Remarks on fixed points, maximal elements, and equilibria of generalized games
The more players, the more difficult it is to find a solution to the game. This situation is similar to physics, where the more particles we have, the more difficult it is to solve the corresponding \(N\)-particle problem; for \(N\approx 10^{23}\) atoms in a solid body, the problem is computationally intractable. Physicists have long developed a good approximate technique for solving such problem: namely, instead of considering the actual \(N\)-body system with a large finite \(N\), they approximate it by a continuous medium model in which there is infinitely many (continuum) independent points. This transition from a large finite number to infinite number of elementary subsystems drastically reduces the complexity of describing the physical system. Similarly, in game theory, researchers have successfully used a model with infinite number of players to describe games with a large number of players (e.g., a market). The problem with this approach is that most game theory results rely on fixed point theorems for mappings in locally convex vector spaces, while the set of strategies for games with infinitely many players is often not locally convex. The authors show how the existing fixed point theorems (and related results) can be generalized to non-compact sets of (not necessarily locally convex) topological vector spaces, and how these extended results help in the analysis of games with infinitely many players.
scientific article; zbMATH DE number 1295265