On a memory game and preferential attachment graphs

TL;DR

This paper analyzes the asymptotic distributions of parameters in a memory game, linking it to preferential attachment graphs and providing new proofs for known results in graph degree distributions.

Contribution

It extends previous work by deriving limiting distributions for memory game parameters and connecting these to preferential attachment graph models with simplified proofs.

Findings

Game length converges to a normal distribution.

Waiting time for first match converges to a Rayleigh distribution.

Number of lucky moves converges to a Poisson distribution.

Abstract

In a recent paper Velleman and Warrington analyzed the expected values of some of the parameters in a memory game, namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively. We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by--product of our methods we obtain simpler proofs (although without rate of convergence) of some of the results of Pek\"oz, R\"ollin, and Ross on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs. For proving that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multi--type generalized P\'olya urn model.