The context signaling game is an extension of the signaling game as introduced by Lewis (1969). Its additional features are contextual cues, and, optionally, sender and receiver costs. In the following I will introduce a minimalist variant of a context signaling game. This variant involves four states, four signals and four response actions: \(T = \{t_1, t_2, t_3, t_4 \}\), \(S = \{a, b, c, d \}\), \(R = \{r_1, r_2, r_3, r_4 \}\). Furthermore, there is a set of contextual cues \(\Gamma = \{\gamma _1, \gamma _2\}\) which indicate what kind of states are possible. Here, \(\gamma _1\) tells the receiver that \(t_1\) or \(t_2\) is the case, whereby \(\gamma _2\) indicated that either \(t_3\) or \(t_4\) is the case.
The game is played between a sender and a receiver. It is assumed that both players act according to pure strategies. A sender strategy \(\sigma \in \mathcal {S}\) is defined as function \(T \rightarrow S\) and determines for every state what kind of signal is used by the sender. A receiver strategy \(\rho \in \mathcal {R}\) is defined as function \(S \times \Gamma \rightarrow R\) and determines for every possible combination of contextual cue and signal what kind of response action the receiver chooses.
Moreover, different strategies can come with different costs. The sender might pay for the precision (Santana 2014) or complexity (Rubinstein 2000; Deo 2015) of her strategy. The receiver might pay for the case that he has to process additional contextual cues. Consider the following example of animal alarm calls: If a call means ‘attack by a bird of prey’ OR ‘attack by a snake’, the receiver is aware of danger, but still has to scan the environment to choose the right response action, whereby an unambiguous alarm call does not require such a scan.Footnote 10 I will assume sender and receiver costs to be minute with respect to the value of communicative success, and I will also allow each to be zero. The sender costs are defined by a sender cost function \(c_s: \mathcal {S} \rightarrow \mathbb {R}_0^+\), and the receiver costs are defined by a receiver cost function \(c_r: \mathcal {R} \rightarrow \mathbb {R}_0^+\). Concrete costs for different strategies will be determined further down, where I will explore various values.
The utilities for sender and receiver can be defined for every state and with respect to strategy pairs, assuming the standard utility function, which yields 1 when the state corresponds to the response action (else 0). Note that such a match is given when the response action \(r_j\) (as result of the receiver strategy \(\rho \)) and the state \(t_i\) of the sender (who uses strategy \(\sigma \)) have the same index, thus when \(i=j\) in \(r_j = \rho (\sigma (t_i))\). By integrating sender and receiver costs, the sender utility function for playing \(\sigma \) against \(\rho \) in state \(t_i\) is given as:
The receiver utility function for playing \(\rho \) against \(\sigma \) in state \(t_i\) is given as:
$$\begin{aligned} U_r(t_i,\sigma ,\rho ) = \left\{ \begin{array}{c@{\quad }l} 1-c_r(\sigma ) &{} \text {if } i=j \text { in } r_j = \rho (\sigma (t_i))\\ -c_r(\sigma ) &{} \text {else} \end{array} \right. \end{aligned}$$
Given these definitions, one round of the game looks as follows:
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1.
A state \(t\in T\) is chosen with probability \(\nicefrac {1}{4}\) and revealed to the sender.
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2.
The sender chooses a signal s according to her strategy \(\sigma \)
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3.
The receiver receives a context strategy pair \((\gamma , s)\), whereby \(\gamma \in \Gamma \) is the contextual cue, thus either \(\gamma _1\) or \(\gamma _2\) , depending on state t
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4.
The receiver chooses a response action r according to his strategy \(\rho \)
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5.
Both players receive payoffs according to utility functions \(U_s\) and \(U_r\)
As already mentioned, the utility functions define how well a sender and a receiver strategy work for a particular state t. In the evolutionary analysis I will study the expected utility, namely the average utility of a sender and a receiver strategy over all states \(t \in T\). These expected utilities are defined by the following two EU functions for sender (\(EU_s\)) and receiver (\(EU_r\)):
$$\begin{aligned} EU_s(\sigma ,\rho )= & {} \sum _{t \in T} \frac{1}{4} \times U_s(t,\sigma ,\rho )\\ EU_r(\sigma ,\rho )= & {} \sum _{t \in T} \frac{1}{4} \times U_r(t,\sigma ,\rho ) \end{aligned}$$
Santana (2014) studied the version of the context signaling game introduced in this paper with respect to two particular strategy types. Each type can be exemplified by a strategy pair, of which one forms a signaling system (Fig. 4a), and the other forms an ambiguous system (Fig. 4b).
In the following I will call the first strategy type perfect signaling (PS), and the second one full ambiguity (FA).Footnote 11 Now lets assume that strategy \(\sigma _1\) might be more costly than strategy \(\sigma _2\) due to precision and/or complexity: \(c_s(\sigma _1) = \epsilon _s \ge 0\) and \(c_s(\sigma _2) = 0\). Furthermore, let’s assume that strategy \(\rho _1\) is cost-free, since the receiver does not need to access contextual cues, whereby strategy \(\rho _2\) might involve some costs: \(c_r(\rho _1) = 0\) and \(c_r(\rho _2) = \epsilon _r \ge 0\). For this scenario Table 1 shows the expected utility (EU) table for all combinations of sender and receiver strategies.
Table 1 EU tables over the strategies \(\sigma _1\) and \(\sigma _2\) versus the receiver strategies \(\rho _1\) and \(\rho _2\) An important concept in evolutionary game theory is evolutionary stability (Maynard Smith and Price 1973), since an evolutionarily stable strategy (ESS) has an invasion barrier and is e.g. resistant to drift. Note that for an asymmetric game, such as the EU table of Table 1, the concepts of an ESS and of a strict Nash equilibrium coincide (Selten 1980). In other words, to detect evolutionarily stable strategy pairs in Table 1, one has to find those whose sender utility is the unique maximum in the column, and whose receiver utility is the unique maximum in the row at the same time. Note that the PS strategy pair \(\langle \sigma _1, \rho _1 \rangle \) fulfills this condition and is evolutionarily stable, if \(\epsilon _r > 0\). For the same reason, the FA strategy pair \(\langle \sigma _2, \rho _2 \rangle \) is evolutionarily stable, if \(\epsilon _s > 0\).Footnote 12
Santana (2014) studied the case where \(\epsilon _s > 0\) and \(\epsilon _r = 0\). In such a scenario, the FA strategy is an ESS, but the PS strategy is not. Figure 5a shows the changes in population states under the replicator dynamics (Taylor and Jonker 1978) for \(\epsilon _s = 0.03\) and \(\epsilon _r = 0\). As the figure shows, the strategy pair \(\langle \sigma _1, \rho _1 \rangle \) (bottom left corner) is not evolutionarily stable, since the receiver population can drift between \(\rho _1\) and \(\rho _2\). Once the receiver population uses (almost) entirely \(\rho _2\), the sender population is attracted by \(\sigma _2\), reaching the ESS \(\langle \sigma _2, \rho _2 \rangle \) (top right corner). This completes the potential drift from PS \(\langle \sigma _1, \rho _1 \rangle \) to FA \(\langle \sigma _2, \rho _2 \rangle \).
Santana adduced this scenario for his argument, namely that ambiguity has an evolutionary advantage over perfect signaling. But his conclusion should be taken with a grain of salt. He argues that \(\epsilon _s\) can be expected to be arbitrarily small. But note that when \(\epsilon _s\) approaches 0, the basin of attraction of the FA strategy approaches 0. Thus, although the FA strategy is still evolutionarily stable, its basin of attraction is minute. Therefore, its stability advantage over PS can be expected to be minute when it comes to evolutionary dynamics with noise, such as mutation. I will underpin this point in the next section.
Moreover, Santana considered only one of four possible scenarios. And only in this scenario does FA have an evolutionary advantage over PS. For example, when \(\epsilon _s = 0\) and \(\epsilon _r = 0\), then none of both strategy types is an ESS, and a drift can happen in both directions. Moreover, when \(\epsilon _s = 0\) and \(\epsilon _r > 0\), Santana’s scenario is reversed and PS is the only ESS of the given EU table. Finally, when \(\epsilon _s > 0\) and \(\epsilon _r > 0\), both strategies are ESS and have an invasion barrier. Figure 5b shows the respective changes of population states under replicator dynamics for \(\epsilon _s = 0.03\) and \(\epsilon _r = 0.03\). As the figure shows, a drift from \(\rho _1\) to \(\rho _2\) is not possible anymore.
When sender and receiver costs are non-zero, PS and FA are both ESS. In such a scenario a more refined selection criterion for evolutionary stability can help to point out which of both strategies has an evolutionary advantage over the other: stochastic stability (cf. Foster and Young 1990). The idea is as follows. We assume that the evolutionary dynamics is non-deterministic due to noisy mutation: the mutation rate changes randomly. If we wait long enough, every ESS will eventually be invaded, no matter how high its invasion barrier is. Thus, given two ESS \(s_i\) and \(s_j\), there is a non-zero probability \(p_{ij}\) that the system switches from \(s_i\) and \(s_j\), as well as a non-zero probability \(p_{ji}\) for the reverse switch. Now, if and only if \(p_{ij} > p_{ji}\), then \(s_j\) is the only stochastically stable strategy (and \(s_i\), if and only if \(p_{ij} < p_{ji}\)), and the probability to stay with \(s_j\) approaches 1 when the mutation rate approaches 0.
To study the stochastic stability of strategy pairs, I will first introduce the pairwise expected utility \(EU_p\), which represents the expected utility for playing a strategy pair \(\langle \sigma ,\rho \rangle \) against \(\langle \sigma ',\rho ' \rangle \). \(EU_p\) is based on the idea that an agent is sender and receiver each with probability \(\frac{1}{2}\). Here, an agent plays her sender strategy \(\sigma \) against another agent’s receiver strategy \(\rho '\) with frequency \(\frac{1}{2}\), and her receiver strategy \(\rho \) against another agent’s receiver strategy \(\sigma '\) with frequency \(\frac{1}{2}\). Therefore, \(EU_p\) is defined as follows:
$$\begin{aligned} EU_p(\langle \sigma ,\rho \rangle , \langle \sigma ',\rho ' \rangle ) = \frac{1}{2} \cdot EU_s(\sigma ,\rho ') + \frac{1}{2} \cdot EU_r(\sigma ',\rho ) \end{aligned}$$
(1)
\(EU_p\) produces symmetric EU tables. The EU table for all possible strategy combinations of \(\sigma _1\), \(\sigma _2\), \(\rho _1\) and \(\rho _2\) is depicted in Table 2.Footnote 13
Table 2 EU tables over all possible strategy pair combinations of sender strategies \(\sigma _1\) and \(\sigma _2\) and receiver strategies \(\rho _1\) and \(\rho _2\) (row player utilities) Note that for non-zero sender and receiver costs, both strategy pairs \(\langle \sigma _1,\rho _1 \rangle \) and \(\langle \sigma _2,\rho _2 \rangle \) are evolutionarily stable, whereas \(\langle \sigma _1,\rho _2 \rangle \) and \(\langle \sigma _2,\rho _1 \rangle \) are not (proof in Appendix A.1). But which strategy pair it stochastically stable? The PS type \(\langle \sigma _1,\rho _1 \rangle \) or the FA type \(\langle \sigma _2,\rho _2 \rangle \)? The answer: it depends on the relationship between sender and receiver costs. More concretely, it can be shown that the following proposition holds (proof in “Appendix A.2”)Footnote 14:
Proposition 1
Given the EU table of Table 2. If \(\epsilon _s > \epsilon _r\) then strategy pair \(\langle \sigma _2,\rho _2 \rangle \) is stochastically stable, whereas if \(\epsilon _s < \epsilon _r\) then strategy pair \(\langle \sigma _1,\rho _1 \rangle \) is stochastically stable.
To conclude, my analysis completes the one by Santana (2014) in that it indicates all the reasonable scenarios of sender and receiver cost combinations and the impact on evolutionary stability aspects. Santana pointed to a scenario where FA has an evolutionary advantage over PS, and I argue that this advantage is minute when the proposed sender costs are minute. Furthermore, I presented a scenario where it can be exactly the other way around. Moreover, I showed that when non-zero sender and receiver costs are involved, then FA and PS both are evolutionarily stable. I proved that in such a scenario one or the other strategy type is stochastically stable if and only if it involves less costs than its counterpart. However, this analysis is still preliminary, since it considers only two strategy types. Note that there might be further strategy types—next to FA and PS—that ensure perfect communication and are evolutionarily relevant. Therefore, it is a straightforward next step to explore the whole strategy space of the context signaling game.