Bayesian Rating

Question

Suppose we're in a situation where we have a website and people rate products on that site. The total number of people on the site is $n$. The total number of people who rated a product $i$ is $c_i$, and the arithmetic mean of the rating of $i$ is $k_i$. The max rating is $m$.
$$k: 0 \lt k \le m$$ $$c: 1 \le c \le n$$ $$c, n: c, n \in \Bbb Z^+$$    
The website I have in mind is a novel site. It happens that when novels are sorted by rating, a single novel $i$ with $c$ of $1$ and $k$ of $m$ would appear at the top of the list if you sort according to rating — I am very displeased and wanted to suggest an alternative rating using a Bayesian algorithm. I don't have any idea what such algorithm would look like though.
   
I do know some properties the algorithm would have. Let $b_i$ be the Bayesian rating of $i$.
*$b_i = k_i$ $\forall$ $ i: c = n$
*$b_i \ge k_i$ $\forall$ $ k_i: k_i \lt \frac{m}{2}$
*$b_i \le k_i$ $ \forall$ $ k_i: k_i \gt \frac{m}{2}$
*$b_i \to k_i \iff c \to n$
   
All help is appreciated. I actually want to use such Bayesian rating for an E-commerce site I'm going to work on later, so I'm interested in it beyond just having more consistent novel ratings on my favourite novel site.

Answer 1

This proposal solves the problem but does not use probability.

I decided to name my sorting method 'Popularating sort'. My idea for the ratings is simple: rav=[average of all the site's product's ratings]. Then for a product where rat=[the rating of that product] and amnt=[the amount of people who rated that product], then the products would be sorted by (rat-rav)*amt^x. 0<=x<=1. The higher x is, the more popularity matters, the lower it is, the more the rating matters. This effectively treats new products as rated average.