**TABLE OF CONTENTS**

How do I start the Conkers Simulation?

What if the simulator fails?

How do the buttons work?

What are the settings?

What is in the score table?

How are the contests simulated?

Why simulate a Conkers season?

Conkers or bonkers? You decide.

Can I see and copy the program?

Copyright, disclaimer and contact.

**How do I get started?**

**1.** Click the top button, observe the table, click the next button, observe the table, then click the top button again and so on for 10 or 20 observations.

**2.** Click the links above for the __settings__ and the __score table__ to find out about the various experiments which can be simulated.

You are probably using an incompatible browser. Therefore switch to Netscape 7 (or later) or Internet Explorer 6 (or later). The arithmetical parts of the simulator were tested on Netscapes 3, 4.5 and 7 and on Internet Explorer 6 and 7.

The function of each button is explained as follows.

__Get next players__- The next pair of opponents is selected at random from the alphabetical list of group members and displayed in the third column of the
**Table**(see below). The button is then automatically disabled and the following button is enabled in stead. __Play the game(s)__- One game is simulated according to the
**Playing procedure**(see below) followed by a further nine simulations if '10 games per click' has been selected in the**Settings**panel (see also below). __Clear scores__- The third, fourth and fifth columns of the Table are all cleared together with the contents of the box labelled 'Total number of games played'. The first and second columns are unaffected. This button is useful if you wish to start a new series of games without upsetting alterations that you have made to the second column.
__Reset all__- The contents of the entire table plus the buttons plus the settings panel are all restored to their original state. The effect is the same as reloading the entire Web page but it is quicker and more convenient.
__Help__- This help file is displayed in its own window. The window can be moved by clicking its title bar and dragging it to a new position on the screen.

The function of each setting is explained as follows.

__1 game per click__- This setting affects the action of the
**Play the game(s)**button. It enables you to see precisely how the scoring pattern develops in the fifth column __10 games per click__- This setting is convenient for observing the scoring pattern over a large number of games such as one or two hundred.
__Each plays all__- The action of this setting is different in kind to the two settings above where players are selected at random from the Member's group, Here, instead of random selection, each player is selected to systematically contest every other player in the group. Although this setting produces a much less realistic simulation of the traditional conker season, it eliminates one of the two random elements leaving only that of the contests themselves. Therefore the effects of the remaining random element can be more clearly observed. In addition, the simulation of an each-plays-all tournament is of some interest in application to the more sophisticated sports such as tennis and football. Note that, with this setting, one click of the 'Play the game(s)' button produces one complete set of each-plays-all contests. In general, if there are
**n**group members, the number of contests will be**n * ( n - 1 )**. __Winner takes 100%__- This default setting simulates the English tradition where a winning conker accumulates 1 point for the win plus
__all__the points that the losing conker may have previously accumulated. For example, if a conker holding 3 points beats a conker holding 5 points then its new accumulated score is:5 + 1 + 3 = 9. __Winner takes 50%__- This setting simulates a rule similar to the one above except that the winning conker only accumulates 50% of the loser's accumulated points. If the loser had an odd number of points then one point is subtracted before halving the remainder, thereby avoiding fractional points. Note that this setting is not appropriate for the real game of conkers because the losing conker is always destroyed. However, the option is included here in case a conkers-style tournament is used for an entirely different game such as tennis, golf or croquet. The general effect of this rule is to somewhat smooth out the extreme points distribution which tends to occur with the traditional rule.
__Winner takes 0%__- This setting simulates a rule whereby the winning conker accumulates just one point. For example, if a conker holding 3 points beats a conker holding 5 points then its new accumulated score is:
1 + 3 = 4. This rule produces an entirely different distribution of points to the two rules described above. Like the traditional 100% rule, this rule can be applied to real conker games.

__'Group member' column__- Each cell in the column contains a letter which identifies a particular simulated conker player. The column as a whole represents the group who play throughout the season. For example, 10 children play each other in a school yard for two weeks.
__'Member's strength' column____The meaning.__The numbers in this column represent the playing strengths of the members. The playing strength is a combination of all the factors which influence the outcome of the game. Such factors include the selection and preparation of the conker in addition to the skill and strength in wielding it during the contest. The numbers influence the relative probable outcome of the game. For example, a player of strength five has a 5:1 chance of beating a player of strength one and a player of strength five who contests a player of strength three has a 5:3 chance of winning.

__Changing the numbers.__A blank cell represents a member who is not playing, so you can delete all the numbers except two in order to observe more clearly the probabilities of winning. A number of any size may be used provided that it fits in its cell box.__'Players in this game' column__- This column displays the two players for the current game namely 'PL1' and 'PL2'. In each game, the two players are selected automatically and entirely at random from the group of active players. The only difference between PL1 and PL2 is that difference (if any) between their respective strengths.
__'Result W or L' column__- This column displays the outcome of the most recent game. W represents the winner and L represents the loser.
__'Member's score' column__- This column represents the current accumulated score for each active playing member's conker. A blank indicates that the member has not yet played and a zero indicates the player lost his/her last game. The distribution of the scores down this column is the simulation's main focus of interest, at least from a numerical point of view. The cell at the foot of this column contains the current total number of games played, as it says.
__The extra columns__- The two extra columns on the right of the main Score table do not form part of the traditional scoring system but they have been added in order to provide further information about the participation and performance of the group members. The 'Total wins' column shows the number of wins that a member (not a conker) has accumulated and the 'Total games' column shows the number of games that the member has played. The results in these columns can be compared with the numbers in the 'Member's strength' column. Note that the season's ultimate champion is not necessarily the member who has won the most games; the champion is the member who owns the conker with the most accumulated points.

__The calculations__- This simulation contains only one significant calculation and it is very simple. The calculation consists of evaluating the solutions of a pair of simultaneous equations ( 1 and 2 below ) to arrive at the probability of winning a contest. Consider two players A and B where:

**s**strength of A ( see Score table )_{A}=

**s**strength of B ( see Score table )_{B}=

**p**probability of A beating B_{AB}=

**p**probability of B beating A_{BA}=

then:

**p**..... ( 1 )_{BA}= s_{B}/ s_{A}* p_{AB}

**p**..... ( 2 )_{AB}+ p_{BA}= 1.0

Substituting 1 in 2 gives the probability of a win for A as:

**p**..... ( 3 )_{AB}= 1 / ( 1 + s_{B}/ s_{A})

and the probability of a win for B as:

**p**..... ( 4 )_{BA}= 1 - p_{AB}

During the simulation procedure, in a function called 'contest()',**p**is evaluated using equation 3 above. Next, a pseudo random number between zero and one is obtained from the Javascript 'Math.random()' function. If this random number is less than or equal to_{AB}**p**then the winner is A, otherwise the winner is B. Therefore, in a particular contest, if_{AB}**p**is less than 0.5 then B is likely to win, if_{AB}**p**is equal to 0.5 then A and B are equally likely to win and if_{AB}**p**is greater than 0.5 then A is likely to win._{AB} __Some limitations of the simulation__- In this simulation, a 'season' is considered to be a fairly large number of games, say 100 to 200. During this session a player's strength remains constant (unless altered by the user). However, during a real conker season, a players strength is likely to change relative to the strengths of some other members of the group. In particular, an initially mediocre player might become very confident and proficient and finish the season dominating the entire group. Another limitation is that the simulation assumes that all the members remain active throughout the session, whereas, in a real conker season, some members are likely to abandon the group, especially the weaker members and especially towards the end of the season. A further limitation of the simulation is that, for any one contest, two opponents are selected entirely at random from the group of members, so all members are equally likely to play. In reality, however, some members are likely to play more often than other members. In particular, the stronger members are likely to play the most games. There are probably other limitations; perhaps you can think of some.

**Why simulate a 'Conkers season' ?**

__The game.__ In English tradition, the game of conkers involves two opponents each equipped with one large seed (conker) taken from the spikey tough fruit of the horse chestnut tree (*Aesculus hippocastanum*, see 'Picture' paragraph below). The conker is pierced and threaded with about half a meter of strong string with a large knot at one end. Each player firmly holds the string at the end furthest from the knot and the conker. The conker can then be either passively dangled or violently swung in an arc shaped trajectory. The players then take it in turns to either dangle or swing their conkers. The swinger attempts to strike his conker against his opponent's in such a way as to break the dangled conker from its supporting string. The player left with a suspended conker is the winner of the game and, typically, the loser will select a new conker in the hope of better future luck.

__The scoring system.__ Winning conkers accumulate points and they are labelled accordingly. For example, a conker with five points is known as a 'five-er', one with thirteen points is a 'thirteen-er' and so on. Traditionally, the rule for gaining new points is as follows: __one point for winning the current game plus all of the loser's accumulated points__. For example if a three-er beats a five-er its new total of points is 3 + 1 + 5 = 9, so the three-er becomes a nine-er. It is important to note that it is not the player that accumulates points, it is the player's conker.

__The context of the games.__Traditionally conkers is played during one season of two or three weeks per year. The season corresponds to the time when the horse chestnut seeds (conkers) fall from their trees in the early autumn. Typically the conker players are groups of 8-12 year old boys who challenge each other in school playgrounds. Eventually, interest in the game wanes and when nobody wants to play anymore the season has ended. The season's champion is the player who happens to own the conker with the greatest number of accumulated points. Note that, although unlikely, the season's winning conker may have won only a single contest provided, of course, that it defeated the previous holder of the highest score.

__The simulation.__ Here, the focus of interest is not upon the social phenomenon of conkers nor upon the playing tactics nor upon the Newtonian mechanics of colliding masses; rather the emphasis is upon the developing progression of scores throughout the season. In particular, this simulation is an aid to studying the following topics.

**A. **Whether the conkers season might serve as a model for other sports where a very loosely organised and unsupervised tournament is sometimes appropriate. For example, a group of neighbourhood friends decide that they would enjoy an informal tennis tournament throughout the forthcoming season but most of the players do not wish to commit themselves to a fixed number of games with pre-selected members of the group. A conkers-style tournament might be suitable.

**B.** Whether other simulations might serve as aids to studying other types of tournament. For example, how long should a ladder-style tournament be run and should it be divided into sub-ladders? What are the effects of too many draws in a knockout tournament. How should points be allocated in a league tournament?

**C.** To what extent are Javascript and similar Web based languages sufficiently flexible and robust to support simulations.

__Picture__. A fine colour photograph of horse chestnut fruits and seeds may be found at:

__The copyright__. The copyright of this Conker Season Simulator belongs to Ian E Currah ( 2003 ) of Stratford-upon-Avon, UK. **You may copy** this simulator for non-commercial purposes **provided that** you copy this note along with it, thereby acknowledging the author. In addition to this proviso, you are also requested to provide a link to the author's Website at http://www.currah.co.uk/.

__Disclaimer__. The author of this Conker Season Simulator cannot take responsibility for the uses to which it may be put by other people. Therefore, if you have more than a casual interest in the results of this Simulator, you should supplement them by studying the relevant reputable publications and by consulting recognised experts in sports and tournament design.

__Contact__. You may contact the author at the Website mentioned above. Please report any faults that you may find in the program's performance.

To move this window, place your mouse pointer on the title bar, hold down your mouse button and drag the window to a new position. To resize this window to full screen dimensions, click the middle button on the title bar. To partially resize this window, locate your mouse pointer exactly on the left-hand or bottom edge, then hold down your mouse key and drag.