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Subject: sell potion probability and reward analysis rss

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sonny sonny
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unless stated otherwise, i will always assume that you know nothing about one of the 2 ingredients and only vary what you know about the first ingredient.

(1) if you know nothing, there's 7 possible results: + and - in each of the 3 colors and soup.

chances for soup is 1/7 (your first ingredient gives soup with exactly one of the other 7)
chances for the other 6 results are equally distributed, so its also 1/7 each.

therefore chances for sucessful selling if you know nothing, are as follows:
exact match: 1/7
at least correct sign: 3/7
at least soup: 4/7
anything: 7/7

everything else that happens (information gained and VP lost) does not depend on your promise. so lets calculate your average result (probability times gold on success). numbers in () or [] is when selling at (-1) or ((-2)) or [+1].
exact match: 4/7 (3/7) ((2/7)) [5/7]
at least correct sign: 9/7 (6/7) ((3/7)) [12/7]
at least soup: 8/7 (4/7) [12/7]
anything: 7/7 [14/7]

in all cases you get most average money when promising at least a correct sign. you always get by far the least average money when promising an exact match. for the rest of this post i will not even consider promissing an exact match unless i know the result (it's never a good idea if you have any uncertainty). i will not give the result for "anything", because it's always 100 % and i will not show the numbers for selling at a lower (or higher) price. for lower prices it's also always best to promise a correct sign. for higher prices - as you can see above - it's the other way round, but that means you lose 3 points instead of 1 for selling a wrong potion and that's the primary concern then.

of course in some situation you absolutely need the money and can't take risks. in this cases you can at least calculate the probability of failing if you want to gamble. but usually you should try to avoid this situation and can go for the best overall result.

so, when you know nothing, you gain 9/7 gold, you lose 4/7 VP and chances to get solid information is 2/7 (1/7 each for exact match and soup). you can probably deduct something from other results later.

now lets have a look at how to improve your chances. lets say,
(2) you know one correct sign (example: any + when the adventurer wants a +). since you know a result, you (likely) also know one ingredient that does not give soup. under this assumption:

chances for soup is 1/6 (you know it's one of the remaining 6)
chances to for + on your known color: 1/6 (if it gives green + with one ingredient, there is exactly one other that gives the same result)
chanced for + or - with the other 2 color: equally distributed, so 1/6 each.

therefore chances for successful selling and average result are:
at least correct sign: 3/6 -> 9/6
at least soup: 4/6 -> 8/6

again, the best average money is promising a correct sign. you gain 9/6 gold, you lose 1/2 VP and you gain information of uncertain quality, unless you had an exact match or soup (1/6 each). so all in all that's only slightly better than the first case.

by now (if you didn't before) you know how to calculate the probabilities, so lets look at more examples quicker.

(3) if you know one + and one - (and 2 ingredients that are no soup), chances and money are almost the same as (1)
at least correct sign: 2/5
at least soup: 3/5

but you gain alot more information from the result! there's 2 possible cases:
(a) you sell something possible (from your point of information)
(b) you sell something you know is wrong (selling a green + when you know your ingredient has a green -). in this case it doesn't matter for your chances or results, but it does matter for the information gained.
in the case of (a) if you get at least a correct sign, you know the exact result (it must be either the known or the unknown correct sign, it can't be the color that you know is wrong). chance for that is 2/5. soup (1/5) is an equally good result in terms of information gained, that's in total over 50 % of getting solid information.

(4) if you know 2 colors with correct signs (and 2 ingredients that are no soup):
at least correct sign: 3/5 -> 9/5
at least soup: 4/5 -> 8/5

if you had a wrong result or soup (1/5 each) you got secured information. but that's the bad case. from the good cases you have 1/5 for an exact match and 2/5 for a correct sign, that helps you nothing (until get more information). in the end this has a good chance to give you 3 gold, and if it fails you get at least good information.

(5) if you know a +++ or --- ingredient, you can sell such potions without risking a wrong sign. if you know the opposite (soup) ingredient of just one of the ingredients used, that means you know for sure you have a correct sign and can safely get information about other ingredients. this is very powerful. therefore, if you have a choice, it's better to further investigate ingredients where you know ++ or -- compared to ingredients where you know +-.

so, how can you further improve your chances for successful selling potions?

you can look at others published theories and guess which of their colors is correct (use potion selling to verify it).

you can look at your other known ingredients. if you have tested mostly - results, then mixing 2 ingredients that you know nothing about, have a higher chance to give a + result (also keep this in mind when drinking potions yourself).

if you know the soup result for one of your ingredients, or can otherwise rule out this result it significantly improves your chances. on the other hand, if you have tested an ingredient often (lets say 4 times) and not found a soup result, chances for that are quite high (1/3 in the given example).

especially when you're farther up the VP track it becomes more important to avoid VP loss. even if you don't know a +++ or --- ingredient, it's sometimes possible to know if the potion will be a + or -. for example if A has green and red +, and B has blue +, and i know the soup ingredient for one of them, the mix can only be a +.

if you know all 3 signs of an ingredient for sure, don't forget to deduct from their size for other ingredients that you have tested them with. for example ingredients with large green - always also have a blue -.
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sonny sonny
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yesterday i actually had a situation where i promised an exact match when selling potions altough i did not know result. i knew 4 formula and was testing one of them with one of the others. the possible results (and average gold gained for promising that) were:
exact match: 2/4 (8/4)
soup: 3/4 (6/4)
wrong potion: 4/4 (4/4)

so in this case, promising an exact match was actually the best option, because a correct sign with a different color was not possible. i don't think i ever had this situation before, but i can obviously happen.

[edit:]yesterday my wife had an interesting analysis: if you get a correct sign result when selling green+, you know it must be blue+ or red+. in that case, she already knew 3 other ingredients that had a red+, so her test result must have been a blue+.
 
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Robert Stewart
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A note on your analysis: while getting more gold is obviously always better, all else being equal, there is often a step in your utility function - if you already have 2 gold, and have cubes down to buy a 4-cost artifact and publish, then you're most interested in whether you can get an extra 3 gold or should settle for 2 gold - getting a guaranteed 1 gold is pretty pointless, and getting 4 gold rather than 3 is only worth attempting if there's no increase in risk.

It's still worth knowing how much gold you can reasonably go for, so you can plan for how much you might be able to spend that round (and if you already have enough gold to be able to afford everything you want to do that round anyway, then it is just a case of how much more you can get).

The correct question most of the time is not "how much gold would I end up with if I followed this strategy over a whole bunch of rounds" but "what are my chances of getting X gold this round"
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David desJardins
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rmsgrey wrote:
The correct question most of the time is not "how much gold would I end up with if I followed this strategy over a whole bunch of rounds" but "what are my chances of getting X gold this round"


I don't find that most of the time I need to generate X gold on this particular sell action or the rest of my plans fall apart. And if I do then I have already taken that into account in planning for what I'm going to sell. But most of the time I'm generating gold that I'm not going to spend all of this turn, and the amount that I can carry into future turns affects my chances of winning the game more or less linearly.
 
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sonny sonny
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rmsgrey wrote:
if you already have 2 gold, and have cubes down to buy a 4-cost artifact and publish, then you're most interested in whether you can get an extra 3 gold or should settle for 2 gold - getting a guaranteed 1 gold is pretty pointless, and getting 4 gold rather than 3 is only worth attempting if there's no increase in risk.

The correct question most of the time is not "how much gold would I end up with if I followed this strategy over a whole bunch of rounds" but "what are my chances of getting X gold this round"

1. i covered that for the case when x = 1.

2. in all other cases it's too risky in my oppinion (i guess we don't need to discuss that for x = 4, and my suggestion is to use x = 3 anyways). the big problem is, even if you succeed, you'll often (unless you get a grant) have no gold left for the next round, putting you in a similar situation again.

3. besides for x = 1 (and that's very rare and usually covered by transmuting for the same reasons as explained in 2.) i never place cubes in a way that requires me to be successful on potion selling and i cannot imagine anyone is doing that in the majority of cases. i guess it can be a viable strategy if you're behind and hope for luck to keep up. but in that case you might as well take the lower chances for potion selling.

4. the difference in chances for soup and correct sign in my examples was only about 15-20%. so in you example that's your chance of losing the action cube that you had placed on buy artifacts. even if you value a cube with 1 gold (it's certainly less) that's only 1/7 to 1/5 gold lost on average. you can add that to the expected result in the calculations. if you do that, promising a correct sign will still not give a worse result.

(added more analysis information in first reply above)
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Robert Stewart
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letsdance wrote:
4. the difference in chances for soup and correct sign in my examples was only about 15-20%. so in you example that's your chance of losing the action cube that you had placed on buy artifacts. even if you value a cube with 1 gold (it's certainly less) that's only 1/7 to 1/5 gold lost on average. you can add that to the expected result in the calculations. if you do that, promising a correct sign will still not give a worse result.

(added more analysis information in first reply above)


It's not just the cube you lose, but also (potentially) the artifact you wanted to buy - which has a straight points value, as well as any in-game effects (which are generally going to be more than the value of the gold you spend on it).

As for the question of running at a surplus, if you always have enough gold in reserve at the start of each round to pay for that round's expenses, then you're sacrificing tempo for security - doing the same thing a round later (once you know you have the gold) means, at best, being a round behind on whatever benefits it offers; at worst, letting someone else shut you out. Of course, the downside is that if you don't get the gold you're after, then you miss out on the actions entirely.
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