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Looking for some advice on balancing factions

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KiltedNinja
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Hi everyone,
I'm developing a boardgame and I would really like some opinions on balancing factions.

Some background info; It's squad-based (typically 4-man teams), grid-based movement and dice-based combat. Each soldier has 3 stats; Time units, Hitpoints and Strength. Each stat is a number which has 4 possible values each, meaning that I there are 4*4*4 = 64 possible configurations for the soldiers. Each possible build costs x-amount of points to purchase the soldier at the start of the game.

I posted here a while ago (around a year ago) and got some amazing advice about various aspects of the game (If you're bored; http://www.bgdf.com/forum/game-creation/design-theory/different-factions...) - and I thought I had mostly solved my issues with the factions.

Basically, I have 6 factions, and I've managed to end up with a configuration as follows; when you take every soldier in one faction, you add up all the time units - it equals the same amount for every other faction. Same with Hitpoints, and same with Strength. So this means that each faction has access to the exact same 'amount' of stats, and soldiers, and point-costs (this seems balanced to me).

So - I created a battle-simulator which did the following; Each faction fights each other faction (1 at a time). For each faction.v.faction, I have several Wars. Each War consists of Several Battles, and each Battle consists of several Skirmishes.

Each Skirmish is a random "100 point" Squad from 1st Faction, versus a random 100-point Squad from 2nd Faction.
Each Battle picks new random squads from each of the 2 factions. Percentages of who has won/lost get counted.
Each War allocates the win/lose percentages into data bins.

Now here's the problem; when I take the data and fit it to a gaussian curve, I get results which aren't exactly balanced. Factions 1,2,4 and 5 seem to win 49% of the time. Faction 3 wins 50/51% of the time, and Faction 6 wins only 47% of the time.

Here are the win percentages with the standard deviations (achieved from the gaussian curve fitting).
Faction 1: 49.25 +- 1.92
Faction 2: 49.92 +- 1.82
Faction 3: 50.56 +- 1.50
Faction 4: 49.49 +- 1.84
Faction 5: 49.41 +- 1.49
Faction 6: 47.25 +- 1.39

I guess I just wanted to get your opinions on this, and see what everyone thinks. We're only talking +- 2 or 3% after all.. but there *is* a clear difference. It's also worth noting, that when I do something simple like "add 1 single strength to any of the soldiers in faction 6", then the results skew enormously.

So - do you think this is okay? Should I just remove factions 3+6? The only reason there are 6 factions is so that 6 people can play at the same time - (although there's nothing stopping people from playing the same faction, I guess..) - Is there something else I'm missing? Or am I over-analysing this whole thing...?

Thanks in advance for your time!

FrankM
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These look close, but worth checking

These win probabilities look similar, though it depends on exactly how many skirmishes were involved. The Gaussian fitting is obviously affecting something because the wins don't average out to 50%. Why not use the raw win percentages? Or do about 0.7% of skirmishes end in a draw?

(Quick note: The +/- figure often quoted with poll results is a 95% confidence interval, which is about 1.96 times the standard deviation for large samples.)

You can perform a statistical test to see if the win probabilities are unequal, though one of the assumptions is violated because these trials are not strictly independent of one another. You could get independence by throwing out all head-to-head skirmishes when comparing two factions, though we are testing play balance here rather that taking an econometrics exam. Personally I would leave in the head-to-head skirmishes.

This still leaves 24 or 30 comparisons, and with a 95% confidence interval we expect a false positive in about 1 out of every 20 tests. The method of handling expected screw-ups is called Šidák correction. This raises the bar for detecting a bias on the theory that we only have one set of evidence and we don't want to "falsely convict" the game. Alternatively, you could use raw t-tests and just run the entire simulator a lot of times to see if any persistent differences remain.

Now, suppose there is a persistent difference. For example, maybe Faction 3 is clearly better than Faction 6. If you can do so without eviscerating your theme, try swapping soldiers between those two while maintaining the point balance. Then try the simulator again and see if that helps.

X3M
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I remember that topic.

But looking at those numbers. Take it from me, you can be proud already. They are balancegasm worthy. I am jealous of those numbers.

Question:
How does the big picture look, if you compare the factions in a table?

Meaning,
Faction 1 has a victory % against faction
2=49%
3=51%
4=51%
5=49%
6=50%
Faction 2 has a victory % against faction
1=51%
3=48%
4=49%
5=51%
6=52%
etc.

That was just an example. But what it can show you is which pair of factions is the worst in general.

However, it also could show you a "practical" balance between the factions.

Some time ago, I took a look at a 3 faction RPS system. Where it was widely accepted that faction 1 had 60% victory against 2, but only 40% against faction 3.
Faction 2 and 3 had roughly the same percentages in a circular motion. (if this is not clear, I re-explain, just ask).

While that was a disaster for game play progress. You still can have this effect while close to the 50% victory.
If each faction has about 2 "victories" (51%), 2 losses (49%), and 1 "draw"(50%) all within the SD (standard deviation). Then you have yourself a RPS system regarding victories.

Players will know, which faction to use against what other factions. But even so, 51%. That is a very cool number.

***

I often said, by adding more randomness, the game gets more balanced.

What does that mean?
Another way, to have the percentages closer to 50% is by increasing SD.

You said, you used dice?
I made a change to how my dice looks like. I went from 011233 to 001234. The SD increased by a factor of 1,34. But so did the difference of most simulations decrease. If I had for example, 46 - 54%, this changed into 47 - 53%.

It depends on your mechanics. So don't throw away what you have. Simply change the die roll in a new simulator. And re-run it.

***

I also use Event Cards. These are the same for all players. And "weaker" units actually benefit more from certain cards.

***

Finally.
If you use any combination that a player can has. Would a player truly use any combination?

What does a skilled player do?
What does a godlike computer AI do?
Is there an combination that beats all others? If so, that is bad. But if there is a counter combination. Then there is notching to worry.

FrankM
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Feasible vs likely combinations

X3M wrote:
Finally.
If you use any combination that a player can has. Would a player truly use any combination?

What does a skilled player do?
What does a godlike computer AI do?
Is there an combination that beats all others? If so, that is bad. But if there is a counter combination. Then there is notching to worry.


This is a very good point. If players draw their soldiers randomly without the chance to discard/redraw, then you're fine. If there is a possibility of manipulating the hand, this should be in the simulator.

For example suppose it's a hand of five cards, but the player can discard/redraw up to two. I'd have the simulator just deal seven cards and pick the five "best" by some rubric. Not a perfect emulation of play, but it gets closer to testing the skirmishes more likely to occur in a game.

let-off studios
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1 vs 6

Is there a situation where a score value of 1 is tremendously bad for a given soldier/squad?

Is there a situation where a score value of 6 is overwhelmingly/"unbalancingly" good for a soldier or their squad?

Statistics is one thing, but if you're so concerned about the supremacy of certain factions then maybe the balance issues can be addressed by reducing the extremes. For example, what if there's a minimum score of 2 in each of the three stats? Or what if there was a limit of one of the three stats that could be 6?

You may have attempted/addressed this already, but I'm mentioning it here in the event you've not.

KiltedNinja
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Thanks!

FrankM and X3M - thanks for taking the time to reply, I really appreciate it.

Edit:let-off studios - just noticed your reply also - I hadn't refreshed the page - I'll reply separately!

FrankM wrote:
/..though it depends on exactly how many skirmishes were involved.

The latest simulation run consists of 256 Wars, each having 128 Battles, each having 64 Skirmishes (totaling 2,097,152 Skirmishes). It seems that the percentages vary noticeably somewhere around 100k (i.e. one faction is up, the other is down, and vice versa), and from around 200k upwards the percentages converge (or stabilise, I guess?) on the values I've written in my previous post.

FrankM wrote:
Or do about 0.7% of skirmishes end in a draw?

Yeah that value is about correct - varies between 6.5/7%. Based on your comments, I ran the sim and accounted for the draws, modifying how the percentage is calculated; it increased Faction 2's wins by ~0.5% and decreased Faction 3's similarly. Faction 6 still stuck around the 47% mark though. I think I agree that it makes sense to leave things as they are though, it's useful to know how many times there is a draw.

FrankM wrote:
/..try swapping soldiers between those two while maintaining the point balance.

Yep I tried that previous as well; It turns out that swapping any equal-costing soldier between those 2 factions causes chaos with the stats. I even tried doing a "Put one from 6 into 3, take the one from 3 and put into 2, then take the one from 2 and put him back into 6".... but .. yeah.. all those possible permutations made my brain implode - especially after around 6/7 different attempts, all the stats were still chaotic.

X3M wrote:
I remember that topic.
But looking at those numbers. Take it from me, you can be proud already. They are balancegasm worthy. I am jealous of those numbers.

Thanks man! Yeah it has been a while (I only get time to work on this every few months) - but the progress to get here has only been possible with help from yourself and others!

X3M wrote:
How does the big picture look, if you compare the factions in a table?

Here are the results from the latest run;

1v2 48.2
1v3 47.5
1v4 50.2
1v5 50.6
1v6 50.0

2v1 50.5
2v3 50.7
2v4 49.7
2v5 44.6
2v6 53.5

3v1 51.3
3v2 48.1
3v4 50.4
3v5 49.9
3v6 52.1

4v1 48.6
4v2 49.1
4v3 48.2
4v5 49.0
4v6 51.4

5v1 48.0
5v2 54.2
5v3 48.8
5v4 49.8
5v6 51.7

6v1 48.7
6v2 45.3
6v3 46.7
6v4 47.2
6v5 47.0

X3M wrote:
I made a change to how my dice looks like.

Yeah that's one of the first things I changed. I was previously using D8's with playtesting. On the Attack dice, 5678 is a Hit - And the Defence dice, 678 is a Block. (..the numbers are to be replaced with symbols at some point..) This was fine for physical tests; getting a Hit was 50% and getting a block was close to 40% (around 37% if I remember).
So after running my sims, I then changed it to D10, with 67890=Hit=50%, and 7890=Block=40% - that changed my win/lose percentages significantly - giving me the current figures.

X3M wrote:
I often said, by adding more randomness, the game gets more balanced. ../.. I also use Event Cards.

Yeah I agree - I also have cards, as well as upgrades.

Cards: During the game, players draw random cards which allow them to do things like re-roll dice, modify dice, increase/decrease strength/defence, etc. These can be held for a while (5 card hand limit) and used at the best time.

Upgrades: At the start of the game, players have free choice to buy whatever squad they want, up to the total points allocated by the type of game e.g. 100pt game, 150pt game etc. Upgrades cost up to 3pts each (each soldier can have 1 or 2 upgrades). Each upgrade can boost strength/defence (and some other things) - but they're not permanent and need to be recharged after a couple of uses.

So there are some player-specific choice (upgrades) to help their squad from the start, and random choice with strategic-timing during the game.

X3M wrote:
Would a player truly use any combination?

Well this is the thing, you can buy 4 cheaper soldiers and give them each 1 or 2 useful upgrades, or you can start with 3 expensive soldiers with maybe 2 upgrades between them. So it is possible to use any combination of squad.

Looking at the win percentages from the latest run - does anything here stand out as being broken?

FrankM
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Doing well so far

Impressive work so far.

It does look like Faction 6 is bringing up the rear here, but the most lopsided match-up is actually 2 vs. 5.

Given the number of skirmishes involved, I'd expect additional runs to be fairly stable. That first 100k or 200k is known as burn-in. At two million, you're well past that point. A couple new runs that result in substantially identical numbers would settle the issue once and for all.

Since swapping soldiers seems to be out, it might be more fruitful to tweak the event cards to ever so slightly favor 2 against 5 and 6 against everybody. If the simulation can store the cards played, you might find a clear culprit in these match-ups, in which case you tweak or eliminate the problem card(s) and try again.

If the simulation isn't actually playing event cards then you are stuck fudging things by hand. Just make sure that your Make Faction Six Great Again hat isn't throwing off the balance for everyone else.

X3M
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one step further

I have run further diagnostics in Excel. By subtracting one % of the other %. To see how much surplus each faction has over another. In a weird explanation: Every + is a rock over scissors, every - is a scissors over rock.

Then I added these all up for each faction. And I suggest, removing faction 6 AND 4.

And here is the tldr train of thought:

I know that 2 vs 5 is very extreme. But both are the top 2 of all 6.
Faction 5 has a surplus of 9.6 over faction 2. But faction 2 has for example a surplus of 8.2 over faction 6. If faction 6 had a surplus over another faction, the circle could go round. Unfortunately, it doesn't. Faction 6 is always the weakest.

Here is the list that I got as total strength of each faction:

Faction 1: -0,6
Faction 2: 4,1
Faction 3: 9,9
Faction 4: -1
Faction 5: 11,4
Faction 6: -23,8

Clearly, faction 6 is useless. In fact, I count no surplus for faction 6. It has -5.
When I remove faction 6. We know things will shift a lot:

Faction 1: -1,9
Faction 2: -4,1
Faction 3: 4,5
Faction 4: -5,2
Faction 5: 6,7

We see that faction 5 has lost a lot of leverage. And so has faction 3. In fact, when removing 1 out of 6 factions, you expect the numbers to reduce with 1/6th. But as you can see, the numbers plummet more. Which is a good sign.

But this list only shows the "bucket imbalance". We need to look back at the differences. That is, if we want to look at the choices that players make. In this case, a yes/no question is going to be answered. That is why we need those + and -

5 will still remain strong compared to 2. But has lost 6, which was a +. In fact, if it comes to choosing a faction. Faction 5 has 2+ and 2-. This faction is a properly balanced faction, regarding choices. To get a true balance in choices of factions. You need an equal amount of + and - for each faction. However, a 3/1 or 1/3 is also allowed, since you have an unique choice in that regard. And it still counts as choice.
Having a 0 however means a choice that will never happen.

The list of + and -
Faction 1: +2 -2
Faction 2: +3 -1
Faction 3: +3 -1
Faction 4: +0 -4
Faction 5: +2 -2

Faction 4 clearly shows, it has no value. So, I tried to see, what happens if I remove that faction as well.
And now, the magic begins.

Faction 1: +1 -2
Faction 2: +2 -1
Faction 3: +2 -1
Faction 5: +1 -2

As you can see, there are no "none" choices left.
Of course, faction 2 and 3 look like the better choices. I have the percentages in my table. But it is better to talk in RPS.

Faction 1 beats 5.
Faction 5 beats 2.
Faction 2 beats 1 and 3.
Faction 3 beats 1 and 5.

The last table:
Faction 1:-3,5
Faction 2:-4,7
Faction 3:2,3
Faction 5:5,9

Faction 5 looks like the strongest and faction 2 like the weakest. But faction 2 beats 2 other factions. The weakest faction sir! Beats 2 out of 3 factions. And lets not forget, faction 1 is rather weak too, and this one beats faction 5.

So, marvel at your system while faction 4 and 6 are not to be heard of ever again.

Of course, this math doesn't include other things that you might change. The whole system could be turned up side down, since you are already that close to the 50.

KiltedNinja
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Wow..

X3M wrote:
I have run further diagnostics in Excel.

Wow that's quite epic - I'd be interested to see the excel sheet where you've done this analysis! This is just a very quick reply, gotta get sleep - but I'll reply to this thread properly tomorrow after removing factions 4&6 and see what happens :D Thank you so much for the input!

X3M
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Pm me your email. And I'll

Pm me your email. And I'll send it.
I saved it for that question.
Please keep in mind, I didn't include the SD. That is a secondary list of + and -. Which might soften the discarding of a faction.

It was fast indeed, but I am used to doing those comparrisons. It doesn't come with the explanation in it.
Sometimes looking back in one. I don't even know what I was doing in the first place. So I guess I need to clean up just a tad and punt in some colors and decision making.

I have a 16 hour work day today. So expect the email Saturday (dutch time)

KiltedNinja
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Following up..

FrankM wrote:
Impressive work so far. ../..
If the simulation isn't actually playing event cards then you are stuck fudging things by hand. Just make sure that your Make Faction Six Great Again hat isn't throwing off the balance for everyone else.

Thanks :)
The simulation doesn't include the cards, simply because the choice of when to play the cards is completely player's taste and their opinion of best strategy for them at each moment in time during the game - Also, there are other aspects to the game (dealing with resources) and some of the cards are used for those aspects as well as combat... seemed like a bit of a nightmare to try and write an a.i. to make those kind of decisions!

X3M wrote:
I suggest, removing faction 6 AND 4.

Well, I have removed 4 and 6, after running the sim, results are as follows:

1v2 49.0
1v3 48.4
1v5 50.0

2v1 49.8
2v3 50.6
2v5 50.4

3v1 50.3
3v2 48.2
3v5 49.9

5v1 48.7
5v2 48.4
5v3 48.7

More-or-less what I expected - although it's not as balanced as I'd hoped (I think I expected some number magic to happen and make everything work out shiny)

However... I think that I have resigned myself to removing (at the very least) Faction 6. This means that I have some freedom to use the left-overs from 6 and try some swap-outs with soldiers from the remaining factions.

I'll keep trying and see what happens :)

X3M
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I only went from imbalanced.

I only went from imbalanced. To practical balance.

Perfect balance doesn't exist.

If you want to change the factions by picking the remains of faction 6. I suggest to work on faction 4 only. And with every switch, take note on the changes.

KiltedNinja
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Perfect Balance..

X3M wrote:
Perfect balance doesn't exist.

But I can still hunt for it! :D I'm kidding.. mostly.

Yeah I understand that there can be no true perfect balance.. ..but, I think I will search for a little while longer - just to satisfy myself that I have done my job as much as I can.

Thank you all for the comments, help and advice - it is, again, sincerely appreciated.

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