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The following games have been studied in a formulation similar to Petri nets and chip-firing games. Given a finite collection of baskets, each basket has an infinite number of balls of the same value. First, balls are selected from several baskets and placed on the table. Then, at each step, a ball is selected from the table and replaced with some $2$ balls from some basket. Which basket to take depends only on the balls to be replaced and is decided in advance. Given some $n$, the object of the game is to find the largest possible sum of table values ββ$g(n)$ of $n$ balls.

In this article, we show that the sequence $g(n)/n$ for $n=1,2,\dots$ converges to the growth rate $\lambda$. Moreover, this value $\lambda$ is the rate of constructs called quasi-loops, which are also solutions to fairly simple linear programs. Structure and linear programming are closely related. For example, the linear programming solution gives a quasi-loop of rate $\lambda$ in linear time for the number of baskets, and vice versa, the quasi-loop gives the solution dual linear program. To approximate $\lambda$, we provide a way to test in quadratic time if a given $\lambda_0$ is less than $\lambda$. If the ball values ββare all rational, then we can compute the exact value of $\lambda$ in cubic time using a quadratic time rate testing algorithm and a special conditional binary search that stops. Four proofs of the limit $\lambda$ are given. One just uses relations between baskets, one uses pseudo-loops, one uses linear programming, and one uses Fekete’s lemma (the latest The proof assumes conditions on substitution rules). .

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