When should receivers follow multiple signal components? A closer look at the “flag” model

Animals will benefit from discriminating between 2 types of signalers (such as profitable or unprofitable prey, male or female) if the best response to a given signaler depends on its true identity (e.g. accept if profitable, reject if unprofitable). Yet if the types differ in more than one trait, such as color and pattern, with one being a more reliable indicator of identity than the other, should animals pay attention to the more reliable trait, the less reliable one, both or neither? To address this important question, Rubi and Stephens (2016a) took an economic perspective and introduced a model in which receivers are faced with a decision to “accept” or “reject” signalers that are either of “good” or “bad” type. The signalers exhibit one of 2 colors (C+, C−) and one of 2 patterns (P+, P−), with the sign denoting the version of the trait associated with being good (+) and being bad (−). The authors first summarized their model results, stating that the receiver should “follow the single most reliable component” and ignore the less reliable one—regardless of the exact magnitude of the reliabilities. This “flag” prediction (so called because the plot resembles a diagonal bi-colour flag) was then generally upheld in a carefully controlled experiment, leading the authors to argue “(…) that alternative receiver benefits need to be considered to explain the prevalence of complex signals in nature.” The flag prediction is very surprising given that multimodal signals are common in nature and have frequently been shown to be more effective in eliciting responses by receivers (references in Rubi and Stephens 2016a). For example, some harmless (Batesian) mimics have evolved to resemble defended prey not just in color and pattern, but also in behavior, suggesting that their predators pay attention to more than one trait. The prediction is also at odds with the findings of other mathematical models (e.g. Fawcett and Johnstone 2003). Rubi and Stephens (2016b) have since qualified the flag prediction with a more detailed mathematical treatment, stating that it is contingent on the base rate (frequency) of each type being exactly 0.5 (“maximum uncertainty”), with discriminative strategies other than following a single component being possible when this condition is not upheld. Here, we show that the flag model is a special type of signal detection model, and therefore it is not base rate per se that drives the results but the interaction between base rate and payoff. Once one frames their model in this way, its relationship to other work becomes more apparent. The results we show also highlight the need to explore the effect of different payoffs in the flag model; only by doing this can the model (and associated experiments) truly provide an economic perspective on signal use. Note that following Rubi and Stephens (2016a, 2016b) we refer to “signals” and “signallers.” However, the approach also applies to cues that incidentally provide information about the carrier’s identity, but have not been shaped by natural selection for this reason. Let ρ denote the base rate, i.e. probability in any given encounter that the signaler is “good.” Let receivers gain payoff VAG for accepting good signalers, VRG for rejecting good signalers, VAB for accepting bad signalers, and VRB for rejecting bad signalers. The signals provide clues to signaler identity, in the sense that the receiver upon seeing a certain combination of signal components has certainty k that the focal signaler is “good” (k is therefore the probability that the signaler is good, conditional upon seeing the signal combination). The receiver profits from accepting all signalers with this combination if kVAG+(1−k)VAB>kVRG+(1−k)VRB⁠. Assuming that it is better to accept than reject good items and better to reject than accept bad items, we may write the inequality as k>k*=c/(b+c), in which the benefit of accepting good items is b=(VAG−VRG)>0 and the cost of accepting bad items is c=(VRB−VAB)>0 Binary signals can be added as follows: Let good signalers express C+ with probability pC+ and bad signalers express C− with probability pC− such that Pr(C+|good) = pC+ (> 1 − pC−) and Pr(C−|bad) = pC− (> 1 − pC+). Likewise let Pr(P+|good) = pP+ (> 1 − pP−) and Pr(P−|bad) = pP− (> 1 − pP+). Since cues are binary, then Pr(C−|good) = 1 − pC+ and so on. We assume that ρ, k*, and all conditional probabilities are between 0 and 1. For a compact notation, let S11, S10, S01, and S00 denote the signal combinations P+C+, P+C−, P−C+ and P−C−, respectively. Assuming that signals are conditionally independent, in the sense that good signalers with a certain color are no more likely to have a particular pattern than good signalers with the alternative color (say), it follows, for example, that Pr(S11|good)=pP+pC+ and more generally, Pr(Sij|good)=pP+i(1−pP+)1−ipC+j(1−pC+)1−j and Pr(Sij|bad)=(1−pP−)ipP−1−i(1−pC−)jpC−1−j From Bayes’ rule, we have: It will (on average) be beneficial for the receiver to accept signal combination Sij if k = Pr(good | Sij) > k*, which can be rearranged to ρ (1−k*)/(k*(1−ρ))>Pr(Sij|bad)/Pr(Sij|good) This inequality is recognizable as a central result in signal detection theory (Green and Swets 1988), since the left-hand side is equal to ρb/((1−ρ)c) which is traditionally represented as β (the “cutoff”), and the right-hand side is the likelihood ratio of event Sij for hypothesis “signaller is bad” relative to hypothesis “signaller is good.” In fact, the “flag” model is formally equivalent to a signal detection model in which the signals are pairs of uncorrelated Bernoulli-distributed variables. To further simplify, we note that it follows from Rubi and Stephens’ (2016a) assumptions that the association indices are symmetrical in that pC+ = pC−= pC (≥ 0.5) and pP+ = pP−= pP (≥ 0.5). This is in effect similar to Fawcett and Johnstone’s (2003) assumption of signaler-specific binary cues with symmetric cue-specific error probabilities, so that eC = (1 − pC) and eP = (1 − pP) (although this model differs from a signal detection model in that it involves costly assessment, for example). In Figure 1a–c, we show some outcomes of the symmetric model. The flag prediction—namely that “follow the single most reliable component” is optimal for all magnitudes of reliabilities—holds only when k*= ρ (Figure 1b). Rubi and Stephens (2016a) assumed a base rate of ρ = 0.5 in their model presentation and implemented it experimentally (the artificial butterflies presented to blue jays were good half the time and bad half the time). The authors did not discuss payoffs explicitly or describe their optimality criteria, but from their identification of the flag prediction we can infer that they also assumed b = c. Indeed, in their experiment, 3 food pellets were provided to the birds if they correctly accepted or correctly rejected a given object, while no food was provided if they incorrectly accepted or incorrectly rejected it. Therefore, one might argue that VRB − VAB = 3 − 0 = c and VAG − VRG = 3 − 0 = b, yielding k*=0.5(=ρ) Under this particular coincidence of conditions, the flag prediction holds, and it is re-assuring that the blue jays appeared to behave accordingly—using color when it was more reliable, and pattern when it was the more reliable. However, the flag prediction cannot be considered a general case since it exists only when the cutoff β (=ρb/((1 − ρ)c)) is exactly equal to 1, that is, when the expected payoffs from unconditional rejection and unconditional acceptance precisely match. The signal combinations that should be accepted for different values of β =ρb/((1−ρ)c), see main text for explanation). Here, we vary the reliability of the color in indicating good (pC+ = Pr(C+|good)), and the reliability of the pattern in indicating good (pP+ = Pr(P+|good)) under the symmetry restrictions (pC+ = pC− = pC and pP+ = pP−= pP). The 3 panels (a–c) show conditions where the cutoff β is >1, =1, and <1, respectively, reflecting conditions where the expected payoff from unconditional acceptance is greater than, equal to, or less than the expected payoff from unconditional rejection. We note that the parameters used to represent signal reliability by Rubi and Stephens (2016b) and in these plots and are not the same as those used by Rubi and Stephens (2016a) who instead used Pr(good|P+) and Pr(good|C+). However, when ρ = 0.5 and the cue associations symmetric then the indices are equivalent, so the central plot is identical to the one in Rubi and Stephens (2016a) under the conditions they assumed. Figure 1a and c illustrate the general “non-flag” outcomes of the model (away from the parametric knife-edge k* = ρ). If the reliabilities of both signal components are too low, they should effectively be ignored. Otherwise, if the components differ sufficiently in reliability, the receiver should pay attention to only the most reliable one. However, if the reliabilities are sufficiently similar and neither is too low, 2 other optimal decision rules are possible, namely 1) accept if either desirable signal component is present (i.e. only C−P− is rejected) (Figure 1a) and 2) only accept if both desirable components are present (Figure 1c). These decision rules were also identified by Fawcett and Johnstone (2003) and Rubi and Stephens (2016b). Rubi and Stephens (2016b) explored the 2-signal flag model by varying the base rate ρ while keeping payoffs equal (b = c) and suggested that decision rules 1) and 2) may be optimal only if the base rate ρ was greater or smaller than 0.5, respectively. Thus, they concluded that receivers should tend to require both cues to be present only when desirable signalers were rare. In contrast, our analysis allows payoffs to vary and shows that rules 1) and 2) may be optimal when the cutoff β is greater or smaller than 1, respectively. Consequently, it might be optimal for receivers to require both desirable components to be present regardless of whether desirable signalers are common or rare, consistent with the findings of Fawcett and Johnstone (2003). Payoffs similarly affect the properties of the original 1-signal “flag model” (e.g. McLinn and Stephens 2006) that the 2-signal model builds upon. This simpler model can be derived from the 2-signal model by making one signal component unreliable, say C+/C−, by letting pC = 0.5. Even here the decision to follow or ignore the other component is not simply a question of signal reliability relative to base rate: The exact conditions under which the reliable signal P+/P− should be ignored are defined bypp<ρ (the 1-signal “flag” prediction, McLinn and Stephens 2006) only when k* = c/(b+c) = 0.5. Rubi and Stephens (2016a) tested the validity of the flag model under a specific combination of conditions where receivers should always pay attention to only the more reliable signal (i.e. the flag prediction). However, the flag model is capable of a wider range of outcomes than this. Thus, the empirical findings of Rubi and Stephens (2016a) cannot be taken as evidence that we need to look beyond economic benefits to explain complex signals. Indeed, the range of different outcomes in the flag model suggests the opposite conclusion of Rubi and Stephens (2016a), namely that economic benefits remain entirely capable of accounting for the use of multiple signals by receivers, a conclusion also consistent with other models (Fawcett and Johnstone 2003). In the natural world, k* and ρ will rarely if ever be identical: while ρ can sometimes be close to 0.5 (e.g. it may reflect sex ratio, Rubi and Stephens 2016b), there is no reason for k* (which depends only on costs and benefits) to be equal to ρ. Nevertheless, an interesting extension of Rubi and Stephens (2016a) would be to test whether the birds behave the same for other combinations of ρ, b, and c that also satisfy ρ b/((1 − ρ)c) = 1 thereby yielding the same flag prediction. Even if not ecologically realistic, the test may say something about the ability of the model to predict behavior. Ultimately, the model should be tested for other values of β. Experimental testing of fundamental ideas is demanding, but given that the model yields qualitatively different predictions under ecologically realistic general conditions (k* ≠ ρ) a broader set of test conditions is necessary to understand when and why receivers pay attention to complex signals. Tom Sherratt was supported by a NSERC Discovery Grant while Øistein Holen was supported by the Research Council of Norway (project number 249987/F20).

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.