Author Topic: Wealth Optimization - because optimizing NP problems was a good idea?  (Read 2665 times)

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Squirrelloid

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On the gleemax forums I've extensively discussed the Negative Difference Engine and its attendant problems.  I've also discussed some algorithms for wealth optimization, but those algorithms do not necessarily translate well into practice (they're ultimately looking at average expectation, something no given character will ever see).

Which leads us to the very practical concern of optimizing wealth acquisition for a given party.  I say party because a party is the unit which acquires wealth, and has to divide that wealth amongst its members.  So you can't just optimize one character, you have to optimize your entire party simultaneously.  I'm going to assume default party sizes (ie, 5 characters).

General Principles
(1) Casting rituals is bad for your wealth.  Don't do it.

(2) Minimizing potion use is good for your wealth.  This implies that certain party compositions are probably better than others (eg, 2 leader parties), and thus even party composition is an arena within which party wealth can be optimized.

(3) You should aim to only replace items that are significantly below your current level.

(4) Items you will never replace are preferable to items you will.  Ie, my Brace of Wands post provides a mechanism by which a wizard can temporarily upgrade his implement for high paragon levels, and keep that implement till the end of epic, thus never selling said item.  (Note that even orb and staff wizards can use this, because nothing about orb or staff mastery requires you to actually cast spells through an orb or staff).

(5) An item you would never use is a 100% loss.  (You should recycle items which will never get used).

(6) A character must be viable at every level.

The effects of 1+2 dominate at low levels, and become less and less significant at higher levels (although excessive potion use or repeated high-level ritual use even in epic will certainly be noticeable).  The effects of 3+4 will dominate high levels, because wealth lost to replacing items will become more and more significant.

Benefits of Wealth Optimizing
If the idea that wealth = power because you can turn wealth into magic items isn't a sufficient justification, consider that every extra item you acquire will either (a) have a useful property or (b) have a daily power.  Properties have no limit on their use aside from your having a limited number of item slots in which to put them.  Daily item powers are limited to 1/tier +1/milestone, which means at epic 5-6 daily item power uses/day should be common.

Items which have properties will be unlikely to be swapped frequently.  But one might imagine a character having a different item set for adventuring than for social interactions (ie, a town set and a dungeon set).  Certainly, having excess wealth would aid in acquiring the appropriate items to do so.

Item daily powers in excess of the number you can use per day obviously aren't being used.  But which ones are being used will vary from day to day.  Thus, more daily item powers, more options.  And many items *only* have a daily power use - such an item can be swapped out between encounters when it has been used, allowing a different item to take its place.  Alternately, weapons or implements with daily powers may have specific uses that having a swap is a viable strategy - most notable with wands for which two useable powers gain +4 to hit against single targets, and thus can be viable after the enhancement bonus of the item itself is considered subpar.  Wands can also have utility powers, with the obvious advantages gained by having them.  Some other implements have secondary powers which don't require the implement in question to have been used to channel a power in order to activate them - these obviously don't care what the item bonus is.  Making a case for any weapons is harder, but I won't rule it out.

Obviously, characters who need multiple types of items from the weapon or implement category are going to be at a disadvantage relative to other characters, and therefore care about wealth optimization solely to try and play the same game as their buddies.  This is especially egregious because the big 3 (weapon or implement, armor, neck item) are going to be your major cost in terms of losing wealth to item recycling - making that the big 4 is a large hit.

Problems with Wealth Optimization
Its no exaggeration to say that this is a *hard* problem.  (NP hard to be exact).  Mathematically its a packing problem.  If that wasn't enough, there are other difficulties:

(1) It is build dependent.  Different builds will want or desire different items, which will have different levels, and thus pack differently into the parcels received.  Some classes or builds are more or less capable of taking advantage of various general principles, meaning there's no well defined maximal optimization.

(2) It requires negotiating with 4 other players about the distribution of 30 levels worth of items.  Either everyone is on board with optimizing this or there will be annoyed buddies or hurt feelings.

(3) Optimal solutions may not involve equal wealth per player (in fact, they likely won't). 

(4) Desirable items aren't distributed uniformly across level.  There will be level N items for some N which you may have a hard time packing items into.  Other levels will be overloaded, and you'll need to find compromise items.  Ie, its not only a packing problem, its a packing problem with highly non-uniform pieces, and they vary in unpredictable ways.  (More supplements with items will help alleviate this, because it will tend to create desirable items at all levels). 

(5) Obviously, some players may be more or less willing to compromise, and this leads to a complex haggling minigame within which various game theory models of behaviour, trust, reputation, etc..., likely apply.  This is *complicated*, and at best at the cutting edge of research in game theoretic topics.

Doesn't the DM Control Treasure?

Quote from: DMG pg 125

Ok, first of all, the best way to know what items will be of interest to your players is to, well, *ask* them.  The DMG then goes on to say you should do exactly that.  The thing about wishlists is you can game them, by limiting the amount of information you give your DM.  Lets say you're level 13, and the players have optimized their treasure progression, and you're supposed to get an Iron Ring of Dwarf Lords in the next level.  If you only tell your DM "I'd really like an Iron Ring of Dwarf Lords" he can (1) give you what you want, (2) give you something else that you're likely to not want at all, and thus defeat the intent of the treasure system.  Ie, the information you communicate to your DM colors what he gives you.

Rather than making individual wishlists, the party may well provide a group wishlist.  (And it can be perfectly reasonable that the party wizard is interested in the tank receiving certain items if the tank is keeping them alive!)

Finally, if there exist available item substitutions, you can provide a list of all the possible items (with or without explicit or clauses).  There are going to be obvious substitutions based on item levels.  A party will ultimately receive 4 level 15 items, and it really doesn't matter (from an overall optimization standpoint) which order they receive them in, so that level 13 character may make his wishlist to include any items the group has decided he should get of levels 14, 15, 16, or 17.  This provides the DM with a diverse wishlist which also adheres perfectly to the party plan.

Another type of substitution if there are two items that fulfill the same function that are equally viable, such as two different +4 wands that a wizard is interested in.  Saying "I'd be interested in this one or that one" as part of his wishlist is quite reasonable.  More complicated substitutions can also be imagined. Lets say Character A would be happy with item a1 (of level x) or item a2 (of level x+1), and Character B would be happy with item b1 (of level x) or item b2 (of level x+1).  They could agree to list both "a1 or a2" on A's list, and "b1 or b2" on B's list.  This gives the DM some real choice (as opposed to illusion of choice) while letting the party optimize its wealth.  And I can construct more complicated substitution schemes, but I think the point is made.

Ie, even if you assume the party doesn't have complete control, they still have some control, and can thus still optimize.  There are even entire areas of applied mathematics interested in similar problems.  Basically, party treasure received is some function f(x,y) determined by f(x) and f(y), where f(x) would be party input and f(y) would be a random factor (relative to the players).  The degree to which the DM doesn't follow f(x) (ie, the less weight f(x) receives relative to f(y)), the less benefit can be obtained by optimizing wealth.  The quoted DMG text is effectively saying "f(x) should receive a strong weighting".  It won't necessarily be 1, but it should be a significant determiner of what the party receives.

Conclusions
4E created the possibility of gaming the wealth acquisition system.  This leads to a rather complicated wealth optimization minigame.  Its also unlikely to be an enjoyable minigame for most people.  However, it can be done, and the possibility of doing so may well color use and enjoyment of the game as a whole, even if done on a micro level (ie, projecting over the next few levels).
« Last Edit: July 18, 2008, 02:36:17 PM by Squirrelloid »
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Squirrelloid

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reserved for links to examples or possibly an example of my own - yeah, I might bother to optimize for a party of 5 characters if I'm bored...
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DemonLord57

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How often do you recommend changing the three essential items? I was thinking only once per 5 levels, (i.e. upgrade every +1) but if you have enough choices that you'll keep then it's obviously more optimized overall (not talking about viability, that's what I'm asking about) to get them as infrequently as possible, since you'll be replacing them.

highbulp

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An intriguing idea. I might argue against the basic assumption that a character earns a finite amount of wealth in the game independent on action, but I'm not sure I have any text to back that up. The DMG certainly doesn't seem to phrase the treasure parcels as "guidelines," which (because of the nature of the game) is what I think they are. But at that point I'm kind of treading on Oberoni.

Can't we also generate a kind of Wealth By Level chart using the DMG's rules for building characters at higher than 1st level? I wonder how that would compare to the finite wealth limit suggested by the parcel system. It's possible the book expects you to use money on potions and rituals and stuff. Not that it would change any of the optimization stuff.

Squirrelloid

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@DemonLord
I'm looking at upgrading the big 3 every 8 levels (approximately) as being optimal.  Ie, you don't need to upgrade every 'plus'.  However, this may depend substantially on build and item reuseability.  A wizard going for a brace of wands may well take a Wand of Mesmeric Hold +4 and a Wand of Force Volley +5 in addition to a +6 implement (possibly an orb or staff), because he'll still be using all of them at the end of the game!

But for a general algorithm, assuming you'll upgrade them every 8 levels instead of every 5 nets you an extra 1mgp or so by 30th level.  Its not a huge difference, but it is a difference (and it will depend on which levels those every 5 are - the 1mgp advantage is a minimal advantage).

@highbulp
The parcel system is most certainly 'fixed'.  You'll note they've gone to great pains to (1) make the parcel system general - it doesn't just cover treasure looted from monsters you stabbed in the face.  It also covers quest rewards, etc... (2) They've removed all other possible wealth-generation schemes from the game.  You can't craft items for profit.  You can't do normal work to earn profit without it being a skill challenge (and therefore earning xp, and all profit is really a parcel). 

(Sidebar: Ultimately, wealth should be fixed, I just happen to think it should be fixed by quantity at level N rather than quantity ever received.  The defenders of 4E on gleemax - you know who I'm talking about - seem to either think that there is secretly some WBL which parcels will automatically arrive at (hah!) OR that any differences between a WBL system scaled properly and the parcel system will necessarily be small (disproven: difference engines allow infinite repeatability of actions that change your state.  But the parcel system is what we have to work with in RAW, so I suppose we might as well work with it).

Now, its possible to generate WBL tables, but it really depends on some assumptions you make.  Ie, the 'new character' method basically assumes that you're only using the treasure acquired in the *last 3 levels* for an organic character.  If you make some assumptions about item-replacement algorithms to get an idea of wealth (and while we can't for a specific character come up with a good general algorithm on item replacement, for a generic character where we're only interested in wealth and not the specific items we certainly can do so), you're looking at 25%-33% more cash at 21st level.  Awaken DM Golem has posted relevant possible WBL tables using just information in the DMG (new character/total received wealth/NPC), which doesn't include any derived versions such as 'total received - estimated expenses', but I've posted those elsewhere.  See my "A 4E WBL Table" thread in the 4E forum of these boards for a table averaged over two algorithms where I describe all my assumptions about how I have handled wealth and how *I believe* wealth should be handled by the game. 

(Sorry for not providing links.  Awaken DM Golem's relevant thread is in gleemax Char Op somewhere, my threads on this matter are generally in the gleemax item + equipment board if I haven't otherwise specified).

Edit: Fixed my spelling of your name ADMG
« Last Edit: July 18, 2008, 06:51:27 PM by Squirrelloid »
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awaken DM golem

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Right here ---> http://forums.gleemax.com/showthread.php?t=1062282
I'm thinking of adding a 4th column to it.


Quote from: DMG
wish lists.
Let's all say it together ... "MONTY HAUL !!"
A non-CO type will have no idea as to what is the better item to have on their "wish" list ; and they might be happier.
I don't see how that is a problem.
A CO type will muck the game up that much more, but hold on.
Why aren't the monsters optimized too?
Why don't the monsters pick up the items and smack the PCs around just a little bit longer.
NPCs that are optimized, will be able to use that wish list of items almost as well as the PC that wants them (plural).


And very nice analysis SQ.
« Last Edit: July 18, 2008, 06:12:49 PM by awaken DM golem »

Squirrelloid

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So, I'm working on an example, and I figured it wouldn't hurt to have some help.  In particular, I'm looking for wishlists for characters.  I've already made one for a wizard, and am working on one for a laser cleric.  My theoretical party also includes MC Hammer (fighter/kensai/demigod), a Bugbear Brutal Scoundrel Rogue/Daggermaster/Trickster, and an Eladrin Tactlord/Battle Captain/Demigod.  So if anyone wants to put together a wishlist, that'd be great.

A wishlist should include:
A minimum of items that need to be replaced.
At least one item for every item slot.

List all levels the item is available at in ().

So if anyone wants to kick out a wishlist for the three remaining classes, that would be a real help, and I could get a functioning example together a lot faster.
The ignorant shall fall to the squirrels. -Chip 4:2