Sunday, August 14, 2011

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 



very unlikely






very likely


 As a rule, ...






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.

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