## 6079 - Voronoi diagrams and Game Theory

### N. Lygeros

Translation: Paola Vagioni

When we examine the Voronoi diagrams or even the Delaunay triangulation, the correlation to Game Theory is not direct. The Voronoi diagrams, as we have proved it in previous studies, include from the start a dynamic due to computational geometry and topology, which allows not only the Delaunay triangulation but an entire topostrategic process. The incorporation of this framework in the field of Game Theory is carried out in a natural way. Consider an empty territory and not necessarily the plane. On this territory two players play alternatively by positioning points. After each movement, the Voronoi diagram of the new sum is calculated. If the territory is finite, then it is meaningful after a few rounds to ask ourselves, which of the players has the greater area by adding the Voronoi cells that he controls. Consecutively, we understand that a problem of tactic exists after each player’s movement and also a general problem of strategy. In this particular situation it is feasible to prove that, except from the case where there is only one round, the second player always wins even if the difference of the total areas of the two players can become the smallest possible but never zero. These elements: framework, field and field of action via Game Theory, can be applied also in real situations and be integrated in topostrategy. A significant example is the Aegean Sea. The complexity of this peculiar structure can be examined in this way and offer mental schemata, which are not direct and appear only in a polemological framework. In any case, the static approach of the Aegean Sea is not sufficient for preparing us and for managing strategically future crises. A further problem in relation to the Aegean Sea is that the structure is given as far as the geography is concerned. In consequence, there is no direct application of the theorem but the most important is that there are key positions even on the initial framework. This means that in the case of the Aegean Sea we have several island categories that are involved for creating a grid. The robustness of this grid is the whole essence of the case. And the Voronoi diagrams via Game Theory promote strategic behaviors, which must be examined in detail and effectively.