To clean or not to clean: The free-rider problem in sequentially shared resources

TL;DR

This paper develops an empirically parameterized evolutionary model of sequentially shared resources to study cleaning behavior under infection risk and social incentives. Using replicator dynamics augmented with a social-pressure potential, it analyzes four strategies and derives infection probability $I$ as a function of strategy mix and infection prevalence $x$. It finds multi-stability and abrupt transitions between altruistic, selfish, and mixed hygiene regimes as $W_{\text{cln}}$ and $W_{\text{inf}}$ vary, with hysteresis and sensitivity to resource access. The framework is designed to be calibrated with behavioral and environmental data and offers policy guidance for public health and digital-security contexts in gyms, co-working spaces, and other shared environments.

Abstract

Shared resources enhance productivity yet at the same time provide channels for biological and digital contamination, turning physical or digital hygiene into a cooperation dilemma prone to free-riding. Here we introduce a game of sequential sharing of common resources, an empirically parameterized evolutionary model of population dynamics in sequential-use settings such as gyms and shared workspaces. The success of the strategies implemented in the model, such as cleaning equipment before or after use, are based on the trade-offs between cleaning costs, contamination risk, and social incentives to mitigate disease transmission. We find that cooperative hygiene can be achieved by lowering the effective costs of cleaning, strengthening pro-social incentives, and monitoring population-level noncompliance. Remarkably, stability of fully altruistic populations is primarily affected by the cleaning costs. In contrast, increasing effective infection costs, for example through punishment, appears less important in this case. The model's evolutionary dynamics exhibit multi-stability, hysteresis, and abrupt shifts in strategy composition, broadly consistent with empirical observations from shared-use facilities. Our framework offers testable predictions and is amenable to quantitative calibration with behavioral and environmental data. Our predictions can be used to inform the design of cost-effective public health and digital security policies.

Figures (5)

• Figure 1: Numerical analysis of stable states and their basins of attraction. Shown are vector fields over the strategy frequency space $(\rho_A,\rho_B)$, assuming that $\rho_{D}=0$ and hence $\rho_{C}=1-\rho_{A}-\rho_{B}$. Blue arrows depict the flows driven by selective forces and social pressure (Eq. \ref{['eq:drhoD_uniqueu']}). Red dots mark stable equilibria. (a) Low infection cost: $W_{\text{cln}}=-4$, $W_{\text{inf}}=-5$. There are three stable equilibria: an altruistic monomorphic $A$ state ($\rho_{A}^*=1$), a selfish monomorphic $B$ state ($\rho_{B}^*=1$), and a mixed state comprising strategies $A$, $B$, and $C$. Background colors (green, blue, coral) denote numerically mapped regions of attraction of the altruistic, selfish, and mixed states, respectively. Most initial conditions converge to the mixed state. (b) High infection cost: $W_{\text{cln}}=-4$, $W_{\text{inf}}=-15$. There are two stable equilibria: an altruistic monomorphic $A$ state ($\rho_{A}^*=1$) and a selfish monomorphic $B$ state ($\rho_{B}^*=1$). The mixed state becomes unstable in this regime and cheaters $C$ go extinct. Background colors (green, blue) denote numerically mapped regions of attraction of the altruistic and selfish states, respectively. Note that most initial conditions converge to the selfish $B$ state; a smaller region leads to the altruistic $A$ state.
• Figure 2: Evolutionary dynamics and fixed point stability in the absence of infection costs. Altruistic ($A$) and selfish ($B$) strategies are symmetric when $W_{\text{inf}}=0$, since $f_A = f_B$ in this case (Eq. \ref{['eq:fD_unique1']}). As in Fig. \ref{['fig:combined_vector_fields']}, we show vector fields over the strategy frequency space $(\rho_A,\rho_B)$, assuming that $\rho_{D}=0$ and hence $\rho_{C}=1-\rho_{A}-\rho_{B}$. Blue arrows depict the flows driven by selective forces and social pressure (Eq. \ref{['eq:drhoD_uniqueu']}). Red dots mark stable equilibria. (a) High cleaning cost: $W_{\text{cln}}=-4$. There are three stable equilibria: an altruistic monomorphic $A$ state ($\rho_{A}^*=1$), a selfish monomorphic $B$ state ($\rho_{B}^*=1$), and a mixed state comprising strategies $A$, $B$, and $C$. Background colors (green, blue, coral) denote numerically mapped regions of attraction of the altruistic, selfish, and mixed states, respectively. Most initial conditions converge to the mixed state. Note that the green and blue regions are equal in size. (b) Low cleaning cost: $W_{\text{cln}}=-2.5$. There are two stable equilibria: an altruistic monomorphic $A$ state ($\rho_{A}^*=1$) and a selfish monomorphic $B$ state ($\rho_{B}^*=1$). The mixed state becomes unstable in this regime and cheaters $C$ go extinct. Background colors (green, blue) denote numerically mapped regions of attraction of the altruistic and selfish states, respectively. Note that $A$ and $B$ regions of attraction are equal in size.
• Figure 3: Mixed-state stability and composition. Here we allow all four strategies to have non-zero frequencies. (a) Stability of the mixed state in the cost parameter space $(W_{\text{inf}},W_{\text{cln}})$. The blue region indicates an unstable mixed state, where cheaters go extinct irrespective of initial conditions. The coral region marks cost combinations that permit coexistence of cheaters with other strategies. The insert shows a zoomed-out view in the larger $(W_{\text{inf}},W_{\text{cln}})$ domain. (b) Strategy frequencies as functions of $W_{\text{inf}}$ along the stability boundary in the inset of panel (a). At each value of $W_{\text{inf}}$, the stability boundary corresponds to the minimum fraction of cheaters allowed in a mixed state. As the boundary is crossed, the fraction of cheaters drops abruptly to zero.
• Figure 4: Abrupt elimination of cheaters as a function of infection and cleaning costs. We observe an abrupt, phase-like transition between non-zero and zero cheater frequencies as the boundary between stable and unstable regions in Fig. \ref{['fig:combined_dynamics']} is crossed. (a) The fraction of cheaters in the population, $\rho_C$, as a function of the infection cost $W_{\text{inf}}$, with the cleaning cost fixed at $W_{\text{cln}}=-6$. Starting from the initial condition $\rho_{A}^0=\rho_{B}^0=\rho_{D}^0=0.1$, we iterate the dynamics either to a fixed point or until cheaters are eliminated and plot the resulting $\rho_{C}$ values (black dots). A sharp transition occurs at $W_{\text{inf}}=-9.8$, matching the boundary between stable and unstable regions in Fig. \ref{['fig:combined_dynamics']}; background colors indicate those regions. (b) The fraction of cheaters in the population, $\rho_C$, as a function of the cleaning cost $W_{\text{cln}}$, with the infection cost fixed at $W_{\text{inf}}=-9.8$. The values of $\rho_C$ for each $W_{\text{cln}}$ are obtained as in (a), using the same initial conditions. A sharp transition occurs at $W_{\text{cln}}=-6$.
• Figure S1: Evolutionary dynamics and fixed point structure as a function of infection prevalence $x$. (a) Same as Fig. \ref{['fig:combined_vector_fields']}a, but with low infection prevalence, $x=0.2$. (b) Same as Fig. \ref{['fig:combined_vector_fields']}a, but with high infection prevalence, $x=0.8$.