### Myopic reallocation improves collective wealth

Beginning from some initial extraction state, agents within a networked population of multiple common-pool resources play an iterated game in which they observe current resource conditions at each round, and incrementally shift their extraction efforts from lower-quality sources toward higher-quality sources in order to maximize their payoffs in the following round (Eq. 3). Agents’ extraction efforts are thus redirected away from over-exploited sources toward less-exploited sources so that the system approaches a steady state in which all sources equally share the burden of over-extraction. In the process, some sources increase in quality, while others are further degraded; nonetheless, the overall result of these reallocations is a net increase in collective wealth.

To show this, we consider an arbitrary initial extraction state, in which the population’s collective extraction effort is (Q=Nlangle overrightarrow{q}rangle). In this state, the initial collective payoff extracted by the population is ({F}_{0}={sum }_{sin mathbf{S}}overrightarrow{q}(s)cdot b(s)) (where we ignore cost terms, since these remain constant under reallocation), and so the population’s collective wealth per unit extraction effort is

$$frac{{F}_{0}}{Q}=frac{sum_{sin mathbf{S}}overrightarrow{q}(s)cdot left[alpha -beta (s)overleftarrow{q}(s)right]}{sum_{sin mathbf{S}}overrightarrow{q}(s)}=alpha -frac{langle beta {overrightarrow{q}}^{2}rangle }{langle overrightarrow{q}rangle }.$$

(4)

Under reallocation dynamics (Eq. 3), this total extraction (Q) is conserved, and the system will approach a steady state in which all sources share a common quality value

$${b}_{f}=alpha -frac{langle overrightarrow{q}rangle }{langle {beta }^{-1}rangle }.$$

(5)

The population’s collective wealth approaches the steady-state value

$${F}_{f}={sum }_{sin mathbf{S}}left[overrightarrow{q}(s)cdot {b}_{f}right]=Q{b}_{f}.$$

(6)

Collective wealth is increased (or at least conserved) if ({F}_{0}le {F}_{f}), or equivalently, if (frac{{F}_{0}}{Q}le {b}_{f}). Using Eqs. 4 and 5, this condition reduces to

$$langle overrightarrow{q}{rangle }^{2}le langle beta {overleftarrow{q}}^{2}rangle langle {beta }^{-1}rangle .$$

(7)

The validity of this inequality is guaranteed by the Cauchy–Schwarz inequality^{29}, (langle XY{rangle }^{2}le langle {X}^{2}rangle langle {Y}^{2}rangle) for random variables (X) and (Y), with the identifications (X=sqrt{beta (s)}overrightarrow{q}(s)) and (Y=sqrt{beta (s{)}^{-1}}). Furthermore, equality occurs if and only if the quantity (beta left(sright)overrightarrow{q}left(sright)) shares the same value for all sources, that is, when initial conditions are already steady states where all sources share a common quality value. Reallocation dynamics thus increase collective wealth for any initial condition where sources vary from one another in quality (see Section S2.1 of the Supplementary Information). This includes Nash equilibrium initial conditions, upon which we will now focus our attention.

### CPR degree heterogeneity leads to greater improvements in efficiency under myopic reallocation

In the unique Nash equilibrium state of a given network^{26}, each agent sets its extraction at each source to the point beyond which further extraction would increase its costs more than it would increase its payoffs, given that all other agents are doing the same. In this state, no agent can increase its payoffs by *unilaterally* adjusting its extraction levels while other agents hold their extraction levels constant. However, when all agents *simultaneously* adapt their extraction levels according to the reallocation update rule (Eq. 3), under which each increase in extraction at one source is matched by an equal decrease at another source, then higher payoffs can be achieved. To quantify the extent to which reallocation alone can help alleviate the “tragedy of the commons” represented by Nash equilibrium, we now apply reallocation dynamics to Nash equilibrium initial conditions on a variety of network types, and compare the population’s collective wealth values before and after reallocation.

When network-structured populations of rational individuals extract benefits from multiple linearly-degrading CPRs, the burdens of over-exploitation tend to fall upon sources in a degree-dependent manner. Myopic reallocation tends to shift these burdens among sources of different degrees, and to distribute the resulting increases in collective wealth among individuals of different degree classes. In order to understand how these reallocations shift extraction pressure and agent payoffs among nodes of different degrees, we use a heterogeneous mean-field approach to derive estimates for these shifts. Under this perspective, the conditions defining Nash equilibrium ((frac{partial f(a)}{partial q(a,s)}=0)) lead us to estimate the expected values for extraction pressure on degree-(n) sources, (langle overrightarrow{q}{rangle }_{n}), by solving a linear system defined by

$$langle overrightarrow{q}{rangle }_{n}=frac{1}{{beta }_{n}}left[frac{n}{n+1}right]left[alpha -sum_{m=1}^{{m}_{mathrm{max}}}{P}_{mathbf{A}}left(mright)frac{m}{langle mrangle }cdot left(frac{gamma m}{[gamma mlangle {beta }^{-1}{rangle }_{m}+1]}left[alpha langle {beta }^{-1}{rangle }_{m}-sum_{{n}^{^{prime}}=1}^{{n}_{mathrm{max}}}{P}_{mathbf{S}}({n}^{^{prime}})frac{{n}^{^{prime}}}{langle nrangle }cdot langle overrightarrow{q}{rangle }_{{n}^{^{prime}}}right]right)right],$$

(8)

with one such condition for each unique source degree (nin {1,dots , {n}_{mathrm{max}}}) represented in the network, where brackets subscripted by agent degree (m) indicate expected values (langle x{rangle }_{m}={sum }_{n=1}^{{n}_{mathrm{max}}}{P}_{mathbf{S}}left(nright)frac{n}{langle nrangle }cdot {x}_{n}) and we have assumed no degree-degree correlations (see the Supplementary Information Section S3 for details). Solving this system numerically (here we use Python 3.7.3 with SciPy 1.2.1^{30}) for each of the 9 network types under consideration by inserting the corresponding ensemble degree distributions ({P}_{mathbf{A}}left(mright)) and ({P}_{mathbf{S}}left(nright)) (Fig. 1), we use the resulting values of (langle overrightarrow{q}{rangle }_{n}) to compute the expected total extraction by a degree-*m* agent (langle overleftarrow{q}{rangle }_{m}) at equilbrium as

$$langle overleftarrow{q}{rangle }_{m}=left(frac{m}{mgamma langle {beta }^{-1}{rangle }_{m}+1}right)left[alpha langle {beta }^{-1}{rangle }_{m}-left(sum_{n=1}^{{n}_{mathrm{max}}}{P}_{mathbf{S}}left(nright)frac{n}{langle nrangle }cdot langle overrightarrow{q}{rangle }_{n}right)right],$$

(9)

from which (langle q{rangle }_{m,n}), the expected equilibrium extraction by a degree-(m) agent from a degree-(n) source, can be computed using the Nash equilbrium condition:

$$langle q{rangle }_{m,n}=frac{alpha }{{beta }_{n}}-langle overrightarrow{q}{rangle }_{n}-frac{upgamma }{{beta }_{n}}langle overleftarrow{q}{rangle }_{m}.$$

(10)

These values are then used to compute the corresponding estimated collective wealth (i.e. the sum of all agent payoffs, (F=sum_{ain mathbf{A}}f(a))) and wealth equality (as quantified by Gini index (G)) attained at Nash equilibrium, as well as the subsequent shifts that are brought by myopic reallocation dynamics toward steady states. These values are shown in Fig. 2 for a range of values of the cost parameter (gamma), which quantifies the influence of diminishing marginal utility. The expected changes in extraction pressure for sources of different degrees, as well as the changes in agent fitness expected for agents of each degree class, are illustrated for each network type for cost-free extraction ((gamma =0)) in Fig. 3, and similarly for a representative case of costly extraction ((gamma =0.2)) in Fig. 4. The estimates presented here correspond to a *uniform capacity* scenario where all CPRs degrade in proportion to the total amount of extraction exerted upon their users. However, we find that qualitatively similar results also hold for a *degree-proportional capacity* scenario in which sources degrade in proportion to the total extraction *per user* that they receive (see Section S4 in the Supplementary Information).

**Figure 2**

Estimates of (**a**) Ratio of total collective wealth of equilibrium (“Eq”) states relative to efficient (“Ef”) states, ({F}_{mathrm{Eq}}/{F}_{mathrm{Ef}}); (**b**) increase in efficiency from equilibrium to steady states (“SS”), (({F}_{mathrm{SS}}-{F}_{mathrm{Eq}})/{F}_{mathrm{Ef}}); (**c**) Gini index of equilibrium states ({G}_{mathrm{Eq}}); and (**d**) decrease in Gini index from equilibrium to steady states, (({G}_{mathrm{Eq}}-{G}_{mathrm{SS}})), all as functions of cost parameter (gamma). Results shown correspond to a *uniform capacity* scenario with (alpha =beta =1).

**Figure 3**

Estimated shifts in extraction patterns due to reallocation dynamics from Nash equilibrium (“Eq”) to steady states (“SS”) under cost-free extraction: (**a**) Change in total extraction pressure (Delta langle overrightarrow{q}{rangle }_{n}=langle overrightarrow{q}{rangle }_{n,mathrm{SS}}-langle overrightarrow{q}{rangle }_{n,mathrm{Eq}}), as a function of source degree (n); and (**b**) change in expected agent fitness, (Delta langle f{rangle }_{m}=langle f{rangle }_{m,mathrm{SS}}-langle f{rangle }_{m,mathrm{Eq}}) as a function of agent degree (m). Results shown correspond to a *uniform capacity* scenario with (alpha =beta =1) and (gamma =0). Note that results for all network types sharing a common source degree distribution type (“D”, “N”, or “PL”) are overlapping.

**Figure 4**

Estimated shifts in extraction patterns due to reallocation dynamics from Nash equilibrium (“Eq”) to steady states (“SS”) under costly extraction: (**a**) Change in total extraction pressure (Delta langle overrightarrow{q}{rangle }_{n}=langle overrightarrow{q}{rangle }_{n,mathrm{SS}}-langle overrightarrow{q}{rangle }_{n,mathrm{Eq}}), as a function of source degree (n); and (**b**) change in expected agent fitness, (Delta langle f{rangle }_{m}=langle f{rangle }_{m,mathrm{SS}}-langle f{rangle }_{m,mathrm{Eq}}) as a function of agent degree (m). Results shown correspond to a *uniform capacity scenario* with (alpha =beta =1) and (gamma =0.2).

In Nash equilibrium states of the *uniform capacity* scenario, sources with fewer users (i.e. lower degree) experience lower extraction pressure. Since all networks under comparison here share an equal number of edges, networks having greater heterogeneity among source degrees—and thus a greater abundance of low-degree sources—suffer less over-exploitation overall, and so tend to operate more efficiently at equilibrium (Fig. 2). As agents then shift their extraction away from over-burdened, lower-quality sources toward higher-quality sources, these systems approach steady states where their multiple CPR sources all share a uniform quality value. In this way, steady states of reallocation dynamics qualitatively resemble Pareto efficient extraction states, which are characterized by uniform quality among all CPR sources (though, unlike these steady states, optimal efficiency also requires uniform extraction levels among all agents regardless of degree; see Section S3.2 in the Supplementary Information). The resulting shifts in efficiency (Fig. 2b), source extraction pressure (Figs. 3a and 4a), and agent payoffs (Figs. 3b and 4b) are more pronounced for networks having greater heterogeneity among CPR source degrees due to the greater initial discrepancies among source quality values that these networks support at Nash equilibrium. When simulations of reallocation dynamics from equilbrium are performed on individual networks (see Section S6 in the Supplementary Information), then the shifts in extraction pressure and agent payoffs observed are often more exaggerated than those estimated here. Since the heterogeneous mean-field perspective treats all sources of a common degree as a single class, it does not distinguish higher-order differences among nodes that share the same degree. As a result, the model predicts *no* shifts under reallocation dynamics for networks in which all sources share a common degree, i.e. delta-function (“D”) source degree distributions, for example. However, on actual networks of this type, reallocation dynamics nonetheless *do* increase collective wealth by equalizing differences in quality among sources.

When extraction is costly ((gamma >0)), agent degree heterogeneity also plays a secondary role to source degree heterogeneity in determining equilibrium efficiency and the effects of reallocation dynamics (Figs. 2 and 4). Diminishing marginal utility motivates agents to moderate their overall extraction levels; all sources affiliated with any given agent will be affected by its tendency to reduce extraction, and the extent of this reduction will depend in turn on each source’s degree, the degrees of its other users, and so on. Higher agent degree heterogeneity is thus predicted to slightly increase equilibrium efficiency due to the presence of higher-degree agents that reduce their extraction per source by larger amounts than do lower-degree agents. While the overall gains in collective wealth expected to be achieved by way of reallocations are thus slightly reduced by the presence of these higher-degree agents, greater agent degree heterogeneity is also associated with *faster times of convergence* toward steady states, since high-degree agents are able to simultaneously shift efforts directly between a large number of sources, and so to more rapidly equalize source quality values (see Section S5.1 in the Supplementary Information).

### Myopic reallocation from Nash equilibrium reduces wealth inequality

Since reallocation dynamics increase collective wealth, many—if not all—agents will attain improved payoffs under reallocation dynamics from suboptimal states like Nash equilibrium. We now turn our attention to *how* these increases in collective wealth are distributed throughout a population with respect to agent degree. Under the heterogeneous mean-field approach, we estimate that the shift in expected payoffs due to reallocations from Nash equilibrium are given by

$$Delta langle f{rangle }_{m}=mleft[left(frac{1}{langle nrangle }left[langle frac{n{b}_{n}}{{beta }_{n}}rangle {b}_{f}-langle frac{n{b}_{n}^{2}}{{beta }_{n}}rangle right]right)-upgamma langle overleftarrow{q}{rangle }_{m}left(frac{1}{langle nrangle }left[langle frac{n}{{beta }_{n}}rangle -langle frac{n{b}_{n}^{2}}{{beta }_{n}}rangle right]right)right],$$

(11)

where ({b}_{n}=alpha -{beta }_{n}langle overrightarrow{q}{rangle }_{n}) (see Section S3.1.3 in the Supplementary Information). When extraction is cost-free ((gamma =0)), the increased payoffs brought about by reallocation dynamics are expected to affect each edge in a uniform way, on average, and thus tend to be shared among agents of all degree classes in proportion to their degree (m). This is reflected in the linear increase of expected agent payoff with respect to degree (Fig. 3b), and also in the lack of change in the expected Gini index predicted for all network types under cost-free ((gamma =0)) extraction (Fig. 2d). However, when extraction is costly ((gamma >0)) and diminishing marginal utility acts to disincentivize increased extraction for higher-degree agents, the overall efficiency (Fig. 2a) and equality (Fig. 2c) of equilibrium states are increased from those observed under cost-free extraction. In these cases, reallocation dynamics also tend to increase the *equality* of the population’s wealth distribution, as reflected in the decreasing—and eventually negative—shifts in payoffs expected for agents of increasingly high degree (Fig. 4b), and also in the expected reductions in Gini index (Fig. 2d), caused by reallocation dynamics. This occurs because diminishing marginal utility motivates high-degree agents to exert less overall extraction effort *per source* at Nash equilibrium than do lower-degree agents. In the steady states subsequently reached under reallocation dynamics, all sources share a uniform quality value; each agent’s total extracted benefits then becomes strictly proportional to the overall magnitude of its extraction effort. Higher-degree agents end up receiving a smaller payoff *per source* than do their lower-degree counterparts in steady states. As Eq. (11) suggests, agents with higher initial extraction levels (langle overleftarrow{q}{rangle }_{m}) will experience a lower (and possibly even negative) shift in payoff per source (Delta langle f{rangle }_{m}/m) as a result of reallocations. This levelling-out of degree-based payoff inequities has its most pronounced effects at intermediate levels of the cost parameter (here, for values of (gamma approx .35), as shown in Fig. 2d). In simulations performed on specific networks, we find that reallocation dynamics lead not only to increased collective wealth, but also to increased *equality*, even on networks with homogeneous, “delta-function” (“D”) source degree distributions, although the heterogeneous mean-field approach predicts no such shift. Networks of other types similarly tend to undergo greater increases in equality than those predicted here due to higher-order types of heterogeneity not captured by the model (see Section S6 in the Supplementary Information).

Source: Ecology - nature.com