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Social Cost in Pandemic
as Expected Convex Loss
Rikiya Takahashi
(Rikiya.Takahashi@gmail.com)
*This material exhibits the author's personal opinion by mathematical
thinking, and does not reflect the views by the author's employer.
Summary
● Low reproduction number R0 calibrated under
homogeneous-agent assumption (e.g., SIR)
cannot justify relaxing lockdown.
● The social cost is expectation of some convex
function of #infections/deaths, integrated over a
power-law (often heavy-tailed) distribution.
– Unstably diverging cost, unless tail is truncated
– Even with truncation, still unexpectedly high cost
● Public policy should focus on how to cut the tail,
e.g., identify & contain super spreaders, instead
of uniformly tightening/relaxing all people.
Outlines
● 1. Why #infections/deaths obey power-law
– Heterogeneity across people
● 2. Why power-law is problematic even if the mean
(average) is stable empirically and/or theoretically
– Convex cost similar to diverging high-order moment
● Conclusion: risk mitigation policy should focus on
cutting the tail rather than uniform control
1. Heterogeneity to Power-Law
Marginal is not the distribution by avg. parameter
● Let Y be the number of additionally infected
persons by one spreader whose rate is θ.
● Marginal probability distribution of Y is not the
distribution by “representative spreader”, i.e.,
● What is worse: marginal implies more catastrophe!
– For risk management, you must never use the
distribution by a representative person.
∫ p(Y∣θ) p(θ)d θ≠ p(Y∣E[θ])
1. Heterogeneity to Power-Law
Parametric Example
● Each spreader has a thin-tailed exponential
distribution
– For simplicity here we forget that Y is an integer.
● Heterogeneity across spreaders is given by a
gamma distribution
● Then marginal is a Pareto distribution that is
heavy-tailed when α<=2
● By contrast, distribution by representative is
still thin-tailed exponential
p(Θ=θ)=Gamma(θ;α ,β)≡
βα
Γ(α)
θ
α−1
exp(−βθ)
p(Y = y∣Θ)=exp( y;Θ)≡Θexp(−Θ y)
p(Y = y)= α
β (1+ y/β)
−α−1
q(Y = y)=exp(y ; α
β
)
1. Power-Law: Stupidity of Watching
Representative in Crisis
● Interestingly, median of the marginal is lower
than representative rate. But let's see >99%-tiles...
– E.g., when α=1.5 and β=1.5 (average spread=1)
● Model forecast by a calibrated parameter (e.g.,
median=0.39) with homogeneous model leads
wrong sense of safety and underestimation of
crisis, due to wrong extrapolation in the tails.
Median 99%-tile 99.9%-tile
Pareto Marginal 0.39 13.9 67.7
Representative Exponential 0.69 4.6 6.91
1. Heterogeneity to Power-Law
Related Work
● Heterogeneity of spreaders as a stylised fact
– E.g., negative-binomial + branching process
(Lloyd-Smith+, 2005))
● Pareto distribution of death toll, whose
exponent leads even infinite mean unless
truncated (Cirillo & Taleb, 2020)
N. Becker and I. Marschner, The effect of heterogeneity on the spread of disease, Stochastic
Processes in Epidemic Theory, 90-103, 1990.
J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp, and W. M. Getz, Superspreading and the effect
of individual variation on disease emergence, Nature, 438, 355–359, 2005.
P. Cirillo and N. N. Taleb, Tail Risk of Contagious Diseases, to appear in Nature Physics, 2020.
2. Not Linear but Convex Cost
● Some still say “Heavy-tail, so what? The
average death rate is small in reported stats.”
– In the Pareto example, mean is just ~2.0.
● Error #1: when heavy-tailed, empirical mean is
highly erroneous as the estimate of true mean.
● Error #2 (more essential): what the society pays
is not the mean but expected convex loss.
– Not constant but increasing cost per infection
● Fatal: similarly to calibration of high-order
moments, the cost can be unexpectedly large
even if the tail is truncated at entire population.
2. Not Linear but Convex Cost
Considerable Example Functions
● Piecewise-linear
– Higher death rate
when exceeding the
maximum capacity
● Power function
– Huge patients spoil
the work efficiency of
essential workers
2. Not Linear but Convex Cost
Diverging High-Order Moments
● Example: Y = # of infections obeys a Zeta
distribution and cost is quadratic
– When α=2.5, mean exists as E[Y]=1.947 but E[Y2
] is infinite
– Finiteness requires (Power exponent) – (Moment order) >1.
● Monte Carlo example in action
– Mean is well behaving and converging, but
– Avg[Y2
] diverges as sample size increases !
p(Y )∝1/Y α
N=1,000 N=10,000 N=100,000 N=1,000,000
Empirical Avg of Y 1.82 2.17 1.98 1.95
Empirical Avg. of Y2
1.01x103
2.29x105
5.74x105
2.77x106
Conclusion
● If people behaviours are too heterogeneous,
number of additionally infected persons obeys a
power-law distribution, potentially with heavy tail.
● Due to the capacity constraint in hospital and
essential workers, the society must pay a convex
cost w.r.t. #infections and/or deaths.
● Convex cost over power-law distribution can be
catastrophic. Cutting the tail, with a strong focus
on heterogeneity across people, is essential.
– Prohibition of large-population gathering, travel
restriction on too-active or gregarious persons, etc.
Ref1. More Accurate Model of p(Y)
● If you want Y to be strictly an integer, then we
can consider beta-binomial distribution.
– Let n be a large fixed number, say n=10,000
– Each spreader has a binomial distribution
– Heterogeneity across spreaders is given as
– Then we get the beta-binomial marginal as
p(Y = y∣Θ)=Bi( y ;Θ)≡
n!
y!(n− y)!
Θ
y
(1−Θ)
n− y
p(Θ=θ)=Beta(θ;α , p0)≡
Γ(α)
Γ(α p0)Γ(α(1− p0))
θ
α p0
(1−θ)
α(1− p0)
p(Y = y)=
Γ(α)Γ(n+1)
Γ(α+n)
⋅
Γ( y+α p0)
Γ(y+1)Γ(α p0)
Γ(n− y+α(1− p0))
Γ(n− y+1)Γ(α(1− p0))
Ref1. More Accurate Model of p(Y)
● Beta-binomial behaves likes a truncated power-
law distribution
– By using Gautschi's inequality,
– Approximately
– When α<1, it behaves like a truncated power-law
distribution whose exponent is
● Corresponding to the case that majority of people do not
spread virus but some people are super spreaders.
C (α , p0)≡Γ(α p0)Γ(α(1− p0))
Γ(α+n)
Γ(α)
( y+1)
α p0−1
(n− y+1)
α(1− p0)−1
<C (α , p0) p(Y = y)< y
α p0−1
(n− y)
α(1− p0)−1
p(Y = y)∝
{ y
−(1−α p0)
,if y≪n
(n−y)
−(1−α(1− p0))
,if y≃n
1−α p0>0
Ref1. More Accurate Model of p(Y)
● Less value of α leads more catastrophe
Rare but
very-high-
prob. People
when α=1
P.d.f. & c.d.f. When avg. infection prob. p0 = 10%
Ref2. Equivalence with
Forced Selling of Options
● Decomposition of convex power function
as an interpolation of piecewise linear functions
● So the cost estimation is regarded as option
pricing under power-law with many strike prices
– Unfortunately we cannot buy this option portfolio.
We are forced to be only in the seller side.
= + + +...

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Social Cost in Pandemic as Expected Convex Loss

  • 1. Social Cost in Pandemic as Expected Convex Loss Rikiya Takahashi (Rikiya.Takahashi@gmail.com) *This material exhibits the author's personal opinion by mathematical thinking, and does not reflect the views by the author's employer.
  • 2. Summary ● Low reproduction number R0 calibrated under homogeneous-agent assumption (e.g., SIR) cannot justify relaxing lockdown. ● The social cost is expectation of some convex function of #infections/deaths, integrated over a power-law (often heavy-tailed) distribution. – Unstably diverging cost, unless tail is truncated – Even with truncation, still unexpectedly high cost ● Public policy should focus on how to cut the tail, e.g., identify & contain super spreaders, instead of uniformly tightening/relaxing all people.
  • 3. Outlines ● 1. Why #infections/deaths obey power-law – Heterogeneity across people ● 2. Why power-law is problematic even if the mean (average) is stable empirically and/or theoretically – Convex cost similar to diverging high-order moment ● Conclusion: risk mitigation policy should focus on cutting the tail rather than uniform control
  • 4. 1. Heterogeneity to Power-Law Marginal is not the distribution by avg. parameter ● Let Y be the number of additionally infected persons by one spreader whose rate is θ. ● Marginal probability distribution of Y is not the distribution by “representative spreader”, i.e., ● What is worse: marginal implies more catastrophe! – For risk management, you must never use the distribution by a representative person. ∫ p(Y∣θ) p(θ)d θ≠ p(Y∣E[θ])
  • 5. 1. Heterogeneity to Power-Law Parametric Example ● Each spreader has a thin-tailed exponential distribution – For simplicity here we forget that Y is an integer. ● Heterogeneity across spreaders is given by a gamma distribution ● Then marginal is a Pareto distribution that is heavy-tailed when α<=2 ● By contrast, distribution by representative is still thin-tailed exponential p(Θ=θ)=Gamma(θ;α ,β)≡ βα Γ(α) θ α−1 exp(−βθ) p(Y = y∣Θ)=exp( y;Θ)≡Θexp(−Θ y) p(Y = y)= α β (1+ y/β) −α−1 q(Y = y)=exp(y ; α β )
  • 6. 1. Power-Law: Stupidity of Watching Representative in Crisis ● Interestingly, median of the marginal is lower than representative rate. But let's see >99%-tiles... – E.g., when α=1.5 and β=1.5 (average spread=1) ● Model forecast by a calibrated parameter (e.g., median=0.39) with homogeneous model leads wrong sense of safety and underestimation of crisis, due to wrong extrapolation in the tails. Median 99%-tile 99.9%-tile Pareto Marginal 0.39 13.9 67.7 Representative Exponential 0.69 4.6 6.91
  • 7. 1. Heterogeneity to Power-Law Related Work ● Heterogeneity of spreaders as a stylised fact – E.g., negative-binomial + branching process (Lloyd-Smith+, 2005)) ● Pareto distribution of death toll, whose exponent leads even infinite mean unless truncated (Cirillo & Taleb, 2020) N. Becker and I. Marschner, The effect of heterogeneity on the spread of disease, Stochastic Processes in Epidemic Theory, 90-103, 1990. J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp, and W. M. Getz, Superspreading and the effect of individual variation on disease emergence, Nature, 438, 355–359, 2005. P. Cirillo and N. N. Taleb, Tail Risk of Contagious Diseases, to appear in Nature Physics, 2020.
  • 8. 2. Not Linear but Convex Cost ● Some still say “Heavy-tail, so what? The average death rate is small in reported stats.” – In the Pareto example, mean is just ~2.0. ● Error #1: when heavy-tailed, empirical mean is highly erroneous as the estimate of true mean. ● Error #2 (more essential): what the society pays is not the mean but expected convex loss. – Not constant but increasing cost per infection ● Fatal: similarly to calibration of high-order moments, the cost can be unexpectedly large even if the tail is truncated at entire population.
  • 9. 2. Not Linear but Convex Cost Considerable Example Functions ● Piecewise-linear – Higher death rate when exceeding the maximum capacity ● Power function – Huge patients spoil the work efficiency of essential workers
  • 10. 2. Not Linear but Convex Cost Diverging High-Order Moments ● Example: Y = # of infections obeys a Zeta distribution and cost is quadratic – When α=2.5, mean exists as E[Y]=1.947 but E[Y2 ] is infinite – Finiteness requires (Power exponent) – (Moment order) >1. ● Monte Carlo example in action – Mean is well behaving and converging, but – Avg[Y2 ] diverges as sample size increases ! p(Y )∝1/Y α N=1,000 N=10,000 N=100,000 N=1,000,000 Empirical Avg of Y 1.82 2.17 1.98 1.95 Empirical Avg. of Y2 1.01x103 2.29x105 5.74x105 2.77x106
  • 11. Conclusion ● If people behaviours are too heterogeneous, number of additionally infected persons obeys a power-law distribution, potentially with heavy tail. ● Due to the capacity constraint in hospital and essential workers, the society must pay a convex cost w.r.t. #infections and/or deaths. ● Convex cost over power-law distribution can be catastrophic. Cutting the tail, with a strong focus on heterogeneity across people, is essential. – Prohibition of large-population gathering, travel restriction on too-active or gregarious persons, etc.
  • 12. Ref1. More Accurate Model of p(Y) ● If you want Y to be strictly an integer, then we can consider beta-binomial distribution. – Let n be a large fixed number, say n=10,000 – Each spreader has a binomial distribution – Heterogeneity across spreaders is given as – Then we get the beta-binomial marginal as p(Y = y∣Θ)=Bi( y ;Θ)≡ n! y!(n− y)! Θ y (1−Θ) n− y p(Θ=θ)=Beta(θ;α , p0)≡ Γ(α) Γ(α p0)Γ(α(1− p0)) θ α p0 (1−θ) α(1− p0) p(Y = y)= Γ(α)Γ(n+1) Γ(α+n) ⋅ Γ( y+α p0) Γ(y+1)Γ(α p0) Γ(n− y+α(1− p0)) Γ(n− y+1)Γ(α(1− p0))
  • 13. Ref1. More Accurate Model of p(Y) ● Beta-binomial behaves likes a truncated power- law distribution – By using Gautschi's inequality, – Approximately – When α<1, it behaves like a truncated power-law distribution whose exponent is ● Corresponding to the case that majority of people do not spread virus but some people are super spreaders. C (α , p0)≡Γ(α p0)Γ(α(1− p0)) Γ(α+n) Γ(α) ( y+1) α p0−1 (n− y+1) α(1− p0)−1 <C (α , p0) p(Y = y)< y α p0−1 (n− y) α(1− p0)−1 p(Y = y)∝ { y −(1−α p0) ,if y≪n (n−y) −(1−α(1− p0)) ,if y≃n 1−α p0>0
  • 14. Ref1. More Accurate Model of p(Y) ● Less value of α leads more catastrophe Rare but very-high- prob. People when α=1 P.d.f. & c.d.f. When avg. infection prob. p0 = 10%
  • 15. Ref2. Equivalence with Forced Selling of Options ● Decomposition of convex power function as an interpolation of piecewise linear functions ● So the cost estimation is regarded as option pricing under power-law with many strike prices – Unfortunately we cannot buy this option portfolio. We are forced to be only in the seller side. = + + +...