## 17 August 2009

### Stochastic Duels: Introduction

Blog ... Math ... Statistics ... Games ... check ... I haven’t done much of that lately, and it’s time I got back on track. There is a topic I've been sitting on for a while now, that of Stochastic Duels. I started writing the post, but the post keeps getting longer and longer; First to the point where I needed to break it up into two parts, then I realized that still wasn't enough and I ought to do it as a series.
So a series it is, with this post serving as the introduction, table of contents, summary of topics, and anything else that occurs to me along the way. This post will change as I update it this the latest installments, and as I figure out exactly where I am going with this.

Stochastic Duels: Introduction

[image GameSignal, where they got it from I have no idea]

Definitions and History: A stochastic process is a series of events that are at least partly random. There may also be a predictable part which is useful to understand, but primarily interested in the random part and how it applies to games. I took this from a paper I found titled "The One-on-One Stochastic Duel: Parts I and II", by C.J.Ancker, Jr. (1979) -

The stochastic duel was first formulated and analyzed by G. Trevor Williams and the author [C. J. Ancker, Jr.], in a paper presented on 1U May 1966 at the 4-th Annual ORSA meeting in Washington, D.C. (1)
I don't have access to the 1966 paper, but I've been able to find a number of Ancker's papers from the 70's and 80's. Some of this is my source material, and I'll be working parts out for myself as well.
The term Stochastic Duel was first used by Ancker and Williams to describe a simplified combat model, where two opponents repeatedly (and simultaneously) fire at each other until one (or both) of them is hit. In this very simple sort of game it’s not too hard to work out the math describing it. The game is not intended to be realistic, but rather to demonstrate the principles involved. Once a basic understanding is established, the game can be made much more complex and realistic.

Contents:

Introduction (your fingers are soaking in it now).

Part 1: A simple stochastic duel.
The simplest form of this game considers only the probabilities of hitting the opponent, and can be represented as a geometric series.
Related topics: Dueling for total damage.

Part 2: A stochastic duel with differing ranges.
How to represent the advantage of unequal ranges.
Related topics: Lanchester's Laws

Part 3: A stochastic duel with hit probabilities dependent on range.
Related topics: Death March, movement paths,

Part 4: Range and movement in stochastic duels.

To be determined: To complete my purpose, my will need to extend these simple "shoot until you hit" games to "shoot and accumulate damage" games such as Battletech and Squadron Strike. That should be good for a few more parts in the series (I think), and I've already written about some of this before.

I should have Part 1 up tomorrow.