the part of the Wikipedia page you’re referring to about fixed-point theorems and graphs is likely discussing the application of fixed-point theory in game theory. Fixed-point theorems, like Kakutani’s and Brouwer’s, are used in game theory to prove the existence of equilibria (such as Nash equilibria).
In game theory, a fixed point represents a situation where each player’s strategy is optimal given the other players’ strategies. For example, the fixed point in a Nash equilibrium ensures that no player can unilaterally improve their outcome by changing their strategy. This is fundamental in many economic models, where these theorems are used to show that certain equilibria exist under various assumptions.
The graphs described are often used to visualize these relationships and help in understanding how players’ strategies evolve and stabilize at equilibrium points, which corresponds to a fixed point in the mathematical model.
This connection between fixed-point theorems and economic/game-theory models, particularly as applied in proving the existence of Nash equilibria, is discussed further in references on mathematical economics and game theory.
https://ja.wikipedia.org/wiki/%E3%82%B2%E3%83%BC%E3%83%A0%E7%90%86%E8%AB%96
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