Enquist’s model [33] is a symmetric game of aggressive communication where two animals fight for the same resource. The players can be weak or strong, where q and 1-q give the frequencies of weak and strong individuals respectively. Each player knows its own state; however, the state of the opponent remains hidden. The game can be divided into three stages (see Fig. 1): (i) Nature decides the state of each player, (ii) each player can choose between two signals A or B, these signals are assumed to be free of production cost; finally (iii) each player can choose between three actions: flee, attack or attack conditionally. Conditional attack implies that the player waits for the opponent to withdraw and it attacks only if the other player stays to fight. Let V denote the value of the contested resource, this resource cannot be divided between the contestants. There are four cost parameters associated with fighting behaviour in the model. Enquist [33] assumed that a strong individual can always beat a weak individual, where C SW , and C WS denote the cost of fighting for strong and weak player respectively. It is further assumed that the cost suffered by the weak player is larger than the cost paid by the strong one, hence the following relation holds: C WS > C SW . Weak or strong individuals fighting between each other has an equal chance to win a fight, where C WW and C SS denote the expected costs. Overall, we assume the following relations between these costs: C WS > C WW , C SS > C SW . There are three cost parameters associated with fleeing in the model. Let F f denote the cost of fleeing and F A , denote the cost of attacking a fleeing opponent. Finally, let F P denote the cost of waiting when the opponent attacks unconditionally. It is usually assumed that these costs are small compared to fighting costs, i.e.: C sw > F A , F P [43].

Fig. 1 Schematic description of the Enquist game [33]. Stage zero: Nature picks a state for the contestants; this stage is hidden from other players. Stage one: each contestant picks a signal, A or B. Stage two: each contestant picks a behaviour as a response to the signal: Flee (F), Conditional Attack (CA) or Attack (At) Full size image

Two more important considerations are involved: the source of variation and the strategy set available to the players. Enquist’s [33] model makes sense only if there is a polymorphism of weak and strong individuals. Enquist’s model [33] assumes a fixed 0.5 ratio between weak and strong individuals, whereas both Hurd’s model (1997) and later Számadó’s [24] first model consider a fixed ratio between 0 and 1. Számadó’s [24] second model further relaxes of this assumption by having a ratio that can change during the course of evolution. Szalai & Számadó’s model [25] followed Számadó’s [24] second model, in which the frequency of the alleles that regulate the ratio of strong to weak individuals is an evolutionary variable. In contrast Helgesen et al. [42] have used a fixed ratio of weak vs. strong individuals that is randomised for every play. Thus, whereas the models of Enquist [33], Hurd [43], and Helgesen et al. [42], and Számadó [24] first model, assume an exogenous explanation for the polymorphism of weak and strong individuals, Számadó’s [24] second model and Szalai & Számadó’s model [25] provide an endogenous explanation. Notably, the model implemented by Szalai & Számadó [25] is not a choice-of-state model, as has been erroneously claimed by H13: the chance of playing ‘strong’ or ‘weak’ is regulated by the alleles of a gene; thus, it is not up to individual choice. Because Szalai and Számadó [25] investigated a model with an endogenous explanation, here, I also implement the same version.

The original model consists of only two global strategies (honesty vs. cheating; Enquist, [33]. Enquist investigated the following honest global strategy denoted S (Enquist, [33]; p. 1155):

“If strong, show A; if the opponent also shows A attack and if the opponent shows B, repeat A and attack only if it does not withdraw immediately. If weak show B and give up if the opponent shows A and attack if the opponent shows B.”

The evolutionary stability of this honest global strategy S was investigated against a simple cheating type in which weak individuals show A instead of B. This cheating strategy was not explicitly defined by Enquist; the corresponding global strategy can be written up as follows (Számadó [24]; p. 222):

“Display always A in the first round, regardless of strength; then in the second round if strong attack unconditionally if opponent shows A or wait until opponent flees if it has shown B; if weak withdraw if opponent signals A or wait until opponent flees if it has shown B.”

Figure 2 shows a schematic representation of the potential strategies and gives Honest Strong as an example. Each individual has 7 genes. The first one encodes the strength of the individual (weak or strong). The next three encode behaviour when weak: (i) signal when weak (A or B), (ii) response to signal A when weak (flee, conditional attack, attack), (iii) response to signal B when weak (flee, conditional attack, attack); finally, the last three genes encode behaviour when strong: (v) signal when strong (A or B), (vi) response to signal A when strong (flee, conditional attack, attack), and finally (vii) response to signal B when strong (flee, conditional attack, attack).

Fig. 2 Schematic representation of the coding of the behaviour of individuals. Each individual has seven genes: the first gene represents the state of the individual (weak or strong), the next three and the last three encodes the behaviour of the individual depending on whether the state of the individual is weak or strong respectively. Out of these three genes the first gene gives which signal to use; the second gene encodes which behaviour to use as a response to signal A; and finally, the last gene encodes which behaviour to use as a response to signal B. W: weak; S: strong; F: Flee; CA: Conditional Attack; At: Attack. Asterisks denote silent regions which do not influence the behaviour of the given individual. The “genom” of an individual playing the Honest Strong strategy is at the bottom as an example Full size image

Szalai and Számadó investigated eight strategies, see Fig. 3 for representations and Additional file 1: Appendix 1 for definitions of these strategies. There are, however, more than eight strategies in Enquist’s game, and the full set has been investigated by Helgesen et al. [42] (see Additional file 1: Appendix 2). It is important to note that altough on paper there are 324 (18 × 18) possible pure strategies in the model, most of these strategies are redundant if the actual behaviour of any individual is examined. Because individuals are weak or strong for life, in the current implementation of the model, therefore half of their genes will be never expressed (see Figs. 2 and 3). When classifying the behaviour of the individuals, these inactive genes can be safely ignored, thus greatly simplifying the analysis. Notably, these inactive alleles are still present (even if they are not used for classification), and they can be turned on by mutation. Accordingly, I will consider only 36 strategies in the further analysis (see Additional file 1: Appendix 2).

Fig. 3 Schematic representation of the eight behavioural strategies that were used in the Szalai and Számadó model [25]. W: weak; S: strong; F: Flee; CA: Conditional Attack; At: Attack. Asterisks denote silent regions which do not influence the behaviour of the given individual Full size image

Szalai and Számadó [25] investigated only 8 strategies; however, they investigated more than 10,000 parameter combinations (see Table 1). Hamblin & Hurd [41] investigated all of the possible 324 strategy combinations; however, they investigated only a small fraction of the possible parameter space (12 parameter combinations, see Table 2). Additionaly, the two groups -Szalai and Számadó [25] and Helgesen et al. [42]- have used slightly different versions of the pay-off matrix; this difference is most noticeable at the flee vs. flee option (for comparison of pay-offs see Table 3). Here, I investigate both versions. Here I also change the genetic representation of the strategy set from Szalai & Számadó [25] to the one suggested by Helgesen et al., [42] (see further details in Fig. 2 and Additional file 1: Appendix 2) to allow the full strategy set to evolve.

Table 1 Szalai and Számadó [25] parameter space Full size table

Table 2 Helgesen et al., [42] parameter space Full size table

Table 3 Combined payoffs matrix Full size table

The feasibility of the evolvability of honest and cheating equilibria is assessed by individual based simulations. Here I investigate the effect of differences in (i) pay-offs and (ii) the effect of initial composition on the evolutionary trajectories of these populations, using the extended strategy set as suggested by Helgesen et al. [42] while keeping the other modelling assumptions the same as those in Szalai and Számadó [25] (i.e., number of fights and source of variation). To investigate the effect of initial strategy distribution, I use two different setups: either (i) seeding the population randomly from all the possible 36 strategies, or (ii) using the eight strategies used by SS09 to seed the initial population in order to compare the effects of switching from 8 strategies to the full strategy space.

All in all, I investigate evolvability with the following four different setups: (i) the initial population consists of random strategies drawn from the full set using the SS09 pay-offs; (ii) the initial population consists of random strategies drawn from the full set using the H13 pay-offs; (iii) the initial population consists of eight strategies used by SS09 using the SS09 pay-offs; and finally, (iv) the initial population consists of eight strategies used by SS09 using the H13 pay-offs. I use the full strategy set in all of these investigations as suggested by H13 (i.e. any of the possible strategies can evolve even if they are not present in the initial distribution), and I investigate parameter regions from the SS09 and H13 studies.

Of the vast parameter space investigated by SS09 I investigate only those sections where main signalling equilibria evolved in the original study (see Additional file 2: Dataset 1). Szalai and Számadó have [25] found six such equilibria: (i) Honest-strong, Honest-weak, which is the traditional honest signalling outcome (SS09 code: 3; current code: <30,2>); (ii) Honest-strong, Liar-strong, this is an “all-strong” honest signalling outcome where strong individuals signal differences in intentions (fight vs. flee) with the use of the signal (SS09 code: 5; current code: <30,20>); (iii) Honest-strong, Honest-weak, Liar-weak, which is the “traditional” cheating scenario (SS09 code: 11; current code: <30,2,14>); (iv) Honest-strong, Liar-strong, Liar-weak, which can be viewed as an “all-strong” cheating scenario in which the weak strategy imitates one of the strong ones (SS09 code: 13; current code: <30,20,14>); (v) Honest-strong, Honest-weak, Liar-strong, Liar-weak, which is a “full-scale” cheating scenario (SS09 code: 15; current code: <30,2,20,14>);.and finally, (vi) Honest-strong, Honest-weak, Liar-weak, Coward, this is an “all-strong” cheating scenario with cowards (SS09 code: 27; current code: <30,2,14,8,17>). The “current code” gives the code of pure strategies (according to Additional file 1: Appendix 2) supporting the given polymorphic equilibrium. These parameter regions are denoted by the code of the strategy combination used by SS09 (code3, code5, etc.). Out of these parameter regions 500–500 parameter combinations were drawn randomly and 10 independent runs were made with each combination. All in all, 3000 parameter combinations were investigated from the SS09 study. See Additional file 2: Dataset 1 for the parameter combinations and results of the SS09 study; and Additional file 3: Dataset 2 for the details of the 6 parameter regions described above. Finally, I investigate the evolvability of mixed cheating with the H13 parameter range as well (see Table 2), using the same modelling assumptions and same variation in pay-offs and initial strategy distributions as for the SS09 parameter space. See Table 4 for a comparison of the main differences between the two studies and for the general setup of the current study. Further details of the computer simulations are described in Additional file 1: Appendix 3, and Additional file 4: Table S1 summarises all of the investigated scenarios.