A fixed point theorem for discontinuous functions

2009

We provide several generalizations of the main results in Reny (1999) for the existence of both pure and mixed strategy Nash equilibria in discontinuous games. We also provide an example demonstrating that one natural additional generalization is not possible.

2012

In this note we prove an equilibrium existence theorem for games with discontinuous pay- offs and convex and compact strategy spaces. It generalizes the classical result of Reny (1999) [Econometrica 67, p. 1029-1056], as well as the recent paper of McLennan, Monteiro, and Tourky (2011) [Econometrica 79, p. 1643-1664]. Our condition is simple and easy to verify. Importantly, an example

International Journal of Game Theory, 2005

We study whether we can weaken the conditions given in Reny [4] and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy ε−equilibria for all ε > 0. We show by examples that there are: 1. quasiconcave, payoff secure games without pure strategy ε−equilibria for small enough ε > 0 (and hence, without pure strategy Nash equilibria), 2. quasiconcave, reciprocally upper semicontinuous games without pure strategy ε−equilibria for small enough ε > 0, and 3. payoff secure games whose mixed extension is not payoff secure. The last example, due to Sion and Wolfe [6], also shows that nonquasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.

2010

2002

It is well known that an upper semi-continuous compactand convex-valued mapping φ from a nonempty compact and convex set X to the Euclidean space of which X is a subset has at least one stationary point, being a point in X at which the image φ(x) has a nonempty intersection with the normal cone at x. In many circumstances there may be more than one stationary point. In this paper we refine the concept of stationary point by perturbing simultaneously both the set X and the solution concept. In case a stationary point is the limit of a sequence of perturbed solutions on a sequence of sets converging continuously to X we say that the stationary point is stable with respect to this sequence of sets and the mapping which defines the perturbed solution. It is shown that stable stationary points exist for a large class of perturbations. A specific refinement, called robustness, is obtained if a stationary point is the limit of stationary points on a sequence of sets converging to X. It is ...

International Journal of Game Theory, 1976

It is shown that all continuous two-person general-sum games over the square have equilibrium pairs of mixed strategy. A slight extension of Helly's Second Theorem, which may be of some slight interest in its own right, is used. The result is then generalized to all continuous n-person games on the cube.

2016

of the Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Master of Arts (in Mathematics) December 2016 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.

Mathematische Nachrichten, 2014

We prove a topological two-way characterization of the existence of fixed-points, without using linear or convexity structures and provide applications in optimization-related problems. Such a characterization is also demonstrated for a fixed-component point, a slight generalization of a fixed point.

We investigate a generalization of midpoint convexity for multivalued maps, and derive fixed points theorems under this more general assumption without requiring any strong compactness condition. As an application we prove the existence of social equilibria for games with n players.

Journal of Fixed Point Theory and Applications, 2009

We have carried out a thorough study on the existence of discrete fixed points and applications. As a result, we establish several general existence theorems of fixed and zero points on a broad class of subsets of the integer lattice Z n and also on arbitrary discrete sets. Most results have their well-known continuous counterparts. One of the major conditions is that of locally gross direction preservation imposed on the mappings under consideration. Furthermore, we establish several theorems for the existence of a unique discrete zero point. In addition, three applications are discussed. We first study a pure exchange competitive economy with indivisibilities and show the existence of a discrete Walrasian equilibrium. Second, we show the existence of a pure strategy Nash equilibrium in games where the set of each player's strategies is a discrete set and the mixed strategies are not played. Third, we discuss a general Samuelson's spatial price equilibrium model in a discrete setting.