The Fixed Point Index

Abstract

An upper hemicontinuous correspondence from a compact subset of a space to the space is index admissible if all of the function’s fixed points are in the interior of the domain. Translation of the results for the degree gives a function, called the index, assigning an integer to each index admissible function mapping a compact subset of a Euclidean space to the Euclidean space. The index is characterized by three axioms, called Normalization, Additivity, and Continuity. The Euclidean index also satisfies Multiplication, which requires that the index of a cartesian product of two functions is the product of their indices, and Commutativity, which requires that if two functions can be composed in either order, and the two compositions are index admissible, then the indices of the two compositions agree. Additivity implies that the index is a property of the germ of the function at its set of fixed points, and adopting this perspective is simplifying in several ways. Normalization, Additivity, Continuity, and Commutativity characterize a unique index assigning an integer to each index admissible upper hemicontinuous contractible valued correspondence from a compact subset of a locally compact ANR to the ANR. This index also satisfies Multiplication. For a manifold, and also for a finite simplicial complex, the index is the unique function assigning an integer to each index admissible function from a compact subset of the space that satisfies Normalization, Additivity, and Continuity.

Exercises

Computing the fixed point index is usually not straightforward. To a rough approximation there are four methods:

• (a) If the function has regular fixed points (or can be approximated by such a function) compute signs of the determinants of the relevant matrices of partial derivatives.

• (b) Show that a set of fixed points has index zero by presenting a perturbation of the function or correspondence that has no nearby fixed points.

• (c) If the domain is a compact AR, and the indices of all but one component of the set of fixed points are known, then the index of the last component can be inferred from the sum of the indices being +1.

• (d) Present a homotopy between the given function or correspondence and a function or correspondence for which the index of the relevant set of fixed points is already known.

The following problems present some instances of these methods.

13.1

Prove that if $a < b$ and $f : [a, b] \rightarrow \mathbb {R}$ is index admissible, then $\varLambda _\mathbb {R}(f) \in \{-1,0,1\}$.

13.2

Prove that if $C \subset \mathbb {R}^2$ is compact and contains, the origin is in its interior, $f : C \rightarrow \mathbb {R}^2$ is index admissible, and $f(x) \ne \lambda x$ for all $x \in C \setminus \{0\}$ and $\lambda > 1$, then $f(0) = 0$ and $\varLambda _{\mathbb {R}^2}(f) = 1$. (Hint: consider the homotopy $h(x, t) := tf(x)$.)

13.3

In a two player coordination game the two players have the same finite set S of pure strategies. They each receive a payoff of 1 if they both choose the same pure strategy and a payoff of 0 if they choose different pure strategies.

• (a) Describe the set of Nash equilibria.

• (b) Find the index of each Nash equilibrium, and prove this result using the index axioms.

• (c) Prove the result you found in (b) by approximating the best response correspondence with a suitable function and computing the sign of the determinant of the relevant matrix.

13.4

(Rubinstein 1989) General A and General B are commanders of allied forces. General A has received an order that both should attack. She transmits this to General B using an email system that sends an automatic confirmation-of-receipt email to the sender of each email (including confirmation-of receipt emails) that it receives. For each email there is a 10% chance that it is not received, so at the end of the communication phase the two generals have sent $s_A$ and $s_B$ emails respectively, where $s_A \ge 1$ and $s_A - 1 \le s_B \le s_A$.

• (a) Compute the conditional probabilities of having sent the last message, which are $\mathbf {Pr}(s_B = s - 1 | s_A = s)$ and $\mathbf {Pr}(s_A = s | s_B = s)$ for $s \ge 1$.

For each general the payoff when neither attacks is 0, the payoff when both attack is 10, the payoff when she attacks and the other does not is $-10$, and the payoff when she does not attack and the other does is $-5$. A behavior strategy for $C \in \{A, B\}$ specifies a probability $\pi _C(s) \in [0,1]$ of attacking conditional on sending s emails for each $s \ge 1$. (We assume that General B cannot attack without receiving at least one email.)

• (b) For what value of $\pi _B(1)$ will General A be indifferent about whether to attack or not when she has sent only one email.

• (c) What condition must $\pi _B(s - 1)$ and $\pi _B(s)$ satisfy if General A is indifferent between attacking and not attacking when she has sent $s > 1$ emails? What condition must $\pi _A(s)$ and $\pi _A(s+1)$ satisfy if General B is indifferent between attacking and not attacking when she has sent s emails?

• (d) Prove that if $(\pi _A^*, \pi _B^*)$ is a subgame perfect Nash equilibrium and there is some $C \in \{A, B\}$ and s such that $\pi _C^*(s) = 1$, then $\pi _A^*(s) = \pi _B^*(s) = 1$ for all s.

• (e) Prove that for a given $\pi _B$, if $\pi _B(s-1) = 0$ and General A is indifferent about whether to attack when she sends s emails, then she strictly prefers to attack when she sends $s + 1$ emails.

• (f) Prove that there are two subgame perfect Nash equilibria, one in which neither general ever attacks, and one in which both generals always attack.

• (g) Give topologies on the spaces of mixed strategies with respect to which they are convex compact subsets of Banach spaces. Define a best response correspondence that is upper hemicontinuous and convex valued, and whose fixed points are the subgame perfect Nash equilibria. Show that the equilibrium in which neither general ever attacks is an inessential fixed point. Conclude that the equilibrium in which both always attack has index +1.