Measuring fluke and Command and Control

This article describes how to quantify the amount of bad luck (or fluke) that affects the unfolding of a game. It also allow to quantify how much the stochasticity of dice roll dominate the game play (Command and Control).

Command and Control
If a player minimizes the effect of randomness by following a sound strategy, the log likelihood of his actions will be maximized.

Let's consider the following table of values:

Target Number	Log-Odd Ratio	Approx LOR
3	-7.75	-8
4	-5.76	-6
5	-4.37	-4
6	-3.28	-3
7	-2.39	-2
8	-1.51	-1
9	-0.70	-0.5
10	0	0
11	0.76	0.5
12	1.51	1
13	2.39	2
14	3.33	3
15	4.24	4
16	5.61	6
17	7.63	8

It is convenient to use the approximate values as they are easier to add mentally. 

The interpretation of the CC score is that the larger the CC score is, the more likely that on average the player maintain the control of the narrative. This doesn't mean that the strategy is sound, however. A player making many trivial initiative will get a large CC without necessarily achieving its objectives. The CC score thus makes sense only if a player achieved the objectives set for its actor or faction. CC are most comparable among opposing players for a given game. Discrepancies in CC to complete all faction objectives is diagnostic of a game imbalance. Note that simulations are not expected to be balanced, however. A better than average CC for a successful strategy indicate superior gameplay.

Fluke
Almost everyone who played a game at some point and attributed defeat to bad luck (it seems to happen to me all the time). The Fluke score is an objective way to determine how surprising were the negative outcomes for a player or group of players. The procedure is analogous to the CC score, but sums only the LOR for initiatives that failed.

The expectation is that, on average, initiative will fail when their target number is at 10 or lower. In all of these cases, the fluke score is expected to decrease as the game progresses. The fluke score increases only when an initiative with a target number above 10 fails. In the possible case of a sound strategy undermined by "bad luck", the fluke score will be positive by the end of a game.

Over a long enough game, a sound strategy that maximizes CC will maintain a fluke score around 0. The more risks are taken, the more negative the fluke score is expected to grow. If the fluke score is positive, then a player may claim that the same strategy should work if it was to be repeated a large number of time. Let's all keep in mind that there is no such thing as bad luck: and blaming the dice as unfair is an hypothesis with a very low prior probability.

Comparing CC and Fluke

	CC<<0 Poor Control	CC is small Weak Strategy	CC >> 0 Sound Strategy
F << 0	No sound strategy was found OR the scenario is unbalanced.	Set your daring moves better.	Too many wild gambles
F is small	The few sane moves you made should have worked... they didn't.	10,000 monkeys would play like this on average.	Expected outcomes for  a sound strategy on a hard scenario.
F >> 0	Almost impossible to get here.	The dices were not kind, nor was your strategy sound.	You breezed through it, OR the scenario is too easy.

These outcomes were determine with a simulation approach comparing consistent and inconsistent  poor, weak and strong strategies. 10,000 games of 20 initiatives were simulated. Weak strategies had a range of target number centered around 10, poor had the distribution centered slightly below 10 and sound strategies was centered above 10. Consistent play had 5 possible targets values while inconsistent play drew from 7 possible target values. This combination of 6 gameplay style was used to generate the table above.
An unbalanced scenario will cause CC to be negative and large and fluke to be similar to CC. A player consistently relying on long shots will get the same results. The better the strategy, the better is the control of the narrative and this the higher is CC. However, the fluke score should not rise significantly above 0 unless a players proposes exclusively strong initiatives. At this point, the umpire should increase the difficulty for this faction, somehow.

Automaton - playtest 1

This playtest is a first shot at testing the automaton. The setup is simple: it is assumed that the player never address the automaton in any initiative. Three cycles are tested.

Scenario
You are Edward, prince of Wales, heir to the English throne and better known as the "Black Prince". You landed from England with 1000 men-at-arm and their retinues and 500 archers on foot. Overall, this force needs 2000 horses but only 1000 have arrived in good health. You are encamped in the suburb of Bordeaux where your forces are getting restless. Supplies from England are insufficient both in gold and food. There is a shortage of arrows and your knights are complaining of the lack of steel armors from Northern Italy.

Weekly cycle against the Black Prince

1. The food stock possibly become low.
2. The fodder stock possibly become low.
3. The ammunition store possibly become short.
4. The attrition through desertion is very rarely significant.
5. The attrition through the plague is very rarely significant.

Round 1

I break my army into three forces: one will go to Bergerac and the other to Angouleme. I remain in Bordeaux with the administrators, and entrust the two other forces to noble and effective leaders.

OK, reorganizing is a technical task and you are trained in this area (target: 12). There are plenty of good leader to choose from (+1), and the men are itching to leave the camp (+1). However, horses are in short supply (-2) and so are other goods (-2) which are required for military operations (food, fodder, etc.).

Target : 10 roll: 6. A smashing success.

Your forces split into three roughly equivalent groups and take their leaves. All is done on good order, despite the various shortages. 

Round 2 

Great, I send foragers up the Garonne River to collect food and fodder. All proceeds are stockpiled in Gascon's stores. This will work because the Bordeaux valley is very fertile and yet to be touched by raiding armies. 

I assume the planned outcome is that the food and fodder stores will be satisfactorily stocked once this is completed. I accept the fact that the valley is fertile (+1) and unspoiled by raiders in the last few years (+1). However, there are some resistance in the Gascon's camp to plunder within their own land (-2).
Target : 10. roll: 13. Failure. (Delay narrative after the automaton's turn).


Automaton:

1. Food shortage (10), smaller force (-2), pre-existing hunger (+1). roll: 15. Food shortage is over.
2. Fodder shortage (10), understrenght in horses (-2), already in a shortage (+1). roll: 12. Fodder is fine.
3. Ammunition shortage (10), no battles (-2), already short (+1). roll: 15. Ammo stock are OK.
4. Desertion (6), camp life is boring (+1), short in food (+1). roll: 7. Desertion is significant.
5. Death from plague (6), camp life (+1), city nearby (+1). roll: 8. Plague affects the camp.

In large part because the Gascons have opposed, the foraging operation is stalling and you must spend too many good days trying to raise enough supply to furnish the camp. Meanwhile, the plague and desertion are taking too many good men from your ranks. On the up side, fewer people to feed means that the food and fodder stores are now appropriate. Camp life is demoralizing and unsanitary.


This ends the first weekly cycle.

The TOEM Automaton

Motivation
The automaton is a mechanical device against which players have to compete in addition to any competition between players. In fact, it introduces the concept of "cycles" (see below), where each time that a player loses the initiative, a number of initiatives are triggered. It is easy to tune the challenge rating of a game for a player by changing the nature or likelihood of the initiatives built into the automaton.

Definitions

• Automaton : A non-playing entity that triggers a set of initiative each time that a cycle is completed.
• Cycle : All game actions between the time that a player propose an initiative after receiving the conch. The trigger point of the cycle is, however, immediately after a player has to surrender the conch.

Procedures

Setting up the Automaton

Each actor is associated to a cycle. Cycles can be shared if more than one actor are acting under the same kind of constraints. For each cycle, a set of initiative is prepared. All of these initiative are listed in no particular order, and are worded such that the outcome is against the interests of the actor(s). More than one cycle can be setup: the umpire decides which cycle(s) are triggered if there are more than one to chose from for a given actor.

Executing the automaton

The automaton executes all initiatives in a cycle simultaneously. This means that the outcome of one of the cycle's initiative doesn't have an effect on the other even if each initiative has to be resolved in a sequence. The state of the actor is then updated and the conch passed to whoever should own it as per the original rules. 

Creating cycles

Let's use a game setting where a medieval army is raiding the country side (which is what Chevauchée will be all about). There can be a weekly cycle for the travelling army that goes like this:

• The food stock possibly becomes low.
• The fodder stock possibly becomes low.
• The ammunition store possibly becomes short.
• The attrition through desertion is very rarely significant.
• The attrition through the plague is very rarely significant.

This means that food, fodder and ammunition have a base target number of 10 to become low. Attrition becomes significant only on a base number of 6. Of course, the player's actions and the current state of the actor can be used as PROCON. For example: low food stocks is a significant factor in promoting desertion.  Travelling through a large city increase the exposure to the plague, etc.

The game can be made harder by simply increasing the chance of running out of something (e.g.: the food stock is likely to run low), or made easier by decreasing this outcome (e.g.: the food stock rarely become low).

Strategic notes
An automaton is essentially a dedicated opponent to a given actor/faction. It acts as a player that is systematically aggressive such that staying "alive" is a challenge and only the very skilled players will have the leisure to go after other players. The automaton should be tweaked such that a player ignoring key factors of a simulation will not survive. It also acts as a way to shape a game by setting objectives that cannot be ignored. To follow up on the previous example: not only the Black Prince needs to undermine the support for the King of France in Poitou, but he must at the same time maintain his troop fed and in supply, and compensate for the unavoidable attrition of a campaign. 

The Kybosh house rule

A friend is playtesting TOEM in a classroom setting. He came up with the term PROCON to identify the phase in which all are invited to provide PROs and CONs to an argument. More interestingly, there are a few innovations that are worth thinking through further. One of them is what I'll call the Kybosh rule:

The Kybosh house rule

Following the submission of an initiative, the referee tallies the number of PRO and CON that are rejected. When three such PROCONs are rejected, the PROCON phase terminates and the initiative is resolved.

Rationale
This rule tackles the problem arising with hair splitting for PROCON.  The most significant PROCONs are more likely to be itemized first. When the PROCON devolves into a laundry list of unacceptable facts/arguments, this is probably because nothing of value will be added to the discussion.

How to break the game with this rule

1. The conch holder proposes an initiative with a number of PROs and three lousy CONs. The result is that no one gets to contribute valid CONs.  

This can be circumvented by letting everyone talk and resolving all PROCON as a batch. A complete PROCON brainstorm ensures that all who wanted to talk either have talked, or intentionally passed. 

Final draft of proposed house rule

A full round of PROCON implies that all who wanted to contribute are given the chance. First, all PROCON from the conch holder and considered. Then, the referee tallies the number of PRO and CON from other players that are rejected. When three such PROCONs are rejected, the PROCON phase terminates and the initiative is resolved. Further rounds of PROCON may be allowed otherwise.

Multiple outcomes

I'll try to rationalize the rule on multiple outcomes.

The rule
Multiple outcomes and a success.

The number of possible successful outcomes equals the positive difference between the target number and the sum of the dice roll used to determine the success of the initiative. Outcomes becomes true in the order in which they were proposed by the conch holder.

Example: An initiative is made with four outcomes. The target number is 11 and the sum of 3D6 is 9. The first three (11-9) outcomes are said to be true, leaving the last one false. 

The rationale
Two neutral initiative (target=10) have a combined probability of consecutive success of 0.25. If one combines the two outcomes into 1 initiative, the success of both outcomes will have a base target number of 9. It is thus slightly advantageous to tack in two outcomes to one initiative for as long as the base target of the argument is 10. If it is any different because the base target is set by training/complexity (see table in the rules), the initial advantage dissapears. This is so because the difference in probability in the 9-11 range is flatter than for the rest of the spectrum.

The downside is, however, that the logical link between the outcomes can be attacked with CONs arguments/facts. At -2 per cons, any advantage is lost unless the link between both outcomes is rock solid.

More than 2 outcomes makes less sense, unless the conch holder sees the additional outcomes as nice to have bonuses.

DK module, playtest 1

Setting

You are a sturdy dwarf, leader of dwarves and in search for a suitable place to lay the groundwork for a brand new mountainhome. Your name is Hekron, you are an experienced Engineer and Leader. Your company of dwarves includes 5 trained axe-dwarves as well as an experience mason (Morice), an experienced carpenter (Leon) and three experienced farmers (Lia, Ben and Groog). Being versatile, all non-specialists (experienced level) know each other's craft or profession at the trained level. 

The  land that lays before you is mostly inhabited by human and goblinoids. You have located a number of tall and soft chalk cliffs overlooking the stormy sea. The climate is temperate, the season is early spring. The coastline is speckled with tiny hamlets of fishermen, either humans and goblins, living in apathic ignorance of each other. 

The cliffs nearby are likely to contain caves of various sizes. Your men are turning to you for guidance: they are weary of travelling and dry rations are running low.

Statistical Basis for TOEM

The target number of 10 is making the following assumptions:

1. There is two possible outcomes (Success or Failure).
2. In absence of evidence, both outcomes are equally likely.

This thus means, that the most reasonable probability for both outcomes should be equal, or with a probability of 50% each. This is very much in line with the Bayesian framework , where the target number reflects the posterior probability and the absence of supporting (or not) evidence indicate flat prior probabilities.

The role of additional information
Any relevant fact either support or counter the logical connection between and event and its outcome. Unless some validated modeling can be done, the effect of facts on probabilities is somewhat arbitrary and the course of most complexity in games. TOEM look at things from a narrative perspective. If fact X is such that it could either explain why an outcome occurs (or not), then it it considered. All arguments or facts are thus weighing the same on shaping the probability of an outcome. However, as more facts are adding up, the effect of additional arguments becomes smaller such that the probabilities never reach 0% or 100%. This effectively imply that all arguments have an influence on the outcome that is distributed roughly according to a geometric distribution. The facts/arguments don't need to be ordered, but it is assumed that some are more influential than others: the way the probability change as facts accumulate abstractly ranks them. In TOEM, the ultimate objective is to generate a reasonable narrative, not to model event the way a numerical calculation would.

Here is an example. In the Harpoon 4.5 ruleset, the probability to damage a ship with 2nd generation counter measures using a 3rd generation, sea skimming missile has a net probability of X%. This probability is compounded by the ship's point defense system and the generation of this system's sensor specs. Finally, in the eventuality of a hit, the chance that a vital system is knocked out depends on the missile DP and the ship point value. Finding the exact probability of this happening can be done with the Harpoon 4.5 appendices and rules, but figuring out whether this happens requires modeling the engagement on a 30 sec impulse basis until the missile has reached the target. With TOEM, the following initiative would be made:

Event: I fire at the opponent using my state-of-the-art SSM missile and disable its ship. Outcome: The ship is now dead in the water, performing firefighting activities and treating injured personel. This is going to happen because my missiles are more sophisticated that the ship's defense (+1), using a sea-skimming approach which makes it harder to detect on time (+1). My missile is designed to disable or sink ship of this size (+1). 

The target number is thus 10+3 = 13, or a probability of  84% of success. I roll 10: a clear success.

It is important to note that although there is a ruleset to find precise probabilities for modern naval combat, this is not the case if the situation calls for determining the reaction of a crowd to a certain event. TOEM provides thus a systematic way to assess the likelihood of "fuzzy" situations.

Using prior information to set the base number
There is no reason why a base of 10 should be selected if there exist information helping in setting it to another base value. Here is a copy of the table:

TGT number	Prob (%)	Expected consecutive successes	Narrative levels	Task Simple	Task Technical	Task Complex
3	0.46	1/217	Desperate
4	1.8	1/55	Desperate
5	4.6	1/22
6	9.3	1/10	very unlikely
7	16	1/5
8	26	1/3	unlikely
9	38	1/2.6
10	50	1	possibly	Naive
11	63	1.7	likely
12	74	2.8	very likely	Trained
13	84	5
14	91	10	As a rule, ...	Experienced
15	95	18
16	98	29		Elite
17	99.5	Indefinite	Certainly
18	100	Indefinite

Method A - Probabilities
The second column provides the % probability for all target number. If the probability of an event is known, it is best to select the smallest target number which includes the probability. For example, if  there is a chance in 4 to draw a Spade from a deck of cards, then use the target number 8 which is valid for events of probability over 16% and up to 26%.

Method B - Narrative Levels
Certain words have a specific meaning in a TOEM game. The 4th column lists them with the base target number that they correspond to. This method is simple and works for events that are otherwise impossible to quantify.

Method C - Expected numbers
This is more esoteric, but it can do the trick in some cases. The 3rd column shows the number of expected consecutive success with this target number in average. This information is redundant for target number under 10, but is helpful to set a target number of likely events. For example, if on average one lake out of six contain a shellfish toxin, one would expect to find 5 lakes on average before finding one with the toxin. The initiative: "I eat the mussel from this lake and they are free of toxin." should have a base target value of 13.

Method D - Training and task difficulty
This is a variant of the narrative level which prove to work well during playtesting. The last three columns show the target number for three task complexity against 4 skill level: naive/untrained, trained, experiences and elite. Complex tasks are difficult to achieve, but this simply implies that these task are likely to surrender the initiative while an actor/faction is attempting to complete it.