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Theory and Decision

Erschienen in:

29.09.2016

Pool size and the sustainability of optimal risk-sharing agreements

verfasst von: Francesca Barigozzi, Renaud Bourlès, Dominique Henriet, Giuseppe Pignataro

Erschienen in: Theory and Decision | Ausgabe 2/2017

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Abstract

We study a risk-sharing agreement where members exert a loss-mitigating action which decreases the amount of reimbursements to be paid in the pool. The action is costly and members tend to free-ride on it. An optimal risk-sharing agreement maximizes the expected utility of a representative member with respect to both the coverage and the (collective) action such that efficiency is restored. We study the sustainability of the optimal agreement as equilibrium in a repeated game with indefinite number of repetitions. When the optimal agreement is not enforceable, the equilibrium with free-riding emerges. We identify an interesting trade-off: welfare generated by the optimal risk-sharing agreement increases with the size of the pool, but at the same time the pool size must not be too large for collective choices to be self-enforcing. This generates a discontinuous effect of pool size on welfare.

Vorheriger Artikel The efficiency of crackdowns: a lab-in-the-field experiment in public transportations

Nächster Artikel A note on a recent paper by Dagsvik on IIA and random utilities

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We refer the reader to footnote 4 and to the end of Sect. 8 for a discussion about the consequences of relaxing the assumption of uniform wealth in the pool.

For a brief overview see http://www.igpandi.org/Group+Agreements/The+Pooling+Agreement, consulted in November 2015.

Contracts with fixed premium are typically offered by traditional for profit companies and correspond to policies where the risk is transferred, in full or in part, to the insurer.

The specification of identical wealth for all individuals simplifies the analysis. It allows us to focus on agents with identical risk perceptions. Heterogeneity would call for further assumptions on risk aversion (whether it is increasing or decreasing in wealth), on the collective objective of the pool (how the agreement aggregates utility of different members) and on the negotiation process. In the conclusion we provide a discussion on how relaxing such assumption would affect the analysis.

As an example, suppose that individuals are facing the risk of illness. Prevention is the loss-reducing action implying an ex ante utility cost and having a benefit in terms of a lower health shock in the case of illness. In Ehrlich and Becker (1972), a consumer’s action decreasing the amount of a possible loss is called a self-insurance measure.

Clearly, with a non-linear contract paying \(-\infty \) if the loss is higher than the one associated with the desired action, the efficient action would be implementable.

This represents a major difference with respect to the standard insurance policy with fixed premium that we analyze in Appendix 9.5.

In this sense this is also the symmetric Pareto efficient agreement.

From an analytical point of view, full coverage is not optimal anymore in the risk-sharing agreement with free-riding because the Envelope Theorem cannot be applied in this case. See, in particular, Eqs. (14) and (27) in Appendix 9.1 and 9.4, respectively.

Lee and Ligon (2001) analyze a risk-sharing agreement where a non-contractible self-protection action is available to policyholders (i.e., in the case of an action that decreases the probability of the loss). Using “Cournot conjectures” they show that full coverage is optimal. Our result differs from theirs since we solve the problem using the concept of Nash equilibrium. This implies that, when choosing the reimbursement level q in the first stage, individuals anticipate the effect of such a coverage on the choice of other members of the pool.

In fact, under full coverage \(\left( q=1\right) ,\) we would observe a positive action in the risk-sharing agreement with free-riding (as mentioned below Eq. 6), whereas we would observe no-action in the case of a linear policy with fixed contribution.

Simulations’ files are available upon request.

Simulations with CRRA are available upon request to the authors.

We will study the sustainability of the optimal risk-sharing agreement with respect to individual deviations by members of the pool. Hence we do not study a cooperative game which would instead require the specification of a value function for each possible coalition of agents and the analysis of collective deviations.

Note that, by focusing on full commitment on the policy, our model is mostly suited to describe formal risk-sharing agreements. See Sect. 7 for a real-world example.

In this sense the number of periods over which the game is repeated is indefinite (see Roth and Murnighan 1978).

See Neyman (1999) for a model in which uncertainty on the duration of the game is interpreted as a small departure from the common knowledge assumption on the number of repetitions.

Such a Grim trigger strategy perfectly fits a non-competitive environment, where there is no choice for individuals except that of staying in the pool. We refer the reader to the end of this section for a discussion about exclusion as a punishment strategy, and about punishment when an outside option exists, for example, because profit and nonprofit organizations compete in the market.

With respect to the setting analyzed in Friedman (1971), our model has two specific features: (a) players do not directly observe the action of other players but they infer deviation through the contribution required for membership and from their knowledge of the number of claims; (b) detection only occurs if the loss of the deviator realizes, that is with probability p.

In Fig. 2, the graphs showing the left and the right-hand side of (10) are not represented for the pool size \(n=1\) because of the scale of the picture.

Green and Porter (1984) examine the nature of cartel self-enforcement in the presence of demand uncertainty. In particular, in their setting, demand fluctuations (that are not directly observed by firms) make the detection of deviation difficult to infer. The collusive equilibria are then less likely and unstable industry performances can occur. Reversionary episodes, where price cut is performed by all firms in the cartel as a punishment strategy, can sometimes happen with no firm really defecting, simply because of low demand.

This is possible, for example, if the pool reveals to its members the list of individuals entitled for reimbursement, together with the specific amount paid to each of them.

Notice that, after the exclusion of the deviator, the pool will be composed by \(n-1\) members, so that sustainability of the optimal agreement will be more likely.

One of the objectives of the Affordable Care Act, recently approved in the USA, is to protect policyholders from the insurers’ practice of refusing policy renewal in case of serious health conditions.

Interestingly, long-term contracts are instead offered by standard companies in other insurance markets. For example, front-loaded contracts in life insurance generate a partial lock-in of consumers: contracts that are more front-loaded have a lower present value of premiums over the period of coverage (see Hendel and Lizzeri 2003).

These simulations are available upon request to the authors.

See the Global Mutual Market Share report 2010, available at http://www.icmif.org/mms2010.

Similarly, Kerleau (2009) shows that the market for mutual contracts is characterized by low concentration in France, as the 5 biggest mutuals in 2005 represented only 20 % of the market share and the 30 biggest ones only 44 %.

The beneficial matching between agents characterized by a similar “mission” (or social attitude) has been analyzed by Besley and Ghatak (2005). They show that fewer incentives are required if employer and employee share the same mission.

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Titel: Pool size and the sustainability of optimal risk-sharing agreements
verfasst von: Francesca Barigozzi
Renaud Bourlès
Dominique Henriet
Giuseppe Pignataro
Publikationsdatum: 29.09.2016
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 2/2017
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-016-9573-9

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