Peering Into the Charge Sequence of TF-ON

Two Flags - One Nation (TF-ON) is Norm Smith's grid-based ACW rules (see: Battlefields and Warriors).  In a recent replay from Sound Officers Call's Seminary Ridge game (see Seminary Ridge), rules' questions led to a discussion of the charge sequence.  After Norm's clarifications, I sat down to put pencil to paper intent on investigating the math within.  For those not interested in probability theory and a brief look into computational statistics, you have been warned.

First, the steps of the Charge Phase before entering Close Combat include:

1. Attacker declares charge
2. Attacker takes Capability Test to assess if charge is FULL or HALF-HEARTED
3. Defender takes Capability Test to react
4. If Defender passes test then likely fires on charger
5. Attacker takes Capability Test if receiving at least one hit
6. Attacker closes for Close Combat or is stopped short

Given the above sequence, the Charge Phase will require either two or three unit Capability Tests (one for the defender and either one or two for the attacker) before combatants actually cross swords.

What was I hoping to unearth?  I was in search of a tactic for improving my play!  I wanted to know the probability of an attacker successfully charging into Close Combat against a defender.  If successful, was this charge going in as a Full or Half-Hearted charge?  With the two or three Capability Tests required and a comparison between Modified Capability Levels of attacker and defender, the answer is not immediately obvious or easily computed.

The diagram below illustrates the process flow for the Charge sequence.  In the flow example, the Attacker and Defender's Modified Capability Level is 6.  What is a Modified Capability Level (MCL)?  This is the unit's Capability Level modified by any situational modifiers that may be applicable.  A smaller Capability Level is preferred to a larger Capability Level.  MCL is then used in a unit's Capability Test.  Probabilities shown are based on one sample of 10,000 trials.  Different samples with differing number of trials may yield slightly different results.  The probabilities shown in the diagram are reasonable approximations to the theoretical probabilities for each event.  

The first decision point is whether the attacker's charge goes in as Full or Half-Hearted.  In this example, the split is 73% Full/27% Half-Hearted.  

Next, the defender tests to see if it responds.  To affect the attacker's ability to close, it is not sufficient for the defender to simply pass this Capability Test.  To have any chance at repulsing the attacker, the defender must pass the test and inflict at least one hit on the attacker.  Without this hit, the attacker makes no second Capability Test and goes into Close Combat with certainty.

If the defender is successful in both passing its test and inflicting at least one hit on the attacker then the attacker must make a second Capability Test.  If the attacker passes this test, it closes with the defender.  If not, the attacker is repulsed and no Close Combat ensues.

After computing the outcomes for these assorted possibilities, we see that the attacker is expected to enter Close Combat with a Full Charge about 61% of the time, Half-Hearted Charge 23% of the time, and repulsed 16% of the time.  Is this result surprising?

How do these computations change if the MCL for attacker and defender varies?

First, consider the attacker's probability of failing to close with the enemy as shown in the chart below.  As expected, for any attacker's MCL, the probability that the attacker fails to close decreases as the defender's MCL increases.  Two take-aways from this chart are,

• A smaller attacker MCL is always preferred across all defender MCLs to a larger attacker MCL.     
• The probability trend of failing to close across the span of defender MCL steepens as attacker MCL increases.

When examining the attacker's charge disposition of either Full or Half-Hearted charge, results are interesting here too.  In the example of Attacker MCL=6 and Defender MCL=6, we saw that the attacker attacks at Full Charge about 61% of the time and Half-Hearted Charge about 23% of the time. 

Notice the effect as attacker MCL increases above six.  Both the probability of charging Full and Half-Hearted increase at an increasing rate of change with Half-Hearted increasing at an even greater rate.  Does this make sense intuitively?  Fortunately, it does.  As the defender's MCL increases, the defender is increasingly less likely to pass its Capability Test.  With that, an attacker is increasingly less likely to take a second Capability Test, guaranteeing closure with the enemy.  Unfortunately as an attacker's MCL increases, the attack is more likely to go in as a Half-Hearted charge.  

While many of these results may be intuitive, breaking down the probabilities embedded within a ruleset can provide insight into design philosophy and intent.  What can be inferred from this exercise of digging into the Charge Sequence?

• The first attacker Capability Test to determine either Full Charge or Half-Hearted Charge is independent of defender status.  I think this an interesting point for debate.
• A smaller attacker MCL is always preferred across the range of defender MCLs to a larger attacker MCL.  
• As Defender MCL increases, the probability of charging home increases.
• As Attacker MCL increases, the probability of charging in Half-Hearted increases at an increasing rate.

Will this information change my style of play?  It may.  Charging with an attacker MCL above six, may give pause to reconsider.  What will be interesting, for sure, is to put this analysis into practice on the gaming table to validate decision science at work.