Introduction

Purpose of the Study

Just before Christmas of 1998, when my wife was scrambling to buy gifts for all the nieces and nephews, she stumbled upon a game called mancala on the shelf at Wal-Mart.  She purchased it for eight-year-old Tyler, but in the week following Christmas, the adults found themselves playing it more than the children.  For three Christmases, now, I have found myself being drawn into this game, engaging in fierce competition with the in-laws.  The game seems deceptively simple; the rules are straightforward enough for even a young child to understand  (Exhibit 1 contains the game rules).  The longer I have played, though, the more I have discovered that the game really does invite deep thought and complex strategy.  In the intellectual vacuum that the holidays often are, I have begun each year to attempt a more thorough analysis of the game, but alas, holidays always end too soon and the game goes back on the shelf for another year.  The course in Game Theory provided me with a toolkit for analyzing mancala in ways I would not have been able to do beforehand, and the paper requirement provided the perfect excuse to pursue the study.

In simplest terms, my goal was to come to a better understanding of the game.  First, I wanted to discover whether mancala was a “trivial” game (meaning that at the beginning of the game, either the first player has a strategy such that she can guarantee a win or a draw, no matter what moves the other player might make, or the second player has such a move), and if so, to explore the implications of this discovery.  I planned to use a computer to attempt to solve game tree.  Second, I wanted to identify general strategies for playing the game well, human versus human, neither player having access to information that might be derived from the first step.  Finally, I wanted to study the relationship between the two approaches to playing.  For example, if a player had access to (and played according to strategies implied by) the entire solved game tree, would her plays make sense according to strategies derived for normal real-time play?  In the end, the various approaches to strategy converged in a very interesting way.

Because I wanted the satisfaction of going through the mental exercise myself, I chose not to consult any outside sources.  I have been made aware that mancala computer programs have been written by others, and that at least one book about mancala has been published.  However, the value of the study to me was not in having the answer, but engaging in the process which would lead to it.  Therefore all the findings presented here are the result of my own analysis and my collaboration with one partner.

Collaboration

On March 16-18, after the study was well underway, I was paid a very serendipitous visit by my cousin Rick, a professor of electrical and computer engineering at a major university.  Though I have seen him only on three brief occasions in the past fifteen or so years, we have both drunk deeply from a common gene pool, and share a great enthusiasm for math, logic, games, computers and such.  It was not difficult for Rick to find himself drawn into the game with me while he was here, and to take a personal interest himself in exploring the kinds of questions I was asking.  I shared with him my thoughts on trying to tackle the game with a computer, and outlined a couple of approaches I had thought about.  Two days after he returned home, he emailed me the result of Version One of his mancala program: he was hooked.  Since he could program in one hour the code I could write in a day, and since he had tools for writing a much faster-executing program (a significant factor, given the hundreds of hours of processor time we would eventually consume in analyzing the game), he offered to take on the programming responsibility, freeing me to focus on analysis.  Conveniently, he was at the beginning of his spring break, and has spent dozens of hours in the past three weeks writing and rewriting computer code for the project.  I gratefully acknowledge his contribution.