Explaining the evolution of honest signalling has been a long-standing problem in research on animal [3] and human communication [4]. Zahavi’s Handicap Principle [5, 6] (HP) has been the leading theoretical paradigm for honest signalling since it was reportedly validated by Grafen’s ‘strategic handicap’ model [1]. The HP predicts that signals must be costly and reduce survival at the evolutionary equilibrium — hence the label ‘handicap’ — in order to be honest. This idea is often claimed to provide a general principle to explain why signals are honest, and it has been widely accepted, although some have questioned its generality [7]. There is no consensus for how to define, model or test the HP, which is often confused with other models known as ‘costly signalling theory’, because handicaps and signalling costs have never been clearly defined, and it has never been shown how signalling costs *per se* enforce honesty [7,8,9,10].

Mathematical signalling games have greatly improved our understanding of honest signalling [3, 11, 12], as they have clarified the logic of honesty in conspecific interactions, including aggression [13], mate choice [1], parent-offspring conflict [2, 14, 15], and interspecific interactions, such as plant-herbivore [16], plant-pollinator [17], aposematic displays [18], and predator-prey [19] relations. They originated from economic signalling games [20] and have been used to analyse the stability of honest signals in a variety of human social interactions [21]. While these models have proved to be useful, identifying the costs of signalling at the honest evolutionary equilibrium (equilibrium cost function) in such models is far from trivial when the signallers’ quality varies continuously [1, 2]. As a consequence, it is difficult to compare the outcome of these models and make any general conclusions.

The so-called ‘strategic handicap’ model [1] is the most influential model of honest signalling, and it critically assumes that signallers differ in their quality and bear differential marginal costs for producing a signal. This is a plausible but widely misinterpreted model because it is very different from the HP. [7] Unlike the HP, honesty in this model does not depend upon the absolute costs of signalling and signals are efficient rather than wasteful. Moreover, honest signals are selectively favoured in the model despite of their costs; not because they are costly. This model has nevertheless provided an important step towards analysing fitness trade-offs for honest signalling, but the steps used to obtain the equilibrium cost function are difficult to replicate, hence the mathematics have been described as ‘brilliant but arcane’ [9].

The complexity of signalling games has been widely under-estimated, as it has been generally overlooked that finding an equilibrium cost function requires solving a double optimization problem [14, 22], one for the receiver as well as one for the signaller, and that the optimal solution for the signaller depends on the receiver’s optimum. This complexity is daunting and it has been circumvented by ignoring the receiver’s optimization problem (Additonal file 1: sections 1–4 gives a detailed description of the steps to solve this problem; while Additional file 1: sections 5–7 provides a more detailed discussion [1, 2, 14, 22,23,24,25,26,27,28,29,30];). This issue cannot be resolved until the optimization problem of the signaller and the receiver are both evaluated.

This double optimization problem has an infinite number of possible solutions [22, 31], and no general solution has ever been provided in analytical form. The lack of a clear methodology for deriving solutions to address this double optimization problem contributes to widespread misinterpretations of the HP and Grafen’s strategic choice model (see [7]) and left the critiques of these ideas difficult to understand [8, 9] and the debates unresolved. It has been known for 20 years that such signalling games have an infinite number of honest equilibria [22, 31], and yet the nature and implications of these equilibria have remained unexplored due to the complexity of this problem. Consequently, the conditions for honest communication in signalling games are still unclear and controversial, and the field has stagnated due to being entwined in the erroneous and confusing handicap paradigm [7].

Here, we provide a novel and general approach for determining stable equilibria in continuous signalling games, and for calculating equilibrium signal cost functions, as a continuation of previous theoretical developments [23]. We examine signalling models with *additive fitness functions* (when signal costs and benefits are measured in the same currency, such as fitness), and also *multiplicative fitness functions* (such as when signals have survival costs that influence their reproductive benefits) [32]. First, we describe an asymmetric signalling model of animal communication, and we aim for a general approach that will apply to any signalling context, given that certain, broad conditions are met. Then, we provide solutions for games with additive or multiplicative fitness functions. We provide a formal proof of the conditions for stability being independent of equilibrium signal cost. Our general formula specifies the full, infinite set of trade-off solutions of the double optimization problem. Furthermore, we show that an infinite number of cost-free and negative cost equilibria exist in these models. The discovery of these previously unknown and evolutionarily stable equilibria shows how new approaches and interpretations can be used to investigate signalling games in general.

Our approach does not require prior knowledge or assumptions about the shape of the potential solution, and hence it is applicable to any signalling model. We apply our method to calculate stable equilibria in classic signalling models, including Grafen’s model of sexual signals [1], Godfray’s signal-of-need model [2] for parent-offspring signalling games, and for the signalling model of Bergstrom et al. [22]. We explain how our results provide testable predictions regarding cost-free and beneficial (negative-cost) honest signals at equilibrium, and how these could support (or refute) our results and their generality. Finally, we discuss the shortcomings of equilibrium models and how signalling theory fits into the larger framework of life-history theory and Darwinian evolution.

### Models of signalling games

Signalling games are mathematical models used to analyse how individuals (*signallers)* attempt to influence the decisions of others (*receivers)* by producing signals (action at a distance, see Fig. 1). *Signals* are often strictly defined as traits that provide information about some aspect of a signaller not directly observable, such as size or sex; otherwise signals are unnecessary [20]. Signalling games are usually described as conflicts over a resource, because some of the first models were contests over food and territories. Indeed, from an evolutionary perspective, a receiver’s body and behaviour can be viewed as resources over which signallers compete to exploit for their own benefit [33]. Signalling games can be symmetric or asymmetric concerning information, resources, and options (strategies) available to the players. In symmetric games, players have the same information sets, resource availability and strategies at the beginning of the game, whereas in asymmetric games, players do not share the same information, resources, or strategic options.

Games can be symmetrical or asymmetrical in many respects, though it was information asymmetries between signallers and receivers that have mainly attracted the interest of biologists [34] and economists [20, 35]. Asymmetrical information is common in nature and thus asymmetrical signalling games have often been used to investigate how individuals resolve a wide variety of interactions and types of *conflicts* (e.g. genomic, sexual, parent-offspring, and other intra- and inter-familial conflicts). In these models, signallers use signals to persuade a receiver to take some action, which can include mating [1], feeding [2], other forms of parental investment [36], committing suicide [37, 38], and performing other actions that may or may not be in the receiver’s interest. Asymmetrical signalling games have been used to model intra-genomic conflicts and molecular signals between cells within the body [39, 40]. They have also been used to model a variety of interspecific interactions, including predator-prey [19, 41], host-parasite [42,43,44], plant-herbivore [16], plant-pollinator [17] and aposematic displays [18]. They are also used to understand and address the spread of misinformation and disinformation in human societies, which is arguably one of the most important problems facing our species [45,46,47].

Here, we focus on games with asymmetries in access to both information and resources. In asymmetric games, receivers possess a resource and can decide whether to share it with signallers or not. For example, young chicks attempt to persuade their parents to feed them by producing begging calls [2]. In discrete models, receivers can either give away the entire resource or keep it for themselves [11, 16, 19, 48,49,50], whereas in continuous models, receivers can share some *portion of the resource* (*z*) [1, 2, 14, 22, 23, 31, 32, 51]. Receivers are assumed to share the resource in a way that maximizes (inclusive) *receiver fitness* (*w*_{R}), but the potential benefits depend upon obtaining reliable information from signallers about what they offer in exchange. The problem is that receivers often have incomplete information about signallers or what they have to offer (information asymmetry). In the case of mate choice, females assess the potential benefits of mating with males that differ in social status, health, resources, or other aspects of *quality* (*q*); however, male quality cannot be directly assessed by the receiver, otherwise there is no need for signals. The signaller can influence the receiver’s decision by its signal, which may or may not reliably reveal the quality *q* of the signaller, to *ask for the resource amount* (*a*) that should maximize *signaller fitness* (*w*_{S}) (see Fig. 1). A signal is ‘honest’ if it provides receivers with reliable information about the signaller’s quality, allowing the receiver to make adaptive decisions. Alternatively, the signal can be useless or deceptive, so that signallers manipulate the receivers to share more than an amount *z* that is in their adaptive interest. Like previous honest signalling models, we investigate the conditions under which signals provide reliable indicators of quality *q* (for details, see the ‘Methods’ section and Additional file 1:sections 1–3).

Theoretical models have previously shown that honest signals are evolutionarily stable at an *honest equilibrium* if the following conditions are met [14, 22]: (*i*) the signal reveals the signaller’s actual quality *q* (signals are honest), so that the receiver can respond adaptively; or (*ii*) the signaller only asks for the amount *a* of a resource that receivers benefit by sharing (shared interest), so that the conflict between the receiver and signaller is removed at the honest equilibrium (*a* = *z*). The mathematical formulation of these conditions is detailed in the ‘Methods’ section.

The standard theoretical approach used for resolving conflicts of interest and to find stable equilibria in asymmetric signalling games is to introduce a *cost function* that transforms the signaller’s fitness function *w*_{S}, so that the optimal amount of resource *a* acquired by the signaller corresponds to the optimal amount of resource that the receiver shares *z* (see Fig. 1), and the optima of signaller and receiver, namely, *w*_{S} and *w*_{R}, then coincide. This step is crucial but missing from many previous models. It is not enough to find the optimum of *w*_{S}, but *w*_{S} must be transformed by using a function traditionally referred to as a ‘cost function’ in which max(*w*_{S}) = max(*w*_{R}). Here, we will refer to this transformation as a trade-off function (*T*). The function *T* transforms the benefit into the actual fitness so it is actually a trade-off function. Accordingly, the signallers’ fitness *w*_{S} is determined by the relation between the benefits *B* and trade-offs *T*, and without any trade-offs, *w*_{S} = *B*. In additive models (e.g. [2]), *B* and *T* are summed, whereas in multiplicative models (e.g. [1]), they are multiplied to yield the fitness *w*_{S}.

We use the term ‘trade-off function’ because the term ‘cost function’ is unnecessarily restrictive to the positive domain (counter-intuitively, the cost value is positive rather than negative), and moreover, it does not represent the full set of possible solutions, as we demonstrate below. We also avoid the term ‘cost function’ because it has generated much confusion, and we provide a more detailed explanation in the ‘Discussion’ section. Our key insight is that this transformation, regardless of its label, does not necessarily represent an absolute cost, whereas it is always defined by a trade-off *sensu* life-history theory.

We construct the most general class of trade-off functions that obey the conditions of honest signalling for both additive and multiplicative fitness functions (see the ‘Methods’ section and Appendices 2–3). Lastly, we apply our method to well-known models of honest signalling (Appendix 4), demonstrating its general applicability.