I have the following definitions of continuity of the probability function: For an Increasing Sequence of Sets:Suppose that $B_{1} \subseteq B_{2} \subseteq B_{3} \subseteq \cdots$. Then, $P\left(\lim_{n \to \infty}B_{n} \right) = P\left( \cup_{i=1}^{\infty} B_{i}\right) = \lim_{n \to \infty}P(...

I've recently been introduced to Laplace transforms, and my understanding so far is that it's a continuous analogue to a Summation of a power series, that maps injectively a function $f(t)$ to another function of a new variable $s$, $F(s)$. My question though is that if we take $$\int_0 ^\infty ...

We have a $2$-player game and each player has $n$ strategies. The payoffs for each player are in range $\left[0,1\right]$ and are selected at random. Show that the probability that this random game has a pure deterministic Nash equilibrium approaches $1-1/\mathrm e$ as $n$ goes to infinity. Can ...