Sperner\u27s Lemma, The Brouwer Fixed Point Theorem, the Kakutani Fixed Point Theorem, and Their Applications in Social Sciences

Can a cake be divided amongst people in such a manner that each individual is content with their share? In a game, is there a combination of strategies where no player is motivated to change their approach? Is there a price where the demand for goods is entirely met by the supply in the economy and there is no tendency for anything to change? In this paper, we will prove the existence of envy-free cake divisions, equilibrium game strategies and equilibrium prices in the economy, as well as discuss what brings them together under one heading. This paper examines three important results in mathematics: Sperner’s lemma, the Brouwer fixed point theorem and the Kakutani fixed point theorem, as well as the interconnection between these theorems. Fixed point theorems are central results of topology that discuss existence of points in the domain of a continuous function that are mapped under the function to itself or to a set containing the point. The Kakutani fixed point theorem can be thought of as a generalization of the Brouwer fixed point theorem. Sperner’s lemma, on the other hand, is often described as a combinatorial analog of the Brouwer fixed point theorem, if the assumptions of the lemma are developed as a function. In this thesis, we first introduce Sperner’s lemma and it serves as a building block for the proof of the fixed point theorem which in turn is used to prove the Kakutani fixed point theorem that is at the top of the pyramid. This paper highlights the interdependence of the results and how they all are applicable to prove the existence of equilibria in fair division problems, game theory and exchange economies. Equilibrium means a state of rest, a point where opposing forces balance. Sperner’s lemma is applied to the cake cutting dilemma to find a division where no individual vies for another person’s share, the Brouwer fixed point theorem is used to prove the existence of an equilibrium game strategy where no player is motivated to change their approach, and the Kakutani fixed point theorem proves that there exists a price where the demand for goods is completely met by the supply and there is no tendency for prices to change within the market