The paradox of productivity during quarantine: an agent-based simulation
TThe paradox of productivity during quarantine: an agent-based simulation
Peter Hardy, Leandro Soriano Marcolino, and Jos´e F. Fontanari School of Computing and Communications, Lancaster University, Lancaster, Lancashire, LA1 4WA, UK Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo,Caixa Postal 369, 13560-970 S˜ao Carlos, S˜ao Paulo, Brazil
Economies across the globe were brought to their knees due to lockdowns and social restrictionmeasures to contain the spread of the SARS-CoV-2, despite the quick switch to remote working.This downfall may be partially explained by the “water cooler effect”, which holds that higherlevels of social interaction lead to higher productivity due to a boost in people’s mood. Somewhatparadoxically, however, there are reports of increased productivity in the remote working scenario.Here we address quantitatively this issue using an agent-based model to simulate a workplace withextrovert and introvert agent stereotypes that differ solely on their propensities to initiate a socialinteraction. We find that the effects of curtailing social interactions depend on the proportion ofthe stereotypes in the working group: while the social restriction measures always have a negativeimpact on the productivity of groups composed predominantly of introverts, they may actuallyimprove the productivity of groups composed predominantly of extroverts, which offers then anexplanation for the paradox of productivity during quarantine.
Introduction. – The interest on the trade-off be-tween work productivity and social interaction is not apresent-day fad, as attested by this verse of the 19thcentury writer Maria Edgeworth: “All work and no playmakes Jack a dull boy, All play and no work makes Jacka mere toy” [1]. In fact, this is a staple subject of socialpsychology that addresses the influence of loneliness or,more generally, of mood on human cognitive function,often with discrepant findings (see, e.g., [2, 3]). Recentlockdowns and social restriction measures that isolatedworkers from their peers have placed these issues at theforefront of public attention, with news outlets remindingtheir customers that they do not need to be productive,and to expect a reduction in overall productivity duringthis time. But why is it that productivity would decreaseduring a period that one would think people have moretime on their hands than ever before?Typical answers to this question are in line with theso-called “water cooler effect” that has been highlighted(and satirized) in how effective placing an inanimate ob-ject for people to congregate around can stir up casualconversations, with many psychologists believing thatthis can also increase company productivity [4]. How-ever, this is not an obvious conclusion since for a singularworker the extra down time communicating may actuallydetract from its productivity as a whole.Here we address quantitatively the productivity andsocial interaction issue using an agent-based model tosimulate a workplace scenario where the agents exhibittwo social stereotypes, viz., extroverts and introverts,that differ solely on their propensities to initiate a con-versation. Social distancing is modeled by controllingthe number of attempts an agent makes to find a con-versation partner and the motivation to work (mood) isassumed to increase with the time spent talking and de-crease with the time spent alone. Moreover, the instanta-neous productivity of a lone agent increases linearly withits motivation to work.We find that the effect of curtailing social interactions depends on the proportion of the different stereotypesin the group. For instance, while social restriction mea-sures always have a negative impact on the productiv-ity of groups composed predominantly of introverts, theymay actually improve the productivity of groups com-posed predominantly of extroverts. In addition, within asame working group the productivity of the two stereo-types is affected differently by those measures. Theseresults may explain the paradoxical findings that pro-ductivity is increased in some remote work experiments[5].
Model. – We model the dynamics of a group of N agents that interact socially and work during a 480-minute (eight-hour) workday. We first make a strongassumption that for all agents they are more productivewhen they are motivated, and their motivations increasewhen they participated in some form of social interaction[6–8]. In this framework, motivation has both an emo-tional dimension, in the sense that talking with peersimproves the mood of the agent and helps it to focus onthe task, and an informational dimension, in that talk-ing with peers may result in the acquisition of valuableinformation that can help the agent to consummate itstask.We assume that the motivation of agent k = 1 , . . . , N isdetermined by the integer parameter L k = 1 , , . . . , L max and that at the beginning of the day, the agents’ motiva-tions are the lowest, i.e., L k = 1. In addition, we assumethat the instantaneous productivity of an agent at time t = 0 , , . . . , T is P k ( t ) = L k − k is not talkingand P k ( t ) = 0 if agent k is talking. Since we measuretime in minutes, we set T = 480. Hence, at time t = 0the instantaneous productivity is zero for all agents.The only way an agent can increase its motivation and,consequently, its productivity, is by engaging in socialinteraction. We introduce two different social stereotypesour agents can be, viz., extroverts and introverts, whichdetermine their propensities to seek and engage in socialinteraction. In particular, if agent k has stereotype l , a r X i v : . [ c s . M A ] A ug where l = e for extroverts and l = i for introverts, theprobability that it instigates a conversation is p lk = (cid:0) T /τ l − L k (cid:1) / (cid:0) T /τ l − (cid:1) (1)for L k ≤ T /τ l and p lk = 0, otherwise. Here τ e and τ i are parameters measured in minutes that are necessaryto make eq. (1) dimensionally correct. Throughout thispaper we set τ e = 1 without loss of generality, but set τ i > p ek ≥ p ik , i.e., that theextroverts are more likely to engage in social interactionthan the introverts. In addition, p ek = p ik = 1 for L k = 1so that when the motivation of agent k is at the bot-tom line, it will try to engage in social interaction withcertainty. The propensity to instigate a conversation de-creases linearly with the motivation to work until it van-ishes altogether when the motivation parameter reaches athreshold value that differs for the two social stereotypes.Regardless of its stereotype, once a target agent de-cides to instigate a conversation, it selects a number m of contact attempts, where m = 0 , , . . . is a random vari-able drawn from a Poisson distribution of parameter q . Ineach contact attempt, a peer is selected at random amongthe N − d is givenby a random integer selected uniformly in { , ..., D } . Wenote that a conversation involves two agents only and theagent that is approached by the target agent is obligedto accept the interaction, regardless of its motivation andstereotype.To complete the model we need to give a prescrip-tion for changing the motivation parameter L k with k = 1 , . . . , N , which is the central factor in the deter-mination of the social behavior of the agents. We as-sume simply that the motivation of agent k increases byone unit for each minute it spends in a conversation anddecreases by one unit for each minute it spends workingalone. Moreover, we set L k = 1 as the lower bound of themotivation parameter so that a lone agent k at the mo-tivation bottom line L k = 1 cannot have its motivationreduced any further.We note that our prescription to change the motiva-tion of the agents implies that the motivation parameteris bounded from above by L max = 480, so that the proba-bility that an extroverted agent instigates a conversation,eq. (1), is never zero during the eight-hour workday. Thisproduces a qualitative distinction between the two stereo-types since the extroverts are, in principle, always willingto initiate a social interaction (i.e., p ek > T /τ i . Throughoutthis paper we set τ i = 5 but note that this choice is im-material, provided that T /τ i is sufficiently distinct fromthe upper bound T /τ e to justify the existence of twostereotypes.In summary, we implement the synchronous or parallelupdate of the N agents as follows. At t = 0 we set L k = 1 and P k = 0 for all agents k = 1 , . . . , N . The updateprocedure at time t begins with the selection of a randomorder of update of the N agents, so that at the end of theprocedure all agents are updated and we can incrementthe time from t to t + 1. Then we check the status of theagent to be updated – the target agent – to determine ifit is participating in a conversation or not. In case it is,we increment its motivation parameter by one unit. Incase it is not, we decrement its motivation parameter byone unit and test whether it initiates a conversation ornot using the probabilities given in eq. (1) in accord withthe stereotype of the target agent. The simulation endsat t = T . Results. – We consider working groups composedof N e extroverts and N i = N − N e introverts and fo-cus mainly on the mean cumulative productivity of thestereotypes at time t = 1 , . . . , T ,Π l ( t ) = 1 N l (cid:88) k ∈S l π k ( t ) (2)where π k ( t ) = 1 t t (cid:88) t (cid:48) =0 P k ( t (cid:48) ) (3)and the superscript l = e, i specifies the social stereotypesof the agents, as before. Here the sum over k is restrictedto the subset S l of agents with stereotype l , whose car-dinality is N l . We recall that P k (0) = 0 for all k . Themean cumulative productivity of the whole group is sim-ply Π w = (cid:0) N e Π e + N i Π i (cid:1) /N , where we have omittedthe dependence on t for simplicity. It is also of interest toknow the mean motivation of the stereotype subgroups,Λ l ( t ) = 1 N l (cid:88) k ∈S l L k ( t ) (4)with l = e, i as before. Then the mean motivation of thewhole group is Λ w = (cid:0) N e Λ e + N i Λ i (cid:1) /N .Our main interest here is Π l ( t ) and Λ l ( t ) evaluated atthe end of the day, i.e, at t = T . These quantities areaveraged over 10 independent runs for each setting ofthe model parameters. In particular, in the following wevary the fraction of extroverts η = N e /N and the meannumber of contact attempts q , but fix the group size to N = 100 and the maximum duration of a conversationto D = 20. As already pointed out, we set τ e = 1 and τ i = 5.To better appreciate the assumptions of the model, fig.1 shows typical time dependences of the motivation andmean cumulative productivity of two agents with distinctstereotypes for a single run. The time intervals wherethe mean cumulative productivity decreases correspondto the periods when the agent is participating in a socialinteraction and are associated with the increase of its mo-tivation. We note that, whereas the cumulative produc-tivity is a non-decreasing function of t , the mean cumu-lative productivity, eq. (3), decreases with increasing t in e i L t ei π t FIG. 1. Motivation (upper panel) and mean cumulative pro-ductivity (lower panel) of typical extrovert ( e ) and introvert( i ) agents for a group of size N = 100 with equal ratio of ex-troverts and introverts, i.e., η = 0 .
5. The maximum durationof a conversation is D = 20 and the mean number of contactattempts is q = 2. the time intervals where the instantaneous productivityof the agent is zero. In addition, the motivation decreasesin the periods where the productivity increases. Most ofthe time in this run, the introvert agent exhibits a highermean productivity than the extrovert agent, despite itslower motivation. In fact, the motivation parameter ofthe introvert stabilizes and fluctuates around the value L = T /τ i = 96, whereas the motivation parameter of theextrovert shows a tendency to increase linearly with t onaverage. It is easy to understand the behavior patternof the introvert’s motivation if we note that an introvertagent with L ≥ T /τ i will not attempt to instigate anyconversation and so the tendency of its motivation is todecrease only, except for the events when another agentengages it in a conversation.Figure 2 shows the influence of the fraction of extro-verts η on the motivation and mean cumulative produc-tivity of the agents at the end of the day. This figurereveals some interesting features. First and foremost,the insertion of extroverts in groups composed predomi-nantly of introverts results in a significant increase of themean productivity of the introverts, whereas the meanproductivity of the extroverts is barely changed. Toomany extroverts, however, cause a decrease on the pro-ductivity of both stereotypes, although they considerablyboost the motivation of the introverts. In addition, thereis an optimum group composition ( η ≈ . q on the optimum compositionof the group. For instance, for small q , say q = 0 .
5, itis difficult to find conversation partners to increase theagents’ motivations and so the highest total productivityis achieved by all-extroverts groups, since this stereotype
50 100 150 200 250 300 350 0 0.5 1 e iw Λ η
34 35 36 37 38 39 40 41 0 0.5 1 ei w Π η FIG. 2. Effect of the fraction of extroverts on the mean moti-vation (left panel) and on the mean cumulative productivity(right panel) at the end of the day of the subgroups of extro-verts ( e ) and introverts ( i ), and of the whole group ( w ). Thetotal number of agents is N = 100, the maximum durationof a conversation is D = 20 and the mean number of contactattempts is q = 2.
26 28 30 32 34 36 38 40 42 0 0.2 0.4 0.6 0.8 10.51241000 Π w η FIG. 3. Mean cumulative productivity of the whole group atthe end of the day as function of the fraction of extroverts formean number of contact attempts q = 0 . , , , N = 100 and themaximum duration of a conversation is D = 20. seeks for partners more frequently than the introverts.As q increases, the odds of finding a conversation part-ner increases as well, and so it becomes advantageous toinsert a few introverts in the group. For very large q , sothat the target agent attempts to contact all the otheragents in the group, the maximum total productivity oc-curs for η ≈ .
47. For groups composed predominantlyof introverts ( η < . q and η lead always tohigher total productivity. However, the scenario is morecomplicated for groups composed predominantly of ex-troverts. In fact, for η > . q = 1and q = 2 for η ≈ q in more detail. In the context of the 2020 coro-navirus pandemic, this is the leading parameter of themodel since, at least for those adhering to social dis-tancing norms, the mean number of attempts to interactsocially was considerably curtailed during the pandemic.Accordingly, fig. 4 shows that a moderate decrease of q actually increases the productivity of the group for η > .
5. For instance, for η > . q = 6 to q = 1 results in a productivity increase of about10%, which is in agreement with the experiments of ref.[5]. We note that in both our simulations and those ex-periments the boost in productivity is modest, thoughaltogether unexpected.
30 32 34 36 38 40 0 1 2 3 4 5 60.90.80.70.60.5 Π w q FIG. 4. Effect of the mean number of attempts to establishsocial interaction on the mean cumulative productivity of thewhole group at the end of the day for the fraction of extro-verts η = 0 . , . , . , . .
9, as indicated. The totalnumber of agents is N = 100 and the maximum duration ofa conversation is D = 20. It is interesting to look at how the productivities ofthe two stereotypes within a same working group are af-fected by the social restrictions. Figure 5 shows that, for η = 0 .
5, a moderate decrease of q reduces the motiva-tion of both stereotypes as well as the mean productivityof the introverts, but increases the mean productivity ofthe extroverts. This finding may explain sporadic per-sonal reports of heightening productivity in the quaran-tine. Further decrease of q leading to a scenario wheresocial interactions happen very rarely results in a sharpdrop of the mean productivity of both stereotypes.A word is in order about the effects of the parameters N and D that determine the group size and the maxi- mum duration of a conversation, respectively. Reducingthe group size has no effect on our results and our choiceof fairly large groups ( N = 100) is so as to smooth outthe variation of the fraction of extroverts η . However,increase of the parameter D has a strong effect on theproductivity of the extroverts. For instance, for longconversations, say D = 60, all-introverts groups yieldthe highest total productivity regardless of the number ewi Λ q ei w Π q FIG. 5. Effect of the mean number of attempts to establishsocial interaction on the mean motivation (left panel) and onthe mean cumulative productivity (right panel) at the end ofthe day of the subgroups of extroverts ( e ) and introverts ( i ),and of the whole group ( w ). The total number of agents is N = 100, the maximum duration of a conversation is D = 20and the fraction of extroverts is η = 0 . of contact attempts. Our choice D = 20, which corre-sponds to conversations of average duration ¯ d = 10, aremeant to model a “water cooler talk” scenario. Conclusion. – In the spirit of sociophysics [9, 10] andcomputational social science [11], our agent-based modelassumes only linear relationships between the quantitiesrelevant to study the influence of social restriction mea-sures on the productivity of working groups. In thatsense, it offers a proof of concept that, somewhat para-doxically, quarantining and social distancing may boostthe productivity of extroverted people.* * *The research of JFF was supported in part by GrantNo. 2020/03041-3, Funda¸c˜ao de Amparo `a Pesquisa doEstado de S˜ao Paulo (FAPESP) and by Grant No.305058/2017-7, Conselho Nacional de DesenvolvimentoCient´ıfico e Tecnol´ogico (CNPq). [1] Edgeworth M.,
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