Evolutionary dynamics of collective index insurance

Abstract

Index-based insurances offer promising opportunities for climate-risk investments in developing countries. Indeed, contracts conditional on, e.g., weather or livestock indexes can be cheaper to set up than conventional indemnity-based insurances, while offering a safety net to vulnerable households, allowing them to eventually escape poverty traps. Moreover, transaction costs by insurance companies may be additionally reduced if contracts, instead of arranged with single households, are endorsed by collectives of households that bear the responsibility of managing the division of the insurance coverage by its members whenever the index is surpassed, allowing for additional flexibility in what concerns risk-sharing and also allowing insurance companies to avoid the costs associated with moral hazard. Here we resort to a population dynamics framework to investigate under which conditions household collectives may find collective index insurances attractive, when compared with individual index insurances. We assume risk sharing among the participants of each collective, and model collective action in terms of an N-person threshold game. Compared to less affordable individual index insurances, we show how collective index insurances lead to a coordination problem in which the adoption of index insurances may become the optimal decision, spreading index insurance coverage to the entire population. We further investigate the role of risk-averse and risk-prone behaviors, as well as the role of partial correlation between insurance coverage and actual loss of crops, and in which way these affect the original coordination thresholds.

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Acknowledgments

The authors would like to dedicate this paper to Mats Gyllenberg on the occasion of his 60th birthday. We also thank Avinash Dixit for his insightful comments. J.M.P. and F.C.S. acknowledge financial support from FCT-Portugal through Grants EXPL/EEI-SII/2556/2013, PTDC/MAT-STA/3358/2014, and PTDC/EEI-SII/5081/2014, and by multi-annual funding of CBMA-UM and INESC-ID (under the Projects UID/BIA/04050/2013 and UID/CEC/50021/2013) provided by FCT-Portugal. S.A.L. acknowledges financial support from National Science Foundation Grants EF-1137894, GEO-1211972 and Project “Green Growth Based on Marine Resources: Ecological and Socio-Economic Constraints (GreenMAR)”, funded by Nordforsk.

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Correspondence to Simon A. Levin.

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Appendix: When actual loss and insurance coverage do not fully overlap

Appendix: When actual loss and insurance coverage do not fully overlap

Here we follow (Clarke 2011) and investigate the effect of introducing two different probabilities instead of the risk r employed in the main text: p, the probability that harvest is lost; q, the probability that any index insurance is activated (as defined previously already in terms of the risk r).

Depending on which one of the four possible states we are in, namely :

  • (0, 0)—(no Loss, no insurance activated)

  • (0, I)—(no Loss, insurance activated)

  • (L, 0)—(Loss, no insurance activated)

  • (LI)—(Loss, insurance activated)

we can write down the following payoff tables for Cs and Ds:

Ds
State Probability Payoff
(0,0) \(p_{0,0}\) b
(0,I) \(p_{0,1}\) b
(L,0) \(p_{1,0}\) 0
(L,I) \(p_{1,1}\) 0
Cs
State Probability Payoff
   \(n_C \ge M\) \(n_C <M\)
(0,0) \(p_{0,0}\) \(b-c_c \) \(\gamma (b-c)+(1-\gamma )b\)
(0, I) \(p_{0,1}\) \(b(1+\alpha )-c_c\) \(\gamma \left[ {b(1+\alpha )-c} \right] +(1-\gamma )b\)
(L, 0) \(p_{1,0}\) \(-c_c \) \(\gamma (-c)\)
(LI) \(p_{1,1}\) \(\alpha b-c_c \) \(\gamma (\alpha b-c)\)

In the simplest scenario, we may assume that the probability p of occurrence of a loss is statistically independent from the probability q that the index I of the II is activated. Then we can write for the probabilities of occurrence of each of the 4 states:

$$\begin{aligned} p_{0,0}= & {} (1-p)\times (1-q)\\ p_{0,1}= & {} (1-p)\times q\\ p_{1,0}= & {} p\times (1-q)\\ p_{1,1}= & {} p\times q \end{aligned}$$

Such a statistical independence is unlikely, however. Thus, the joint probability distribution will not, in general, be the product of the 2 marginal probability distributions. Following (Clarke 2011), we investigate a symmetric joint probability distribution, defined in terms of a new parameter u, which corresponds to the probability that an individual will incur a Loss but without the II be activated (\(p_{1,0}\)). In terms of u, the probabilities now read:

$$\begin{aligned} \begin{aligned} p_{0,0}&= 1-q-u&p_{0,1}&= q-p+u \\ p_{1,0}&= u&p_{1,1}&= p-u \\ \end{aligned} \end{aligned}$$

Again, we can write for the fitness difference between Cs and Ds

$$\begin{aligned} f_C (x)-f_D (x) =\Delta +\left[ {\Omega -\Delta } \right] B_{N,M} (x) \end{aligned}$$

with \(\Delta =\gamma c\big (\frac{p_{0,1} +p_{1,1} }{m}-1\big )\)and \(\Omega -\Delta =c \big (\gamma +\frac{p_{0,1} +p_{1,1} }{m}-s_C \big )\).

The expressions for \(\Delta \) and \(\Omega -\Delta \)depend only on the combination \(p_{0,1} +p_{1,1} =q\), despite the fact that, individually, both \(p_{0,1} \) and \(p_{1,1}\) depend on p, q and u. Since \(p_{0,1} +p_{1,1} =q\), irrespective of whether the p and q distributions are independent or not, this means that the results we obtain for the original model also hold for this “extended model” provided that the original risk r is replaced by the probability q.

\(\square \)

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Pacheco, J.M., Santos, F.C. & Levin, S.A. Evolutionary dynamics of collective index insurance. J. Math. Biol. 72, 997–1010 (2016). https://doi.org/10.1007/s00285-015-0939-3

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Keywords

  • Index insurance
  • Collective action
  • Evolutionary game theory
  • Non-linear returns

Mathematics Subject Classification

  • 91-XX
  • 91A22
  • 91A06
  • 91A40
  • 91B18
  • 91B30