# Expectation value of m parallel games

Easy example to start: You throw a $$n$$-sided dice until your lucky number shows. This is a Bernoulli process with $$p=1/n$$, your expected number $$E$$ of throws is $$n$$.
Now imagine you play this game $$m$$ times in parallel. Stop if any dice shows your lucky number, i.e. first win wins the whole game. (Feel free to discuss also the last win case :-) It's still a Bernoulli process with $$p'=1-(1-1/p)^m, E'=1/p'$$.
Now generalize. The game $$G$$ is defined by fixing some winning probability $$p(i)$$ for each move $$i$$, we only assume $$E$$ shall be finite. Play $$m$$ copies of $$G$$ parallel, again first win in a copy ends the game. What can one say about $$E'$$?
1. $$E'\le E$$. (trivial)
2. $$E'\ge E/m$$ ? (tempting :-)
3. Let $$p(i)$$ stem from some well-known probability distribution, say, instead of the geometric from above a Poisson or whatnot. Surely $$E'$$ already has been computed for many of these distributions?

• What is $i$? Time index or index of a parallel game? – kodlu Apr 28 at 14:35
• In the general case, your question amounts to the following: you have a positive-integer-valued random variable $X$ (with $\mathbb{E}X=E$) and you take a random variable $Y$ which is the minimum of $m$ independent copies of $X$ (with $\mathbb{E}Y=E'$). – James Martin Apr 28 at 22:04
• Property 2. doesn't hold in general. e.g. let $m=2$, and let $X$ take value $1$ with probability $2/3$ and value $100$ with probability $1/3$. Then $Y$ takes value $1$ with probability $8/9$ and value $100$ with probability $1/9$. You get $E=34$ and $E'=12$. – James Martin Apr 28 at 22:10
• @kodlu: time index. I (and James) assume the games (including their $p_i$) are identical – Hauke Reddmann Apr 29 at 11:34
• @JamesMartin: E'=12>11.333...=34/3=E/m exactly as I supposed – Hauke Reddmann Apr 29 at 11:38