Abundant resources can trigger reduced consumption: Unveiling the paradox of excessive scrounging

Significance

In game-theoretic scenarios, pursuing self-interest is known to occasionally yield suboptimal outcomes. This paper delves into a related, yet distinct, phenomenon, exploring how increasing payoffs for any given strategy profile can result in declining equilibrium payoffs as players adapt to the changing incentive structure. This phenomenon is demonstrated in two basic games: one concerning group foraging and the other workplace cooperation. In both games, we show how improving the efficiency production at the individual level can amplify the inclination to free-ride to such an extent that more than offsets the increase in individual efficiency, ultimately resulting in reduced outcomes at both the group and individual levels. Our results provide insights into the mechanisms underlying the emergence of free-riding behavior.

Abstract

In ecological contexts, it is conventionally expected that increased food availability would boost consumption, particularly when animals prioritize maximizing their food intake. This paper challenges this conventional wisdom by conducting an in-depth game-theoretic analysis of a basic foraging model, in which animals must choose between intensive food searching as producers or moderate searching while relying on group members as scroungers. Our study reveals that, under certain circumstances, increasing food availability can amplify the inclination to scrounge to such an extent that it leads to a reduction in animals’ food consumption compared to scenarios with limited food availability. We further illustrate a similar phenomenon in a model capturing free-riding dynamics among workers in a company. We demonstrate that, under certain reward mechanisms, enhancing workers’ production capacities can inadvertently trigger a surge in free-riding behavior, leading to both diminished group productivity and reduced individual payoffs. Our findings provide intriguing insights into the complex relationships between individual and group performances, as well as the intricate mechanisms underlying the emergence of free-riding behavior in competitive environments.

Working under improved conditions is often expected to lead to enhanced performance. However, in groups composed of self-interested players, this expectation does not always hold. For example, if the punishment cost is decreased in a chicken game, then it is easy to verify that the payoff of players at a symmetric equilibrium decreases. This may be perceived as intuitive since taking the punishment cost to infinity leads both players to fully adopt the cooperating strategy, which yields a higher payoff than any other mixed strategy (see SI Appendix, Theorem 2 for a detailed explanation). A less intuitive example is Braess’s paradox, which reveals that adding a road to certain transportation networks can paradoxically increase traffic latency at equilibrium (1, 2). Our study demonstrates that this phenomenon can also occur in cooperative productive contexts, emphasizing the impact of free-riding behavior.

Productive groups often consist of individuals who benefit from their own resource generation or findings, and enjoy the added advantage of reaping the rewards of others’ contributions (3–7). For example, workers in a company may receive performance-based bonuses as a reward for their productivity, while also benefiting from the collective production of their peers, through stock shares or other profit-sharing mechanisms. Similarly, in the realm of joint research projects, the success of the endeavor contributes to the collective prestige of the researchers, yet those who make substantial contributions often receive heightened recognition and prestige. Likewise, in group foraging scenarios, animals that first discover food patches often have an opportunity to feed before other group members join in, granting them the ability to directly consume a portion of the food they found and secure a larger share (8–11).

Within such productive group contexts, the pervasive occurrence of free-riding becomes apparent (3, 4, 6, 12–14). Free-riding refers to individuals exploiting collective efforts or shared resources without contributing their fair share. In team projects, for instance, free-riders neglect their assigned tasks to avoid costs or risks while still benefiting from the project’s overall success (5, 15). This phenomenon is also remarkably prevalent in foraging contexts, where animals opportunistically engage in scrounging or kleptoparasitism, feeding off prey discovered or captured by others (11, 14, 16–20).

The framework of Producer-Scrounger (PS) games is a widely used mathematical framework for studying free-riding in foraging contexts (6, 7, 19, 21, 22). In a PS game, players are faced with a choice between two strategies: producer and scrounger. The interpretation of these strategies varies according to the context, but generally, a producer invests efforts in order to produce or find more resources, whereas a scrounger invests less in producing or finding resources, and instead relies more on exploiting resources produced or found by others. Based on the rules of the particular PS game, specifying the production and rewarding mechanisms, each animal chooses a strategy and the system is assumed to converge into an equilibrium state, where each animal cannot improve its own calorie intake by changing its strategy (23).

This paper investigates how changes in individual production capacity, which can be influenced by, e.g., environmental conditions or by changes in the individual’s resource-producing abilities, affect the resulting performances in equilibrium configurations. The first PS game we consider aims to model a scenario consisting of a group of foraging animals, with each animal striving to maximize its own food intake. Intuitively, as long as the group size remains unchanged, one may expect that even if it may trigger more opportunistic behavior (24, 25), increasing food abundance should ultimately improve consumption rather than diminish it. Likewise, within a productivity-based reward system in a company, one may expect that enhancing individual productivity levels would boost group productivity and subsequently increase workers’ payoffs, despite a possible increase in free-riding behavior. However, our findings uncover a more nuanced reality, unveiling a pronounced detrimental effect of free-riding behavior and emphasizing that the existence of such a positive correlation between individual productivity and payoffs is strongly contingent on the specific characteristics of the setting.

Results

The general game-theoretic phenomenon we study concerns scenarios where increasing payoffs for any given strategy profile results in declining equilibrium payoffs as players adapt to the changing incentive structure. To facilitate comparisons across different parameter settings, we ensure that the games we examine are symmetric and have a unique evolutionary stable strategy (ESS) (Materials and Methods and SI Appendix, Lemma 14). We consider a parameter $\gamma$ that governs the entries of the game’s payoff matrix in a nondecreasing manner. For example, in Eq. 1 which describes the payoff matrix of one of our games, the entries are all nondecreasing, affine functions in $\gamma$. Increasing $\gamma$ thus results in a new game whose payoffs are at least as large for every fixed strategy profile. However, as players adapt to the new situation, their equilibrium state may also change. We say that a game admits a reverse-correlation phenomenon if a continuous increase in $\gamma$ leads to a (strict) decrease in the players’ payoff, when evaluated at ESS (Materials and Methods).

To obtain intuition, we first study the reverse-correlation phenomenon in 2-player games. We prove that any 2-player game that exhibits this phenomenon must be a chicken game (SI Appendix, Theorem 2). A weak form of the phenomenon may still occur in other games than chicken. Specifically, in some cases, increasing $\gamma$ changes the order between payoffs, which may lead to a decline in payoffs at ESS, but this decline occurs only at the specific point where the order changes. In contrast, we show that to obtain a continuous decline, the game must be chicken. In addition to this necessary condition, we also provide a sufficient condition for the occurrence of this phenomenon in 2-player chicken games (SI Appendix, Theorem 2). Specifically, we note that increasing any of the payoffs obtained by a “defector” while fixing the payoffs of a “cooperator” leads to reduced payoffs at equilibrium.

Next, we turn to investigate our main games of interest, namely, PS games. We consider two types of PS games: a Foraging game involving animals searching for food and a Company game involving a group of workers in a company. Our main objective is to analyze the effects of changes in individual production capabilities on players’ payoffs, evaluated at equilibrium configurations.

We begin with the Foraging game, which is a generalization of the classical PS game in refs. 21 and 22. The main difference with the classical model is that our model considers two types of food, instead of a single type as previously assumed. As we show, this model yields the reverse-correlation phenomenon, for example, $n=3$, despite the fact that its 2-player version is not a chicken game.

The Foraging Game.

To illustrate our model, consider a scenario involving a group of animals engaged in fruit picking from trees (Fig. 1). Each animal aims to maximize its fitness, which is determined by the amount of food it consumes. The trees in this scenario contain both low-hanging fruit, accessible to both producers and scroungers, and high-hanging fruit, which can only be reached by producers. When an animal picks fruit (either low-hanging or high-hanging), it retains a portion for its own consumption (let’s say 70%), while the remaining fruit falls to the ground. Producers pick both low-hanging and high-hanging fruits. Scroungers, instead of picking high-hanging fruit, focus both on picking low-handing fruits and on scanning the ground for fallen fruit. Fallen fruit (of either type) is distributed equally among all scroungers and the animal that originally obtained it.

Fig. 1.

An illustration of the biological scenario that serves as an inspiration for the Foraging game. Producers (in blue) can grab high-hanging fruits (in green) as well as low-hanging fruits (in yellow). Scroungers (in red) are restricted to low-hanging fruits. Whenever some fruit is picked, a fraction is eaten directly by the animal, and the remaining falls to the ground, and is then consumed by the animal and the scroungers.

More precisely, consider $n\ge 2$ animals, where each of which needs to choose to be either a producer or a scrounger. We assume that a producer finds an amount of food corresponding to $F_{\mathcal{P}}=1+\gamma$ calories, where, adhering to the trees example above, 1 corresponds to the amount of high-hanging fruit and $\gamma$ is a parameter that governs the animal’s access to low-hanging fruit. In contrast, a scrounger directly finds only low-hanging fruit, corresponding to $F_{\mathcal{S}}=\gamma$ calories. After finding food (of any type) consisting of $F$ calories, the animal (either producer or scrounger) consumes a fraction $s\in [0,1]$ of what it found (called the “finder’s share”) and shares the remaining $(1-s)F$ calories equally with all scroungers. See Fig. 2 for a schematic illustration of structure of the foraging game.

Fig. 2.

The definition and key elements of the game are illustrated on a group of five animals—two producers and three scroungers. Each scrounger finds food consisting of $\gamma$ calories (yellow, corresponding to low-hanging fruit) and each producer finds food of $1+\gamma$ calories, where 1 (green) corresponds to the high-hanging fruit and $\gamma$ to low-hanging fruit. A portion of $s=70\%$ of the food found (corresponding to both low-hanging or high-handing fruits) is directly consumed by the finding animal, and the remaining $1-s=30\%$ portion (corresponding to fruit that fell on the ground) is equally shared between the animal and all scroungers. Scrounger-to-scrounger food exchanges cancel one another, making the situation equivalent to the case that each scrounger fully consumes the low-hanging fruit it finds. Hence, the figure depicts the latter case, since its illustration is simpler to draw and think about.

The payoff of a player is defined as the capacity of its calorie intake. Hence, for each $0\le k\le n$, the payoff of each pure strategy in the presence of exactly $k$ producers in the population is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\pi }_{\mathcal{P}}^{(k)}& =s(1+\gamma )+(1-s)\frac{1+\gamma }{1+n-k},\hfill \\ \hfill {\pi }_{\mathcal{S}}^{(k)}& =\gamma +k(1-s)\frac{1+\gamma }{1+n-k},\hfill \end{array}\end{array}$$ [1]

where the second equation follows since scrounger-to-scrounger interactions compensate each other, and hence, can be ignored in the expression of the payoff. Note that the classical model in refs. (21, 22) is retrieved by setting $\gamma =0$, which essentially implies that there is only one type of food, namely, high-hanging fruit.

We study what happens to the payoffs of players at equilibria configurations, denoted by ${\pi }_{\star }$, as we let $\gamma$ increase. This increase captures the case that the low-hanging fruit becomes more abundant in the environment.

Note that for each fixed $k$, both ${\pi }_{\mathcal{P}}^{(k)}$ and ${\pi }_{\mathcal{S}}^{(k)}$ are increasing in $\gamma$. Hence, simply increasing $\gamma$, without changing the strategy profile, necessarily results in improved payoffs. However, allowing players to modify their strategies after such a change may potentially lead to enhanced scrounging at equilibrium, especially since increasing the low-hanging fruit seems to have a somewhat more positive effect on the scrounger’s payoff than on that of producers (Eq. 1). This, in turn, can have a negative impact on the payoffs. Nevertheless, as mentioned earlier, one might expect that this negative effect would be outweighed by the overall improvement in fruit availability, resulting in an increase in consumption rather than a decrease. This expectation may be further enhanced by the fact that in the foraging model, all food that is found is fully consumed by the group members. This intuition becomes apparent when comparing the scenarios with $\gamma =0$ and $\gamma =1$. As $\gamma$ increases from 0 to 1, we can expect an increase in the proportion of scroungers due to the rising ratio of $F_{\mathcal{S}}/F_{\mathcal{P}}=\gamma /(1+\gamma )$. However, even if the system with $\gamma =1$ ends up consisting entirely of scroungers, the average food consumption of a player (which equals 1) would still be at least as large as that of any strategy profile in the $\gamma =0$ case. Nonetheless, as shown here, upon closer examination within the interval $\gamma \in [0,1]$, a different pattern is revealed.

We combine numerical results (Figs. 3 and 4) with mathematical game-theoretical analysis (SI Appendix, Theorem 3) to disclose a reverse-correlation phenomenon in the Foraging game. Specifically, for the case of $n=3$ players, we prove in SI Appendix, Theorem 3 that for any finder’s share $s<1/2$, there exists an interval of values for $\gamma$ over which the reverse-correlation phenomenon occurs. Interestingly, the reverse-correlation phenomenon occurs for $n=3$ even though its 2-player variant is not a chicken game. Indeed, the reverse-correlation phenomenon does not happen for $n=2$. Instead, we prove in SI Appendix, Theorem 3, that the game yields a weak form of reverse correlation (Materials and Methods), where there exists a critical value of $\gamma$ at which ${\pi }_{\star }$ decreases locally. Essentially, by increasing $\gamma$ the game changes from being a Harmony game (in which cooperating is the only ESS) to a Prisoner’s dilemma game (in which defecting is the only ESS), where the transition occurs when $\gamma =\frac{1+s}{1-s}$.

Fig. 3.

Foraging game with $n=4$. The probability of being a producer $p_{\star }(\gamma )$ and the payoff ${\pi }_{\star }(\gamma )$, at equilibria, for three values of the finder’s share $s$. For each of these $s$ values, ${\pi }_{\star }$ is decreasing over a large interval of $\gamma$, effectively illustrating the reverse-correlation phenomenon.

Fig. 4.

Foraging game with $n=4$. The relationship between the payoff at equilibrium ${\pi }_{\star }$ (color scale), the abundance of low-hanging fruits $\gamma$, and the finder’s share $s$. Once again, it illustrates the reverse-correlation phenomenon, while highlighting its independence from specific values of $s$.

Numerical results for $n=4$ players show a noticeable decline in the payoffs at equilibrium as $\gamma$ increases over a relatively large subinterval of $[0,1]$ (Figs. 3 and 4). For greater values of $n$, they also appear to reveal a reverse-correlation phenomenon, but to a lesser extent, and only for smaller values of $s$ and $\gamma$. See SI Appendix, Fig. S3 for an illustration of the phenomenon in the case $n=7$.

The Company Game.

We consider a PS game aiming to model a scenario with a group of $n\ge 2$ workers of equal capabilities who collaborate to produce a common product for a company. (Alternatively, by replacing the salary received by a worker with a notion of prestige, the game can also capture a scenario where a group of researchers collaborate in a research project.)

Each worker is assigned a specific part of the project and can choose between two pure behavioral strategies. A producer pays an energetic cost of $c>0$ units and with probability $p$ produces a product of quality $\gamma$ (otherwise, with probability $1-p$, it produces nothing). In contrast, a scrounger pays no energetic cost and with probability $p$ produces a product of lower quality ${\gamma }^{\prime }=a\gamma$ for some given $0\le a<1$ (similarly, with probability $1-p$, nothing is produced). Let $I=\{1,2,\dots ,n\}$, and let $q_i$ denote the quality of the product made by worker $i$, for $i\in I$, with $q_i=0$ if no product is made by this player. We define the total production as

$$\begin{array}{c}\hfill \mathrm{\Gamma }={\displaystyle \sum _{i\in I}}{q}_i.\end{array}$$ [2]

Note that there is no synergy effect, as the total production is simply the sum of all individual contributions. We assume that the salary ${\sigma }_i$ of player $i$ is proportional to a weighted average between the quality of the products made by the workers, with more weight given to $q_i$. In fact, by appropriately scaling the system, we may assume without loss of generality that the salary is equal to this weighted average. Formally, we set:

$$\begin{array}{c}\hfill {\sigma }_i=s{q}_i+\frac{1-s}{n-1}{\displaystyle \sum _{j\in I\setminus i}}{q}_j,\end{array}$$ [3]

for some $s\in [1/n,1]$. Note that $s=1$ implies that the salary each worker receives is identical to the quality of its own production, and $s=1/n$ represents equally sharing the quality of the global product between the workers.

Next, we aim to translate the income salary of a player into his payoff using a utility function, denoted by $\varphi (·)$. These quantities are expected to be positively correlated, however, the correlation may in fact be far from linear. Indeed, this is supported by the seminal work by Kahneman and Deaton (26) which found that the emotional well-being of people increases relatively fast as their income rises from very low levels, but then levels off at a certain salary threshold. To capture such a correlation, we assume that $\varphi$ is both monotonically nondecreasing, concave and bounded. Note that this can also be seen as an assumption that the workers are risk-averse. In addition, the payoff of worker $i$ is decreased by its energetic investment. Finally,

$$\begin{array}{c}\hfill {\pi }_i:=\varphi ({\sigma }_i)-c_i,\end{array}$$ [4]

where the energetic investment $c_i=c>0$ if $i$ is a producer and $c_i=0$ if $i$ is a scrounger. See Fig. 5 for an illustration of the semantic structure of the game.

Fig. 5.

Illustration of the Company game’s definition for the case of 2 players: a producer and a scrounger. The production $q_i$ of player $i$ is 0 with probability $1-p$, and otherwise, it is $\gamma$ if the player is a producer, and $a\gamma <\gamma$, otherwise. The salary ${\sigma }_i$ is a weighted average of the production of both players with more weight given to $q_i$. The utility function $\varphi$ (Inset) aims to capture the impact of salary on payoff. Finally, the payoff is ${\pi }_i=\varphi ({\sigma }_i)-c_i$, where $c_i$ is equal to $c$ if $i$ is a producer, and 0, otherwise.

The question of whether or not the system incurs a reverse-correlation phenomenon turns out to depend on the model’s parameters, and, in particular, on the function $\varphi (x)$. For example, when $\varphi :x\mapsto x$ (i.e., the case that the salary is converted entirely into payoff), there is no reverse-correlation phenomenon (SI Appendix, Fig. S1 and section 3A). However, for some concave and bounded functions $\varphi (x)$ the situation is different.

We combine mathematical analysis with numerical computations considering the function (see Inset in Fig. 5):

$$\begin{array}{c}\hfill \varphi (x)=1-exp(-2x).\end{array}$$

Our mathematical analysis proves the presence of a reverse-correlation phenomenon for the case of two workers (SI Appendix, Theorem 7). Interestingly, this result holds for every $s<1$, demonstrating that the reverse-correlation phenomenon can occur even when the payoffs of individuals are substantially biased toward their own production compared to the production of others. Our numerical results consider the case of $n=4$ workers and reveal (Figs. 6 and 7) that for certain parameters, letting $\gamma$ increase over a range of values results in a reduction in payoffs in equilibrium, thus indicating a reverse-correlation phenomenon. Moreover, as $\gamma$ increases over a range of values we also observe a substantial reduction in total production at equilibria.

Fig. 6.

Company game with $n=4$, $\varphi :x\mapsto 1-exp(-2x)$, $a=p=\frac{1}{2}$, and $s=0.6$. The graph depicts the payoff ${\pi }_{\star }(\gamma )$ and the total production ${\mathrm{\Gamma }}_{\star }(\gamma )$, as well as the probability of being a producer $p_{\star }(\gamma )$ at equilibria, as a function of individual capabilities $\gamma$ and for several values of $c$. It displays the reverse-correlation phenomenon over a certain interval that depends on $c$.

Fig. 7.

Company game with $n=4$, $\varphi :x\mapsto 1-exp(-2x)$, $a=p=\frac{1}{2}$, and $s=0.6$. The relationship between (A) payoff ${\pi }_{\star }$, and (B) total production ${\mathrm{\Gamma }}_{\star }$ at equilibrium, as a function (color scale) of individual capabilities $\gamma$ and the cost $c$ for production.

While the general shape of the utility function $\varphi (x)=1-exp(-2x)$ is justifiable, the function itself was chosen somewhat arbitrarily. To strengthen the generality of our results, we also provide in SI Appendix, Fig. S2 numerical results supporting the reverse-correlation phenomenon under another type of nondecreasing, concave, and bounded utility function, specifically,

$$\begin{array}{c}\hfill \varphi (x)=min(1,x).\end{array}$$

A Necessary Condition for General n.

Finally, we identify a necessary condition for the emergence of a reverse-correlation phenomenon in arbitrary PS models, holding regardless of the population size. Specifically, we prove (SI Appendix, section 4) that a reverse-correlation phenomenon can occur only if the definition of the producer’s payoff is sensitive to the fraction of scroungers in the population.

An interesting consequence of this condition is that a seemingly minor change in the definition of the Foraging game can prevent the occurrence of the reverse-correlation phenomenon. Recall that in this game, when an animal finds food, it consumes a fraction $s$ of it (the finder’s share), and the remaining $1-s$ fraction falls to the ground and is then equally shared between the animal and all scroungers. If the game is changed such that when a producer finds food, it only consumes the finder’s share and does not eat at all from the food that falls on the ground (i.e., only the scroungers eat from it), then the game stops satisfying the aforementioned necessary condition. Indeed, in this case, the payoff of a producer would always be $1+\gamma$ irrespective of the number of scroungers. Hence, the modified game does not exhibit a reverse-correlation phenomenon, regardless of the parameters involved.

Discussion

The reverse-correlation phenomenon corresponds to a (strict) decrease in payoffs assessed at equilibrium, resulting from nondecreasing changes in the entries of the payoff matrix. In the case of PS games, these changes represent improved production abilities at the individual level.

Another measure of interest in PS games is the total production, defined as the sum of production over all players (Eq. 2). Observe that in the Foraging game, since the animals eventually consume all food found by the group, the total production (i.e., the total food found) at equilibrium is proportional to the payoff ${\pi }_{\star }$, and hence their dynamics are similar. This implies that whenever an increase in $\gamma$ results in a decrease in payoff at equilibrium (indicating a reverse-correlation phenomenon), the same increase in $\gamma$ also leads to a decrease in total production at equilibrium. In contrast, in the Company game, production is not fully represented in the payoffs, since some of it is “lost” when translating salaries into utilities. Additionally, the distinction between payoffs and production is further emphasized due to the energetic cost incurred by producers, which is reflected in their payoffs. Despite this distinction, as observed in Figs. 6 and 7B, the measure of total production also exhibits a decrease across a range of $\gamma$ values. This phenomenon may carry particular importance for system designers, such as the company’s principal, as it challenges a fundamental assumption underlying bottom–up approaches, namely, that as long as the system naturally progresses without external disruptions, improving individual performances should lead to enhanced group performances.

We demonstrated the reverse-correlation phenomenon on two basic game theoretical models. As evidenced by these games, the occurrence of this counterintuitive phenomenon is highly contingent on the specific details of the game. For example, the Foraging game considers two types of food: low-hanging and high-hanging fruit [instead of just one type as considered in the classical game in refs. 21 and 22). Only producers have access to high-hanging fruit, while both producers and scroungers can access low-hanging fruit. Similarly to the classical model, when an animal finds food, it consumes a portion $s$ of it and the remaining $1-s$ portion is equally shared between this animal and all scroungers. The reverse-correlation phenomenon emerges as the abundance of low-hanging fruit increases. Intuitively, it may appear that the fact that increasing the low-hanging fruit has a more positive effect on the scrounger’s payoff than on that of producers (Eq. 1) may play a role in the emergence of the reverse-correlation phenomenon. However, this is also the case if one modifies the model so that the remaining $1-s$ portion is shared only between the scroungers, and in this variant, as we showed, the system no longer exhibits a reverse-correlation phenomenon. Hence, while at first glance this change may appear minor, it has a profound impact on the dynamics.

In the Company game, a key aspect of the model concerns the choice of the utility function, which captures the relationship between salary and payoff. Inspired by the work of Kahneman and Deaton (26), we focused on nondecreasing, concave, and bounded utility functions $\varphi (x)$. Within this family of functions, we identified two that exhibit a reverse-correlation phenomenon. Intuitively, such a nonlinear function $\varphi (x)$ creates the situation that scroungers benefit more from an increase in $\gamma$. However, the occurrence of the phenomenon depends on the details of $\varphi (x)$ rather than on its general shape. Indeed, as we show on SI Appendix, Fig. S4, the function $\phi :x\mapsto log(1+10x)/3$, which has a similar shape to $1-exp(-2x)$ does not enable this phenomenon.

To conclude, this paper uncovers an arguably counterintuitive phenomenon that can emerge in productive group contexts involving rational players. It reveals that under certain conditions, increasing individual production efficiency can lead to diminished payoffs and overall group production, due to a significant rise in free-riding behavior. Importantly, this phenomenon is contingent upon the details of the setting, in a way that is not always fully understood. This highlights the necessity for additional theoretical and empirical examination. Overall, our findings yield insightful perspectives into the intricate interplay between individual and group performances, as well as the potential pathways through which free-riding behavior can emerge, leading to detrimental outcomes.

Materials and Methods

We consider two types of PS models, for which we combine analytic with numerical investigations. In both models, we assume that both producers and scroungers are able to produce, but that producers are expected to produce more. In our models, the payoffs and total production are positively correlated with the number of producers. We consider a parameter $\gamma$ that is positively correlated to the expected production capacities of both producers and scroungers. To check what happens as individual capabilities improve, we increase $\gamma$ and observe how the payoff and total production measures change, for configurations at equilibria.

We focus on the strong definition of equilibria, known as evolutionary stable strategy (ESS), using the standard definition as introduced by Maynard Smith and Price (23). Specifically, given a PS game, let ${\pi }_{q,p}$ denote the expected payoff of a player if it chooses to be a producer with probability $q$, in the case that all $n-1$ remaining players are producers with probability $p$. We say that $p_{\star }\in [0,1]$ is an ESS if and only if for every $q\in [0,1]$ such that $q\ne p_{\star }$,

• (i)either ${\pi }_{p_{\star },p_{\star }}>{\pi }_{q,p_{\star }}$,
• (ii)or ${\pi }_{p_{\star },p_{\star }}={\pi }_{q,p_{\star }}$ and ${\pi }_{p_{\star },q}>{\pi }_{q,q}$.

To be able to compare instances with different parameters, we make sure that for every value of $\gamma$, the game we consider always has a unique ESS, termed $p_{\star }(\gamma )$. In such a case, we write ${\pi }_{\star }(\gamma )={\pi }_{p_{\star }(\gamma ),p_{\star }(\gamma )}$ the payoff at the ESS, and omit the parameter $\gamma$ when clear from the context.

In our rigorous analysis, presented in SI Appendix, we prove the existence and uniqueness of the ESS, for the corresponding scenarios. To obtain numerical results, we determine the ESS with simple procedures that take the values of $p$ and $q$ as inputs and calculate ${\pi }_{q,p}$. Then, we search for the specific value of $p$ that satisfies (i) ${\pi }_{1,p}={\pi }_{0,p}$, (ii) for every $q<p$, ${\pi }_{1,q}>{\pi }_{0,q}$ and (iii) for every $q>p$, ${\pi }_{1,q}<{\pi }_{0,q}$, which together are sufficient conditions for $p$ to be the unique ESS (SI Appendix, Lemma 14).

All codes were implemented in Python. For further details and access to the code, please refer to ref. 27.

We say that the system incurs a reverse-correlation phenomenon if increasing $\gamma$ over a certain interval yields decreased payoff when evaluated at (the unique) ESS. In other words, this means that ${\pi }_{\star }(\gamma )$ is a decreasing function of $\gamma$ over this interval.

In some cases, ${\pi }_{\star }(\gamma )$ may be decreasing locally at a particular point, but not over an interval. Specifically, we say that the system incurs a weak form of the reverse-correlation phenomenon if there exists a point ${\gamma }_0$, where ${\pi }_{\star }(\gamma )$ is discontinuous, such that there exist two points ${\gamma }_0^{-},{\gamma }_0^{+}$ where ${\gamma }_0^{-}<{\gamma }_0<{\gamma }_0^{+}$, and ${\pi }_{\star }({\gamma }_0^{+})<{\pi }_{\star }({\gamma }_0^{-})$.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Code and simulation results’ data have been deposited in GitHub https://github.com/RobinVacus/reverse_correlation (27).