Greg Jones
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Has anyone done a thorough mathematical analysis of how ToEE or any of the D&D board games are balanced?

I would like to make some scenarios and want to be able to analyze them to ensure that they are balanced well without having to play test them an unreasonable amount of time to ensure that they are well balanced.

I also think that if some mathematics were employed to assess the balance of the game then home rules and variants could be analytically assessed to see how they impact the balance of the game.

Anyways, I think this might be a fairly large undertaking and I don't want to reinvent the wheel if someone has already done all the work.
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Greg Jones
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Re: Analysis of Balance
Well I guess I am just going to do it.
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Alan Stewart
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Re: Analysis of Balance
zerosum79 wrote:
Well I guess I am just going to do it.
I think you're gonna have to. I have no recollection of anyone else trying this. We'd all love to see your findings though!
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Greg Jones
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Re: Analysis of Balance
So I actually spent the last day doing this. It was a challenge and I have had to make a lot of assumptions but I feel like its good for a first attempt. I'll post it tonight with my spreadsheet once I make a few more tweaks and have time to write up what will be a long post.
 
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Erin B
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Re: Analysis of Balance
Looking forward to it.
 
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Greg Jones
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Re: TOEE: Analysis of Balance First Pass
So I decided to do the analysis of how different elements of ToEE contribute to balance in the overall game. The reason I wanted to do this is that composing an adventure based off of the original ToEE D&D module by Gary Gygax.

In the original module players go from lowly level 1 up to around level 10 so I wanted to keep this sense of a longer character progression by creating a set of house rules that also level characters from 1-10. I also have added some variants into my house rules on how often event cards are played that should allow for scenarios that are much shorter or much longer than the average 9-12 tile 1hr game to make this ToEE module much more exciting than just playing a bunch of 9-12 tile sections with a boss at the end of each. So because I am making so many changes that muck around with the balance of the game I wanted to do a thorough analysis of the way the game is balanced. That way hopefully when I am designing scenarios later in the module for the higher level characters, the balance is close to right and don't require heavy play testing to adjust.


With that said a crazy amount of assumptions and a significant amount of mathematical calculation have gone into this analysis. For those who are not mathematically inclined I present the high level overview in the form of a graphic.



At its core ToEE and the other D&D Adventure System board games are a battle of attrition. You start with a certain number of Hit Points and try to get through on average 10.5 tiles before you take enough damage to kill a character. Revealing a new tile can create three different kinds of damage: Damage from monsters, damage from traps, and damage from events. Each of these must be analyzed individually and then combined to get an average amount of damage per tile. I assume that the game is balanced when there is a 50% chance of likelyhood that after 10.5 tiles the combined averages of these three types of damage are equal to the total pool of hit points that the party has. I calculate balance 5 different times based on the number of adventurers in the party (Column A). If the game is properly balanced then it should be equally likely to win using any number of characters.

Column B shows the average AC for a party of the size specified. I have made the assumption that the Fighter must be used playing with just one player since he has the highest AC, HP, and damage (up to 3HP when attacking adjacent opponent). Columns C and D are the HP and Surge values you would get on average from a party of that size. You will notice that as you go from 1 to 5 players on each of these stats you converge on the average stats of combining all 5 characters.

Column E shows the amount of damage that a party can do per round using at will attacks. This number is only important to show that a party of any size should on average be able to dispatch an average villain in one attack. This becomes very important when changing initiative rules since if the party gains initiative they will have a much better chance of not taking damage from a monster.

Column F shows the total number of HP that a party has to work with including two surge tokens.

Column G shows the number of HP that I assume remain in the rest of the party once the first player has died and no surges remain.

Columns H, I, J are where the majority of the calculation is really done. They show the chance of taking damage from a monster, trap, or event. Traps are a relatively straightforward calculation but monsters are a bit trickier and events are really challenging. I will break each of them down separately in a different post. It's important to note that because the monsters and traps are relatively constant, the events are how the game makers really balance the game across different sized parties. Some of these cards can do huge amounts of damage to a large party. For example Rage of Imix can do up to 10 damage (two per person on the tile) but only does 2 for a party of 1. These big event cards are really what take up huge chunks of HP from a large party and you can see that they grow progressively bigger doing up to 3x as much damage for every tile played with a 5 person group as opposed to a 1 person group.

Column K and shows the amount of damage caused by playing a tile. It is just the combination of adding all damage from traps, monsters, and events. Column shows the total damage for laying 10.5 tiles (the average adventure length).

Finally Column M shows that each of the five party sizes are balanced. This is the total HP of the party minus the amount that must remain to win and then minus the total damage done per dealing 10.5 tiles. If this column comes out to 0 the game is perfectly balanced and players should have a 50% chance to win. As you can see the result is balanced to within about 0.5HP for each party size.
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Greg Jones
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Re: TOEE: Analysis of Balance First Pass
I am not really sure how to upload a version of my spreadsheet so I am just going to put a dropbox link here for now if anyone wants to download and play with it.

This is the latest version:
https://www.dropbox.com/s/05hcj2pk2wjm54k/Model_ToEE_V1.5.xl...

I've removed a lot of errors and created a spreadsheet that balances across party sizes then shows how the balance gets distorted by leveling from 1-10 on my leveling system (which is similar to other system's 1-5 leveling)
 
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Greg Jones
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Re: TOEE: Analysis of Balance First Pass
Implications of looking at the spreadsheet.

First off lets say that a party moves to my equivalent of Level 10. This is basically a level 5 in terms of stats for most other leveling systems I have seen on the forums.

The way I do this is for every odd level I add 1AC, 2HP, 1 surge.

For a 10th level fighter the stats would be: AC21, HP18, Surge9

Notice now that total pool of HP of a party of 1 fighter would go from 20 to 36. Without some adjustments to the difficulty of the adventure it would be hard for a standard 12 tile game to consume all 36HP of the 10th level fighter and the player would win virtually 100% of the time. So the next step is to figure out how many tiles, additional monsters, and events would be necessary to re-balance an adventure for level 10.

Another thing this spreadsheet does is allows someone who is making a house rule like initiative to assess their impact. You would need to look at how the monster damage section is calculated and rework the equation to account for the probability that hero gets to attack first (rather than always second) and the chance of killing the monster before they will get a chance to attack. This will reduce the average damage that a monster will do per tile and force altering of other rules or addition of tiles to keep the game challenging.

Anyways, feel free to ask questions as to how some of the calculations are done if you can't figure it out from the spreadsheet equations. I did my best to annotate some of the major assumptions in the spreadsheet.
 
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Erin B
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Re: TOEE: Analysis of Balance First Pass
It's all really fascinating, I just don't see how you can be sure of your calculations when so much depends on assumptions and the randomness of rolling a die or drawing cards. Then again, I do see how the difficulty of an adventure lies in how much HP a party has vs. how many tiles you have to venture through, so I guess I at least understand that much of the balancing act. When you level up your heroes, and thereby give them more HP, then you either have to change the distance to the goal, the toughness of the monsters, or the difficulty of the events. Okay, I think I understand the concepts here.
 
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Greg Jones
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Re: TOEE: Analysis of Balance First Pass
I didn't really go into how the calculations are performed but I can give an example.

Based on the geometry of the tiles there there are 20 traps that would need to be disabled to traverse 32 tiles. That means that there is 20/32 traps per tile.

There are 44 trap markers, 22 of which are blank. 22 of the others cause an average of 2.62 damage. So when drawing a trap marker on average you will face an damage of 1.31. You have a 50% chance of disarming so there is now only 0.659hp dealt per trap disarmed. With 20/32 chance of having to disarm a trap per tile the total damage of traps per tile is 0.411hp.

It does get more complicated with combat and even more complicated with events. However. I can also show my reasons for thinking a lot of these assumptions are correct.
 
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Jan Bárta
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Re: TOEE: Analysis of Balance First Pass
zerosum79 wrote:

There are 44 trap markers, 22 of which are blank. 22 of the others cause an average of 2.62 damage. So when drawing a trap marker on average you will face an damage of 1.31. You have a 50% chance of disarming so there is now only 0.659hp dealt per trap disarmed. With 20/32 chance of having to disarm a trap per tile the total damage of traps per tile is 0.411hp.

It does get more complicated with combat and even more complicated with events. However. I can also show my reasons for thinking a lot of these assumptions are correct.


These trap numbers look a bit too harsh. My own calculation shows that the average damage per trap token is 1.18 dmg. 22×0, 6×1, 6×2, 6×3, 4×4 => (6+12+18+16)/44 = 52/44.
Otherwise, it is a very nice work - if I find the time to do it, I will look through your file and check other assumptions. Maybe after work I am also curious - how did you perform the correction on multiple Monster cards in play? That scales like x^2 in difficulty and is not so uncommon to have.
 
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Greg Jones
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Re: TOEE: Analysis of Balance First Pass
Thanks for pointing this out. There was an error in my formula. In fact there are likely to be multiple errors which is why I am so glad to present this on here. I threw it together in a day to get a quick result up but am still building it out.

Ill continue to post revisions.

As far as dealing with monsters though, the total number of monsters on the tiles exactly equals 32 so I assumed that the designers had still used the old 1 monster per tile and that monsters are dispatched prior to exploring the next edge. That may or may not be a good assumption but I am nowhere near looking at the affects of multiple monsters yet.

 
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Jan Bárta
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Re: TOEE: Analysis of Balance First Pass
zerosum79 wrote:

As far as dealing with monsters though, the total number of monsters on the tiles exactly equals 32 so I assumed that the designers had still used the old 1 monster per tile and that monsters are dispatched prior to exploring the next edge. That may or may not be a good assumption but I am nowhere near looking at the affects of multiple monsters yet.


Well, I used to think that there are 32 Monster symbols - but my Excel spreadsheet was not counting the last tile in the list. There are 33 symbols (11 tiles with 0 symbols, 11 with 1 symbol, 8 with 2 symbols and 2 tiles with 3 symbols), i.e. ~1.031 Monster per tile. The average number of traps is exactly 1.25 per tile (18×0, 8×2, 6×4 traps).
When I play ToEE, one Hero usually tries to destroy some Monster in play (preferably double Monster card, then if it is his/her card) AND then explore. Your assumption about exploration may or may not be true based on the tactics used.

Furthermore, you assume that any Monster can be destroyed by 2 attacks in average - that may be true for the basic Monster deck. But once the Empowered Cultists or Elementals start to appear... it will be considerably more At-Will attacks. The average number of HP the Monsters take from Heroes (and vice versa) depends on the actual Monster deck composition. This may be the way to further modify the difficulty and make a Gygax:ToEE campaign. Starting with skeletons, zombies and hordes of goblins, followed by cultists, then elementals and much stronger cultists of the Great Temple, ogres, hordes of bugbears ...
 
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Greg Jones
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Re: TOEE: Analysis of Balance First Pass
Hi Jan,

I am glad to chat with someone who also finds this subject interesting. Did you make your own spreadsheet annotating the contents of the game? It seems like you have some of the work done already.

Jenceslav wrote:
zerosum79 wrote:

As far as dealing with monsters though, the total number of monsters on the tiles exactly equals 32 so I assumed that the designers had still used the old 1 monster per tile and that monsters are dispatched prior to exploring the next edge. That may or may not be a good assumption but I am nowhere near looking at the affects of multiple monsters yet.


Well, I used to think that there are 32 Monster symbols - but my Excel spreadsheet was not counting the last tile in the list. There are 33 symbols (11 tiles with 0 symbols, 11 with 1 symbol, 8 with 2 symbols and 2 tiles with 3 symbols), i.e. ~1.031 Monster per tile. The average number of traps is exactly 1.25 per tile (18×0, 8×2, 6×4 traps).
When I play ToEE, one Hero usually tries to destroy some Monster in play (preferably double Monster card, then if it is his/her card) AND then explore. Your assumption about exploration may or may not be true based on the tactics used.


I just took the 32 basic tiles out and counted the monster locations. I got 32. I'll have to go back and look again. I did not count any of the larger tiles so my count for traps is probably off since if you start on that tile you are guaranteed to have to go through an extra trap.

My assumption on events was that you would have 0.5 events/tile due to black tiles and somewhere around another 0.5 events per tile due to not exploring a new edge. It turns out that in this spreadsheet you get somewhere around 0.9 which was close to my initial assumption.

Jenceslav wrote:

Furthermore, you assume that any Monster can be destroyed by 2 attacks in average - that may be true for the basic Monster deck. But once the Empowered Cultists or Elementals start to appear... it will be considerably more At-Will attacks. The average number of HP the Monsters take from Heroes (and vice versa) depends on the actual Monster deck composition. This may be the way to further modify the difficulty and make a Gygax:ToEE campaign. Starting with skeletons, zombies and hordes of goblins, followed by cultists, then elementals and much stronger cultists of the Great Temple, ogres, hordes of bugbears ...


I only did the basic game, no boss battles and no tougher opponents because I figured this should be balanced against the level 1 players. I totally agree with your point about eventually having to significantly modify the monster deck. I also want there to be lots of monsters as the campaign characters get better. However since that skews against smaller parties there will also have to be a way to rebalance with more events.
 
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Jan Bárta
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Re: TOEE: Analysis of Balance First Pass
zerosum79 wrote:
Hi Jan,
I am glad to chat with someone who also finds this subject interesting. Did you make your own spreadsheet annotating the contents of the game? It seems like you have some of the work done already.


Interesting? Yes. Well, the boardgame got me interested, because I am a fan of Troika's ToEE game (never played pen&paper D&D, only the Czech spin-off). And the mechanics aren't too strategical, which is good - with all respect to Descent 2nd ed. fans, I don't like that game at all . My Hero moves 1, 2, 3, 4, 5 steps, now the 6th - and the MasterVillain(TM) plays some hideous trap. And then some 2×2 large Merriod passes through several 1-wide openings. Uhm, I will pass.
So, I gathered all the information that I could get before I bought ToEE So I have done some of the work - mainly Monster and Tile list, including all previous AS games (and DC sets). Then some basic evaluations white/black <=> traps and monsters, number of exits on all tiles. Nothing more.

Ad Monster symbols - I counted all my scanned tiles just to be certain. And it's really 33. No Monster symbols on the double tiles, neither traps (except the Start Tile).
 
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Greg Jones
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Re: TOEE: Analysis of Balance First Pass
Hello All,

I've updated the spreadsheet to reflect my changes to the model over the last couple days. I've become a little obsessed with this.

The major change that I made is to update a lot of the events to try to make the event damage more accurate and also to create a balance across party size, then extrapolate to how that balance changes when the party stats are altered with my leveling system.

As you can see it becomes significantly easier as the party gets stronger (as expected). So the next steps will be to start adding in the stronger monster and event cards to extrapolate their damage. Once all thats done I am going to try to make the different cards and monsters sortable and selectable in the spreadsheet so that you can swap in and out cards and monsters to rebalance at higher levels.
 
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Greg Jones
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Re: TOEE: Analysis of Balance First Pass
Oh and Jan, I forgot to add the extra monster encounter space. That will not make a huge difference but I will need to add it in at some point.

 
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Greg Jones
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Re: TOEE: Analysis of Balance (update Second Pass)
A couple interesting things to note.

1. Addition of an initiative system to this game does not appear to affect the balance much. Assuming players get the initiative half the time you would reduce the damage due to monsters to roughly half of the ~ 6 points per game at level 1.

2. This would be pretty easy to balance out by:

a) simply removing some of the 0 damage traps from play. (I also have a rule variant for later in games that I am calling "chain reaction" where crossing one trap automatically triggers the trap next to it, doubling the equivalent damage.
b) Increasing monsters per tile based on a die roll.
c) Add an extra tile or two.
d) Remove 1 surge.

3. Addition of rules that change the amount of encounter cards per game drastically change the balance of the game. Unfortunately this also happens in a very nonlinear way affecting larger parties more than smaller parties. To really design effectively for rules that reduce encounters you have to also change the deck to make the mix of cards that are harder on big parties more likely to occur. This is doable.
 
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Greg Jones
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Re: TOEE: Analysis of Balance (update Second Pass)
I think this will be the last entry into this topic. I have updated the spreadsheet with one final revision that adds a ton of functionality.

The model is now complete showing how parties of sizes 1-5 heroes level form 1-10.

I have created tabs that have filterable tables with monsters and events from both the basic and advanced deck. This will allow for filtering of specific combinations of cards to create customized adventures that are perfectly balanced. I have set all statistics so that cards filtered out do not contribute to the overall stats. That way you can create customized sets of cards an analyze how they affect the balance of the game.

As you can see from the model everything actually works out pretty smoothly. I expect that any minor errors in formulas or guesswork in frequencies of things happening is pretty much in the noise.

One major assumption that I would like to get statistics on is the frequency of drawing an event card when no new edge is explored. Right now I have it set as 0.85/tile. There is 0.5/tile just based on the number of black triangles. I assume that this number which was mostly derived from trying to balance the HP. I would like to see whether when playing the game in practice whether this frequency holds up.

Anyways, Ill leave the spreadsheet up as long as drop box allows. I hope that its useful, and Ill probably add the WOA event cards and monsters to it eventually.

Best Regards,
zerosum79
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