Games and Random Search

Abstract

We use dynamics of measures, i.e. iteration of the operators from measurable space to space of probabilistic measures on this space, to model and prove properties of random search algorithms. Specifically using this technique in the context of Game Theory we show that stochastic better response dynamics, where players in the potential game perform their moves independently choosing the random strategy improving their outcome, converges in stochastic sense to playing strategies near equilibrium.

A Appendix

Following is the proof of measurability of the function f defined in the proof of Theorem 8. We will infer it from two lemmas.

Lemma 1

Let P be the finite set, and for each $i \in P$ $S_{i}$ be a compact metric space with metric $d_{i}$. For each i by $\sigma _{i}$ we will denote $\sigma \text {-algebra}$ of Borel sets and $\mu _{i}$ be a regular probabilistic measure on $S_{i}$. Let $S=\prod \limits _{i \in P}$ be a metric space with metric d:

$$ d(x, y) = \sum \limits _{i \in P}d_{i}(x_{i},y_{i}). $$

which gives a product topology in S. We denote by $\sigma$ the $\sigma \text {-algebra}$ of Borel set in S, and take measure $\mu =\bigotimes \limits _{i \in P}\mu _{i}$. Let $V:S \rightarrow \mathbb {R}$ be continuous, $\alpha \ge 0$ and function $g_{i}: S \rightarrow \mathbb {R}$ be defined as follows:

$$ g_{i}(s)=\mu _{i}(\iota ^{-1}_{i,s}(U_{V(s)+\alpha })) $$

where $U_{t}=V^{-1}((t,\infty ))$. Under this assumptions we have: $g_{i}$ is lower semi-continuous

Proof

We will show, that $g^{-1}_{i}((\rho , \infty ))$ for $\rho \in \mathbb {R}$ is open. Let’s assume that $g^{-1}_{i}((\rho , \infty )) \ne \emptyset$ (if it is empty, it is open trivially) and take $s \in g^{-1}_{i}((\rho , \infty ))$. From definition we have:

$$ \mu _{i}(\iota ^{-1}_{i,s}(U_{V(s)+\alpha })) > \rho $$

Since $\mu _{i}$ is regular, there is a compact set K such that $K \subset \iota ^{-1}_{i,s}(U_{V(s)+\alpha })$ and $\mu _{i} > \rho$ Obviously, $\iota _{i,s}(K)$ is compact so V takes it’s extrema on K. Thus, there is $\epsilon > 0$ such that, for each $x \in \iota _{i,s}(K)$ we have $V(x) > V(s)+\alpha +\epsilon$ For each $x \in \iota _{i,s}(K)$ then, there is open open neighborhood $O_{x}$ of x such that: $O_{x} = \prod \limits _{j \in P}O^{j}_{x}$, where $O^{j}_{x}$ is neighborhood of $x_{j}$ in $S_{j}$ and for each $y \in O_{x}$ we have $V(y) > V(s)+\alpha +\frac{\epsilon }{2}$ From the compactness, we can over $\iota _{i,s}(K)$ by finite number of these neighborhoods:

$$ \iota _{i,s}(K) \subset \bigcup \limits _{j \in \{0,\ldots ,n\}}O_{x_{j}}=\bigcup \limits _{j \in \{0,\ldots ,n\}}O_{j} $$

where last equality is introduced to simplify further referencing these sets. Now let $O_{s} = V^{-1}((-\infty , V(s)+\frac{\epsilon }{2}))$. It is open neighborhood of s in S. Finally let’s construct another open neighborhood of s following way:

$$ O = O_{s} \cap ((\bigcap \limits _{j=0, \ldots , n}O^{-i}_{j})\times S_{i}) $$

where $O^{-i}_{j} = \prod \limits _{l \in (P-\{i\})}O^{l}_{j}$

We will show, that $O \subset g^{-1}_{i}((\rho , \infty ))$.

Let’s take $s' \in O$. For each $x \in K$ we have: $\iota _{i, s'}(x) = (x, s'_{-i})$ where $s'_{-i} \in \bigcap \limits _{j=0, \ldots , n}O^{-i}_{j}$, thus belongs to all $O^{-i}_{j}$. Since $O_{j}$ was cover of $\iota _{i,s}(K)$, there is specific j, such that $(x,s_{-i}) \in O_{j}$. So, also $(x, s'_{-i}) \in O_{j}$. Now, we have:

$$ V(\iota _{i,s'}(x))> V(s)+\alpha +\frac{\epsilon }{2}>V(s')+\alpha $$

thus $K \subset \iota ^{-1}_{i,s'}(U_{V(s'}+\alpha$ and since $\mu _{i}(K) > \rho$ we have $O \subset g^{-1}_{i}((\rho , \infty ))$. This completes the proof.

Lemma 2

Let (X, d) be a metric space, $f:X\rightarrow \mathbb {R}$ and $g:X\rightarrow \mathbb {R}$ be two lower semi-continuous functions such that, for each $x \in M$ we have $0 \le g(x) \le f(x) \le M$. Let $\phi :X\rightarrow \mathbb {R}$ be defined as

$$ \phi (x) = {\left\{ \begin{array}{ll} 1 &{} g(x) = f(x) \\ \frac{g(x)}{f(x)}&{} otherwise \\ \end{array}\right. } $$

then $\phi$ is measurable in the sense of the Borel $\sigma \text {-algebra}$

Proof

We will show, that $\phi$ is Baire class 2 function (so measurable with regard to Borel’s $\sigma \text {-algebra}$. Let’s take sequence $(\epsilon _{n})_{n \in \mathbb {N}}$, such that: $\epsilon _{n}\underset{n \rightarrow \infty }{\longrightarrow }0$ for each $n \in \mathbb {N}$ we have $\epsilon _{n} > \epsilon _{n+1}$.

Obviously functions $f_{n}=f+\epsilon _{n}$ and $g_{n} = g+\epsilon _{n}$ are also lower semi-continuous, and what’s more strictly positive ($>\epsilon _{n}$). From Baire theorem on lower semi-continuous functions [8] there are pointwise increasing sequence of continuous functions $(f_{n,i})_{i \in \mathbb {N}}$ and $(g_{n,i})_{i \in \mathbb {N}}$ such that pointwise $f_{n,i}\underset{i \rightarrow \infty }{\longrightarrow }f_{n}$ and $g_{n,i}\underset{i \rightarrow \infty }{\longrightarrow }g_{n}$. By taking

$$\tilde{f}_{n,i}(x) = max\{f_{n,i}(x), \frac{\epsilon }{2}\}$$

and

$$\tilde{g}_{n,i}(x) = max\{g_{n,i}(x), \frac{\epsilon }{2}\}$$

we can assure, that sequences are bounded below from 0 and taking

$$\tilde{\tilde{f}}_{n,i}(x) = max\{\tilde{f}_{n,i}(x), \tilde{g}_{n,i}(x)\}$$

and

$$\tilde{\tilde{g}}_{n,i}(x) = min\{\tilde{f}_{n,i}(x), \tilde{g}_{n,i}(x)\}$$

we can assure that $\tilde{\tilde{g}}_{n,i} \le \tilde{\tilde{f}}_{n,i}$. Now, we have $\frac{\tilde{\tilde{g}}_{n,i}}{\tilde{\tilde{f}}_{n,i}}\underset{i \rightarrow \infty }{\longrightarrow }\frac{g_{n}}{f_{n}}$ so $\frac{g_{n}}{f_{n}}$ is of first category as a pointwise limit of continuous functions.

We can also check (by easy calculation), that for each $x \in X$ $\frac{g_{n}(x)}{f_{n}(x)} \ge \frac{g_{n+1}(x)}{f_{n+1}(x)}$. Thus $\frac{g_{n}}{f_{n}}$ is pointwise convergent equal to 1 whenever $f(x)=g(x)$, so convergent to $\phi$. Thus $\phi$ is of second Baire category.

Let us now recall definition of function f from Theorem 8 (we will use notation and assumptions from Sect. 4 and proof of Theorem 8):

$$ f(s)=A(s)(U_{V(s)+\epsilon }) $$

We have for each $i \in P$ that functions $f_{i}(s)=\tilde{\mathcal {V}}_{i}(s)(U_{i,s})=\mu _{0}^{i}(\iota ^{-1}_{i,s}(U_{V(s)}))$ and $g_{i}(s)=\tilde{\mathcal {V}}_{i}(s)((U_{V(s)+\epsilon '}))=\mu _{0}^{i}(s)(\iota ^{-1}_{i,s}(U_{V(s)+\epsilon '}))$ both are lower semi-continuous by Lemma 1, and what’s more satisfy conditions of Lemma 2. So, $\mathcal {V}_{i}(U_{V(s)+\epsilon '})$ is measurable for each i and $f(s)=A(s)(U_{V(s)+\epsilon '})=\sum \limits _{i \in P}\alpha _{i}\mathcal {V}_{i}(U_{V(s)+\epsilon '})$ is measurable as a linear combination of the measurable functions.