# Networks, Games and Risk

Decentralized risk-sharing markets are markets for risk exchange in which a pool of individuals agree to mutually insurer each other, without recourse to a centralized insurance provider. Some important problems to examine in these markets are the following:

- The coalitional stability of the pool, or the formation of risk-sharing networks (subcoalitions) within the pool.
- The Pareto-efficiency of allocations along risk-sharing networks.
- The structure of allocation mechanisms within networks, that is, the mappings that transform feasible allocations into other feasible allocations within networks.

Examining these problems requires an interdisciplinary approach, drawing from economic theory, insurance and actuarial science, game theory, and related fields of applications. It is the aim of this workshop to bring together researchers from various fields, to discuss open problems in the theory of decentralized risk-sharing along networks, as well as potential interdisciplinary approaches to tackle these problems.

**Monday, December 18th, 2023** (Montréal, QC)

The event will be free, but registration is mandatory.

## Agenda

Time | Speaker | Title |
---|---|---|

8:30-9:00 | registration | |

9:00-10:30 | Renaud Bourles |
Altruism in networks |

10:30-10:45 | coffee | |

10:45-11:45 | Federico Bobbio |
Dynamic fundamentals in the Stable Matching |

11:45-13:00 | lunch | |

13:00-14:00 | Vincent Boucher |
The sources of homophily in social networks |

14:00-15:00 | Leonie Baumann |
Strategic Evidence Disclosure in Networks and Equilibrium Discrimination |

15:00-15:15 | coffee | |

15:15-16:00 | Fallou Niakh |
Risk sharing and Taxation |

16:00-16:45 | Philipp Ratz |
Convex Order and Linear Risk Sharing on Networks |

## Location

## Speakers and Abstract

**Renaud Bourles** (Centrale Marseille, Aix-Marseille School of Economics, and Institut Universitaire de France, France)

Altruism in Networks: inequality, risk-sharing and public goods

“We provide the first analysis of the risk-sharing implications of altruism networks. Agents are embedded in a fixed network and care about each other. We explore whether altruistic transfers help smooth consumption and how this depends on the shape of the network. We find that altruism networks have a first-order impact on risk. Altruistic transfers generate efficient insurance when the network of perfect altruistic ties is strongly connected.We uncover two specific empirical implications of altruism networks. First, bridges can generate good overall risk sharing, and, more generally, the quality of informal insurance depends on the average path length of the network. Second, large shocks are well-insured by connected altruism networks. By contrast, large shocks tend to be badly insured in models of informal insurance with frictions.We characterize what happens for shocks that leave the structure of giving relationships unchanged. We further explore the relationship between consumption variance and centrality, correlation in consumption streams across agents, and the impact of adding links.”

**Vincent Boucher** (Université Laval, Québec, Canada)

The sources of homophily in social networks

“We explore the sources of homophily (meetings vs preferences) in social networks. Using variation in meeting opportunities, we show that a large part of the realized homophily is due to the meeting probabilities. Public policies highlight a tradeoff between the homophily and density of the equilibrium network”

**Federico Bobbio** (Université de Montréal, Montréal, Canada)

Dynamic fundamentals in the Stable Matching

“We address the dynamic facets in the primitives of the many-to-one stable matching problem. This problem is based on a bipartite graph where vertices on opposite sides are connected via edges. We further enrich each vertex with an ordinal ranking (i.e., preference list) over the incident edges. We also assume that each vertex has a capacity, i.e., a maximum number of vertices on the other side to which it can be matched. Thus, the primitives in this framework are the preferences of vertices and the capacities. Our investigation sees as a natural application the school choice problem and the hospital-resident matching. First, using the school choice model as a motivating example, we explore the joint optimization of increasing school capacities and achieving one-side-optimal stable matchings within an expanded market. We devise an innovative mathematical programming formulation that models stability and capacity expansion, and we develop an effective cutting-plane method to solve it. Real-world data from the Chilean school choice system validates the potential impact of capacity planning under stability conditions. We show that this problem is not immune from strategic manipulation, however, we also prove that in practice this manipulation is difficult to perform. Our findings reveal the NP-completeness of the decision problem of allocating optimally extra capacities, and the inapproximability of the ensuing optimization problem, even under strict preferences and disjoint capacity allocations. These results pose significant challenges for policymakers seeking efficient solutions to pressing real-world issues. Second, we delve into stable matching under dynamic priorities, primarily focusing on the school choice problem. We introduce a model that accounts for sibling priorities, necessitating novel stability concepts. Our research identifies scenarios where stable matchings exist, accompanied by polynomial-time mechanisms for their discovery. However, in some cases, we also prove the NP-hardness of finding a maximum cardinality stable matching under dynamic priorities, shedding light on challenges related to these matching problems. Collectively, this research contributes to a deeper understanding of dynamic capacities and priorities within stable matching scenarios and opens new questions and new avenues for tackling complex allocation challenges in real-world settings.”

**Leonie Baumann** (McGill University, Montréal, Canada)

Strategic Evidence Disclosure in Networks and Equilibrium Discrimination

“A group of agents with ex-ante independent and identically uncertain quality compete for a prize, awarded by a principal. Agents may possess evidence about the quality of those they share a social connection with (neighbours), and themselves. In one equilibrium, adversarial disclosure of evidence leads the principal to statistically discriminate between agents based on their number of neighbours (degree). We identify parameter values for which an agent’s ex-ante winning probability is monotone in degree. All equilibria that satisfy some robustness criteria lie between this adversarial disclosure equilibrium and a less informative one that features no snitching and no discrimination.”

**Fallou Niakh** (CREST, ENSAE, Institut Polytechnique de Paris, France)

Risk sharing and Taxation

“We consider an economy composed of regions that wish to be hedged against a disaster risk by using multi-region catastrophe insurance. This method disperses risk among high- and low-risk regions and ensures profitability for the insurance company. However, for natural risks, the insurer has a non-zero probability of insolvency. When a disaster occurs and the losses are below the coverage limit, then the regions are fully compensated by the insurer. On the other hand, if they exceed the coverage limit, we consider that the central government decides to share the residual claims among the regions in the form of taxation. In this study, we propose a theoretical framework for regional participation in collective risk-sharing through tax revenues by accounting for their disaster risk profiles and dependencies between them.”

**Philipp Ratz** (Université du Québec à Montréal, Montréal, Canada)

Convex Order and Linear Risk Sharing on Networks

“The peer-to-peer (P2P) economy has been growing with the advent of the Internet, with well known brands such as Uber or Airbnb being examples thereof. In the insurance sector the approach is still in its infancy, but some companies have started to explore P2P-based collaborative insurance products (eg. Lemonade in the U.S. or Inspeer in France). The actuarial literature only recently started to consider those risk sharing mechanisms, as in Denuit and Robert (2021) or Feng et al. (2021). In this paper, describe and analyse such a P2P product, with some reciprocal risk sharing contracts. Here, we consider the case where policyholders still have an insurance contract, but the first self-insurance layer, below the deductible, can be shared with friends. We study the impact of the shape of the network (through the distribution of degrees) on the risk reduction. We consider also some optimal setting of the reciprocal commitments, and discuss the introduction of contracts with friends of friends to mitigate some possible drawbacks of having people without enough connections to exchange risks.”