  The following game in a similar formulation to Petri nets and chip-firing
games is studied: Given a finite collection of baskets, each has an infinite
number of balls of the same value. Initially, a ball from some basket is chosen
to put on the table. Subsequently, in each step a ball from the table is chosen
to be replaced by some $2$ balls from some baskets. Which baskets to take
depend only on the ball to be replaced and they are decided in advance. Given
some $n$, the object of the game is to find the maximum possible sum of values
$g(n)$ for a table of $n$ balls.
  In this article, the sequence $g(n)/n$ for $n=1,2,\dots$ will be shown to
converge to a growth rate $\lambda$. Furthermore, this value $\lambda$ is also
the rate of a structure called pseudo-loop and the solution of a rather simple
linear program. The structure and the linear program are closely related, e.g.
a solution of the linear program gives a pseudo-loop with the rate $\lambda$ in
linear time of the number of baskets, and vice versa with the pseudo-loop
giving a solution to the dual linear program. A method to test in quadratic
time whether a given $\lambda_0$ is smaller than $\lambda$ is provided to
approximate $\lambda$. When the values of the balls are all rational, we can
compute the precise value of $\lambda$ in cubic time, using the quadratic time
rate test algorithm and the binary search with a special condition to stop.
Four proofs of the limit $\lambda$ are given: one just uses the relation
between the baskets, one uses pseudo-loops, one uses the linear program and one
uses Fekete's lemma (the latest proof assumes a condition on the rule of
replacements).
