3.1 The Lewis Signaling Game

Reference QuineQuine ([1936] 1976) famously argued that linguistic conventions could not emerge naturally. The idea is that one cannot assign meaning to a term without using an established language. In Convention, David Lewis first employs game theory to chip away at this argument (Reference LewisLewis, 1969). In particular, he introduces a highly simplified game to show how terms might attain conventional meaning in a communication context. Philosophers have often called this the Lewis signaling game, though it is also referred to as the common interest signaling game, or just the signaling game. As we will see, there are actually a large family of games that go under this heading, but let us start with the basics.

Suppose that a mother and a daughter regularly go on outings. Sometimes it is rainy, in which case they like to walk in the rain. If it is sunny, they have a picnic outside. To figure out whether to bring umbrellas or sandwiches, the daughter calls her mother ahead of time to see what the weather is like. The mother reports that it is either “sunny” or “rainy,” and the daughter makes her choice by bringing either umbrellas or a picnic.

We might represent a scenario like this with the following game. There are two possible states of the world – call them state 1 and state 2 – which represent the weather being sunny or rainy. A player called the sender (the mother) is able to observe these states and contingent on that observation to send one of two signals (“sunny” or “rainy” – call them signal a and signal b). A player called the receiver, the daughter, cannot observe the world directly. Instead, she receives whatever message is sent and can take one of two actions (act 1 and act 2) – bringing umbrellas or a picnic. Act 1 is appropriate in state 1 and act 2 in state 2. If the receiver takes the appropriate action, both players get a payoff of 1 (just like the mother and daughter get a payoff for a picnic in the sun and umbrellas in the rain), otherwise they get nothing. In this game, two players want the same thing – for the action taken by the receiver to match the state observed by the sender. (This is why the game is a common interest signaling game.) The only way to achieve this coordination dependably is through the transmitted signal.

Figure 3.1 shows the extensive form of this game. This game tree should be read starting with the central node. Remember from the last section that play of the game can go down any of the forking paths depending on what each agent chooses to do. In order to capture the fact that one of two states obtains independent of the players’ choices, we introduce a dummy player called Nature. Nature (N) first chooses state 1 or state 2. Then the sender (S) chooses signal a or b. The receiver (R) has only two information sets since they do not know which state obtains. These, remember, are the nodes connected by dotted lines. The receiver, upon finding themselves at one of these sets, chooses act 1 or 2. The payoffs are listed at the terminal branches. Since payoffs are identical for both actors, we list only one number.

Figure 3.1 The extensive form respresentation of a Lewis signaling game with two states, two signals (a and b), and two acts.

Lewis observed that there are two special sorts of Nash equilibria of this game often called signaling systems. These are the sets of strategies where the sender and receiver always manage to coordinate with the help of their signals. Either signal a is always sent in state 1, and signal b in state 2, and the receiver always uses these to choose the right act, or else the signals are flipped. Figure 3.2 shows these two signaling systems. Arrows map states to signals and signals to acts. Notice that the left arrows in each diagram represent the sender’s strategy, and the right ones the receiver’s.

Figure 3.2 The two signaling systems of the Lewis signaling game. In the first, signal a represents state 1 and causes act 1, while signal b attaches to state 2 and act 2. In the second system, the meanings of the two signals are reversed.

What Lewis argued is that at each of these equilibria, players engage in a linguistic convention – a useful, dependable social pattern which might have been otherwise. (In Section 7 we will say a bit more about this concept.) For Lewis, these conventions might arise through a process of precedent setting or as a result of natural salience. If so, this could explain how conventional meaning might emerge naturally in human groups.

But one might worry that while Lewis gestures toward a process by which signals can obtain conventional meaning, he does not do much to fill in how such a process might go. Furthermore, we have seen that conventional meaning has emerged throughout the natural world. Can Lewis’s analysis address vervet monkeys? We do not suppose that their alarm signals were developed through a process of precedent setting between members of the group. Furthermore, Lewis requires that his players have full common knowledge of the game and expectations about the play of their partner for an equilibrium to count as a convention. These conditions will not apply to the animal world (and sometimes not to the human one either). Let us now turn to explicitly evolutionary models of this game.

3.2 The Evolution of Common Interest Signaling

Reference SkyrmsSkyrms (1996) uses evolutionary game theory to strengthen Lewis’s response to Quine. He starts by observing that in the simple signaling game described above, the only ESSs are the two signaling systems (Reference WärnerydWärneryd, 1993).Footnote ¹⁴ This is because each signaling system achieves perfect coordination, and any other strategy will sometimes fail to coordinate with those playing the signaling system. In addition no other strategy has this property – each may be invaded by the signaling system strategies (Reference WärnerydWärneryd, 1993; Reference SkyrmsSkyrms, 1996).Footnote ¹⁵ So in an evolutionary model, then, we have some reason to think the signaling systems will evolve.

As Skyrms points out, true skeptics might not be moved by this ESS analysis. In particular, they might worry that there is no reason to think that either one signaling system or the other will take over a population. If so, groups might fail to evolve conventional meaning. To counter this worry, he analyzes what happens when a single, infinite population evolves to play the signaling game under the replicator dynamics. We observed in Section 2 that these dynamics can represent both cultural evolution and natural selection. If we see meaning emerge in this model, then, we have good reason to think that simple signals can emerge in the natural world as well as the social one. Using computer simulation tools, Skyrms finds that one of the signaling systems always takes over the population. Subsequently, proof-based methods have been used to show this will always happen (Reference Skyrms and PemantleSkyrms and Pemantle, 2000; Reference HutteggerHuttegger, 2007a).Footnote ¹⁶ As Skryms concludes for this model, “the emergence of meaning is a moral certainty” (93).

As we will see in a minute, this statement must be taken with a grain of salt. But this “moral certainty” nonetheless has explanatory power. Signaling is ubiquitous across the biological and human social realms, and now we see part of the reason why. Despite Quine’s worries, signaling is easy to evolve. It takes very little to generate conventional meaning from nothing.

This is not the end of the story, though. Subsequently, philosophers and others have developed a deep understanding of common interest signaling, deriving further results about this very simplest signaling model and a host of variants. In addition, the signaling game has been co-opted in numerous ways to address particular research questions. Let us first look at what else we have learned about the signaling game, and then move on to some applications. Much of the literature I will now describe is discussed in more detail in Reference SkyrmsSkyrms (2010b).Footnote ¹⁷

The simple game we have been discussing is sometimes called the 2 × 2 × 2 game, for two states, two signals, and two possible actions. But we can consider N × N × N games with arbitrary numbers of states, signals, and actions. When N > 2, these models have a new set of equilibria called partial pooling equilibria. The sender sends the same message in multiple states, and upon receipt of this partially pooled message the receiver mixes between the actions that might work. Figure 3.3 shows what this might look like for the 3 × 3 × 3 game. After receipt of signal b, the receiver does act 2 with probability x and act 3 with probability 1 − x. These are mixed strategy equilibria, where the actors mix over behaviors.Footnote ¹⁸ The only ESSs of these games are the signaling systems, not these partial pooling equilibria. These equilibria, however, do have basins of attraction under the replicator dynamics (Reference Nowak and KrakauerNowak and Krakauer, 1999; Reference HutteggerHuttegger, 2007a; Reference PawlowitschPawlowitsch et al., 2008). And for larger games, simulations show an increasing likelihood that a system of partial communication will emerge instead of perfect signaling (Reference BarrettBarrett, 2009). In other words, the moral certainty of signaling is disrupted when even a bit of complexity is added to the model.

Figure 3.3 A partial pooling equilibrium in a 3 × 3 × 3 signaling game. The sender uses b in both states 2 and 3. Upon receipt of b, the receiver mixes between acts 2 and 3.

Another, even smaller change can disrupt this certainty. Consider the 2×2×2 model where one state is more probable than the other. The more common one state is, the smaller the benefits of signaling. (When one state occurs 90 percent of the time, for example, the receiver can be very successful by simply taking the action that is best in that state.) As a result, babbling equilibria, where no information is transferred because sender and receiver always do the same thing, evolve under the replicator dynamics with greater likelihood the more likely one state is (Reference HutteggerHuttegger, 2007a; Reference SkyrmsSkyrms, 2010b).

There is a methodological lesson here. Early analyses of ESSs in com mon interest signaling games, as in Reference WärnerydWärneryd (1993), indicated that perfect signaling is expected to evolve. But, as we have just noted, more complete dynamical analyses show the ESS analysis insufficient – it fails to capture the relevance and stability of babbling equilibria and partial pooling equilibria.

There is also a philosophical lesson. The fact that these small changes lead to situations where perfect signaling does not emerge seems to press against the power of these models to help explain the ubiquity of real-world signaling. But we should not be too hasty. Philosophers have looked at changes to these more complex models that increase the probability of signaling. These include the addition of drift and mutation to the models (Reference HutteggerHuttegger, 2007a; Reference Hofbauer and HutteggerHofbauer and Huttegger, 2008), correlation between strategies – where signalers tend to meet those of their own type (Reference Skyrms and PemantleSkyrms and Pemantle, 2000; Reference Huttegger, Zollman, Okasha and BinmoreHuttegger and Zollman, 2011a; Reference SkyrmsSkyrms, 2010b), and finite populations (Reference PawlowitschPawlowitsch, 2007; Reference Huttegger, Skyrms, Smead and ZollmanHuttegger et al., 2010). Furthermore, even in cases where evolved communication is not perfect, quite a lot of information may be transferred (Reference BarrettBarrett, 2009). In other words, our simple signaling models retain their power to help explain the ubiquity of real-world signaling, even if in many cases perfect signaling does not emerge.

Let’s look at some more variants that extend our understanding of signaling. Sometimes more than two agents communicate at once. Bacterial signaling is usually between multiple senders and multiple receivers, for instance. And vervet monkeys send alarm calls to all nearby group members. Reference BarrettBarrett (2007) and Reference SkyrmsSkyrms (2009) consider a model with two senders, one receiver, and four states of the world, where senders each observe different partitions of the states. (That is, one sender can distinguish states 1–2 from states 3–4, and the other can distinguish odd and even states.) These authors show how signaling systems can emerge that perfectly convey information to the receiver.Footnote ¹⁹ Reference SkyrmsSkyrms (2009, Reference Skyrms2010b) considers further variations, such as models where one sender communicates with two receivers who must take complimentary roles, where a receiver listens to multiple senders who are only partially reliable, where actors transmit information in signaling “chains,” and dialogue models where the receiver tells the sender what information they seek and the sender responds. In each case, the ease with which signaling can emerge is confirmed.

Some creatures may be limited in the number of signals they can send. Others (like humans) may be able to easily generate a surfeit of terms. In an evolutionary setting, games with too many signals often evolve to outcomes where one state is represented by multiple terms or “synonyms”. In these states, actors coordinate just as well as they would with fewer signals. This is because each signal has one meaning, even if a state may be represented by multiple signals (Reference PawlowitschPawlowitsch et al., 2008; Reference SkyrmsSkyrms, 2010b).

Without enough signals, actors cannot perfectly coordinate their action with the world. This is because they do not have the expressive power to represent each state individually. For this reason, Reference SkyrmsSkyrms (2010b) describes this sort of situation as an “information bottleneck.”Footnote ²⁰ At the optimal equilibria of such models, actors may either ignore certain states or else categorize multiple states under the same signal. Reference Donaldson, Lachmann and BergstromDonaldson et al. (2007) point out that in some cases actors have payoff reasons for choosing some ways of categorizing over others, in which case these ways may be more likely to evolve. For example, while the ideal response to predatory wolves and dogs might be slightly different for a group of monkeys, it still might make sense to use one alarm call for these highly similar predators if monkeys are limited in their number of signals. We will come back to this theme in Section 3.3.

One thing actors without enough signals might like to do is to invent new ones – this should help them improve coordination. To add signals, though, involves a change to the structure of the game itself. Reference SkyrmsSkyrms (2010b) uses a different sort of model – one that explicitly represents learning – to explore this possibility. Such models have provided an alternative paradigm to explore the evolution of signaling, one that has often created synergy with evolutionary models. Let us now turn to this paradigm.

3.3 Applying the Model

We have now seen that a great deal of theoretical progress has been made understanding the Lewis signaling game and the evolution of simple, common interest signaling. Let’s turn to how philosophers of biology have applied the model to deepen our understanding of specific phenomena in the biological world. We will not be able to discuss all the literature that might be relevant. Rather, the goal is to see some examples of what the common interest signaling game can do.

4.1 Costs and Conflict of Interest

The first thing to observe is that even the mantis shrimp do not have completely opposing interests. In particular, neither of them wants to fight, meaning that instead, they have a partial conflict of interest. The most classic case of partial conflict of interest signaling in biology comes from mate selection. In peacocks, for example, the peahen shares interests with high quality males because they would both like to mate. But while low-quality males would like to mate with her, she would not choose them. High-quality males could send a signal of their quality to facilitate mating, but low-quality males would be expected to immediately co-opt it. Analogous situations crop up in human society. The classic case is one of a business trying to hire. They share interests with high-quality candidates but have a conflict with low-quality ones. How can the business trust signals of quality it receives, given that low-quality candidates want to be hired?

Both biologists and economists figured out a solution to the signaling problem just introduced. In biology, Amotz Zahavi introduced what is called the handicap principle, which was later modeled using signaling games by Alan Grafen (Reference ZahaviZahavi, 1975; Reference GrafenGrafen, 1990). The insight is that communication can remain honest in partial conflict of interest cases if signals are costly in such a way that high-quality individuals are able to bear the cost, while low-quality individuals are not. In an earlier model, economist Michael Spence showed the same thing for the job market case described above (Reference SpenceSpence, 1978). If a college degree is too costly for low-quality candidates to earn, it can be a trustworthy signal of candidate quality. To be precise, in these models, signaling equilibria exist where high-quality candidates/mates signal, and low-quality ones do not because of these costs.

For the purposes of this Element, I will describe a very simple version of a partial conflict of interest signaling game (borrowed from Reference Zollman, Bergstrom and HutteggerZollman et al. (2012)). Figure 4.1 shows the extensive form. Nature plays first and selects either a high-type (type 1) or low-type (type 2) sender. Then the sender decides to signal or not. Upon observing whether a signal was sent, the receiver chooses action 1 (mate, or hire) or action 2 (do not mate, do not hire). The sender’s payoffs are listed first – they get a 1 whenever the receiver chooses action 1, minus a cost, c, if they signaled. We assume that c₁ < c₂, or high-type senders pay less to signal. Receivers get 1 for matching action 1 to a high-type sender or action 2 to a low-type sender. This corresponds, for example, to a peahen mating with a healthy peacock or rejecting an unhealthy one.

Figure 4.1 The extensive form respresentation of a partial conflict of interest signaling game.

Consider the strategy pairing where high types send, low types do not, and receivers only take action 1 in response to a signal. This is not an equilibrium when costs are zero. This is because if a receiver is taking action 1 in response to the signal, low types will switch to send the signal. Once both high and low types send, there is no longer any information in the signal, and receivers should always take whatever action is best for the more common type.

Suppose instead that c₁ < 1 < c₂. High types gain a benefit of 1 – c₁ from signaling if receivers are then willing to choose action 1 as a result. Low types cannot benefit from signaling, since the cost of the signal is more than the benefit from being chosen as a mate/colleague. Thus there is a costly signaling equilibrium where the signal carries perfect information about the type of the sender, and receivers and high types get a payoff as a result. Reference GrafenGrafen (1990) showed that this type of equilibria are ESSs of costly signaling models.

It has been argued that costly signaling equilibria can help explain a variety of cases of real-world signaling. The original application for biology, as noted, is to the sexual selection of costly ornamentation. The peacock’s tail, for example, imposes a cost that only healthy, high-quality peacocks can bear. Contests, such as those between mantis shrimp, are another potential application (Reference GrafenGrafen, 1990). If it is costly to develop expensively bright meral spots, this may be a reliable indicator of strength and fitness. Another potential application regards signaling between predator and prey. Behaviors like stotting, where, upon noticing a cheetah, gazelles spring into the air to display their strength, may be costly signals of quality.Footnote ³⁵ And as we will see in Section 4.4, similar models have been applied to chick begging.

There are a few issues, though. In many species, when the actual costs paid by supposedly costly signalers are measured, they come up short. They do not seem to be enough to sustain honest signaling (Reference Caro, Lombardo, Goldizen and KellyCaro et al., 1995; Reference Silk, Kaldor and BoydSilk et al., 2000; Reference GroseGrose, 2011; Reference Zollman, Bergstrom and HutteggerZollman et al., 2012; Reference ZollmanZollman, 2013). This has led to widespread criticism of the handicap theory.

There are theoretical issues with the theory as well, which arise because costs are generally bad for evolution. Reference Bergstrom and LachmannBergstrom and Lachmann (1997) point out that some costly signaling equilibria are worse, from a payoff standpoint, than not signaling at all. And once one does a full dynamical analysis, it becomes clear that the existence of costly signaling equilibria is insufficient to explain the presence of real-world partial conflict of interest signaling. This is because costly signaling equilibria tend to be unlikely to evolve due to the fitness costs that senders must pay to signal. These equilibria have small basins of attraction under the replicator dynamics (Reference WagnerWagner, 2009; Reference Huttegger and ZollmanHuttegger and Zollman, 2010; Reference Zollman, Bergstrom and HutteggerZollman et al., 2012). Is the costly signaling hypothesis bankrupt? Or does it nonetheless help explain the various phenomena it has been linked to? Let us now turn to further theoretical work that might help circumvent these worries.

4.2 The Hybrid Equilibrium

By the early 1990s, economists had identified another type of equilibrium of costly signaling models (Reference Fudenberg and TiroleFudenberg and Tirole, 1991; Reference GibbonsGibbons, 1992). In a hybrid equilibrium, costs for signaling are relatively small. But even small costs, as it turns out, are enough to sustain signals that are partially informative.

Let’s use the game in Figure 3.1 to fill in the details. Suppose the cost for low types, c₂, is constrained such that 0 < c₂ < 1, and the cost for high types is less than this, c₁ < c₂. Under these conditions, there is a hybrid equilibrium where (1) high types always signal, (2) low types signal with some probability α, (3) the receiver always chooses action 2 (i.e., do not mate) if they do not receive the signal, and (4) the receiver chooses action 1 (i.e., mate) with probability β upon receipt of the signal. Figure 4.2 shows this pattern.

Figure 4.2 The hybrid signaling equilibrium.

The values for α and β that sustain this equilibrium depend on the details of the costs and the prevalence of high types in the population. If x is the proportion of high types, α = x/(1 – x) and β = c₂ (Reference Zollman, Bergstrom and HutteggerZollman et al., 2012). These values are such that low types expect the same payoffs for sending and not sending (and thus are willing to do both with positive probability), and high types expect the same payoffs from taking either action given the signal. Unlike the costly signaling equilibrium, only partial information is transferred at the hybrid equilibrium. Upon receipt of the signal, the receiver cannot be totally sure which state obtains. But she is more certain of the correct state than she was before observing the signal and can shape her behavior accordingly.

In some ways, the hybrid equilibrium seems inferior to the costly signaling one – in the latter perfect information is transferred. In the former, high types and receivers are only somewhat able to act on their shared interests. From an evolutionary perspective, though, the hybrid equilibrium proves very significant. Unlike the costly signaling equilibria, low signaling costs mean that hybrid equilibria are often easier to evolve in costly signaling models (Reference WagnerWagner, 2013b; Reference Huttegger and ZollmanHuttegger and Zollman, 2010; Reference Zollman, Bergstrom and HutteggerZollman et al., 2012; Reference Kane and ZollmanKane and Zollman, 2015; Reference Huttegger and ZollmanHuttegger and Zollman, 2016). This might help explain how partial conflict of interest communication is sustained throughout the biological world. Small costs help maintain information transfer, without creating such an evolutionary burden that communication is not worthwhile. Furthermore, these low costs seem more in line with empirical findings in many cases.Footnote ³⁶

4.3 When Honesty Met Costs: The Pygmalion Game

Hybrid equilibria can evolve and guarantee partial information transfer. There is another solution, though, to the problem posed by the costs of signaling. Philosophers of biology have argued that, in fact, the very setup of the costly signaling game builds into it an unrealistic assumption.

Aside from signal cost, there is another way to guarantee information transfer in partial conflict of interest scenarios. This is for high types to have a feature or signal that is unavailable to low types, or unfakeable. Suppose humans were able to innately read the quality of job market candidates, for instance. There would be no need for expensive college degrees to act as costly signals, because it would be obvious which candidates to hire. A feature that correlates with type in this way is sometimes called an index rather than a signal (Reference Searcy and NowickiSearcy and Nowicki, 2005).Footnote ³⁷

Reference Huttegger, Bruner and ZollmanHuttegger et al. (2015) point out that while biologists have traditionally thought of costly signaling and indexical signaling as alternative explanations for communication in partial conflict of interest cases, this distinction is a spurious one. They introduce a costly signaling model called the pygmalion game where, when low types attempt to signal, they do not always succeed. With respect to stotting, this would correspond to fast gazelles having a higher probability of jumping high enough to impress a cheetah than slow ones. In other words, the signal is partially indexical. As they point out, at the extremes, their model reduces to either a traditional costly signaling model – if low and high types are both always able to signal perfectly – or to an index model where only high types are able to send the signal at all.

Between these extremes is an entire regime where costs and honesty tradeoff. As they show, the harder a signal is to fake, the lower the costs necessary to sustain a costly signaling equilibrium. The easier the signal is to fake, the higher the costs. In these intermediate cases where signal costs are not too high, the signaling equilibria often have large basins of attraction. So if signals are even a bit hard to fake, the costs necessary to sustain full, communicative signaling might not be very high, easing the empirical and theoretical worries described earlier in the section.Footnote ³⁸

4.4 Altruism, Kin Selection, and the Sir Philip Sydney Game

As mentioned earlier, the costly signaling hypothesis may also explain elements of chick begging. Reference Maynard-SmithMaynard-Smith (1991) introduced the Sir Philip Sydney game to capture this case.Footnote ³⁹ In this model, chicks are needy or healthy with some probability. They may pay a cost to signal (cheep), or else stay silent. A donor, assumed to be a parent, may decide whether to pay a cost to give food conditional on receipt of the signal. If chicks do not receive a donation, their payoff (chance of survival) decreases by some factor that is larger for needy chicks. Notice that this model is analogous to the partial conflict of interest models described thus far, but with a notable difference – healthy and needy chicks do not pay differential costs to signal but instead receive differential benefits.

In this model, there is never a direct incentive for the receiver to donate food, since this only leads to a direct cost. Any such donor would have to be an altruist in the biological sense, i.e., willing to decrease their fitness in order to increase the fitness of a group member. But, as Maynard-Smith points out, in cases where donor and signaler are related, donation may be selected via kin selection. He uses a heuristic for calculating payoff in this model, which supposes that the donor (receiver) gets a payoff themselves when they donate to a needy chick related to them (more on this shortly). As Maynard-Smith shows, if there is total relatedness between donor and recipient, the signaling equilibrium is an ESS – only needy individuals send the signal, and the receiver donates in response. If there is partial relatedness, signal costs can ensure a signaling equilibrium as well. Only needy individuals are willing to pay the costs to signal, and the receiver is only willing to donate upon receipt of the signal. As Reference Johnstone and GrafenJohnstone and Grafen (1992) point out, the higher the relatedness, the lower the cost necessary to protect signaling. As such, relatedness provides another solution to worries about costly signaling by increasing shared interests between signalers.

Furthermore, Reference Bruner and RubinBruner and Rubin (2018) give a methodological critique of the heuristic used by Reference Maynard-SmithMaynard-Smith (1991) to analyze the Sir Philip Sydney game that further addresses these worries. Maynard-Smith’s heuristic calculates inclusive fitness, which takes into account how an individual influences the fitness of group members who share genetics. In particular, Maynard-Smith calculates payoffs to an individual and adds to this the payoffs of relatives weighted by relatedness. Although it is widely recognized that this is not the correct way to calculate inclusive fitness, the heuristic is used because it is simple and captures the basic idea that inclusive fitness should take influence on relatives into account.Footnote ⁴⁰ What Bruner and Rubin show is that under more proper calculations, honest signaling emerges in the Sir Philip Sydney game even without costs, for partially related individuals. In other words, their results more strongly support the claim that relatedness can sometimes solve the costly signaling problem.Footnote ⁴¹

4.5 Measuring Conflict of Interest

While this entire section has focused on partial conflict of interest signaling, we have not given a clear definition of what conflict of interest is. Complete common interest means that sender and receiver perfectly agree about which outcomes of the game are preferable to which others. In conflict of interest scenarios, this is no longer true. But while it is clear that interests can conflict to greater or lesser degrees, it is not entirely clear how this should be measured. One typical claim in signaling theory is that signaling cannot occur without some common interest and that the more shared interests there are, the easier it is to generate signaling. Without a measure of conflict, though, these claims are hard to assess.

Reference Godfrey-Smith and MartínezGodfrey-Smith and Martínez (2013) wade in. They consider a suite of randomly generated 3×3×3 signaling games, and for each one, calculate the degree of common interest between sender and receiver. They compare preference orderings over outcomes and generate a measure between 0 and 1 that ranks how different these are. They also calculate, for each game, whether at least one equilibrium exists where information is transfered and where there are no signal costs whatsoever. Surprisingly, they find that for some games with zero common interest information transfer is possible at equilibrium, though this is more likely the greater the common interests are.Footnote ⁴² Reference Martínez and Godfrey-SmithMartínez and Godfrey-Smith (2016) use replicator dynamics models to derive similar, but explicitly evolutionary, results – that common interest strongly predicts whether a population playing a game is likely to evolve communication, but sometimes this happens despite zero common interest.

This finding is dependent on their measure, though. If they took zero common interest to correspond to a zero-sum game (where more for you is always less for me), the finding would not hold. Surprisingly, though, Reference WagnerWagner (2011) does show how in a zero-sum model, information can be transfered in evolving signaling populations. This is not equilibrium behavior but occurs as a result of chaotic dynamics.Footnote ⁴³ What we see here is that theoretical work on the meaning of conflict may have important consequences for signaling theory. There is certainly room for further elaboration of this concept.

The costly signaling hypothesis has been beset by critics, but philosophers of biology have helped contribute to the literature showing why costs may nonetheless help sustain partial conflict of interest signaling. Even low costs can work for partial information transfer, especially when signals are partially indexical or when organisms are related. Furthermore as we saw, philosophers of biology are contributing to conceptual work defining what conflict of interest signaling is, and, relatedly, assessing claims about the relationships between conflicts, costs, and information transfer.

The next section continues with work on the signaling game but turns toward more philosophical issues related to signal meaning.

5.1 Information Theory

A good place to start, in thinking about theories of meaning and content, is with the theory of information. Information theory, beginning with the influential work of Claude Shannon, focuses on questions regarding how much information can be sent over various channels, and how best to package it (Reference ShannonShannon, 1948). This theory also gives a way to measure information working off of probabilistic correlations.

It might sound strange to talk about measuring information, but we can think of it in the following way. The more something reduces the uncertainty of an observer, the more information it contains. Consider a random coin flip. Before observing the coin, you are uncertain about which face will be up. There is a 50/50 probability of either heads or tails. Upon observing the coin, your uncertainty is reduced. You are now certain that it is, say, heads. Consider a different case, where you are flipping a biased coin that comes up heads 99 percent of the time. Now even before you flip, you’re almost certain that it will come up heads. Once you observe that the coin is, indeed, heads, your uncertainty is reduced, but only by a bit. Alternatively, consider throwing an unweighted die. There are six sides, each of which will turn up with probability 16.6 percent. Now there is more uncertainty than in the unbiased coin case. With the coin you knew it would be either heads or tails, but now it could be any of six outcomes. When you observe that the face is “four,” you reduce your uncertainty more than with the coin.

Reference ShannonShannon (1948) introduced the notion of entropy to capture the differences between these cases. Entropy tells us how much uncertainty is reduced, and thus how much information is conveyed through some channel on average. A channel is some variable that can be in multiple states – a flipped coin, a die, or else a monkey alarm, or a job resume stating whether a candidate attended college. If the probability of each possible state i is p_i, the entropy of a channel is

${\displaystyle \sum _i-p_i{\log }_2(p_i)}.$(5.1)

Let’s break this down. For each possible state of the channel, we take its probability. In the case of the coin, p_H = .5 and p_T = .5. Then we multiply this by the negative log of that probability and add it up. In the coin case, –log₂(.5) = 1. So the entire formula here yields .5 ∗ 1 + .5 ∗ 1 = 1. When we observe the coin, we gain one bit of information.

Why the logarithm part? The answer is that it has nice properties for measuring information. The log of 1 is 0. Intuitively, if something has a probability of 1, i.e., is entirely certain to happen, our uncertainty is not reduced at all from observing it. On this measure, it carries no information. As the probability of an event decreases toward 0, the negative log becomes larger and larger. This, again, makes intuitive sense: if something is highly unlikely, we learn a lot from observing it. The more unlikely the event is, the more information it carries on this measure. We could use another logarithm, if we wanted, which would simply change the unit of information from bits to some other measure. (And, to be clear, the negative log is used simply so that the measure will be positive.)Footnote ⁴⁴

Let’s try this for the biased coin. The probabilities are p_H = .99 and p_T = .01. If we take the negative log of these, we get –log₂(.99) = .014 and –log₂(.01) = 6.64. Notice that we get very little information from heads and quite a lot from tails. The entire formula is thus .99 ∗ .014 + .01 ∗ 6.64 = .08. This is much less than the expected information in the equiprobable coin, tracking our intuition that we don’t learn much in this case (even though we learn a lot in the unlikely case that tails comes up). For the die roll the probability is .166, and the negative log of that is 2.6, so the entropy is 2.6, tracking the intuition that we expect to learn more.

How does this all connect up with signaling and communication? We can use information theoretic measures to tell us not just how much information is transmitted on average via some channel but how much that channel tells us about some other distribution. If our channel is vervet alarm calls, for example, we can measure how much these alarms tell us about the actual state of the world vis-à-vis predators.

As Reference SkyrmsSkyrms (2010a, Reference Skyrms2010b) shows, the signaling game framework provides a nice way to do this, using what is called the Kullback–Liebler divergence.Footnote ⁴⁵ This measure can tell us for any two distributions of events, such as states of predators in the forest, and alarm calls given, how different the second is from the first. Let’s transition to the language of signaling games to understand this measure better. Let p_i be the prior probability of the state and $p_i{}^{\mathrm{sig}}$ be the probability of that state after the signal arrives. The measure is

${\displaystyle \sum }_i{p}_i^{\mathrm{sig}}{\log }_2(p_i^{\mathrm{sig}}/p_i).$(5.2)

Here is one way to think about this formula. In trying to figure out how much information is in the signal, we want to know how much it changes the probability of the state. So we can take the ratio of the new and old probabilities and take the log of that, ${\log }_2(p_i{}^{\mathrm{sig}}/p_i)$. This gives us the information in the signal about state i. Then we can average this over all the states to get the full information in the signal about all possible states.

Let’s apply this. Suppose I am flipping an unbiased coin. Also suppose that after I observe it, I turn on either a red light for heads or a blue light for tails. We know the initial probabilities are p_H = .5 and p_T = .5. Let us consider the information in the red light. If I am a perfect observer, when the light is red the new probabilities are $p_H^{\mathrm{sig}}=1$ and $p_T^{\mathrm{sig}}=0$. Then we can fill in the following values: 1∗log₂(1/.5)+0∗log₂(1/.5). This is equal to 1∗1+0∗–∞, or to 1. If the signal carries perfect information about the state, the amount of information in it is equal to the entropy of the state.

Suppose that I am an imperfect observer and that when the state is heads, I flip the red light switch 90 percent of the time, and the blue light 10 percent, and vice versa when the state is tails. Now when we observe the red light, we cannot be certain that the state was heads. Instead, the probability the state is heads is $p_H^{\mathrm{sig}}=.9$ and for tails $p_T^{\mathrm{sig}}=.1$ So now we get .9∗log₂(.9/.5)+.1∗log₂(.1/.5) or .9 ∗ .85 + .1 ∗ (–2.32) = .532. There is less information about the state when the signal does not correlate perfectly with it.

We can use this kind of measure to calculate what we might call natural information. For instance, there is an old adage that “where there is smoke there is fire.” If we like, we can figure out what the probability is that there is fire at any point, p_F, or no fire, p_¬F. We can also figure out, given that there is smoke, what the probabilities are that there is or is not fire. From this, we can calculate how much information the smoke carries about fire.

We can also use this measure on acquired meaning. Reference SkyrmsSkyrms (2010a, Reference Skyrms2010b), in particular, uses this measure within the signaling game context. In a signaling model, the probabilities of the states, and of signals being sent given any state, are perfectly well defined. This means that we can measure the information in the signals about the states at any point. As Skyrms points out, we can watch signals go from carrying no information to carrying nearly full information about a state as a simulation progresses.

5.2 Semantic Content

Many have expressed dissatisfaction with the idea that we can say everything we’d like to about the meaning of signals using just information theory. Consider vervet signals. We can point out that there is a probabilistic correlation between the presence of a leopard and the vervet leopard alarm call. In this sense there is a measurable quantity of information in the signal about leopards. But we might also want to say that the signal has semantic content, i.e., that its content is inherently about something, much in the way that human language is. In this case, it seems natural to say that the content in the signal is very similar to the content in the human sentence “There is a leopard!” (In fact, in their landmark articles about vervet signaling, biologists Dorothy Cheney and Robert Seyfarth describe the vervets as engaged in “semantic communication” [Reference Seyfarth, Cheney and MarlerSeyfarth et al., 1980a, Reference Seyfarth, Cheney and Marler1980b].)

In the case of human language, one can appeal to intentions to specify semantic content (Reference GriceGrice, 1957, Reference Grice1991). For example, when I say “duck!” it might either be that I intend the listener to know there is a duck there or am telling him to duck. My intentions define the content. In the case of vervets (or bacteria) it is not clear that they have intentions of this sort. But we still would like to say their signals have semantic meaning in many cases. The worry is that information theory gives us no good way to describe this sort of content. In other words, it can tell us how much, but not what about.

Contra these worries, Reference SkyrmsSkyrms (2010a, Reference Skyrms2010b) uses the signaling game framework, and information theory, to define semantic content in signals. In many ways, this account bears similarities to that of Reference DretskeDretske (1981), who uses information theory to develop an account of mental content and representation. On Skyrms’s view, the content of a signal is a vector specifying the information the signal gives about each state. For states i, the vector is

$<{\log }_2(p_1^{\mathrm{sig}}/p_1),{\log }_2(p_2^{\mathrm{sig}}/p_2),\dots ,{\log }_2(p_n^{\mathrm{sig}}/p_n)>.$(5.3)

This is just the Kullback–Liebler divergence measured for each state the signal might tell us something about. To give an example, consider me flipping coins and turning on light switches again. Suppose there is an unbiased coin and perfect observation. The red light has the content < 1, –∞ >. Consider a similar case with a six-sided die and six perfectly correlated lights. The content in the yellow light (only and always sent in state 4) would be < –∞, –∞, –∞, 2.6, –∞, –∞ >. As Skyrms notes, this account is more general than traditional accounts of propositional content. On his account, we can sometimes think of these vectors as carrying propositional content, i.e., having the content of a proposition like “the coin flip was heads.” The negative infinity entries tell us which states do not obtain, and the proposition is the conjunction of the remaining entries, i.e., “the state is 1, 4 or 6.” When no states are ruled out, the signal has semantic informational content, just not propositional content.Footnote ⁴⁶

Reference Godfrey-SmithGodfrey-Smith (2011) suggests an amendment to this account. As he points out, we usually think of the content in a signal as about the world, not about how much the probability of a state was moved by a signal. His suggestion is that the proper content of a signal should be the final probabilities. So in the case of the die, the content would be < 0, 0, 0, 1, 0, 0 >, telling us that after observing the yellow light, it is perfectly certain the roll was four. This has a nice property. Say that we know ahead of time that the weather will be sunny and receive a signal that specifies the weather is sunny. On Skyrms’ account, this message has no content because it does not change probabilities, but on Godfrey-Smith’s account, it does.

Reference BirchBirch (2014) raises a problem for content in both views. As he points out, a key feature of semantic content is that it can be misleading or false. We can say “there is a leopard!” when there is no leopard, but nonetheless the statement still means that there is a leopard. But on these accounts, if a signal is sometimes sent in the “wrong” state, it also means that state, because it increases the probability of that state.

To see the problem more clearly, let’s focus on propositional content. Signals will only be propositional on the Skyrmsian accounts when they are never sent in the wrong state. Consider a capuchin monkey who gives an alarm call in order to sneak food while dominant group members are hiding. It seems natural to say that the propositional meaning of the call is still “there is a leopard” and that the call is simply misleading. On the Skyrmsian view, though, the signal cannot have propositional content because it does not exclude a state. Instead, it would actually mean, to some degree, that there is not a leopard as well as that there is.

This is sometimes called the problem of error, and similar objections have been made to information theoretic accounts of mental content.Footnote ⁴⁷ Birch points out that there are a few ways forward. One is to appeal to notions of function to ground content – the teleosemantic approach. Then, if a signal is used in a context that did not help select for it, or stabilize it in an evolutionary process, we can say it is misleading. (More on teleosemantics shortly.) The other is to specify fidelity conditions for a signal that define its content and allow one to say when it is being used in a misleading way. The trick is to do this in a non arbitrary way.

Birch argues that evolutionary models of Lewis signaling games give a very natural way to do this. A population may be in a state that is close to a separating equilibrium. If so, on his account, the meaning of a signal is specified by its propositional content at the nearest equilibrium. So if the population is close to a state where an alarm is always sent when there is a leopard, the meaning of the alarm is leopard. When some individuals in the population send the alarm and no leopard is present, the fidelity criteria specified by the equilibrium do not hold, and thus the signal can be false.

There are various issues with defining closeness of equilibria. Reference Skyrms and BarrettSkyrms and Barrett (2018) make another suggestion, which is that fidelity criteria might instead be defined by a fully common-interest interaction or “sub game” that stabilizes signaling. For example, capuchin monkeys will not usually be in a context where false alarms allow them to steal food. Under normal circumstances, they share interests that stabilize alarm signaling. These interactions define fidelity criteria. On the other hand, the contexts where interests diverge such that the subordinate capuchin is tempted to send a false signal do not define fidelity criteria.Footnote ⁴⁸

Reference Shea, Godfrey-Smith and CaoShea et al. (2018) move in a similar direction. They do not hunt for a fully common-interest interaction on which to ground signal meaning. Instead, they separate informational content, which can be defined in a Skyrmsian manner, from functional content. Functional content, for them, exists in cases where a signal is improving the payoff of sender and receiver over a non signaling baseline and corresponds to the states where the signal is improving pay off in this way. In other words, the functional contents track the states that are part of selecting for a signal. Their functional content is also a vector (weighted to sum to 1) and so, like the other measures we have seen, ranges in degrees.

These suggestions share some features with a very influential branch of philosophical thought, to which I now, briefly, turn.

5.3 Teleosemantics

The teleosemantic program has a long, rich philosophical history. It is beyond the purview of this Element to survey this literature, but it is worth taking a little space to discuss.

The key idea behind teleosemantics is that signals derive their content from some sort of selective process.Footnote ⁴⁹ In the case of evolution by natural selection, the content of a signal is intertwined with its function. If an alarm signal was selected to fulfill the function of warning group members in the presence of a leopard, its content comes from this process. The content helps explain why the signal has persisted through selection.

Reference MillikanMillikan (1984), in her influential account, uses a framework that shares many properties with the Lewis signaling game. She refers to senders and receivers as “producers” and “consumers,” and she argues that by dint of evolutionary processes whereby signals play a role in promoting successful behavior, these signals derive semantic content. For Millikan, much of this content has a particular character, which is that it both indicates some state of the world and suggests a responsive action (Reference MillikanMillikan, 1995). In the leopard alarm case, we might say that the signal has a combined content of “there is a leopard” and “run up a tree.” She calls this pushmi-pullyu meaning, after the llama-like creature of Dr. Doolittle with heads on both ends. From the perspective of the Lewis signaling game, this is an apt suggestion. In the basic models, signals at equilibrium are correlated both with state and action, so it is natural to think of them as possessing pushmi-pullyu content. As we will see in the next section, variations of the model can pull this content apart.Footnote ⁵⁰

Relatedly, (Reference HarmsHarms (2004a, Reference Harms, Müller, Pradeu and Schäfer2004b) has argued that there is no need to suppose that animal signals have propositional content. Instead, he proposes that signals of the type modeled in the Lewis game, with pushmi-pullyu meaning, have “primitive content.” This simple content emerges as a convention develops for sender and receiver. As he points out, it is hard to translate this content to language – the phrase “there is a leopard” is full of extra meaning and implications, for example. Instead, primitive content, although it can be described with human language, is really content of a different sort, referring to state–act pairs.

5.4 Indicatives and Imperatives

As we have just seen, Millikan and Harms describe meaning with two kinds of content, one referencing the world and the other referencing the action that ought to be performed given the state of the world. These two types of content are sometimes called assertive and directive, or else indicative and imperative. Reference LewisLewis (1969), in fact, drew attention to this distinction as well, by making a divide between “signals-that” and “signals-to.” As Reference MillikanMillikan (1995) points out, some human phrases have pushmi-pullyu meaning. She gives a few examples, such as “The meeting is adjourned,” which prompts people to leave while also making an indicative statement.Footnote ⁵¹ We have other phrases and terms where the content is mostly of one sort or the other. “Duck!” or “put your shoes on!” or “don’t do that!” are all imperative-type statements, while “I am sleepy” or “that cat is fat” are indicative. Animal signals sometimes seem to share these distinctions. For California ground squirrels, one signal is usually given for aerial predators, which are far away and take time to approach, and another for ground predators, which are usually closer and require immediate action. However, the squirrels reverse the signals for aerial predators that are close and ground predators who are far. Thus the content of the signals does not nicely map to predator species. Rather, one is more like “a predator is present” and the other is more like “run!”

Reference HutteggerHuttegger (2007b) uses the signaling game to try to pull apart these two sorts of meaning. One way to drive a wedge is to consider who is deliberating (Reference LewisLewis, 1969). If the sender deliberates over what to send, and the receiver does not deliberate over how to act, the signal can be thought of as imperative, and vice versa. Huttegger looks at a version of the Lewis game where in some states it is better for either the sender or receiver to deliberate and where their strategies allow them to choose whether to do so. As he shows, well-coordinated systems of behavior evolve with signals that have one type of content or the other.

Reference SkyrmsSkyrms (2010a, Reference Skyrms2010b) as noted, gives a measure of the information content in a signal about a state. I neglected to mention that he also gives a measure of the content about a act. This corresponds to how much the signal changes the probability that each possible act is performed. Reference ZollmanZollman (2011) points out that Huttegger’s model does not break informational asymmetry in the way we would want in pulling apart indicative and imperative meaning. Each type of signal carries full information about act and state at Huttegger’s equilibrium. Zollman instead gives a model where a sender communicates with two receivers who must coordinate their acts. The sender can either send one message indicating the state, and the receivers figure out which action they must take, or else the sender can send two messages indicating the two ideal acts for the receivers. This version of the model indeed breaks informational symmetry between indicative and imperative signals.Footnote ⁵²

In Section 3 we saw how Lewis games can be applied to understanding internal signaling. Reference Martínez and KleinMartínez and Klein (2016) combine this insight with work on the indicative/imperative distinction to argue that pain is an internal signal with mostly imperative content. This is because pain, again, has an informational asymmetry. A similar pain can attach to many causes but directs just a few actions. It thus carries more information about acts than about states.

5.5 Deception

When a capuchin monkey sends an alarm call in order to snatch food from a dominant group member, is it “lying”? Is it deceiving? In the case of human language, we can again define deceit with respect to intention. The other day my daughter told me she had not drunk her sister’s kombucha.Footnote ⁵³ She had. And she intended me to not know that. Most animals do not have the cognitive capacity to intend to mislead in this way. Still, the capuchin case sounds like deception. Another classic case is that of firefly signaling. The genus Photuris has coopted the mating signals of the genus Photinus to lead the latter species to their deaths. Unsuspecting fireflies head toward the signal, anticipating finding a willing mate, and are instead eaten. Isn’t the Photuris signal deceptive?

Like the rest of this section, debates over deception in the animal kingdom are widespread and go well beyond the signaling game framework. Again, we focus on signaling games and on the philosophical literature. As we will see, though, even in this smaller literature, there is a great deal of debate. Readers may want to look at Reference Maynard-Smith and HarperMaynard-Smith and Harper (2003) and Reference Searcy and NowickiSearcy and Nowicki (2005) for some wider perspective.

Before discussing deception, we will want to consider a related concept: misinformation. This is because many theorists define deception as a special case of misinformation. For Reference SkyrmsSkyrms (2010b), misinformation occurs whenever a signal increases the probability of a state that does not obtain, or decreases the probability of the true state. Reference Fallis and LewisFallis and Lewis (2019) argue that this is too broad a notion, because sometimes misinformation in this sense increases the overall accuracy of beliefs (on a host of measures offered by formal epistemologists). Instead, we should use a good measure of accuracy (which one is still under hot debate) to define misinformation. Most theorists agree that this is not enough for deception, though, and add conditions related to payoff, typically that the sender benefits, the receiver is hurt, or both.

Deception for Reference SkyrmsSkyrms (2010b), for example, occurs when there is misinformation that also increases the payoff of the sender at the expense of the receiver.Footnote ⁵⁴ Skyrms’s account, though, fails to differentiate between two possible sorts of “deception” (Reference Godfrey-SmithGodfrey-Smith, 2011; Reference Franke, Aloni, Franke and RoelofsenFranke, 2013a; Reference McWhirterMcWhirter, 2015; Reference Fallis and LewisFallis and Lewis, 2019; Reference MartínezMartínez, 2019). In some cases, a sender can use ambiguity to manipulate receiver behavior. For instance, the sender might send a in both state 1 and state 2, but in no other states, in order to make the receiver choose an option the sender prefers. (For space reasons, I do not give a concrete example, but see Reference SkyrmsSkyrms [2010b]; Reference Godfrey-SmithGodfrey-Smith [2011]; Reference Franke, Aloni, Franke and RoelofsenFranke [2013a]; Reference McWhirterMcWhirter [2015], Reference Fallis and LewisFallis and Lewis [2019].) In such a situation, the probability of a non actual state is increased by the signal, and in a way that systematically advantages the sender and disadvantages the receiver. But the two are still involved in mutually beneficial equilibrium behavior, and furthermore, the signal can be thought of as half true in the sense that it correctly excludes states other than 1 and 2.Footnote ⁵⁵ This looks quite different from the case where Photuris co-opts the cooperative signaling of Photinus. In the latter case, enough signaling by Photuris would make the signaling behavior collapse. This leads to a family of proposals from Reference Godfrey-SmithGodfrey-Smith (2011), Reference FrankeFranke (2013b), Reference Shea, Godfrey-Smith and CaoShea et al. (2018), and, ultimately, Reference Skyrms and BarrettSkyrms and Barrett (2018). These proposals are approximately that the content of signals should be defined by their “sustaining” uses – the ones that keep the behavior going on an evolutionary scale – or else their common interest uses, or beneficial uses, and deception, in contrast, should be limited only to “unsustaining” uses, or ones that undermine the signal. For Reference Shea, Godfrey-Smith and CaoShea et al. (2018), for example, deception occurs when a signal is sent with false functional content (in the sense outlined earlier) to the benefit of the sender.Footnote ⁵⁶

Reference McWhirterMcWhirter (2015) worries about a different issue, which is that we often think about deception as requiring a deviation from expected behavior. For him, deception occurs only when an actor misuses a signal in a state, in the sense that they send it more often than the rest of the population, in a way that benefits themselves and hurts the receiver. On this definition, a signal cannot be universally deceptive but can only be deceptive with respect to some population level baseline. Reference Fallis and LewisFallis and Lewis (2019), contra this, point to some natural cases where behavior used across a population can look like deception. They also point out that an individual could send a signal that differed from the standard usage, to their benefit, but that improved the the epistemic state of the receiver. Is this deception?

More recently, several authors have argued that the common requirements that deception benefit a sender and harm a receiver need not hold. Reference BirchBirch (2019) gives a good example. The pied babbler trains young with a purr call emitted during feeding. The call eventually serves to cause approach behavior and begging in the young. But adults subsequently use the same call to summon young away from predators or toward new foraging sites. This looks deceptive but actively benefits the young, potentially at the cost of adult safety when a predator is present. Reference Artiga and PaternotteArtiga and Paternotte (2018) propose that to tackle such cases, deception should be defined in terms of function: a deceptive signal is one whose function is to misinform (and where the signal was selected for this purpose). Reference BirchBirch (2019) points out that this account has the same problem with misinformation that Reference SkyrmsSkyrms (2010b) sees – ambiguous signals are also misinformative and so can count as deceptive. Instead, for Birch, deception is strategic exploitation of receivers by senders: the sender exploits a receiver behavior that evolved for the benefits it yields in some other state of the world. It does so by increasing the probability (from the receiver’s point of view) that the state obtains. This exploitation need not benefit the sender nor hurt the receiver. As is probably clear, there is not a clear consensus on what deception looks like in the animal world, just as there is not a clear consensus on content. In the case of deception, though, part of the issue seems to be that we generally ground judgments of what is deceptive in terms of human behavior. It may be that there is no neat, unitary concept underlying these judgments. As such, there may be no single definition of biological deception that captures everything we might want it to do.

As we have seen, there are deep issues related to meaning in animal signals that are not particularly easy to resolve. This discussion ends the part of the Element about signaling. The second part of the Element, as mentioned, focuses on prosociality. Before moving on, it is worth mentioning one current (and fascinating) debate in philosophy of biology that centers around the question of what sort of information is carried in the genome. Does it carry semantic content? Some other kind of content? We will not discuss these questions here for space reasons, but interested readers can look at Reference Maynard-SmithMaynard-Smith (2000), Reference Godfrey-Smith, Sterelny and ZaltaGodfreySmith and Sterelny (2016), Reference SheaShea (2011), and Reference Bergstrom and RosvallBergstrom and Rosvall (2011). And with that, on to part 2.

6.1 The Prisoner’s Dilemma

Two prisoners are being asked to rat each other out. If they both stay silent, they each go to prison for one year. If one rats on the other, the rat goes free, while their partner is sent away for five years. If they both rat, though, they are both implicated, and each goes to jail for three years. This scenario induces a payoff structure where each prisoner has a clear preference over outcomes. We can capture these preferences using payoffs in a game.

Consider the payoff table in Figure 6.1. Here we have labeled staying silent as “cooperating” and ratting as “defecting.” (This follows tradition, but the reader should note, as will become clear, that cooperation here specifically refers to altruistic behavior rather than mutually beneficial behavior.) Two cooperators get a payoff of 2, while two defectors get a payoff of 1. For this reason, mutual cooperation (staying silent) is preferable to mutual defection (ratting). But notice that despite this, each actor is always incentivized to defect. If their parter is cooperating, defection gets a payoff of 3 (going free), while cooperating only gets a payoff of 2 (one year of jail time). If their partner is defecting, defection gets a payoff of 1 (three years in prison) while cooperating gets a payoff of 0 (five years in prison). This is why the game is a dilemma – despite the benefits of mutual cooperation, there is always a temptation to defect. In fact, the only Nash equilibrium of this game is for both players to defect.

Figure 6.1 A payoff table for the prisoner’s dilemma. Payoffs for player 1 are listed first.

Notice that the cooperation strategy is altruistic in the sense described above. By playing cooperate, a player always raises their partner’s payoff while lowering their own. Thus we can use this framework to say more precisely why the evolution of altruism is a puzzle: in standard evolutionary game theoretic models, populations always evolve to a state where all actors defect.Footnote ⁵⁹

6.2 Evolving to Cooperate

The main thrust of the literature mentioned at the beginning of this section is to figure out why, despite the fact that we should expect defection to evolve, vampire bats are vomiting in each others’ mouths every night. Clearly something is missing from the model. That something is correlated interaction, meaning an increased likelihood of interaction between players who take the same strategies. Imagine if cooperators only met cooperators and defectors only met defectors when playing the prisoner’s dilemma. Then cooperators would get a payoff of 2, and defectors 1, and cooperation would straightforwardly be expected to evolve. As it turns out, it is tricky to generate scenarios with perfect correlation of this sort, but there are many mechanisms which generate enough correlation to get altruism off the ground. Let’s discuss these in turn.

6.3 Signaling and Altruism

There is another sort of mechanism that can lead to the evolution of altruism in the prisoner’s dilemma. I separate this section because this mechanism returns to themes from the first part of the Element. We now look at the relationship between signaling and prosociality.

6.4 The Evolution of Spite

In yellow rumped caciques, adolescent males sometimes harass fledglings to such a degree that the fledglings die. In doing so, these adolescent males put themselves in danger of retaliation from adult females and also waste time and energy.

We have now seen how correlation between strategies can lead to the evolution of altruistic behavior. This observation can be flipped, though, to explain the evolution of something a bit darker – spite. Spiteful behavior occurs when individuals pay a cost not to benefit another individual but to harm another individual. After illustrating how kin selection might lead to altruism, Reference HamiltonHamilton (1964), and independently George Price, realized that a similar mechanism might lead to the selection of spite (Reference HamiltonHamilton, 1970; Reference PricePrice et al., 1970; Reference Hamilton, Eisenberg and DillonHamilton, 1971). In particular, spiteful behavior toward individuals of less than average relatedness might be selected for. Since these individuals tend not to have spiteful genes themselves, harming them can help spite out-compete the population average (Reference GrafenGrafen, 1985; Reference Lehmann, Bargum and ReuterLehmann et al., 2006; Reference West and GardnerWest and Gardner, 2010).

It is perhaps clearer to reframe this observation in terms of anticorrelation, where, more generally, the harms of spite fall disproportionately on those who are not, themselves, spiteful. This sort of anti-correlation, as in the case of altruism, might be generated multiple ways: through greenbeards (Reference West and GardnerWest and Gardner, 2010; Reference Gardner and WestGardner and West, 2010), local interaction (Reference Durrett and LevinDurrett and Levin, 1994), group selection (Reference SkyrmsSkyrms, 1996), and even due to the anticorrelation generated by the fact that individuals do not interact with themselves (Reference HamiltonHamilton, 1970; Reference Smead and ForberSmead and Forber, 2013).Footnote ⁶⁷ Reference Smead and ForberSmead and Forber (2016) and Reference Forber and SmeadForber and Smead (2016) find that, paradoxically, spite can promote altruism by stabilizing mechanisms for recognizing altruistic and selfish types.

Correlation between strategies can lead to the evolution of altruistic behavior, despite its costs. There are many and varied ways this can happen, though. And disagreements still exist over which of these are plausible or more likely to explain real-world altruism. As we will see in the next section, altruism is not the only prosocial game in town, and many of the same mechanisms that promote altruism can help promote the evolution of other sorts of prosocial behavior.

7.1 The Stag Hunt

Two hunters would like to capture as much game as possible. They have two options strategy-wise – to hunt for stag or to hunt for hare. If both hunters go for the stag, they are able to take it down together. If either hunts hare, they will be able to catch the small prey alone. But if one hunts for hare, the remaining hunter cannot catch a stag by themselves. They get no meat if their partner fails to do their part.Footnote ⁶⁹

The payoff table for this game is shown in Figure 7.1. As we can see, if both players choose stag, they get a payoff of 3. If either chooses hare, they get 2. And hunting stag while your partner hunts hare yields 0. This game, unlike the prisoner’s dilemma, has two Nash equilibria. There is a prosocial equilibrium where both players hunt stag, and a loner equilibrium where both hunt hare.

Figure 7.1 A payoff table for the stag hunt. Payoffs for player 1 are listed first.

We can take an evolutionary model of the stag hunt to represent a human group who, over time, develop strategies for dealing with cooperative scenarios.

For this reason, authors like Reference BinmoreBinmore (1994, Reference Binmore2005) and Reference SkyrmsSkyrms (2004) have taken this game as a simple representation of the social contract. Hare hunting represents an individualistic state of nature, lacking in social trust, and stag hunting the emergence of successful contracts of cooperation and mutual benefit.Footnote ⁷⁰

It might seem obvious what should emerge in the stag hunt game. There is no dilemma, and the stag equilibrium yields a higher payoff. So we should always see stag hunting, right? Things are not so simple. First of all, both equilibria are ESSs, so either should be stable to invasion. Furthermore, hunting stag is more risky, in this game, than hunting hare.

This extra riskiness means that under typical modeling assumptions stag hunting is less likely to evolve. Hare hunting has larger basins of attraction under the replicator dynamics, because of its more dependable payoffs. Figure 7.2 shows this. For the game in Figure 7.1, a population that is more than two-thirds stag will go to stag hunting, and any proportion less than that will go to hare hunting. Furthermore, in models that incorporate mutation and stochasticity, the prediction is typically that hare hunting, rather than stag hunting will emerge (Reference Kandori, Mailath and RobKandori et al., 1993). If we imagine a population with random mutations and perturbations, the key observation here is that it is easier to go from stag to hare than from hare to stag.

Figure 7.2 The phase diagram for a replicator dynamics model of the stag hunt.

So in light of this model, how do we explain the widespread success of humans at engaging in joint action? And, likewise, how do we explain wolves who engage in joint hunting? Or bacteria who act together to protect themselves from a hostile environment?

Well, first, we might alter the payoffs a bit. The payoffs for the stag hunt are often constrained to guarantee that hare hunting is risk dominant, meaning in this case that it yields a higher expected payoff given no knowledge of opponent strategy. But we can imagine increasing the payoff to stag hunting until it is so high that stag hunting is risk dominant instead. We can even imagine situations like that in Figure 7.3. We might call this a mammoth hunt. Evolutionary models predict that mammoth is more likely to evolve for this altered model, as is evident in Figure 7.4. Note, though, that hare hunting is still a stable equilibrium. So even for this model, there is a question of how uncooperative populations (states of nature) might evolve to be cooperative (social contract) in the first place.

Figure 7.3 A payoff table for the mammoth hunt – a stag hunt with very high stag payoffs. Payoffs for player 1 are listed first.

Figure 7.4 A phase diagram for the mammoth hunt.

We can also ask, In situations where stag hunting is moderately beneficial, but risky, what sorts of mechanisms might promote the evolution of cooperation? What allows humans (and other animals) to work together despite the risks? A general message, once again, is that correlated interaction can help stag hunting, just as it helps altruism in the prisoner’s dilemma. If stag hunters tend to meet each other with high probability, they do not suffer the costs of their risky prosocial strategy, and so they outperform hare hunters. Again, there are various mechanisms by which this can happen.

7.1.1 Signals and Stag Hunting

Remember in the last section that preplay signaling (secret handshakes or greenbeards or emotions) could play a role in promoting altruism in the prisoner’s dilemma. The idea there was that actors could use signals to coordinate on altruism, despite a temptation to defect. Of course, as we saw, because of the nature of the prisoner’s dilemma there is always selection for actors who learn how to subvert these systems of communication.

What about preplay signaling in the stag hunt? Reference SkyrmsSkyrms (2002b, Reference Skyrms2004) considers a strategy where stag hunters have a secret handshake. They usually hunt hare, but if they receive a special signal, they hunt stag instead. As he points out, this strategy can invade a population of hare hunters. The stag signalers will always get better payoffs by using their signals to coordinate on beneficial joint action.Footnote ⁷¹

This result even holds in a slightly more surprising setting. Consider a version of the stag hunt (sometimes called the assurance game) where hare hunters do slightly better against partners who play stag. This is shown in Figure 7.5. In this case, hare hunters are incentivized to mislead potential collaborators as to their strategy, because doing so improves their payoff. For this reason, Aumann (1990) argues that cost-free signals cannot lead to stag hunting in this game. As Reference SkyrmsSkyrms (2002b, Reference Skyrms2004) shows, though, the logic of the last paragraph holds even for the assurance game – cheap talk can lead to the emergence of joint action because coordinating stag hunters do so well against each other. Hare hunters are incentivized to mislead but even more incentivized to become stag hunters.

Figure 7.5 A payoff table for the assurance game. Payoffs for player 1 are listed first.

7.2 Bargaining

Two partners in a household must figure out who will contribute how much time to keep the household running. While they share interests in that they wish the household to be successful, both would prefer to do less of the work to keep it that way. Aphids and ants are engaged in a mutualism where ants protect the aphids, who produce a sugar and protein rich nectar for the ants. While the arrangement benefits both of them, there are different levels of nectar the aphids could produce. The ants prefer more and the aphids less. Andrew and Amelia have just cleared a forest where they can now graze cattle. Both benefit from the effort, but now there is a question of who gets to graze how many cattle. Each prefers more for themselves, but too many will ruin the field.

These are scenarios of resource division. In particular, they are all scenarios where individuals benefit by dividing some resource, where there are multiple divisions that might be acceptable to each individual, and where there is conflict over who gets how much. The study of such scenarios in game theory goes back to Reference NashNash (1950), who introduced a bargaining problem to represent them. Subsequently, bargaining games have been widely studied in economics and the other social sciences, and applied to cases ranging from household division of labor, to salary negotiations, to international trade deals.

We will consider a relatively simple version of the Nash demand game here.Footnote ⁷² Two actors divide a resource of value 10. The actors’ strategies consist in demands for some portion of this resource. We will suppose that they can demand either a low, medium, or high amount. Let us constrain these demands so that the medium amount is always for half the resource (5), and the high and low amounts add up to 10. They might be 4 and 6 or 3 and 7. The usual assumption is that if actors make compatible demands, in that they do not over-demand the resource, they get what they ask for. If they make incompatible demands, it is assumed that they failed to achieve a mutually acceptable bargain, and each gets a low payoff called the disagreement point. Figure 7.6 shows this game for demands 4, 5, and 6, and a disagreement point of 0.

Figure 7.6 A simple version of the Nash demand game. Payoffs for player 1 are listed first.

There are three Nash equilibria of this game. These are the strategy pairings where the actors perfectly divide the resource: Low vs. High, Med vs. Med, and High vs. Low. At these pairings, if either player demands more, they overdemand the resource and get nothing. If either player demands less, they simply get less. Notice that we might alter this simple model to include more possible demands – all the numbers from 1 to 9, for example. In each case, any compatible pairing of demands, 1 and 9, or 3 and 7, will be a Nash equilibrium for the reason just given.

Depending on the number of strategies, then, this game can have many equilibria. If we imagine that actors can make a demand for any percentage of the resource, the game has infinite equilibria, for each possible combination that adds up to the total resource. This poses an equilibrium selection problem – what does the game predict vis-à-vis behavior given the surfeit of Nash equilibria? Traditional game theoretic approaches attempt to answer this question via various means (Reference NashNash, 1950, Reference Nash1953; Reference RubinsteinRubinstein, 1982; Reference Binmore, Rubinstein and WolinskyBinmore et al., 1986). Here we will discuss how cultural evolutionary models have been used to discuss this problem, focusing on work in philosophy.

7.2.3 Hawk-Dove and Resource Division

Some of the earliest evolutionary game theoretic models of resource division did not focus on the Nash demand game, but on a model called hawk-dove. Reference Maynard-Smith and PriceMaynard-Smith and Price (1973) present this game as a model of animal conflict (though a version called chicken was developed earlier in game theory). Imagine two spiders would like to occupy the same territory. If they are both too aggressive, they will fight to their mutual detriment. If one is aggressive, and the other passive, the aggressive spider will take the territory. If both are passive, they might split the territory, living closer together than either might prefer. Such a scenario can be represented with the payoff table in Figure 7.7. The hawk strategy represents an aggressive one, the dove a passive one.

Figure 7.7 A payoff table for hawk-dove. Payoffs for player 1 are listed first.

Notice that we can also interpret this game in the following way: two actors divide a resource of size 4. Two doves will divide it fairly, each getting 2. If a hawk and a dove meet, the hawk will get a high payoff of 3 and the dove 1. If two hawks meet, they will be unable to efficiently divide the resource, and will get 0. This interpretation of the game sounds a lot like the Nash demand game, but there are some differences. In hawk-dove, if two actors are engaged in a fair, or passive, interaction, one can always unilaterally change strategies to take advantage of the other. For this reason, there is no fair pure strategy equilibrium. The two pure equilibria are hawk versus dove and dove versus hawk.

There is also a mixed strategy equilibrium where both players get the same expected payoff. This equilibrium, in fact, is the only ESS of the game, which may help explain why damaging animal contests are relatively rare (Reference SigmundSigmund, 2017). In the game shown here, the mixed equilibrium involves both players playing hawk 50 percent of the time and dove 50 percent of the time. When the cost to conflict is high enough, though (i.e., the payoff to hawk vs. hawk is low enough), this equilibrium will involve playing hawk only rarely. The chance that two hawks meet in such a case will be rarer still, meaning that the serious detriments of fighting can be avoided.

But at the ESS hawks sometimes still fight and harm each other, and sometimes doves do not take advantage of their social situation. As in the Nash demand game, mechanisms for anti-correlation can help. Reference Maynard-SmithMaynard-Smith (1982) introduces the notion of an uncorrelated asymmetry – any asymmetry between organisms that can facilitate coordination in a game like hawk-dove. For instance, in the spider example, it may be that one spider is always the current inhabitant of a territory, and one is the intruder. If both spiders adopt a strategy where they play hawk whenever they inhabit and dove whenever they intrude, they will always manage to reach equilibrium. (Maynard-Smith calls this the “bourgeois strategy.”) A population playing the mixed strategy ESS can be invaded by a population that pays attention to uncorrelated asymmetries in this way, and thus avoids inefficiencies. The opposite solution – where the intruder plays hawk, and the inhabitant dove – also works to coordinate behavior (Reference SkyrmsSkyrms, 1996). Evolution can select one or the other, though it tends to select the first, possibly because of extra benefits obtained by keeping possession of familiar property.Footnote ⁷⁷

Reference HammersteinHammerstein (1981) expands this analysis to look at cases where there are multiple asymmetries between players. They might be owners or intruders but also might be larger or smaller, for instance. He shows that generically, either can act as an uncorrelated asymmetry, even in cases where one asymmetry matters to the payoff table of the game, and the other is entirely irrelevant. This happens because often the benefits of coordination outweigh other payoff considerations. In cases, though, where these coordination benefits are less important, the bourgeois strategy may not be stable. Reference GrafenGrafen (1987) points out, for instance, that when territory is necessary for reproduction, desperate organisms may risk everything to oust an owner.

7.3 Bargaining and Joint Production

As mentioned at the beginning of the section, the stag hunt is a particularly good model for joint action. But, of course, when humans want to act together, there are certain questions they have to answer. Who will do what work and how much? Who will reap which rewards? In other words, bargaining, whether implicit or explicit, is necessary for effective joint action. This bargaining can be settled over evolutionary time via the evolution of innate behavior, over cultural time via the evolution of norms, or over the course of one interaction via debate and discussion. But it must be settled. In a literal stag hunt, for example, it is always the case that the hunters must also decide how to divide the meat they have brought in, whether these hunters are humans or wolves. Likewise, mating birds often must first decide whether to mate and then how to divide the labor of raising nestlings (Reference WagnerWagner, 2012).

To address this sort of case, Reference WagnerWagner (2012) analyzes a combined stag hunt/Nash demand game where actors first choose whether to hunt stag or hare. If they hunt stag, they then choose a demand for how much resource they would like and play the Nash demand game. The payoff table for this game is shown in Figure 7.8.Footnote ⁷⁸ Notice that there are four strategies here, Hare, Stag-Low, Stag-Med, and Stag-High. This game again assumes a resource of 10, so that the medium demand is for 5. Instead of choosing values for Low and High, this payoff table uses variables L and H where 0 < L < 5 < H < 10, and where L + H = 10. For the Hare payoff we have S (for “solo,” since H is taken).

Figure 7.8 A payoff table for a combined stag hunt/Nash demand game. Payoffs for player 1 are listed first.

This four-strategy game has four possible Nash equilibria. These are Hare vs. Hare, Stag-Low vs. Stag-High, Stag-Med vs. Stag-Med, and Stag-High vs. Stag-Low. In other words, actors can go it alone or engage in joint production with either fair or unequal divisions of the good produced. Notice that the values of S, L, and H determine just how many of these equilibria will exist in a particular version of the game. In particular, if the value for hare hunting, S, is high enough no one will be willing to get the Low payoff, L. Anyone getting this payoff will instead just switch to solo work. If so, the only remaining equilibria are hare hunting, and stag hunting with an equal division. If S > 5, hare hunting is the only remaining equilibrium. Notice that this model displays two sorts of social risk. First there is the risk of the stag hunt – actors who attempt joint action while their partner shirks will get nothing. Second, there is the risk of bargaining. Even if actors agree on joint action, if they do not make compatible demands, they get the disagreement point.

Reference WagnerWagner (2012) shows, surprisingly, that in this combined model, actors are more likely to evolve stag hunting than they are in the stag hunt and more likely to evolve fair bargaining than they are in the Nash demand game. In particular, there are three stable evolutionary outcomes of this game under the replicator dynamics – all Hare, all Stag-Med, or some Stag-Low and some Stag-High (this is the “fractious” equilibrium that we saw in the Nash demand game). As the payoff for Hare increases, it becomes more and more likely that the fair outcome emerges than the fractious one. Once the payoff for Hare is high enough, it disrupts the stability of the fractious outcome altogether. And, for many values of S, the combined basins of attraction for stag hunting are larger than in the stag hunt alone. There seems to be a positive message here for prosociality – the dynamics of joint production and division of labor tend to yield good outcomes.

Once again, though, anti-correlation can change this picture. Reference Bruner, O’Connor, Mayo-Wilson, Boyer and WeisbergBruner and O’Connor (2015) look at the stag hunt/Nash demand game described here, but again consider models with multiple groups or social categories. As they point out, inequitable divisions of resource commonly emerge between these groups, because, once again, group structure allows actors to break symmetry. One group becomes the group that gets low, and the other the group that gets high. As they also show, this kind of inequity can lead to a dissolution of joint action (or the social contract, if we like). A group that expects to get too little from a joint project may be unwilling to take on the risks of collaboration, and decide to hunt hare instead. This happens even when joint action provides benefits to both sides, because it is also risky.Footnote ⁷⁹

7.4 Norms, Conventions, and Evolution

This section has now considered several models that have been taken to inform the cultural evolution of conventional, and sometimes normative behavior. Over the years, philosophers have provided many accounts of what conventions are. We will not overview these here. But, generally, conventions are patterns of social behavior that solve some sort of social problem but that might have been otherwise. Reference LewisLewis (1969) influentially used games as a way to define conventions. For him, they are equilibria in games with multiple equilibria, along with certain beliefs for the actors.Footnote ⁸⁰ On this sort of account, it is easy to see how the work described in this section, and earlier in the Element, can be taken to inform the cultural emergence of human conventions. Whenever we have a model where actors might evolve to various equilibria, and where we can show that learning processes tend to lead actors to stable, equilibrium patterns, we have a model that might represent the emergence of human convention. This could be a linguistic convention as in Section 3, a convention of adhering to a social contract, a convention for fair treatment, or a convention of unequal resource division. The suggestion is not that all conventions are equilibria in games but that evolutionary game theoretic models provide simple representations of many sorts of emerged social conventions. (Readers interested in more on game theory and convention might look at the work of Peter Vanderscraaf, including Vanderschraaf [Reference Vanderschraaf1995, Reference Vanderschraaf1998a, Reference Vanderschraaf1998b], and of economist Ken Binmore [1994, Reference Binmore2005].)

Social norms are also guides to behavior, but of a different sort. The key factor that makes something a norm is the judgment that actors ought to follow it. Some norms are also conventions. For instance, it is conventional to stand on the right and walk on the left of escalators in London. But many underground riders also feel that everyone ought to conform to this behavior. Anyone who does not risks social disapproval. Other norms are not conventions.Footnote ⁸¹ But, in general, we might ask whether evolutionary game theoretic models can tell us anything about norms. This is especially so given that these models often do tell us about normatively proscribed prosocial behaviors like altruism, cooperation, and fairness.

An evolved equilibrium in a model fails to capture the ought inherent in norms. Actors adhere to the behavior because it is personally beneficial, not because they ought to. Reference BicchieriBicchieri (2005) uses standard game theory to give an influential account of how norms work. For her, norms can transform one game into another one, by making it the case that individuals prefer certain behavior on the understanding that it is expected of them, or on the belief that others will adhere to it, or punish them for non-compliance.Footnote ⁸²

Evolutionary models can also be used to elucidate norms. As Reference Harms, Skyrms and RuseHarms and Skyrms (2008) point out, norms are tightly connected to retribution and punishment. Individuals are often willing to punish norm violators, and it seems that humans have an evolved tendency to do so. As such, understanding the evolution of punishing in strategic scenarios may help us understand our normative behavior. As mentioned in Section 6, costly punishment poses a second-order free rider problem – why should people punish others altruistically? Nonetheless, various extant models argue that altruistic punishment can evolve (Reference Boyd, Gintis, Bowles and RichersonBoyd et al., 2003). Furthermore, ostracizing behaviors like avoiding defectors, or defecting against them in the prisoner’s dilemma, are arguably types of punishment that actually lead to fitness benefits (Reference Harms, Skyrms and RuseHarms and Skyrms, 2008). So evolutionary game theoretic models can shed light on this aspect of normativity.

In this section we looked at models that have been widely applied to the cultural evolution of human conventions, especially those related to mutually beneficial cooperation and resource division. As we saw, these models help explain the emergence of social contracts, norms of fairness, and the ubiquity of unfairness. Some themes from Section 6 were repeated here, including the importance of signaling and of interaction structure to the emergence of prosocial behaviors.