Normalized equilibrium in Tullock rent seeking game

International audienceGames with Common Coupled Constraints represent manyreal life situations. In these games, if one player fails tosatisfy its constraints common to other players, then theother players are also penalised. Therefore these games canbe viewed as being cooperative in goals related to meetingthe common constraints, and non cooperative in terms ofthe utilities. We study in this paper the Tullock rent seekinggame with additional common coupled constraints. We havesucceded in showing that the utilities satisfy the property ofdiagonal strict concavity (DSC), which can be viewed asan extention of concavity to a game setting. It not onlyguarantees the uniqueness of the Nash equilibrium but also of the normalized equilibrium

The evolution of living beings started with prokaryotes and in interaction with prokaryotes

In natural world, no organism exists in absolute isolation, and thus every organism must interact with the environment and other organisms. Next-generation sequencing technologies are increasingly revealing that most of the cells in the environment resist cultivation in the laboratory and several prokaryotic divisions have no known cultivated representatives. Based on this, we hypothesize that species that live together in the same ecosystem are more or less dependent upon each other and are very large in diversity and number, outnumbering those that can be isolated in single-strain laboratory culture. In natural environments, bacteria and archaea interact with other organisms (viruses, protists, fungi, animals, plants, and human) in complex ecological networks, resulting in positive, negative, or no effect on one or another of the interacting partners. These interactions are sources of ecological forces such as competitive exclusion, niche partitioning, ecological adaptation, or horizontal gene transfers, which shape the biological evolution. In this chapter, we review the biological interactions involving prokaryotes in natural ecosystems, including plant, animal, and human microbiota, and give an overview of the insights into the evolution of living beings. We conclude that studies of biological interactions, including multipartite interactions, are sources of novel knowledge related to the biodiversity of living things, the functioning of ecosystems, the evolution of the cellular world, and the ecosystem services to the living beings