CITE THIS NOTEBOOK: Self-organised prevention against adverse bias in risky decision making by Wataru Toyokawa. Wolfram Community MAR 31 2023.
ORIGINAL ARTICLE: Wataru Toyokawa, Wolfgang Gaissmaier, Conformist social learning leads to self-organised prevention against adverse bias in risky decision making eLife 11:e75308 (2022). https://doi.org/10.7554/eLife.75308.
GITHUB: https://github.com/WataruToyokawa/ToyokawaGaissmaier2021
ORIGINAL ARTICLE: Wataru Toyokawa, Wolfgang Gaissmaier, Conformist social learning leads to self-organised prevention against adverse bias in risky decision making eLife 11:e75308 (2022). https://doi.org/10.7554/eLife.75308.
GITHUB: https://github.com/WataruToyokawa/ToyokawaGaissmaier2021
Article Abstract: Given the ubiquity of potentially adverse behavioural bias owing to myopic trial-and-error learning, it seems paradoxical that improvements in decision-making performance through conformist social learning, a process widely considered to be bias amplification, still prevail in animal collective behaviour. Here we show, through model analyses and large-scale interactive behavioural experiments with 585 human subjects, that conformist influence can indeed promote favourable risk taking in repeated experience-based decision making, even though many individuals are systematically biased towards adverse risk aversion. Although strong positive feedback conferred by copying the majority’s behaviour could result in unfavourable informational cascades, our differential equation model of collective behavioural dynamics identified a key role for increasing exploration by negative feedback arising when a weak minority influence undermines the inherent behavioural bias. This ‘collective behavioural rescue’, emerging through coordination of positive and negative feedback, highlights a benefit of collective learning in a broader range of environmental conditions than previously assumed and resolves the ostensible paradox of adaptive collective behavioural flexibility under conformist influences.
Summary
Summary
We assumed a group of n individuals consisting of two different behavioural states: one choosing a safe alternative that contains S individuals; and the other choosing a risky alternative that contains R individuals (n = S + R). We also assumed that there were two different `inner belief’ states, labelled 1 and 2. Individuals who possessed the negative belief, labelled 1, prefer the safe option to the risky option, while those who possessed the positive belief 2 prefer the risky option. A per-capita probability of choice shift from one behavioural alternative to the other was denoted by p. In particular, the probability of shifting to the more preferable option is pH, while that of shifting to less preferable option is pL, where pH > pL.
We assume that the inner beliefs can change by choosing the risky alternative. We define that the per-capita probability of getting higher preference to the risky alternative is “e” (), that is, , and the rest of individuals who chose the risky alternative become a belief state 1, that is, . Assuming , we could approximate a situation where the risky alternative $R$ gives higher payoffs on average than does the safe alternative, and hence for those individuals who choose the risky alternative there is a higher chance to become the belief state 2 than to become state 1.
One possible interpretation of “e” is that it represents a risk premium of gambles. For example, assume a 1-safe-1-risky-armed bandit task with Gaussian noises for the risky arm. The larger the risk premium (i.e. a mean reward of risky option), the more people who chose the risky arm are expected to experience a larger reward than the safe payoff. Note that, if the shape of noise is skewed so that a large bonanza from the risky alternative is a rare event, there exist only a small fraction of risk-choosing-people who get the large payoff. Nevertheless, I assume that the large bonanza for those lucky gambler would induce the positive belief which lasts long, and hence the number of people having the positive belief accumulates.
In the following, I first replicate the hot stove effect by this model. Secondly, I will show how social influence on the choice probability (i.e. pL and pH) rescue the system from getting stuck in the safe alternative.
We assume that the inner beliefs can change by choosing the risky alternative. We define that the per-capita probability of getting higher preference to the risky alternative is “e” (
0≤e≤1
R2eR
R1(1-e)R
e>1/2
One possible interpretation of “e” is that it represents a risk premium of gambles. For example, assume a 1-safe-1-risky-armed bandit task with Gaussian noises for the risky arm. The larger the risk premium (i.e. a mean reward of risky option), the more people who chose the risky arm are expected to experience a larger reward than the safe payoff. Note that, if the shape of noise is skewed so that a large bonanza from the risky alternative is a rare event, there exist only a small fraction of risk-choosing-people who get the large payoff. Nevertheless, I assume that the large bonanza for those lucky gambler would induce the positive belief which lasts long, and hence the number of people having the positive belief accumulates.
In the following, I first replicate the hot stove effect by this model. Secondly, I will show how social influence on the choice probability (i.e. pL and pH) rescue the system from getting stuck in the safe alternative.
Individual Learning Only
Individual Learning Only
Equiliburium Analysis for the simplest case
Equiliburium Analysis for the simplest case
In[]:=
Simplify[Solve[pLS1+pHS2-pH(1-e)R-pLeR0&&(*dR/dt0*)pH(1-e)R-pLS10&&(*dS1/dt0*)pLeR-pHS20&&(*dS2/dt0*)R+S1+S2n&&SS1+S2,{S,R,S1,S2}]]
Out[]=
S,R-e+pHpL+e,S1,S2
n((-1+e)-e)
2
pH
2
pL
(pH+pL)((-1+e)pH-epL)
npHpL
2
pH
2
pH
2
pL
(-1+e)n
2
pH
(pH+pL)((-1+e)pH-epL)
en
2
pL
(pH+pL)(pH-epH+epL)
R* - S* and R*/S* (degree of risk-seeking)
Simplify-e+pHpL+e-Simplify-e+pHpL+e
npHpL
2
pH
2
pH
2
pL
n((-1+e)-e)
2
pH
2
pL
(pH+pL)((-1+e)pH-epL)
npHpL
2
pH
2
pH
2
pL
n((-1+e)-e)
2
pH
2
pL
(pH+pL)((-1+e)pH-epL)
-
n(pH-pL)((-1+e)pH+epL)
(pH+pL)((-1+e)pH-epL)
pHpL
-(-1+e)+e
2
pH
2
pL
The condition of risk-seeking (R* > S*)
Reduce->0&&1>pH>pL>0&&0<e≤1&&0<d≤1&&n>0,eReduce>1&&1>pH>pL>0&&0<e≤1&&0<d≤1&&n>0,eSolve-0,eSolve1,e
n(pH-pL)((-1+e)pH+epL)
(pH+pL)((-1+e)pH-epL)
pHpL
-(-1+e)+e
2
pH
2
pL
n(pH-pL)((-1+e)pH+epL)
(pH+pL)((-1+e)pH-epL)
pHpL
-(-1+e)+e
2
pH
2
pL
0<d≤1&&0<pL<1&&pL<pH<1&&n>0&&<e≤1
pH
pH+pL
0<d≤1&&n>0&&0<pL<1&&pL<pH<1&&<e≤1
pH
pH+pL
e
pH
pH+pL
e
pH
pH+pL
Plot (Fig. 4b; manipulatable)
Plot (Fig. 4b; manipulatable)
In[]:=
R[e_,pH_,pL_,n_]:=-e+pHpL+e;S[e_,pH_,pL_,n_]:=;Rseeking[e_,pH_,pL_,n_]:=-;RSeek[pH_,pL_]:=;Manipulate[Plot[Rseeking[e,pH,pL,30],{e,0,1},PlotLegendsAutomatic,PlotRangeAll],{{pH,0.5},0,1},{{pL,0.1},0,1},SaveDefinitions->True]
npHpL
2
pH
2
pH
2
pL
n((-1+e)-e)
2
pH
2
pL
(pH+pL)((-1+e)pH-epL)
n(pH-pL)((-1+e)pH+epL)
(pH+pL)((-1+e)pH-epL)
pH
pH+pL
Out[]=
Social Learning
Social Learning
simulation - non biased initial distribution
simulation - non biased initial distribution
pH=0.7;stepSize=1/50;length=300;FigSocialLearningSimplest=ParallelTable[slNDSolve=NDSolve[{NS[t]==Ns1[t]+Ns2[t],PR1[t]==(1-c)pL+cNR[t]^f/(NR[t]^f+NS[t]^f),PS1[t]==(1-c)pH+cNS[t]^f/(NR[t]^f+NS[t]^f),PR2[t]==(1-c)pH+cNR[t]^f/(NR[t]^f+NS[t]^f),PS2[t]==(1-c)pL+cNS[t]^f/(NR[t]^f+NS[t]^f),Ns1'[t]-PR1[t]Ns1[t]+PS1[t](1-e)NR[t],Ns2'[t]-PR2[t]Ns2[t]+PS2[t]eNR[t],NR'[t]==PR1[t]Ns1[t]-PS1[t](1-e)NR[t]+PR2[t]Ns2[t]-PS2[t]eNR[t],Ns1[0]5,Ns2[0]5,NR[0]10},{Ns1,Ns2,NR},{t,0,length}];{e,c,f,pL,Max[Flatten[Table[Evaluate[Ns1[t]+Ns2[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]],Min[Flatten[Table[Evaluate[Ns1[t]+Ns2[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]],Round[Max[Flatten[Table[Evaluate[Ns1[t]+Ns2[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]]-Min[Flatten[Table[Evaluate[Ns1[t]+Ns2[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]],0.001],Max[Flatten[Table[Evaluate[NR[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]],Min[Flatten[Table[Evaluate[NR[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]],Round[Max[Flatten[Table[Evaluate[NR[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]]-Min[Flatten[Table[Evaluate[NR[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]],0.001],Max[Flatten[Table[Evaluate[NR[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]]-Max[Flatten[Table[Evaluate[Ns1[t]+Ns2[t]/.slNDSolve],{t,Ceiling[2length/3],length}]]]},{e,1/2,1,stepSize},{c,0,3/4,1/4},{f,{1,2,10}},{pL,0.0,pH-stepSize,stepSize}];(*,{d,0,1,1/50},{l,0,1,1/50},{s,0,3/4,1/4},{f,1,3,1}];*)(*SetDirectory["~/Dropbox/wataru/papers/RiskySocialLearning/populationDynamics"]*)Export["FigSocialLearningSimplest.csv",Round[Flatten[FigSocialLearningSimplest,3],0.0001]]
Simulation - biased initial population distribution
Simulation - biased initial population distribution
In this simulation, the population starts from the biased state where the majority of people are choosing S.
Time evolution - biased initial distribution
Time evolution - biased initial distribution
Numerical simulation for Fig. 5
Numerical simulation for Fig. 5