Social efficiency deficit deciphers social dilemmas

What do corruption, resource overexploitation, climate inaction, vaccine hesitancy, traffic congestion, and even cancer metastasis have in common? All these socioeconomic and sociobiological phenomena are known as social dilemmas because they embody in one form or another a fundamental conflict between immediate self-interest and long-term collective interest. A shortcut to the resolution of social dilemmas has thus far been reserved solely for highly stylised cases reducible to dyadic games (e.g., the Prisoner’s Dilemma), whose nature and outcome coalesce in the concept of dilemma strength. We show that a social efficiency deficit, measuring an actor’s potential gain in utility or fitness by switching from an evolutionary equilibrium to a social optimum, generalises dilemma strength irrespective of the underlying social dilemma’s complexity. We progressively build from the simplicity of dyadic games for which the social efficiency deficit and dilemma strength are mathematical duals, to the complexity of carcinogenesis and a vaccination dilemma for which only the social efficiency deficit is numerically calculable. The results send a clear message to policymakers to enact measures that increase the social efficiency deficit until the strain between what is and what could be incentivises society to switch to a more desirable state.

The dilemma strength parameters follow straight from their definitions To calculate the social efficiency deficit (abbreviated SED), we recognize that the social optimum is fully cooperative, 14 in which case the expected per-capita utility is Π SO = R. The evolutionary equilibrium, by contrast, is fully defecting, 15 meaning that the expected per-capita utility is Π NE = 0. The SED is the difference between these two utilities, thus Finally, defining DS ≡ D g = D r , we see that DS ∝ SED −1 , i.e., dilemma strength is inversely proportional to the H = (k + 1) (R − P ) − T + S (k + 1) (k − 2) , [5] which in the case of the Donor-Recipient game defined by Eq.
[2] turns into which, for a fixed ∆>0, increases with R and decreases with k > 2. The payoff matrix is Following the definition of the dilemma strength parameters [8b] As long as D g = D r > 0, the game here is also an instance of the Prisoner's dilemma, implying that exactly the same 49 argument as the one used to derive Eq.
[4] still holds. Thus, we again have We see from Eq.
[6] that for a fixed ∆, there may be k > 2 such that SED = R ≥ ∆ (k − 1), yielding H ≥ ∆. At this 51 moment D g = D r ≤ 0 and the game exits the Prisoner's Dilemma domain. Unlike the Prisoner's Dilemma, in which the payoff order is T > R > P > S, in the Chicken game, the fact that being 54 exploited by a defector is better than getting punished for mutual defection demands a payoff order T > R > S > P . If

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x under this payoff order denotes the fraction of cooperators in a population of actors, then the replicator dynamics (3) 56 leads to where the expected utility of a randomly chosen actor is π = xπ C +(1 − x) π D , while the expected utility of cooperators 58 is π C = xR + (1 − x) S, and the expected utility of defectors is π D = xT + (1 − x) P . The equality of expected utilities 59 π C = π D yields a dimorphic equilibrium Rewriting the game payoffs in the spirit of the Donor-Recipient game in Eq.

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The Public Goods Games are a multiplayer generalization of the Donor-Recipient game. This is best seen by starting 77 from the unilateral payoff matrix for the latter game is then respectively given by To interpret these equations, let us focus on the first one. The equation states that a cooperator happens to be 91 in a group with j other cooperators with the binomial probability G−1 j x j (1 − x) G−1−j , and then earns the payoff r j G − 1 c . The sum runs across all possibilities, i.e., from j = 0 to j = G − 1 other cooperators. A similar reasoning 93 applies to the second equation, with a distinction that here a defector happens to be in a group with j cooperators.
This identity shows that if r < G (resp., r > G), then under the replicator dynamics in Eq.
[10] defectors (resp., 96 cooperators) prevail. The cost c determines the speed of convergence. The expected utility is maximized when everyone 97 cooperates, implying that Π SO = π C (1) = (r − 1) c . If r > G, then Π NE = Π SO and SED = 0. Put alternatively, 98 when the return factor r is large enough, the dilemma disappears. If, however, r < G, then Π NE = π D (0) = 0 and One immediate difference between economic and biological public goods is that the benefit of receiving a diffusible 106 factor, instead of being proportional to, saturates with increasing concentration (6). Mathematically, this corresponds 107 to replacing the linear benefit r j G c in Eqs.
[21] with a saturating benefit function b = b (j), e.g., the logistic function (6) where 0 < k ≤ G is an inflection point at which the benefit function has the steepness s > 0. The difference in utilities 109 in Eq.
[22] now becomes Intuition suggests that if we repeat an experiment G − 1 times with a success probability of x, then the number of 111 successes should be j ≈ (G − 1) x, or even j ≈ Gx as G → ∞. This would further suggest that [26] Rewriting the benefit function as where h ≡ k G , 0 < h ≤ 1, is a fractional inflection point, yields and finally [29] The last equation allows for up to two internal equilbria x * 1,2 on the domain 0 < x < 1 under the condition 0 < c < s 4 . It can be shown that, as a consequence, one of five different types of dynamics 117 arises (6): • Type A dynamics occurs when x = 0 is the stable equilibrium, x = 1 is the unstable equilibrium, and there are 119 no internal equilibria.

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• Type B dynamics occurs when x = 0 and x = 1 are unstable, and there is one stable internal equilibrium 121 0 < x * 2 < 1.

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• Type C dynamics occurs when x = 0 is stable, x = 1 is unstable, and there are two internal equilibria 123 0 < x * 1 < x * 2 < 1 among whom x * 1 is unstable and x * 2 is stable.

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• Type D dynamics occurs when x = 0 and x = 1 are stable, and there is one unstable internal equilibrium 125 0 < x * 1 < 1.

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• Type E dynamics occurs when x = 0 is unstable, x = 1 is stable, and there are no internal equilibria.

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The outlined analytical approximation for the difference of expected utilities given in Eq.
[29] is useful to qualitatively 128 understand the types of evolutionary dynamics that arise in Public Goods Games with the saturating benefit function In simulations, however, we used the exact expression given in Eq.
[25], but only after normalizing We examined the effects of three key parameters on the cooperation frequency and the SED. The first of the three To explore the usefulness of the SED in a vaccination dilemma, we relied on an existing setup (11) that assumes an 175 infinite well-mixed population in which the evolution of decision making is coupled with a periodic outbreak of a 176 seasonal flu-like disease. A vaccination campaign is assumed to precede the flu season. Once the flu season starts, 177 spreading is governed by a susceptible-infectious-recovered (SIR) process. Thereafter, actors compare how they fared 178 during the epidemic against the performance of their peers, and probabilistically imitate one peer (12, 13). The 179 probability of imitating the peer increases when the peer fared better than the actor. Conversely, the probability 180 of imitating decreases when the actor fared better than the peer. Imitating a worse performer is improbable, but 181 possible. The vaccine is assumed to be imperfect in that it fails to protect a fraction of vaccinated actors at random.

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Beside payoffs associated with each of the four outcomes, we also need the corresponding probabilities of occurrence.

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These can be calculated based on the fact that the final epidemic size (i.e., the fraction of recovered actors at the end 193 of the SIR process), R (x, ∞), is given by the probability of being without immunity, 1 − xe, and the probability of 194 contracting the disease conditional on being non-immune, where x is the fraction of vaccinators, 0 < e ≤ 1 is the vaccine efficacy, and R 0 is the basic reproduction number (14). In addition to the payoffs π (i) and the corresponding probabilities p (i) , we need one last bit of information that 202 mutually connects the four possible outcomes. Specifically, we define that actors who experienced outcome (i) during 203 the past epidemic season, imitate their peers who experienced outcome (j) with the probability where κ is the irrationality of selection in the sense that the larger the value of this parameter, the lower is the 205 influence of the payoff difference on decisions to imitate. The dynamic equation for the fraction of vaccinators (i.e., 206 cooperators) is no longer given by the replicator equation, but rather by a mean-field equation of the form At last, we are in a position to define the SED. The expected utility of vaccinators (i.e., cooperators) is and the expected utility of free riders (i.e., defectors) is have Π SO = max x π (x) in the social optimum and Π NE = π (x * ) in the evolutionary equilibrium x * obtained from 211 Eq.
[34]. If, in addition, the system is bistable for given values of key parameters, here c R and e, the equilibrium 212 fraction of cooperators depends on the initial condition x 0 , i.e., x * = x * (x 0 ). In such a case we perform additional 213 averaging across all possible initial values 0 < x 0 < 1, i.e., Π NE = 1 0 π [x * (x 0 )] dx 0 . The SED is finally given by [37] Fig. S1. Cancer metastasis is a cellulo-social phenomenon analyzable using the social efficiency deficit. Cancer cells benefit from the production of diffusible factors as a form of public good. However, such production is costly, making producer cells susceptible to free riding. A, B, Density of producer cells decreases with the increase of the production cost c and the diffusion range G. Interestingly, changing the fractional inflection point h of the benefit function in Eq.
[27] from a small value of h = 0.3 to a large value of h = 0.7 results in similar producer densities. This is unexpected because the parameter h quantifies the difficulty of conferring benefits via diffusible factors, and therefore should negatively affect cell cooperation. Note that the results in these plots are averaged over all possible initial conditions. Capital letters A -E denote the five types of dynamics defined in the text. C, D, Although the producer density stays the same irrespective of the h value, small-h and large-h situations are very different when examined using the social efficiency deficit. In particular, cancer cells under the C-type dynamics change from being fairly efficient to being rather inefficient in the sense of underproducing diffusible factors in the evolutionary equilibrium relative to the optimum. E, F, Bevacizumab-like drugs are designed to inhibit diffusible factors, which in the model corresponds to increasing the value of the parameter h. The social efficiency deficit, however, reveals that increasing the parameter h is unlikely to have more than a limited effect on cooperation among cancer cells; the cell system is indeed pushed towards the less efficient C-type or D-type dynamics, yet producer cells may still persist. The gradient field suggests that even when increasing the value of the parameter h is effective, this should be accompanied with simultaneous increases of the parameter c , which in medical terms would translate to therapies that target producer cells specifically.