Essential Sets of Fixed Points

Abstract

A fixed point of a continuous function is essential if nearby functions have nearby fixed points. Kinoshita established this chapter’s central result, which asserts that for any finite partition of the correspondence’s fixed points into compact sets, one of the cells of the partition is essential. It follows that a minimal essential set is essential. The proof strategy is to suppose not, so that for each cell of the partition there is a perturbation of the correspondence that has no nearby fixed point, and to combine these perturbations with the given correspondence, using convex combination with locally varying weights, to produce a convex valued correspondence without a fixed point. The Fan–Glicksberg fixed point theorem, which is the generalization of the KFPT to infinite dimensional domains, guarantees that such a correspondence does have a fixed point, so this is a contradiction. We also study weakenings of essentiality defined by requiring robustness with respect to some class of perturbations of the given function or correspondence, which arise in the definitions of some refinements of Nash equilibrium.

Exercises

We outline certain concepts and results from the theory of refinements of Nash equilibrium. It is assumed that the reader knows the basic elements of the theory of Nash equilibrium as they are laid out in Sect. 15.9. We do not discuss the conceptual significance of these concepts and results; relevant background can be found in, for example, Selten (1975), Myerson (1978), Kohlberg and Mertens (1986), and Myerson (1991).

Let $G = (S_1, \ldots , S_n, u_1, \ldots , u_n)$ be a given strategic form game: $S_1, \ldots , S_n$ are nonempty finite sets of pure strategies , and $u_1, \ldots u_n : S \rightarrow \mathbb {R}$ are functions, where $S = S_1 \times \cdots \times S_n$ is the set of pure strategy profiles . For any nonempty finite set X let

$$\varDelta (X) := \{\, \mu : X \rightarrow [0,1] : \sum \mu (x) = 1 \,\}$$

be the set of probability measures on X. For each $i = 1, \ldots , n$ the set of mixed strategies for agent i is $\Sigma _i := \varDelta (S_i)$, and the set of totally mixed strategies is

$$\Sigma ^\circ _i := \{\, \sigma _ \in \Sigma _i : \sigma _i(s_i) > 0\, \text {for all}\, s_i \in S_i \,\}.$$

The sets of mixed strategy profiles and totally mixed strategy profiles are $\Sigma := \Sigma _1 \times \cdots \times \Sigma _n$ and $\Sigma ^\circ := \Sigma ^\circ _1 \times \cdots \times \Sigma ^\circ _n$ respectively.

The functions $u_i$ are understood to be von-Neumann–Morgenstern utility functions, and we extend $u_i$ to $\Sigma _i$ by taking expectations: $u_i(\sigma ) := \sum _{s \in S} (\prod _i \sigma _i(s_i)) u_i(s)$. For each i let $BR_i : \Sigma \rightarrow \Sigma _i$ be agent i’s best response correspondence: $BR_i(\sigma ) := \text {argmax}_{\tau _i \in \Sigma _i} u_i(\tau _i,\sigma _{-i})$, and let $BR : \Sigma \rightarrow \Sigma$ be the best response correspondence: $BR(\sigma ) := BR_1(\sigma ) \times \cdots \times BR_n(\sigma )$. A fixed point of BR is a Nash equilibrium.

Let $Q^H$ be the set of possible payoffs ${\tilde{u}}= ({\tilde{u}}_1, \ldots , {\tilde{u}}_n)$ for games with the pure strategy sets $S_1, \ldots , S_n$. We endow $Q^H$ with the Euclidean topology derived from the obvious identification with $(\mathbb {R}^S)^n$. For ${\tilde{u}}\in Q^H$ and $\sigma \in \Sigma$ let $BR^{\tilde{u}}_i(\sigma ) := \text {argmax}_{\tau _i \in \Sigma _i} {\tilde{u}}_i(\tau _i,\sigma _{-i})$ for each i, and let $BR^{\tilde{u}}(\sigma ) := BR^{\tilde{u}}_1(\sigma ) \times \cdots \times BR^{\tilde{u}}_n(\sigma )$.

7.1

A Nash equilibrium $\sigma ^*$ is essential (Wu and Jiang 1962) if, for every neighborhood $U \subset \Sigma$ of $\sigma ^*$, there is a neighborhood $V \subset Q^H$ of u such that for every ${\tilde{u}}\in V$, $BR^{\tilde{u}}$ has a fixed point in U. Give an example of a game with one player that does not have an essential Nash equilibrium.

7.2

(E. Solan and O. N. Solan) We study an example of a game with an isolated totally mixed Nash equilibrium that is not essential. In the game below the first player chooses the top or bottom row, the second player chooses the left or right column, and the third player chooses the left or right matrix. Let $S_1 = S_2 = S_3 = \{a, b\}$ where a represents top or left and b represents bottom or right.

$$\begin{aligned} \begin{array}{|c c|} \hline (1,1, 1) &{} (-5, 0 , 3) \\ (0,3, -5) &{} (0,0,1) \\ \hline \end{array} \qquad \qquad \begin{array}{|c c|} \hline (3,-5, 0) &{} (1, 0 , 0) \\ (0,1,0) &{} (0,0,0) \\ \hline \end{array} \end{aligned}$$

• (a) Show that this game is symmetric in the following sense: if $(r,s, t) \in S$ is a pure strategy profile, then the first player’s payoff at (r, s, t) is the same as the second player’s payoff at (t, r, s) and the same as the third player’s payoff at (s, t, r).

• (b) Show that this game has a single pure Nash equilibrium and no Nash equilibria in which one player plays a pure strategy and another player plays a mixed strategy.

• (c) Show that if the second player plays a with probability $\tfrac{1}{2} + y$ and the third player plays a with probability $\tfrac{1}{2} + z$, then the first player is indifferent between playing a and b if and only if $y - z + yz = 0$.

• (d) Show that (0, 0, 0) is the unique solution of the system of equations $x - y + xy = 0$, $y - z + yz = 0$, and $z - x + zx = 0$.

• (e) In this and the next part we consider the perturbed game obtained by adding $4\varepsilon$ to all of the nonzero payoffs of the first player. Show that if the second player plays a with probability $\tfrac{1}{2} + y$ and the third player plays a with probability $\tfrac{1}{2} + z$, then the first player is indifferent between playing a and b if and only if $\varepsilon + y - z + yz = 0$.

• (f) Show that for small $\varepsilon > 0$ the system of equations $x - y + xy = 0$, $\varepsilon + y - z + yz = 0$, and $z - x + zx = 0$ has no solution.

For each i let ${\hat{Q}}^F_i$ be the set of nonempty polytopes contained in $\Sigma ^\circ _i$, and let $Q^F_i := \{\Sigma _i\} \cup {\hat{Q}}^F_i$. Let ${\hat{Q}}^F := {\hat{Q}}^F_1 \times \cdots \times {\hat{Q}}^F_n$ and $Q^F := \{(\Sigma _1, \ldots , \Sigma _n)\} \cup {\hat{Q}}^F$. We endow each $Q^F_i$ with the topology induced by the Hausdorff metric, we endow $Q^F_1 \times \cdots \times Q^F_n$ with the product topology, and we endow ${\hat{Q}}^F$ and $Q^F$ with the relative topologies induced by their inclusions in this space. For $P \in Q^F$ and $\sigma \in \Sigma$ let $BR^P(\sigma ) := BR^P_1(\sigma ) \times \cdots \times BR^P_n(\sigma )$ where $BR^P_i(\sigma ) := \text {argmax}_{\tau _i \in P_i} u_i(\tau _i,\sigma _{-i})$. A set of Nash equilibria is fully stable (Kohlberg and Mertens 1986) if it is minimal in the class of closed sets of Nash equilibria C such that for every neighborhood $U \subset \Sigma$ of C there is a neighborhood $V \subset Q^F$ of $(\Sigma _1, \ldots , \Sigma _n)$ such that for every $P \in V$ there is a fixed point of $BR^P$ in U.

For each i let ${\hat{Q}}^T_i$ be the set of $P_i \in {\hat{Q}}^F$ such that $P_i = (1 - \varepsilon _i)\Sigma _i + \varepsilon _i{\overline{\sigma }}_i$ for some $\varepsilon _i \in (0,1)$ and ${\overline{\sigma }}_i$ in the interior of $\Sigma _i$, and let ${\hat{Q}}^T := {\hat{Q}}^T_1 \times \cdots \times {\hat{Q}}^T_n$ and $Q^T := \{(\Sigma _1, \ldots , \Sigma _n)\} \cup {\hat{Q}}^T$.

7.3

A mixed strategy profile $\sigma ^* \in \Sigma$ is a perfect equilibrium (Selten 1975) if there are sequences $\{P^r\}$ in ${\hat{Q}}^T$ and $\{\sigma ^r\}$ in $\Sigma ^\circ$ such that $P^r \rightarrow (\Sigma _1, \ldots , \Sigma _n)$, each $\sigma ^r$ is a fixed point of $BR^{P^r}$, and $\sigma ^r \rightarrow \sigma ^*$.

• (a) Prove that a perfect equilibrium is a Nash equilibrium.

• (b) Prove that the set of perfect equilibria is nonempty.

• (c) Prove that the set of perfect equilibria is closed.

7.4

Find a two player game such that there is no Nash equilibrium $\sigma ^*$ such that for any neighborhood $U \subset \Sigma$ of $\sigma ^*$ there is a neighborhood $V \subset Q^T$ of $(\Sigma _1,\Sigma _2)$ such that for every $P \in V$ there is a fixed point of $BR^P$ in U. Make sure your example is minimal with respect to the numbers of pure strategies of the two agents, and show that any “smaller” game has no such equilibrium.

For each i let ${\hat{Q}}^P_i$ be the set of $P_i \in {\hat{Q}}^F$ such that for some ${\overline{\sigma }}_i \in \Sigma ^\circ _i$, $P_i$ is the convex hull of all points obtained by permuting the coordinates of ${\overline{\sigma }}_i$. (Recall that if the coordinates of ${\overline{\sigma }}_i$ are all different, then $P_i$ is a permutahedron.) Let ${\hat{Q}}^P := {\hat{Q}}^P_1 \times \cdots \times {\hat{Q}}^P_n$ and $Q^P := \{(\Sigma _1, \ldots , \Sigma _n)\} \cup {\hat{Q}}^P$.

7.5

A mixed strategy profile $\sigma ^* \in \Sigma$ is a proper equilibrium (Myerson 1978) if there are sequences $\{\varepsilon ^r\}$ in (0, 1) and $\{\sigma ^r\}$ in $\Sigma ^\circ$ such that $\varepsilon ^r \rightarrow 0$, $\sigma ^r_i(s_i) \le \varepsilon ^r \sigma ^r_i(t_i)$ for all r, i, and $s_i, t_i \in S_i$ such that $u_i(s_i,\sigma ^r_{-i}) < u_i(t_i,\sigma ^r_{-i})$, and $\sigma ^r \rightarrow \sigma ^*$.

• (a) Prove that a proper equilibrium is a perfect equilibrium.

• (b) Prove that the set of proper equilibria is nonempty.

• (c) Prove that the set of proper equilibria is closed.

A set of Nash equilibria is stable (Kohlberg and Mertens 1986) if it is minimal in the class of closed sets of Nash equilibria C such that for every neighborhood $U \subset \Sigma$ of C there is a neighborhood $V \subset Q^T$ of $(\Sigma _1, \ldots , \Sigma _n)$ such that for every $P \in V$ there is a fixed point of $BR^P$ in U.

7.6

A fact that is beyond our scope is that the set of Nash equilibria has finitely many connected components. Taking this as known, prove that one of these components contains a fully stable set that in turn contains a stable set.

7.7

A pure strategy $s_i \in S_i$ is weakly dominated if there is a $t_i \in S_i$ such that $u_i(s_i,s_{-i}) \le u_i(t_i, s_{-i})$ for all $s_{-i} \in \prod _{j \ne i} S_j$, with strict inequality for some $s_{-i}$.

• (a) Prove that is $\sigma ^*$ is a perfect equilibrium, then each $\sigma ^*_i$ assigns no probability to any weakly dominated pure strategy.

• (b) Prove that any stable set is contained in the set of perfect equilibria.

• (c) Prove that a fully stable set contains a proper equilibrium.

• (d) Find the fully stable sets and the stable sets of the game below (Fig. 7.2).

Fig. 7.2

A two-by-two game

7.8

Let X be a topological space, and let V be a topological vector space. Recall that for an arbitrary index set I a partition of unity for X is a collection of functions $\{\psi _i\}_{i \in I}$ from X to [0, 1] such that each x has a neighborhood on which only finitely many of the functions are nonzero and $\sum _i \psi _i(x) = 1$. Let $\mathscr {PU}^I(X)$ be the space of such partitions of unity. Let $\mathscr {PU}^I_S(X)$ and $\mathscr {PU}^I_W(X)$ be $\mathscr {PU}^I(X)$ endowed with the relative topologies it inherits as subspaces of $C_S(X)^I$ and $C_W(X)^I$. Prove that the function $(\{\psi _i\}_{i \in I}, \{F_i\}_{i \in I}) \mapsto \sum _i \psi _iF_i$ is continuous as a function from $\mathscr {PU}^I_S(X) \times \mathscr {U}_S(X, V)^I$ to $\mathscr {U}_S(X, V)$ and also as a function from $\mathscr {PU}^I_W(X) \times \mathscr {U}_W(X, V)^I$ to $\mathscr {U}_W(X, V)$.