In game theory, the interactions of multiple players utiling shared public resources have been studied through the public goods game. This framework demonstrates that individuals achieve lower rewards in non-cooperative behavior than in cooperative ones, a phenomenon known as the tragedy of the commons (TOC). Plants also exhibit TOC empirically in the form of root overproliferation when competing for limited belowground resources, such as soil nutrients, decreasing seed production.
  To explore the mechanisms behind TOC in plants, we analyzed the resource allocation schedule of an annual plant with two components: aboveground (shoot) and belowground (root) structures. Plant fitness was measured based on shoot biomass at the end of the growing season. The plant assimilates resources through photosynthesis nad nutrient absorption from the soil, and allocatie those between shoots and roots. The rate of the nutrient absorption is a function of the nutrient concentration in the soil and the plant’s shoot and root biomasses. We investugate the optimal resource allocation schedule by using Pontryagin’s maximum principle in the presence and absence of neighboring plants. By comparing those results, we reveal the machanism causing TOC in the plant.
  In both solitary and competitive scenarios, we found that plants can adopt a singular solution with simultaneous resource allocation to both root and shoot growths, or a bang-bang control with a perfect switch of allocation between root and shoot. In the schedule with the singular control, plants initially allocates resources exclusively to shoots or roots, and subseruently allocates to both evenly. Afterward, plants allocated all resources to shoots. In the case of the bang-bang control, the schadule immediately swiches from exclusive allocation to root to such to shoot. The analysis shows that the presence of competitor extends the simultaneous allocation period in the singuloar condition, and delays switching time of allocation in the bang-bang control, associating with a reduction of the final shoot size (i.e. fitness), an increment of the final root size. The present study shows that the plant can exhibit TOC in the response to the presenc of competitors, in which a prolongation of period allocating to root is a key machanism of TOC.