Monday, September 5, 2011

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
100 0

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.

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

Poor Control
CC is small
Weak Strategy
CC >> 0
Sound Strategy
F << 0

No sound strategy was found
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 >> 0Almost impossible to get here.The dices were not kind, nor was your strategy sound.You breezed through it,
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.

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