Luis R. Izquierdo

Games with endogenous separation are repeated games played in a population where players have the option to leave their current partnerships and continue playing in newly formed partnerships. Arguably, most repeated interactions in real life fall into this category. We present a general framework to analyze equilibria in games with endogenous separation, with a focus on neutrally stable strategies, i.e., stable strategies that are resistant to direct invasion by any alternative strategy or small group of strategies. We provide a definition of neutral stability for these games that guarantees dynamic stability properties. We also introduce path-protecting strategies, which play a role similar to that of trigger strategies in standard (fixed-partnership) repeated games. We provide a constructive proof of existence and a Folk Theorem for path-protecting neutrally stable strategies.

Games with costly endogenous separation are repeated games where players have the option to leave their current partnership (with some cost) and keep on playing in a newly-formed partnership. Players can also be separated by exogenous factors. We study equilibria in these games. As a relevant case of application, cooperation in a repeated Prisoner's Dilemma is compromised in environments where individuals are free to leave their partners, since defectors can exploit cooperators and move on. Previous studies have explored diverse mechanisms to prevent defect-and-leave strategies from taking hold. We show that, for large enough separation costs, not only full cooperation, but actually any symmetric sequence of outcomes can be sustained as a (path-protecting) neutrally stable state, which is Lyapunov stable under the replicator dynamics.

Energy Communities (ECs) have emerged as a key instrument for promoting local renewable energy integration and citizen participation in the energy transition. While their economic performance largely depends on the ability to generate savings through self-consumption and internal energy trading, their long-term viability is strongly influenced by how these savings are distributed among heterogeneous participants. Despite extensive literature on cost allocation methods, there remains a lack of integrated, data-driven approaches that clearly disentangle the sources of savings in ECs and examine how different allocation methods perform under realistic operating conditions. This paper presents a simulation-based analytical framework to quantify the economic savings generated within Energy Communities and to analyse how a set of widely used cost allocation methods distribute these savings among participants. The approach explicitly separates savings due to renewable self-consumption from those arising from internal trading in a Local Energy Market and explores allocation outcomes across a broad range of community configurations. Extensive simulations based on both synthetic and real-world consumption and price data are used to examine community-level savings, individual outcomes, surplus distribution patterns between Net Consumers and Net Producers, and computational tractability. The results show that internal energy trading consistently increases community-level savings, although its contribution is typically modest relative to self-consumption and strongly dependent on contextual factors such as renewable penetration, demand heterogeneity, and price conditions. The analysis highlights important trade-offs between savings generation, surplus distribution, and computational feasibility, underscoring the relevance of context-aware selection of allocation mechanisms. Overall, the proposed approach provides a transparent and reproducible tool for analysing the economic performance of Energy Communities under practical constraints.

Reciprocity is central to explanations of cooperation among unrelated individuals in societies of humans and other animals. Most mathematical analyses of the evolution of reciprocity are based on the repeated Prisoner’s Dilemma (RPD) and typically assume that new strategies are rarely introduced by mutation or analogous cultural processes, that behavioral errors are absent or infrequent, and that agents are bound to interact with the same partner. Here we analyze a version of the RPD in which new strategies are frequently introduced, behavioral errors occur at a substantial rate, and actors may have the option to leave their current partner. In this environment, the usual indeterminacy disappears and the mix of strategies and cooperation levels are quite stable. With the option to leave, cooperation persists at a substantially higher level than without the option to leave. Classical strategies such as Grim, Tit-for-Tat , or Win-Stay-Lose-Shift disappear and are replaced by strategies that sanction cheaters by leaving rather than by retaliatory defection. Beyond a threshold, increasing the number of times partners interact decreases the level of cooperation without the option to leave, but increases it when leaving is possible. Significance Statement: Cooperation is central to human societies, and the repeated Prisoner’s Dilemma has long been used to explain reciprocity. Most models assume that individuals must interact repeatedly with the same partner. We show that when behavioral errors and variability are substantial, allowing individuals to leave their partner fundamentally changes which strategies succeed and how much cooperation is sustained. Classical strategies such as Grim, Tit-for-Tat , and Win-Stay-Lose-Shift are eliminated. Instead, strategies that sanction cheaters by leaving prevail. Leaving both protects cooperators from exploitation and generates positive assortment, as similar strategies interact more often with each other. These results reveal a simple principle: freedom to leave can promote cooperation at the population level.

Energy communities enable prosumers to jointly operate distributed energy resources and thereby generate economic benefits that exceed those achievable individually. A central challenge in their implementation is selecting a Cost Allocation Method (CAM) that distributes these benefits fairly among heterogeneous participants. Although numerous CAMs have been proposed, they are often evaluated under different assumptions, making direct comparison difficult. This paper develops a unified axiomatic framework for assessing CAMs in energy communities and applies it to eight representative methods classified in three families: simple rules, savings-based, and price-based. The framework is built around seven desirable properties capturing principles of fairness, environmental friendliness, and continuity. Our main contribution is a comparative table that positions all methods within a single evaluative space and reveals the structural trade-offs that arise across CAMs. The analysis shows that the Average-Price CAM satisfies the same axiomatic properties as the Shapley method while remaining computationally trivial, making it an attractive practical option. We also show that the Extreme-Price CAM is the only price-based method that ensures the property of Beneficial Group Participation (core stability); however, this method violates other properties related to environmental friendliness and continuity—trade-offs we prove to be unavoidable for price-based rules. Finally, we conjecture that the nucleolus satisfies all seven properties, although its computation is rarely feasible in practice. The proposed framework provides researchers and practitioners with a transparent foundation for selecting and designing cost allocation methods in emerging energy communities.

In populations subject to evolutionary processes, the assortment of players with different genes or strategies can have a large impact on players’ payoffs and on the expected evolution of each strategy in the population. Here we consider assortment generated by a process of partner choice known as selective assortment . Under selective assortment, players looking for a mate can observe the strategies of a sample of potential mates or co-players, and select one of them to interact with. This selection mechanism can generate positive assortment (if there is a preference for players using the same strategy), or negative assortment. We study the impact of selective assortment in the evolution and in the equilibria of a population, providing results for different games under different evolutionary dynamics (including the replicator dynamics).