Monday, 6 June 2011

How I Learned to Stop Worrying and Love the Medium Laser


When I started writing this blog back in the day, two odd years ago (and very odd years they have been indeed, channeling Charles Gray for a moment) I was still working through my thoughts and feelings about the combat values for weapons and usefulness of Catalyst Games battle values for use in pick up games.

For a die hard mech bunny knowing what the "real" in game effects arise from the weapon choices that one makes is a very good thing indeed. If only so that when you design a suboptimal mech design you know that it really is suboptimal, and yet meets the operational expectations you've set out. IMNSHO, too many of the mech designs in BattleTech end up suboptimal for fluff reasons that don't make much sense from a real world operational perspective, because the writer designing the mech has trouble translating the brief into a playable unit that would meet the tasks required of it (Did I just say real world perspective for a game of giant battling robots?).

As such one gets to see an uneven mix of designs, where the truly awful are fluffed away as being just like real life, with insert your example of choice, to justify something that while history shows us could exist, yet adds nothing of value to the experience of playing a game of BattleTech. YMMV.

So, I've come to the conclusion that, for me, the best way to understand the effectiveness of the weapon choices is to use the weapons test format of averaging the damage over 36 rounds of firing to come to a damage inflicted ratio. For those of you new to this blog here is the link back for you to read.
   
I've also tagged these as Number Crunch Geekiness for good measure. After all I'm talking real world perspective for a game of giant battling robots, if that is not the very definition of geekiness, I don't know what is.

Going off tangent for a moment, sort of on an affiliated blog of note, if you don't read Skiltao's Xanga  may I recommend that you at least take a look at his posts that are exploring the unliked weapons and equipment in BattleTech, see here:
 
Well worth reading IMNSHO! Now back on topic, and there is one, no matter how meandering and rambling this post may seem. So my plan, when I have time, is to create an Excel spreadsheet that has all the current weapons in BattleTech that will calculate the damage over 36 results for short, medium and long range, and then I'll look at the results divided by range to create a comparison chart of weapon effectiveness.

Now running into the dark laughing and cackling madly...
  

54 comments:

  1. Pink, thanks for the kind words! I will have a proper reply here and to your most recent comment when I am less busy, but I wanted to say that there must be something in the water, because I started on a similar spreadsheet last night (comparing damage only to tonnage, not range yet, though). I wasn't sure how to handle TN modifiers and different range brackets, so I will be very curious to hear how you pick your base TN.

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  2. I'm going to use iterative deduction and chose 4, 6 and 8 as the base and treat all things as being equal and see what the numbers say.

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  3. It's funny, as I read this I am taking a break from studying for a STAT552 final exam tomorrow (graduate level engineering statistics). The concept (and indeed answer) that you are dancing around is called "expected value" or E[x]. As this is a die game where all the possibilities are known and the problem involves a what is known as a "discrete random variable" the procedure is really quite straightforward.

    Taking a summation of the damage inflicted at each of the 36 combinations of 2d6 for each range of interest will get you a valid result but it is sure doing it the long way. It is accurate statistically to simply multiply the damage the weapon can do by the probability that the weapon will hit at a given range. If you were to sum up the hits from the 36 combination ideally distributed trial you propose and then divide that by 36 you will get the exact same answer as simply multiplying the probability of one hit at that range by the damage that the weapon does at that range.

    For example, take a medium laser at range 5, no other modifiers to hit. Base to hit of 6.

    The explicit 36 combinations yeilds:
    (1*0+2*0+3*0+4*0+5*5+6*5+5*5+4*5+3*5+2*5+1*5)

    The terms multiplied by zero represent the combinations that result in misses. The terms multiplied by 5 represent the hits.

    The direct route:
    Probability of hitting = (5+6+5+4+3+2+1)/36
    = 26/36
    Damage = 5
    Thus expected value = (26/36)*5
    And expressed over 36 rounds of firing with ideal distribution = (26/36)*5*36
    Or simply = 26*5

    Both of these would formally be expressed as:
    E[dam]*36 = 130
    where E[dam] is the expected damage of a medium laser at range 6.

    You could factor the two expressions and find out they are exactly alike or you could copy and paste the two terms into MatLab or Excel to find out that they both equal 130.

    A serious fallacy in your analysis however is the neglecting of the overall ranges that the weapons have. In your method (if I am reading it correctly) a Medium Laser, with 5 damage and 0/3/6/9 range (min/short/medium/long) would score exactly the same as a weapon of the same mass, size and heat that did 5 damage at a range of 0/10/20/30. Clearly a medium laser with a range of 30 hexes does not have the same value as one with a range of 9. If you were to calculate E[dam] for each hex in its range and then sum the expected value for ALL of those hexes you would have a better idea about the relative value of weapons that have similar damage (med laser, AC-5) but different ranges. Essentially you are integrating under the damage curve in piecewise fashion. Even this is not an apples to apples comparison, but it gives a little more insight.

    I've actually done some similar calculations in trying to come up with expected values in order to tinker around with BattleForce'esque old fashioned mass wargames in the ilk of Panzer Blitz and could send you the sheets (or just a sample calc row if you, like me, actually enjoy the exercise and would prefer to do the rest yourself). I went to the bother of calculating Exp[dam] for each hex range, taking into consideration range brackets and minimum ranges then aggregated those ranges into 3, 6 and 9 hex brackets for application to different hex scales.

    Would you be at all interested in the sheets?

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  4. I would be interested in whatever you might like to send me, though given what you've written I should be able to code up an Excel sheet.

    My intuitive deduction was that one will then have to divide the values by the range bands in some way? Not sure how yet.

    However while I agree with you about the expected value, some assumptions will be modified for other weapons when one takes in the number of shots that they are allowed to fire due to ammo constraints, which is a lovely confounding variable.

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  5. L Richardson's post went missing so I'm using the copy sent to me to correct this problem:

    If you were to divide the values by range you end up with the longer ranged weapon being less valuable than the shorter ranged but otherwise identical item. Summation of the expected value at each hex range for various bands of ranges of interest is the way to go that way. If anything it does not value the longer range weapon enough.

    See, if a simple summation is used a weapon having an expected value of .2 for hexes at range 21-25 (ie sum=1) will have the same value as the weapon that has an expected value of .2 for ranges 1-5. Or, as another example a weapon doing 100 damage 50% of the time at range 1 only, ie 50 pts, has the same "value" as a weapon that does 10 pts of damage 20% of the time ( E[d]=2 ) for every hex from 1 to 25. I don't know about you but I would MUCH rather have a weapon with 25 range and 10 damage but only a P(hit) of 0.2 than a 100 damage weapon with a range of 1 and a P(hit) of 0.5. So, you need some kind of markup function for hexes at a longer range. A linear multiplier by range is probably not optimal, but who knows. Intuitively I would ponder a logarithmic function so that the difference between E(5hex)=d and E(20hex)=d is substantially greater than the difference between E(20hex)=d and E(35hex)=d.

    Of note is that my own flat calculations were not an attempt to find some kind of CV but simply aggregating the expected amount of damage at a given range for the sake of some kind of attack value.

    As per ammo constraints, it is a valid point but one that begs the question of why 36 rounds? Aside from being the number of permutations of 2d6 do you have a theoretical basis that it is "normal" for an ammo based weapon to be able to fire for 36 continuous rounds? If the typical engagement takes significantly less than 36 rounds of firing (I don't think I have EVER fired a weapon 36 times in a game), such a test unfairly weighs the results towards energy weapons. Coming up with a more accurate combat value than the published battle value is a fairly complex task.

    And that doesn't even address the issue of the differences in standard deviation between an AC/20 doing d damage in n rounds and an array of SRM's doing the same d in n rounds and how this compares to some arbitrary threshold.... The assertion that doing an average of 20 damage to the head with an AC/20 in x rounds is equivalent to the same E[dam] of a SRM array doing 20 dam to the head in x rounds is provably false.

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  6. Actually, I can see I'm confused about the issues here. Not a statistician.

    So by deduction I would calculate values for all the range bands (non-trivial as they are not all the same), so comparing a weapons that falls into fives and tens versus three, sixes and nines would be difficult.

    I really need to do a stats course so that I can learn all the groovy mathematical shortcuts that would make this project more fun.

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  7. Heh, sorry, I have been living and breathing this stuff for six years now, it is easy to forget how hard it was to stuff it all in my brain in the first place. I'll try and lay a basis out and make sure you follow each part before addressing the next.

    First thing that ought be discarded is the idea of range bands. It is an arbitrary distinction that doesn't need a label. Just think of range, ie, 1,2,3,4 ... R. Each discrete range will have a probability of hitting associated to it for each weapon. Each range individually is a useful data point and can be used as an apples to apples comparison.

    To find probabilities, pick what the "design" condition will be. For simplicity I chose "no modifiers" for my own comparison before.

    So, for an AC/20 the ranges would be

    1;2;3;4;5;6;7;8;9

    And the to-hit numbers would be:
    4;4;4;6;6;6;8;8;8

    Now, it might be more "realistic" in game terms to assume that the typical conditions will be an average +2 modifier, in which case the to-hits would be 6;6;6;8;8;8;10;10;10, but for this description I will stick with the +0 modifier.

    The odds of missing with a 4 to-hit number is 6/36 or 0.167. Odds of missing with a to-hit of 6 is 15/36 or 0.417. Odds of missing with a 8 to-hit are 26/36 or 0.722

    So, expressed as probability of hitting (1-P(miss)) our AC/20 ranges look like

    0.833;0.833;0.833;0.583;0.583;0.583;0.278;0.278;0.278

    Since the AC/20 does 20 damage per hit, this 20 is multiplied through. This results in expected damage of:

    16.7;16.7;16.7;11.7;11.7;11.7;5.55;5.55;5.55

    Which is what I was interested in for my own calculations. I wasn't trying to decide what the weapon is "worth" though, just what id inflicted.

    One way would be to count all ranges as being equally important, which I think we agree is false, but it shows the beginning of the method. If all ranges are equally important you can take the expected values for each hex in the range and simply sum them together. In this case the sum is:

    sum(16.7;16.7;16.7;11.7;11.7;11.7;5.55;5.55;5.55)= 101.667

    This doesn't take into account that being able to dish out 5.55 average damage per turn at range 9 is more valuable than dishing out 5.55 damage per turn at range 3. So, to account for this you could apply some kind of coefficient based on each hex range. The simplest would be linear, with the slope of the coefficient being one. That is to say simply multiply the expected damage by the range to get a value.

    So, with our expected damage of
    16.7;16.7;16.7;11.7;11.7;11.7;5.55;5.55;5.55

    and range coefficients of

    1;2;3;4;5;6;7;8;9

    Multiply the two together and you get
    16.7;33.3;50.0;46.7;58.3;70.0;38.9;44.4;50

    which has a sum of 408.3

    Now, I think that is putting TOO much of a premium on damage at range, but there is no statistically "true" answer to that. The relative value of being able to inflict x damage at greater range is not just a function of probabilities. The map, the speed of the mech it's mounted on, the speed of the target, the playing style of your opponent, the weapons your opponent has ect all have a strong influence on how valuable longer range is. So, the coefficient for range is a tricky one to calibrate. I would tend to think of a ln(range)+1 coefficient myself, but that is rather of off the cuff.

    With me so far?

    At any rate with my final exam done today I have suddenly been "promoted" from being a "promising engineering student" to an "out of work engineer", so of course I am going to go out and celebrate this "progress" by going out and getting right hammered. ; )

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  8. And just for quick comparison, the same calculations for an AC/5

    Ranges : R[range]
    1;2;3;4;5;6;7;8;9;10;11;12;13;14;15;16;17;18

    Probability of hitting, including modifiers for minimum range : P[range]
    0.583;0.722;0.833;0.917;0.917;0.917;0.722;0.722;0.722;0.722;0.722;0.722;0.417;0.417;0.417;0.417;0.417;0.417

    Expected damage : E[range]
    2.9;3.6;4.2;4.6;4.6;4.6;3.6;3.6;3.6;3.6;3.6;3.6;2.1;2.1;2.1;2.1;2.1;2.1

    Summation of expected damage
    sum = 58.61

    Expected damage * range : R[range] * E[range]
    2.9;7.2;12.5;18.3;22.9;27.5;25.3;28.9;32.5;36.1;39.7;43.3;27.1;29.2;31.3;33.3;35.4;37.5

    Summation of expected damage * range
    sum = 491.0

    Oh, and I was indeed curious what happened to my post. : ) I was assuming it was a technical glitch but a little part of me was saying "What? What did I say?"

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  9. L. Richardson said: "Now, I think that is putting TOO much of a premium on damage at range, but there is no statistically "true" answer to that...With me so far?"

    I can follow the maths and the logic, but let's deal with the "there is no true answer statistically", because that is where ascertaining value of choices, or effectiveness of choices would yield the answers to effectiveness of weapons in the game.

    L. Richardson: "The map, the speed of the mech it's mounted on, the speed of the target, the playing style of your opponent, the weapons your opponent has etc. all have a strong influence on how valuable longer range is."

    Or, all the things that I would call the confounding variables in our experiment. So, taking a more accountancy like perspective, let us consider value as being a price to be paid rather than something that has "value" per se (accountancy joke I'm afraid).

    A hypothesis might be that lighter longer range weapons are more effective than heavier shorter range weapons. The null hypothesis would be that speed of the mech is more important than the weight, or range of the weapons carried?

    If we define the experiment as the game called BattleTech, then the thought experiment might be imagine calculating the range of outcome probabilities for each weapon, at all of its ranges, with all the possible modifier combinations, and then charting the outcomes as a number of games with range, speed and damage over number of turns perhaps?

    We would have to play with what we put on our x/y axis, but once we have data we can test the hypothesis.

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  10. Well, once you are talking about experiment you get in to the realm of actually recording the results of games. The probability end of things addresses what is likely to occur rather than drawing inferences on what HAS occurred.

    "all the things that I would call the confounding variables in our experiment."

    Yup. Complications.

    "So, taking a more accountancy like perspective, let us consider value as being a price to be paid rather than something that has "value" per se."

    By this I assume by this you mean an approximate definition akin to "Fair Value Accounting" in trying to compare likely weapon choices by a player. Something really really important here is that the "true" value is probably nonexistant. If you are talking about some kind of "cost" of weapons to compare as a balancing act as a replacement for published "Battle Value" or the short lived "Combat Value" odds are you will never have an answer that will satisfy you. If you were to "perfectly" balance a set of values for some group of players that you experimented on where for equal values on opposing sides there was no difference in equipment picked the results would not be constant. If you experimented on a different group of players with the same method your results would be different. If you experimented on the same group of players some time later odds are you would not get the same result. So many of the "confounding variables" really are just that. They are both "confounding" and far more importantly they are quite variable. Which is the best choice: rock, paper or scissors?

    "A hypothesis might be that lighter longer range weapons are more effective than heavier shorter range weapons."

    One could posit that, but that is putting the cart before the horse. The analysis can be broken down further first.

    "The null hypothesis would be that speed of the mech is more important than the weight, or range of the weapons carried?"

    Hmm... Overreach of the term "Null "Hypothesis". The NH result would be that the range of the weapon has -no- significance on the usefulness of the weapon. A more specific NH would be that range is irrelevant compared to the aggregated mean expected damage at each range. But again, that is getting ahead of ourselves. The term NH means that there is no connection at all between two things, that it really DOESNT have anything to do with the price of tea in China.

    "If we define the experiment as the game called BattleTech,"

    Now by experiment do you mean recording n number of games and reducing the data set or do you simply mean taking a probabilistic approach?

    "calculating the range of outcome probabilities for each weapon, at all of its ranges, with all the possible modifier combinations, and then charting the outcomes as a number of games with range, speed and damage over number of turns perhaps?"

    I assume you mean the probabilistic approach then. Again, including everything from the start is getting ahead of oneself and leads to more complex and further abstracted reasoning. This makes keeping the underlying assumptions consistent rather difficult.

    At this point you really need to ask yourself a lot more specifically what it is you are trying to measure or estimate, and how this is different from the published BV.

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  11. Madness. Sheer madness. Pink, don't you have some terrain to paint? I just finished a MUL and am still rubbing my eyes. The prospect of doing an Excel spreadsheet that encompasses all those variables and yields a number meaningful to us lower-order types? The horror!

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  12. Might be easier than you suspect. The hard part is not the calculation but the underlying assumptions about the decisions that players make. I'm sort of curious where this might go. Your MUL is probably a lot more tedious in that it's just so much data entry. For these sorts of calcs most of the numbers are already neatly in tables. It's just applying the calcs...

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  13. @ Steven Satak: Not painting terrain at the moment as I'm desperately trying to get the OPFOR for the next game painted, but once they are done, then I will have lots more terrain to build.

    @ L. Richardson: Your good, you are keeping me honest and not allowing me to jump from A to Z without at least acknowledging B to Y.

    Not interested in replacing Catalyst Games Battle Values, as I'm not interested in replacing rules per se, because that leads to Cul-de-Sac of stuff that is contradictory to the rule set.

    This is rather like rewriting the Bible to remove the bits one doesn't like.

    Rather what I seek to do is make tools that one can use to understand the dynamic processes that make the game the way it is. So, in my head, and remember a thought experiment is always in one's head, I would imagine a database front end with all the calculations being done in tables, where one can pull up a page with with the weapons on and as you change the range and applied modifiers one would see all the damage totals change.

    Really quite a simple thing to do [how hard can it be ;-)]. Then one could run experiment to compare the results from games played to test the usefulness of the tool.

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  14. Ashley Wrote: "I seek to .. make tools that one can use to understand the dynamic processes that make the game the way it is."

    At this point you need to clarify further what it is you are after. There are many things that make the game the way it is. The question is are you trying to make a tool that will predict outcomes based on the decisions that players currently make or are you trying to make a tool that will help players make better decisions? Whatever you come up with wont be both. Once you have a tool that affects the decisions that players make, evaluations made on the way players make decisions will shift to some degree, possibly seriously.

    Take chess as an example that has no randomness in the game mechanics itself. You could examine a large sample of games where a certain pattern emerged and make an estimation of what the next move by a typical player is likely to be. In doing so you would coach the related moves by the other player. Thing is, if the other player knows the estimation of what the first player is going to do and the first player is aware of this the initial estimation has been altered. This sort of back and forth guessing of "I know that he knows that I know that he knows that I know..." is what separates great players from good ones. The great players are able to go through many layers of a recursive decision probability.

    In this way if you are trying to predict what the result of a given pairing of equipment is going to be you would take one approach. If you are trying to advise what a players choices of equipment or tactics should be and/or, given a particular set of equipment, what their choice of tactics ought to be then you would take another approach.

    Ashley Wrote: "a database front end with all the calculations being done in tables, where one can pull up a page with with the weapons on and as you change the range and applied modifiers one would see all the damage totals change."

    That is a lot more specific and frankly a lot easier. In this case the notions of expected value as discussed before are the primary step off to find mean damage inflicted. The second part of that is to find the effect of different variance on effectiveness.

    So, another key question for you to ask at this point is are you trying to figure out how important a long range weapon vs a short range one or are you simply trying to predict the typical damage being inflicted by a particular weapon at a known set of conditions?

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  15. I would like to be able to answer the how effective a long range weapon versus a short range one, but would accept just knowing the damage inflicted under a particular set of conditions. I would posit that the latter will be necessary to answer the former, but could be wrong on this.

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  16. Yes, expected damage under given conditions is the necessary input into estimation of the effectiveness of damage from a particular weapon at range. There are ways of generalizing this, but even the generalized case is a unit value input into an effectiveness at range function.

    To find usefulness at range you need to make assumptions about usefulness at range. I assume you can see the problem with this. Here is where you need to use your intuition and make assumptions or take data and then make assumptions or analyze the geometry of maps and the design of `Mechs and and then make assumptions. I'll leave you to make your own assumptions on that. ; )

    This is where it comes back to the question of the relative worth of being able to do D damage at range R1 over N rounds vs D damage at range R2 over N rounds. The assumptions you are making are what leads you to some kind of weighted damage function.

    I suspect what is leading you to be curious about all of this in the first place is that you are dissatisfied with informal intuitive estimation, also known as wild-ass guessing.

    If you are wanting to employ an analytical method you need to come up with simpler rubric statements that describe elements of value. For example:
    :: The ability to inflict a unit of damage at a range the target is likely to be at is more valuable than being able to inflict that same damage at a range that the target is not likely to be at.

    From just that statement we can start to make assumptions about how far apart units tend to be. This could be based on raw assumptions or perhaps on recording games and looking at the range units tend to be when they fire. Similarly you could look at maps and pick hexes at random (by die roll for example) and arbitrarily decide the typical ranges to that hex that there is a valid LOS. At any rate you need to find a probability distribution for the opportunities to fire that occur. From there you can start to weigh things against each other.

    There is more than this of course but I will stop here to see if you get the notion at this point. You want to find the likely conditions that a shot is typically taken along with the unlikely conditions and quantify how likely/unlikely each condition is. Easiest starting point on that is range. Follow?

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  17. Oh where have I been hiding? ;-)

    There is really just ONE calculation of Expected Value - you calculate the expected value of *hitting*, and then you multiple by the damage done. The average damage for a Medium Laser (for example) is then 5 times this value, or 8 times for a Large Laser, 10 for a PPC, etc..

    As for range, may I suggest instead a scale of Difficulty. Instead of calculating average damage for short, medium, and long ranges, calculate the probability of hitting where your target number is 2 (always hit), 3, 4, ..., 12, 13 (always miss).
    Adding range to this is a matter of multiplying these values by the number of hexes in each range bracket (3 for Medium Lasers, 5 for AC/5's. etc.. You might come up with a better way of doing these multipliers, as Battletech tends to make combat at shorter ranges much more important than at longer ranges.

    Such multipliers to adjust for range are still a bit arbitrary, and it is hard to know what it means, but there is another way of doing this. Create a "distribution of ranges" similar to what you might have in a typical game, so (for example) maybe 10% of all attacks are at range 1, 30% at range 2-3, 30% at range 4-10, 20% at range 11-18, and 10% at range 19 or more. (This could/should be more detailed.)
    Now use this distribution of ranges just as you were using your "average of 36 shots", and calculate the average damage over this distribution of range. Instead of weighting all short/medium/long range shots equally, you will be weighting by the probability of having a shot at each range. A Small Laser won't have a very high average because many of its shots will be out of range (zero damage), but an AC/2 will always* be in range.

    * well, maybe not always. It depends on how the maximum range of your damage distribution.

    This brings up another matter. You don't play the game the same way with a Small Laser as you do with an AC/2. It may not make sense to use the same range distribution for both.

    You can adjust for such things as the weight of the weapon, and it would be fair to cost the weight of the ammo needed for 36 rounds of fire. It would also be fair to count the weight of the heat sinks needed to keep the weapon operating for 36 continuous turns.

    You can keep adding in more and more details to your calculations, but the more you add, the more the numbers you get out will reflect *your* knowledge of the game and *your* opinion of how to represent it in the calculations.

    I have some old spreadsheets that do some of these calculations. I could dig them out for you if you like, but I suspect you might have more fun doing it for yourself. :-)

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  18. >"You can adjust for such things as the weight of the weapon ..."

    not enough detail ... you can calculate the expected damage of a weapon, and then divide by its weight (possibly including ammo and heat sinks). This give a measure of "average damage per ton", which is useful if you want to learn that you should load up your mechs with as many Medium Lasers as they can carry. ;-)

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  19. @L Richardson: It's good to have another statistician (engineer, whatever) in the house!

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  20. Hi, I came across your site and wasn’t able to get an email address to contact you. Would you please consider adding a link to my website on your page. Please email me back and we'll talk about it.

    Thanks!

    Joel Houston
    JHouston791@gmail.com

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  21. Joel I'll get back to you when I'm at my other computer.

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  22. EastwoodDC: At last the cavalry has arrived ;-) Not a statistician and getting into heavy waters here. I agree with what you are saying, mostly because it feels like you are in my head thinking my thoughts and putting them into statistician speak.

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  23. L Richardson: I'm more of an intuitive heuristic analysis kind of gal, and I'm more likely to look for facts first and find theories that fit, rather than try and come up with a theory and find the facts person. Hence I'm struggling to put into writing my intuitive reasoning here.

    However, my experience of the game is that I kind of know that with a few exceptions (one of which I designed myself for the 3055 TRO) that longer range weapons are proportionately not as effective as shorter ones, for example the medium laser versus the auto-cannon 2. This is usually down to being able to dominate the range band the weapon fires at through mounting it on a high speed platform.

    I know through playing games of BattleTech that the speed-to-weapons-range ratios favour shorter range combat.

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  24. Cavalry? More like Don Quixote. ;-)

    You have been playing Battletech a long time, and are now coming back to examining the game very analytically. I'm a relative newcomer since I only started playing the board game regularly about 10 years ago (but computer version before that). This sort of analysis is how I taught myself the game, or at least to play it well.

    I'll bite: Which 3055 mech did you design?

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  25. Well all the 2Cs, apart from the Jenner were mine, but I was referring to my Kraken. You can blame me for pulse lasers and targeting computers rules having to be re-written, I know that Herb does, and I think it is highly unlikely I will ever be asked to write for CGL.

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  26. I remember that `Mech... You were the one who came up with death by a thousand paper cuts? TEN AC/2 ultras? Really? ; ) Are those cannons or a ERMRM-20 rack?

    Anyhow...

    EastwoodDC Wrote: "There is really just ONE calculation of Expected Value - you calculate the expected value of *hitting*, and then you multiple by the damage done."

    Sounds like we agree almost totally on the proper approach. E[x]= ∑ x*f(x) is where I started with all of this. (my first post) I was just leaving P(hit) in fractional notation so Ashley would quickly be able to see how her experiment of rolling an ideal distribution of 2d6 could be factored out to leave P(hit)*Dam(weapon).

    Eastwood Wrote: "As for range, may I suggest instead a scale of Difficulty. Instead of calculating average damage for short, medium, and long ranges, calculate the probability of hitting ... range... is a matter of multiplying these values by the number of hexes in each range bracket"

    My suggestion above is to eliminate range brackets altogether and look at each hex range as a separate discrete expected value, then sum the number of hexes to get a value. Same advice that you have really, this simply considers range brackets semantic and allows for the effect of minimum ranges directly.

    EastwoodDC Wrote: "multipliers to adjust for range are .. arbitrary, and it is hard to know what it means... there is another way. Create a "distribution of ranges" similar to what you might have in a typical game"

    Again, we are on the same wavelength on this one. (See my most recent post above). Basically create a PDF for the odds of the range being X when a shot is able to be taken across all ranges for which there is a weapon that shoots that far (I used [0,35] hexes as the range of the PDF). The assumptions I was suggesting she make are how she would evaluate the specifics of such a probability distribution. She could use a data based approach, examining the results of actual games, an intuitive approach just recalling how close `mechs tend to fight or a geometric approach looking at typical sight lines on standard maps. At any rate we seem to be talking about the same method.

    As to peoples decisions changing based on their weapons that is why I was asking her if she wanted to predict outcomes based on given weapon loadouts or if she was trying to optimize design decisions.

    Direct mass and associated mass were things I was going to bring up after the significance of E[x] and a PDF weighted E[X] were clear to give a benefit:cost ratio.

    Then I was going to bring up the elephant in the room, the statistical significance of the difference in Variance between an average of 60 points of damage from an AC/2 and an AC/20 vs a given armor value on a location. For that I was going to bring up the Bernoulli process for a binomial outcome. Does that work for you?

    And hey, I enjoy stats. Ironically, it was pondering BT 25 years ago in middle school math that got me interested in stats (and engineering for that matter) in the first place. : )

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  27. L Richardson wrote: "Then I was going to bring up the elephant in the room, the statistical significance of the difference in Variance between an average of 60 points of damage from an AC/2 and an AC/20 vs a given armor value on a location."

    Ah that old friend...

    The Kraken AC2 variant is a good example of pushing the rules envelope to the maximum and watching the game system flaws revealed as one rolls enough dice to see high luck component in combat resolution process.

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  28. LR: I agree that we are in agreement. I was in something of a hurry, and your description was much more detailed.

    Variance is indeed the pachyderm on the premises. It matters, because high variability gives a greater probability of a single "knock-out" shot, but the effects also depend on the toughness of what is being hit.
    A Bernoulli process is just right, and if we want to get fussy you could even account for weapons with variable damage (I started to work that out once, not sure what I ever did with it).

    If you know the average damage and variance of a weapon (or a distribution of weapons!), and the amount of damage the target can absorb before it it destroyed, there is a nice application of the Central Limit theorem for reliability/survival (Birnbaum-Saunders). Check this link, you will like it! http://www.itl.nist.gov/div898/handbook/apr/section1/apr166.htm

    My attempt to apply this to Battletech stalled out due to the complexity of the calculations: http://giantbattlingrobots.blogspot.com/2009/07/150-ways-to-destroy-battlemech.html

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  29. Interesting link. More use to me professionally than BT in fact. I am a civil engineer (well, EIT) and fatigue life is of interest to me.

    Anyhow, yes, the effort to try and totally reduce the game statistically is probably more complex than the precision is worth. A nice luxury of engineering vs mathematics is that "Meh, close enough" is an acceptable answer. If I were a better coder I might be tempted to really take Ashley's idea of an "experiment" to it's extreme. Virtually all of the canonical `mechs are available digitally as data (for the various `mech design programs for example).

    The thought comes to pull a massive Monte Carlo method. For each `mech in the MUL, each weapon of interest and each range of interest find the mean number of shots it takes to knock out the target. Use the textbook threshold of a "large sample" at n=30. This way you would even account for fine details of mech design that the investigator had not considered, like just where to place ammo bins.

    But I digress... and have somewhere to be. Up next, Bernoulli...

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  30. Oh, and a comment on rolling so many dice you are sure to get lucky: It is not that bad of a game system flaw. It is very much a real world effect that a bunch of little bits of damage can be more catastrophic than a single big bomb. Consider the case of cluster bombs, less explosive per mass of payload, less shrapnel per mass of payload too, but the dispersal has a dramatically enhanced on-target effect.

    Just ask the little kids that pick up unexploded bomblets every year...

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  31. > The thought comes to pull a massive Monte Carlo method.

    I've had that thought - but I'll have to save that story for another day.

    Dan

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  32. So, basically where this stands is that the statistician and the engineer agree that expected damage as a function of range is the ideal step off point for what you are looking for. I have a spreadsheet I can send you but I can't seem to find an Email link. Drop me a line at lXeXuXmXiXsXatXgXmXaXiXlXdotXcXoXm (remove the X's to see my email) and I will send you the partial sheet. I have only done the inner sphere autocannon and will leave it to you to do the rest. I have also left it up to you to draw your own conclusions about the relative value of being able to do X damage at N hexes range for all X. And of course what to do with this output is similarly left for you to reason. From this sheet you should also be able to figure out how to run the calcs again for various to-hit modifiers being applied.

    Once you indicate that you are following this method so far (I think you do but I don't want to jump ahead just assuming that I explained it competently) then I can give you some suggestions on how to fire a Bernoulli elephant gun...

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  33. In my other research I came across this passage from a 1944 article on the effectiveness of Sherman tanks vs 72mm and 88mm tank guns.

    "The complexities of military tactics (have) proved for a long time intractable, since even the smallest battle is a bewildering compound of variables, and new methods had therefore to be worked out before there could be any hope of results. In spite of these difficulties, each of the six Operational Research Sections set up at one time or another with the Field Armies achieved a considerable measure of success. But where the future is concerned, it is not so much the results they achieved, however valuable, as the methods they used, that will matter. For the superficial details of battle may be altered in a moment by the introduction of a new weapon, while the underlying principles of warfare scarcely change from one century to the next. "

    Just thought that it seemed appropriate here. Have you been able to open that file?

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  34. Haven't had time. Work has been a bit of a nightmare recently.

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  35. Heh. Lucky you. Know anyone hiring engineers? ; )

    At any rate, take your time. I just hope it is useful to your quest.

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  36. It seems to me this would not so much a way of replacing BV as it would be assessing a new weapon design. Something I've seen a lot of, recently.

    Many of the games out there, the campaigns at least, seem to develop their own weapons and it would be handy to be able to run a given design through Ashley's 'Damage Engine' to see if it balances more or less.

    Is it not possible to use the examples of 'complications' to determine what to leave out when attempting to sum up a variable as a hard number or equation? Since it is so very hard to express as numbers something which nevertheless exisst as a real influence on the game, perhaps some sort of fudge factor for a given type of play?

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  37. You're right Steven. Never meant to imply a replacement of BV, just a better understanding of the relationships between BV, C-Bills, and tonnage, some heuristic tools if you like, to base assumptions on, so as to know what weapons are really effective.

    A sort of mathematical version of the five Ws; what, when, where, whom and why (how) of weapons?

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  38. Howdy. I had a long post typed out but it got lost in the internet... here is an abridged version.

    Basicly, the 'gameyness' of battletech makes your mathmatically model of damage forever flawed. Prime example is a super fast mech like a Grendel with a weapon like a clan er large laser on a flippable arm. A math model and a logical model will tell you that said fast long range mech will never lose to a slower shorter ranged mech. They simply have to keep the Grendel at range 25 and spend a few hundred rounds of battletech at said range, and any opposition will be gone.

    Enter the gamey constraints. First is the maps. Limited map size is a problem. If rolling maps are instituted then the opposing mech must find some hole to hide in, as he can do the math as well and see that if he steps out he will die. I believe the term is 'Turrettech' for this situation?

    Another huge gamey constraint is time. Expected values as above work because they take every possible value into consideration. However, many battle reports involving 12 or so mechs end between turn 5 and 10, and only involve perhaps 3-6 rounds of shooting. In games with time constraints, closing the distance as fast as possible, even on long range mechs, is the only way to do damage. After all, if you only have 10 total turns with the above grendel, and need to do at least 70 damage to the enemy, then the grendel is going to close very quickly to bring to hit numbers into the 5+ range or risk running out of time when the game ends.

    Anyway, my long winded point is that while we know that the king and queen of battletech weapons are the medium laser and PPC, all weapon math involved is going to be very flawed because the gamey nature of battletech means that logical and mathmatical answers to what are good are terrible 'in-game' answers.

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  39. Holy cow! L Richardson and EastwoodDC covered everything I was going to say and then some!

    What I'd really like to do is make a graph, with range on one axis, target numbers on a second, and expected damage on the third. Hard to do in 2D. :(

    Would help me see what decisions to make, in-game, to deal with those gamey constraints though.

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  40. @DevianID -- Also, mechs without hand actuators are really bad at playing Chess - the only move they can make is kick-over-the-table.

    The situation you describe is beyond the scope of what Ashley is trying to do. It's true that you cannot put a good single value on weapons for every gamey scenario that player's might come with, but you *can* come up with good numbers for a specific situation. You *can* get reasonably good numbers (better than BV) for a general scenario that most players would call "a fair game".

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  41. DevianID: Your comments are hardly long winded in comparison to what has gone before. So thank you for contributing, and welcome to my blog.

    As for the game mechanisms of BattleTech, that will have to wait and be addressed in future posts, though I have skirted around the issue in prior posts.

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  42. apologies if my previous comments came across as snippy. It seems I was grumpy when I got up this morning --> I blame it on lack of sleep and not enough coffee.

    Mmmmm ... coffee!

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  43. NP Eastwood. And Ashley I have posted before under BradleyN but I do mostly lurk. Anyway my point was that a graph with expected damage is not what you can expect in a game. And I still disagree that you can find a decent number of worth without making a specific set of assumptions first. Like for example every weapon must be reevaluated at each different gunnery skill versus each other gunnery skill. That does more to set your range than the actual weapons range IMHO. Its just such a complicated topic when you use 2d6 for your distributions.

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  44. I think you are misunderstanding the intent - there is nothing here about measuring weapons vs weapon (or Mech vs Mech), but rather about measuring weapons versus a standard. Further, it is not a fixed standard, but a set of conditions each assigned a probability.

    You can do the same thing to evaluate Mechs, by determining how much damage they might do in a set of conditions. Rick Raisley's Weapon Value in HeavyMetal Pro is an example of this - he adds up all the damage a mech can produce at all ranges (and adjusted a bit for heat, I think). He has essentially assigned equal probability to all possible ranges and added up add the expected damage (tho he doesn't describe it that way). Ashley is doing a very similar calculation.

    Raisley's range distribution isn't very good (IMO), because he assumes all ranges are equally likely. Anyone who plays Battletech knows that short ranges are much more likely then long ones.

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  45. Hey Eastwood, my point I suppose is that you cant accurately compare weapons versus a standard, as each weapon has a different standard not only on different mechs but with different pilots versus different mechs, ect.

    How big is the scope of the weapons breakdown I guess?

    I mean, lets compare the difference in damage between something like an ER medium laser versus standard medium lasers. Is it fair to calculate range at 3 and 4? When comparing the damage of these 2 weapons, its my belief that straight comparisons of damage fail to illustrate the in-game damage such weapons will deal.

    As I stated, it seems ingame players get perhaps 10 rounds to a game before time is up, while in something like megamek you may go 30 rounds in the same time. This makes a huge difference in what each weapon can do. In the short 10 round game range is a much bigger deal, while in a 30 round megamek game there is more time to close into the shorter ranged lasers effective range bracket.

    I dont know, maybe the scope of the weapon breakdown is not that large. However, if it indeed is to try and find accurate BV-like weapon worthwhile values, then I feel the game system shoots you in the foot. Kind of like the pilot skill increases do--because the value of something like a 1 better TN changes at every probability step on a 2d6 distribution, and average worth based on all values just encourages players to stick to the most beneficial area on the 2d6 distribution for larger than expected benefits.

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  46. Well you certainly can't compare against a standard if you keep changing the standard. Don't do that. ;-)

    You can compare the relative value of weapons in a given setting, and by averaging over a set of reasonable conditions (assumptions) you can come up with some pretty good estimates of value. This is considering the question "How much damage can the Weapon do, on average, in a given set of conditions?" These estimate will depend on those conditions, so the numbers Ashley comes up with will differ from the numbers I come up with, reflecting the difference in our opinions of how we ought to do it. But if we both make reasonably good assumptions, then our estimates probably won't be too far apart. Since we are averaging over a number of conditions, it could be that our differences really don't matter all that much.

    If you take a weapon and put it on two Mechs of differing capability, then you are no longer comparing weapons but comparing mechs. Now the question becomes "How much damage can the Mech do, on average, in a given set of conditions?" We can still compare to a standard - it the same calculation with greater complexity (MUCH greater), and it is now a standard for Mechs rather than for weapons.

    Now comparing strategies is a different animal, and you are correct that a single value (per Mech) won't tell you everything you need to know. The previous sort of calculation doesn't apply very well, because it is hard to define a standard for how people play, and it would be pointless to try to put numbers on it.
    What would be helpful is to have several values per mech to give its value at different ranges. You might then select your mechs (based on firepower, range, and mobility) to support the strategy you want to play, or play the best strategy for the mechs you are given.

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  47. Hello,

    I’ve been working on a similar system. Weapons are rated for minimum and maximum range, heat, crits, and ammunition

    Mech are rated on speed, jump, equipment, crits, and total armor tonnage.

    Now I am trying to rate special equipment like C3 Computers.

    A kind of CV rating.

    The CV of each weapon and equipment is combined with the mech ratings to get a final CV.

    These are also used to create a rating for large units of lance and above, to make a quick play large scale battles system.

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  48. Well I hope we see something from you in due course either on one of the forums, or your own blog when you feel ready. I think it is always good to share ideas and the internet is quite a good place to do that.

    BTW: thanks for the comment, and to everyone, thank you all for reading my blog.

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  49. You know, I was looking at how BV was calculated and noted that a lot of what we have been talking about is covered in that calculation. The explicit form can be found at http://www.heavymetalpro.com/bv_calc.htm

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  50. I will check that out. I've not been able to open the xml file you sent me, as it keeps crashing my machine.

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  51. That is odd. What spreadsheet software are you using?

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  52. Excel X for Mac, old program, version 10.1.7.

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  53. Aha... I assumed something a bit more modern.. I will re-code and send...

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  54. A bit more modern... harumph sir.

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