yet this leaves us with an evolutionary puzzle. If there is a partial conflict of interest between sender and receiver, what prevents receivers from increasing the speed at which they adapt to the behavior of the sender? In other words, what explains the absence of an evolutionary arms race between sender and receiver? Partial-conflict-of-interest settings are the exact circumstances to which we would expect the Red Queen hypothesis to apply. We believe the results of this article may form the basis of a novel explanation for this puzzling phenomena. Recall that when the interests of sender and receiver are perfectly aligned, it is actually in the interest of both parties for the sender population to ‘take the lead’ and evolve at the faster rate, as doing so ensures the community is more likely to hit upon a mutually beneficial signaling system.

Yet senders and receivers rarely find themselves engaged exclusively in either common or conflict-of-interest signaling games. As is well known by any parent, not all signaling interactions between relatives are free of conflict. Likewise, agents whose interests are typically thought to be partially opposed, such as two potential mates, may frequently engage in common interest signaling games in contexts unrelated to mating. The point here is that a variety of distinct strategic scenarios can hold between sender and receiver. There is no principled reason to think all interactions will involve the perfect alignment of interests or sizable conflict. Nor is it sensible to assume that communicative strategies will always be fine-grained enough for scenario-by-scenario optimization.

If this is correct, then when a sizable proportion of interactions between sender and receiver involve no or very low conflict of interest, the generational asymmetry result from the previous section may hold to some degree. Both sender and receiver profit when the sender population evolves at a faster rate than the receiver population. Receivers in this context do best to limit how responsive they are to senders so as to ensure the emergence of informative signaling systems in those cases in which their interests do overlap.