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Tuesday, December 19, 2023

Computational Combat Statistics

Well, if there ever was a post title that would cause readers' eyes to gloss over and tempt them to skip a post, today's post might fit the bill. Had I mentioned that this would be an exercise in mathematical simulation, I may have lost everyone.

What am I doing and why?

Motivation
In the first game of Madonna dell'Olmo (see Madonna dell'Olmo), King Emmanuele III automatically holds the initiative on Turn 1.  With Spanish grenadiers defending the redoubt positioned in front of the cathedral, the prospect of storming the position looked daunting.  Given that taking this position is one of the criteria for eventual victory, the cathedral and redoubt represent a key piece of real estate.  With Spanish support nearby, attacking on Turn 1 may offer the best chance at securing the position before the Spanish grenadiers are reinforced.  What the Piedmontese must accomplish is to either destroy the grenadiers or throw them out of Madonna dell'Olmo on Turn 1.  While success may be short-lived, the loss of the Spanish grenadiers and occupation of the cathedral will set back Spanish plans on this wing for at least two turns.  This is valuable time the King could put to good use. 
Turn 1 starting positions.
In last week's game, the King opted to use his first turn initiative to activate other formations and stand pat with Corbeau's Far Left Wing in front of Madonna dell'Olmo.  Is this the only viable strategy?  Consider a few choices of action. 

Options
  1. Do nothing.
  2. Use first fire volleys with Grenz and attack with the grenadiers.
  3. Attack with grenadiers and grenz.  Corbeau does not lead the charge.  
  4. Attack with grenadiers and grenz.  Corbeau leads the charge.   
Only one of the grenz regiments and Sardinian grenadiers can reach the redoubt on Turn 1.  Can one of these four scenario options lead to success using the rules in play?  Can players assess intuitively beforehand if any of these options provides a path to success?  Intuitively, it may seem that (4) may offer the best chance of success but will this be enough to dislodge grenadiers in the heavy cover of the redoubt?  Perhaps combining an assault with preparatory volleys is useful?  

Discrete Event Modeling
Now, if these events required only a single computation, getting an answer may be straightforward.  However, there are many events in play both sequentially and simultaneously.  To get a grip on possible solutions, conditional probabilities are involved.  To accomplish this task, discrete event simulations need to be built and executed.  Once built, these combat models will be executed 500,000 times per scenario to create a large sample size from which to draw inferences.

Let's looks at each one.
1. Do nothing.  No computations needed here.
2. Grenz volley followed by grenadier assault.  This scenario requires first a simulation of volley fire and defender morale tests followed by paired simultaneous close combat attacks and morale tests by both attacker and defender. 
The grenz' first fire bonus offsets the heavy cover penalty of the redoubt.  The results demonstrate that grenz first volley fails to score a hit and resulting enemy loss through morale failure about 42% of the time.
Assuming the grenz fail to make a mark against the defenders, then the subsequent assault yields the following result.  That is, the defender is only forced out about 30% of the time.
What if the grenz manage one hit?  Well, then the results change to,
with the chance of defender retreating jumping from 30% to 38%.  Still, prospects for success are not great.

Is this worth the risk especially given the risk to the attacker where "Attacker Retreat" and "Fight On!" results carry losses to the attacker?  A “Retreat” result signifies an actual retreat or unit destruction.  In either result, the unit vacates the hex.

3. Attack with grenadiers and grenz.  Corbeau does not lead the charge.
This scenario sees the Sardinian grenadiers assault the redoubt without any preparatory volley from the grenz but the grenz join in to support the assault.  Corbeau does not join in to lead the attack.  This scenario requires paired simultaneous close combat attacks and morale tests from both attacker and defender.   
These results show, that on average, the defenders will be ejected from the redoubt about 57% of the time.  Again, the attacker suffers losses through failed morale tests about 40% of the time.

4. Attack with grenadiers and grenz.  Corbeau leads the charge.
This scenario sees the Sardinian grenadiers assault the redoubt without any preparatory volley from the grenz.  The grenz join in to support the assault while Corbeau leads the attack.  Like scenario (3), paired simultaneous close combat attacks and morale tests from both attacker and defender are required.
With Corbeau leading a coordinated attack at the head of both grenadiers and grenz yields success in throwing the Spanish grenadiers out nearly 75% of the time.  For me, that is a risk worth taking.  Of course, the attackers will suffer casualties but the price for gaining the redoubt seems not too high. 

For this situation, a coordinated assault with an attached general provides the greatest payoff.  Of course, if Corbeau is lost or the Sardinian grenadiers are badly damaged, holding the position may be difficult unless reinforced quickly.  Still, aggressive action by the King on Turn 1 could set the direction for the entire battle.  

Without tackling the complicated and intertwined computations underlying the game engine, itself, it can be difficult to "guess" which approach is best.  Even with a correct guess, confirmation is valuable. 

What would I do?  Attack straight away with full force.  Your mileage may vary.

59 comments:

  1. An interesting comparison between the options. Without all your analysis I would most likely have gone to option 2.

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    1. Without crunching the numbers, (2) would have been my initial choice too.

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  2. 'Computational Combat Statistics' is, for me, the kind of click-bait headline that absolutely prevents me from skipping a post!!!!!

    And I wasn't disappointed - thank you. I think the next big advance in game design would be for game designers to do this kind of thing - basically to ensure the calibration of their rules is WAD.

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    1. Noted for future use in deterring post skipping!
      Thanks for your positive encouragement. I agree with your assessment that designers/developers ought to study the combat engine with a bit more scrutiny.

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  3. Very interesting, as it quantifies something about my own gaming behaviour and indeed that of my regular opponent.

    before reading you, I would choose option 2, because it just feels the right tactical thing to do and a proper use of the Grenz / Grenadier relationship.

    Having read the analysis and seeing that 4 is the mathematically more certain outcome …….. I would still go with 2! I can’t help myself, doing what ‘seems’ proper over-rides the maths and I would be hoping that the deeper, complicated math connections, not delved into, would be kind to me and that I might get lucky!

    Also interesting, is that it defines my regular opponent, who would almost certainly choose option 4, as he reads wargame situations in terms of numbers, strengths, ratio’s and likely success etc and will apply this even above period tactics. He sometimes needs tighter rules to keep him on the straight and narrow - especially on Command and Control and retreat paths etc.

    Also, looking at the charts, I can instantly see how you can easily identify and present double jeopardy and its effects where it occurs.

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    1. Norm, thanks for adding in your thoughts on the topic.

      I reckon wargamers (and really humans , in general) are not so adept at assessing risk. Many studies prove this hypothesis out. I know, I should cite at least one to be more convincing.

      Your argument leads me to believe that you view numerical analysis and "proper" period tactics as (often) mutually exclusive. At least so, in this case.

      In a mission critical operation, would "proper" period tactics recommend splitting one's force in the face of the enemy into two halves with one delivering a likely ineffective volley against defenders ensconced in earthworks while the other half assaults this fortified position?

      In this situation, the analysis and period tactics align in that you hit the objective with as much force as possible, as quickly as possible before the enemy can respond or reinforce.

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    2. I have gone for the strengths and assumed roles of the units. Grenadiers assaulting is playing to their strength and Grenz firing is playing to their strength. I would hope that the rules might reflect that and I say that without the subtle knowledge of what is going on immediately around them and the likelihood of what an enemy can do in their next turn.

      I really really don’t mind losing in a game, so I hope I play the period / tactic, rather than looking for ways that the system might deliver success, therein lays my enjoyment and is possibly why I am quite happy with solo play.

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    3. I wager few of us here think much about losing a game. Heck, I have lost (and repeatedly) to many of the readers here. We are all veterans in this hobby and realize that winning and losing matters not so much when you are enjoying a game and camaraderie around a table. It remains the journey and not the end result where the true experience is found.

      As for playing the period/tactic rather than the system, the ruleset defines the system. That system defines the constraints and boundaries in which players must operate. I don't know how you can play within a particular system while using different criteria for how a game should be played.

      Perhaps this is one reason there are so many rules with players constantly in search of the "system" that matches their perceptions and style of play?
      Some never find that system and are never satisfied.

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    4. I think it mainly explains the 'house rule' phenomenon. Sure, there are a few house rules which aim to simplify, a few which re-work mechanics slightly, but in my experience most of them seem to aim to re-calibrate (even if they don't re-calibrate the specific sub-system, the aim is obviously to re-calibrate at the system level).

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    5. Recalibrating existing rules can lead to unintended consequences or a breakdown of the system completely. Game designers often build a system holistically. With sub-systems and processes intertwined, changing one aspect (or variable) can lead to the entire work collapsing.

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  4. We are always guaranteed an interesting read on your posts Jonathan. I am interested in probabilities but never have the math skills to work them out so I am always interested in reading a professional's viewpoint.
    What would be the long term (game wise) impact of losing Corbeau? Would his loss be worth the risk?

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    1. Thanks, Ben! Perhaps I should include a disclaimer on these analyses that readers should not try this at home.

      On the loss of Corbeau, his wing would lose his combat and morale bonuses, and his command would be Out of Command until the CiC rode over to take up command. Of course, the King may not wish or be able to ride to the Far Left Wing's rescue so it could be on its own. Being Out of Command is not a good place to be in combat.

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    2. No chance of me trying this at home Jonathan! 😂

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    3. Once you write the code properly, simulation effort is downhill from there.

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  5. I'm with Norm on this, and would intuitively feel most comfortable with the second option. Is there anything to stop the Grenz peppering away over several turns until they do score a few hits, and then charging with the grenadiers? I suppose it comes down to the time pressure and how the battle is unfolding elsewhere.

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    1. The Grenz may continue peppering away but when the enemy activates, they can bring up reinforcements and pepper away back! The poor Grenz will be out in the open and subject to First Fire from up to three regiments. Ouch! The Sardinian grenadiers could find themselves counterattacked as well.

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  6. Grand review of play and possible outcomes. My favorite tactics include putting muddy fields in between opposing battle lines to see if the attacking force can take the multiple turn punishment. Breastworks are fun and sunken roads work mayhem as well.
    Thanks for ajj the work that went into this post.

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    1. Thanks, Joe! Hopefully, these exercises in probability help to clarify and reinforce sound decision-making choices.
      I am happy to see that this type of post is useful.

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  7. Interesting exercise. Looking at the options before your analysis, my initial "instinctive" choice was 4. With a cooler logical appraisal, option 2 seemed more sensible (I noted however there was no option to include the general).
    One aspect such mathematical analysis cannot determine is the effect on your opponent. I am minded of Napoleon and his morale to the physical quote here.
    An aggressive display will often discombobulate an opponent out of all proportion to the actual effect. This is especially true where the person is prone to vacilation or defensively minded.
    Even if the attack is unsuccessful, it can achieve results in the mind of the opponent.
    Neil

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    1. Neil, having watched your writings for some time now, I am not surprised that you naturally gravitated to the correct choice.

      I agree on the mental to physical ratio and relationship. I have seen many games where the objective to winning is not to break the opponent's army morale but to break the enemy player's morale!

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  8. Yup almost certainly I would have gone with option two, though I would be mindful of how effective the Grenz fire was before committing the Grenadiers. Then is it better to charge regardless rather than stand back and receive further defensive fire?

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    1. That is a good tactic but if the grenz fire is ineffective and the grenadiers do go in, valuable time is lost.

      This is only one particular situation. There are times when shooting is better than charging and vice versa.

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  9. Yes I found your statistical analysis of these four options interesting too Jon! I have to agree with Scotty re the psychological effect on some opponents too (even though it is quite illogical) - it's quite odd, when we consider these are not real men and no lives are actually being lost, the way some people get upset and prevaricate over making a decision, you would think the lives of their own families were at risk!

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    1. Glad you found this exercise interesting, Keith. Some gamers can make these decisions quickly while others take time to have the decision come to them. Both approaches are OK to me.

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  10. Yup I would have opted for number 4. Interesting post Jon. I suspected that he’d have to go for the defenders on turn 1 and the first move I made was to put them on reserve. I bet being attacked before you can attack screws things up a tad?

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    1. Mark, no doubt which approach you would choose! Given the situation that you know well, not acting immediately allows the Spanish to bring reinforcement up to support the church's defenders. That allows for either a defensive stance with first fire, a counterattack, or both! Placing your grenadiers on Reserve with your first activation was cunning and the right move. Had the enemy been adjacent, you could not have done it.

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  11. Interesting analysis Jon 🤪 thankfully I have found with experience ‘balanced’ games rarely go as expected and thinking too much about the statistical likelihood of victory in any given situation would definitely spoil my enjoyment. I am of course reckless in my wargaming so my option is almost always to charge in 👍

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  12. Interesting stuff, thanks Jon! The mark of an instinctvely good general is clearly the ability to 'intuit' the correct answer without having to run through all those calculations - and I'm afraid I didn't manage to do that.
    Sadly I seem to have gone for option 1 on turn 1 - albeit using the attached artillery to bombard ( fairly ineffectually ) the defenders. On Turn 2, I then went for option 4, but was bloodily repulsed.
    One thought, though: in reality, the artillery would have let fly with a few rounds to soften up the defenses before the infantry charged in. The sequence of 'move, then combat' in these rules prevented that happening within one turn - how about a slight rule change to allow guns to fire before other units in the same command move, if desired?

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    1. David, glad you found this an interesting exercise.

      Historically, when the battle commenced in earnest, I suspect there was already some artillery fire going on as well as skirmishing. The MOVE then FIRE sequence allows artillery to move up into position and fire within the same impulse. Within the context of this scenario, if you want a more meaningful preparatory bombardment, you must forego the quick assault on the redoubt. As you note, you tried the preparatory bombardment approach on T1 and it was ineffective. Perhaps having a pre-game, preparatory bombardment would be useful but if we do that, the enemy has time to bring more units up to contest the space.

      While T1 in this scenario may be a special case of wanting artillery to fire before movement, the rules can handle this sequence using RESERVE. Artillery on RESERVE can activate and fire at the end of an enemy's impulse but before you activate your formation and send it onto the attack.

      I try to avoid rules "by exception" since it introduces deviations from the standard game engine that can be forgotten in the heat of battle. I will keep your suggestion in mind and ponder its implications some more.

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    2. Thanks Jon, as time goes on I am realising that your 'RESERVE' rule is really clever, and can be used in many interesting ways!

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    3. RESERVE is an interesting twist but it is not always so easy to put a unit into RESERVE when you want to use it NOW.

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  13. It’s noticeable that there is such a difference between the instinctive feel and the result of mathematical modelling. Please don’t take this the wrong way but often my experience of this in industry is that the model may not reflect reality (or reality as we think it should be!). Is it a case that the short term overwhelming force is given too much weight and longer term factors e.g. loss of the commander are downplayed? I would also say that I have seen models overturn the conventional opinion.

    On the other hand, I had a manager who, whenever something went wrong, said (without irony) “Whatever we do, we must overreact!”

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    1. Thank you for your comment. There is nothing in your response to take exception with so no need to worry about me taking your comment in the wrong way. Differences of thought are useful.

      I spent a career building mathematical models, simulation, and evaluating these models. In that career, if these forecast and inventory planning models did not reflect reality in some fashion, I would have been out of work quickly. Mathematical modeling is always built upon the vision and understanding of stakeholders, subject matter experts, and modelers to quantify the Real World problem. The two are rarely the same.

      The same holds for game designers. These modeling exercises simulate "their" perspective of how warfare worked. Without doing the work, how do we know what the design deems important?

      Back to this game exercise, of course, players armed with this information on attacking with both regiments having Corbeau leading the way must weigh the cost/benefit and risk/reward in each decision.

      Whether short or long term goals drive the decision process within a game, that is a choice the player must make and live with the results.

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  14. CCS? Well there's something to think about, to try and come to grips with.

    I recall struggling with Algebra and Geometry years ago, so I am not one to comment, however . . .

    I wonder what the percentage of success would be were this particular event played out (computed) 10 times, 20 times, 100 times? Would there be a bell curve kind of thing or would much of the result(s) depend on the roll of the dice?

    For some reason, I am reminded of one of the chapters in a book by Malcolm Gladwell. If I recall correctly, he looked into large armies battling small armies. If I also recall, the situation or warfare scenario was not as simplistic as one would have thought, meaning the country with the bigger army did not always win.

    Anyway, certainly a different kind of post.

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    1. Hello Anon,
      The answer to your question is in the post, itself. For each of the four situations, I simulated all of the die rolls and conditional probability (branching) computations half-a-million times. With 500,000 events in each case, each having multiple die rolls, the statistics you see are from large sample sizes. In the long run, over many games, these results posted represent mean results. If you repeated these exercises many times, these are results you are likely to see.

      Different kind of post but in a good way???

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    2. In the likely immortal words of Homer: "D'oh!"

      Anon quickly checks Amazon for new and better glasses. Also conducts a search for better brain cells. Debates whether or not to sign up for a intro class on computational something or other in the new year . . .

      Good would push it to subjective, so I will refrain. Different, while also an opinion, seems more objective, especially given your usual and much admired fare (as indicated by the comments) of painting project updates and frequent/repeated battle reports.


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  15. Well, I thought my analysis of the firepower tables in the Von Reisswitz Kriegspiel was heady stuff, but I bow to your study of the analytics of this tactical problem! War being an uncertain affair (tabletop or no), I would opt for #4--go strong and lead from the front (the general ordering the assault accepting risk to himself being more reflective of the intangibles involved with an actual combat rather than how it might be managed on a tabletop).

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    1. Good to see that you would go all in with everything available. If the situation is critical, best to put the odds in your favor as much as possible.

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  16. An interesting post and a good mix of responses for sure. I would naturally go for Option 2, as to my mind, it best reflects what I've read about these sort of action over the years. But then maybe I'm justifying this stance as it reflects how I play, possibly ignoring those instances when an immediate coup de main carried that day as it were.

    Rather like Norm, I've played against other gamers who have worked out the probabilities of a situation based upon the rules and gone with that, rather what would have generally happened for a given period, so they could be accused of playing the rules rather than the period.

    Of course in this situation and the limit of Turns available, there is only so much preparatory bombardment etc than you could factor in before the game kicks off in earnest.

    It also brings to mind something touched upon in the latest issue of Warning Order! magazine, of how easy is it to recreacte/write historical scenarios. I think you mentioned in the early post on this scenario of how to make both sides attack, given the situation on the table and the wider context of the battle. Often on small things do battles turn, which would be nigh on impossible to recreate for our games, should we even wish to. I remember my gaming chum who, when at staff college and studying the invasion of Crete in WWII, the German 'victory' came down to one such small action that one would not immediately think was that significant.

    Anyway, enough waffling from me! Again another fascinating post and ones to get the grey cells going for sure:).

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    1. Steve, thanks for weighing in with your thoughts on this Tabletop Teaser. While in the spirit of playing the period or not, I am surprised how many stick to the recommendation that one (likely ineffective) volley followed up by a one-to-one attack against an enemy within a redoubt is the best course of action even when the evidence suggests otherwise. Is ordering 600 men to attack 600 men behind earthworks and expecting success really playing the period? Seems madness to me but perhaps these were maddening times.

      Glad you enjoyed the post and discussion. As you note, the discussion has been fascinating.

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  17. A different and stimulating post Jonathan.
    My first inclination would be to use 2. However, the decision has to be adjusted depending on the tactical context. If time is critical and opportunity may evaporate quickly, then 4 is the choice I would have to make. I have a reputation with the Rejects for putting my generals in dangerous positions (and losing them!), so maybe in the heat of the moment my careless character would take over.

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    1. Richard, you assess the problem correctly in that both time and opportunity are fleeting. If you do not strike now, another chance may not present itself later.

      Is this post much different from a Tabletop Teaser of old?

      Perhaps if I do this again, I should set the stage by posing the situation, solicit courses of action from readers, and then present the data analysis?

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    2. Well that sounds like an excellent idea Jonathan. I think that there are more than a few of us who would enjoy them.
      It'll keep you busy during the winter months as well.

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  18. Interesting timing on your excellent post, Jon - I am tumbling through an idea about converting a 2D6 roll into something else and grappling with the probabilities is enough 'mathemagic' for me.


    I would tend to pick option #4, but what is the chance of Corbeau being removed from action? I expect that it is probably still worth the gamble to launch a full-scale assault as not only will the position be taken it throws a jolt into the opposing commander's plan.

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    1. Thanks, Greg! If you ever want a second opinion on your probabilities work, you know where to find me.

      The chance of Corbeau falling in the assault is really quite small. A leader is only susceptible to loss when a unit in his hex is destroyed. The likelihood of unit loss in this attack is very small. If that does happen, there is a 40% chance that Corbeau comes a cropper. Not much to worry about, really.

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  19. Interesting analysis. Like most I think option 2 is the one most players would go for. It feels right for the period to me which, for me is the aim for a friendly/club game. If I was playing a tournament game, I'd be crunching numbers in my head and would probably have gone with option 4 to maximise chance if success. Hence I don't go for tournament play

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    1. Thanks, Neil! Some player decisions are curious in that the choice to bring a one-to-one attack against a redoubt with less chance of success is deemed more historical than a two-to-one attack against the same. I am no tournament player but I might be hesitant to carry out an order in the former situation under battle conditions.

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  20. Seems to prove that the Monday Night go-to approach is a winner.

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  21. Very interesting posting. I love your focus on such details.

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  22. Yeah, I quite enjoy this kind of stuff as it occupied far too much mental space for me during my drives to and away from many of the games you have put on over the years. Running the statistics for various approaches is another way I evaluate to value of a rules set I am considering. If the scenario and rules seem too deterministic than you lose freedom to explore different approaches, if too random than it loses the strategic value of planning. I have been thinking a lot on this of late as I continue to struggle with finding that balance in a scenario to reward force conservation and maintaining a reserve.

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    1. Jake, we ought to schedule a game so that you can see this in practice.

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