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Subject: Evaluating strength of cards rss

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Petri Savola
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Shakespeare was one of the games in EuropeMasters tournament this year and I made a little bit of mathematical analysis for the game. The tournament is now over, so I figured I could finish the analysis and share it with everybody. The idea is to evaluate strength of different persons you can recruit during the game.

In order to evaluate everything we'll have to come up with some value for coin and for this analysis I have assumed that each coin is worth 0.3 victory points. Relative strength of the persons does not seem to change much by changing this value.

Next, we'll have to come up with a value for each quill. Because quill tracks are scored twice in the game, I had to split the analysis for quill values for rounds 1-4 and rounds 5-6. For each track if you have advanced 3 times, you'll gain 1 point in each scoring, so for this purpose each quill is worth 0.33 victory points. If you max the tracks out, you need 6 more steps and you'll gain 5 money (1.5 victory points) from the red track, 2 points (actually 2.67 because you take 2 points from somebody and each point from somebody else is 1/3 to you) from the yellow track and 3 points from the blue track. So if you max out tracks you get 0.28 per quill from red, 0.41 per quill from yellow and 0.44 per quill from blue, in each scoring.

For evaluating the value of red quill I wanted to take average of 3 quills and max quills and got the value 0.31. For yellow quills I wanted to take the average in similar fashion and got value 0.37. For blue quills I just took the value 0.44 directly because the value of blue quills doesn't decrease before you max out. For white quill I gave the same value 0.44 and for black quill I gave the same value as for red, 0.31. On rounds 1-4 all these values have to be doubled.

The value of happiness I tried to evaluate by considering all the possible transitions in the happiness track:

* -1 vp → black quill
* black quill → nothing
* nothing → coin
* coin → white quill
* white quill → 1 vp
* 1 vp → 1 vp

I considered each of these transitions to have same probability and got value 0.33 for happiness. If you don't consider the situation where happiness is useless, you'd get value 0.4, but you often do seem to have those situations where happiness has no value. Value of happiness is not affected by the round, because the sum of those transitions doesn't change even if the quill value changes.

Next, I want to find out value for craft points and start the analysis with cloth. If your costume has sum 6-7, you get 2 coins, with sum 8-10 you get 1 coin and 1 vp, with sum 11-12 you get 2 vp and with sum 13-15 you get 3 vp. From this I made the following simplification:

* token 1 is worth nothing
* token 2 is worth 0.2 vp
* token 3 is worth 0.43 vp
* token 4 is worth 0.67 vp
* token 5 is worth 1 vp

Additionally you do get 1/3 of a quill in each income for each token. For value of this quill I took the average of all 3 quills, which is 0.37. From these numbers I could calculate the value for each token when the quills are included and finally I normalized this value by dividing by the size of the token to get the following normalized craft point values for each token (turns 1-4 / turns 5-6):

* 1: 0.50 / 0.12
* 2: 0.35 / 0.16
* 3: 0.31 / 0.19
* 4: 0.29 / 0.20
* 5: 0.30 / 0.22

So small tokens seem to be great early and big tokens are better on rounds 5-6, which does match with my intuition also. Here we don't take into account that you will not have enough slots for the tokens if you only use small tokens and you won't get all those tokens from the pool either. By taking average of these normalized values we get cloth craft point value 0.35 for rounds 1-4 and 0.18 for rounds 5-6.

Analysis for rooms is similar. In order to get points from building rooms you need to build 16 rooms to get all 5 points, so each room has value 0.31. By taking the room bonuses you get into account we get the following values for each room:

* 1: 0.31
* 2: 0.61
* 3: 0.65
* 4: 0.98
* 5: x

Now, in order to calculate value for room 5, we have to first figure out rough value for each craft point, which we don't know yet because we didn't complete the analysis for rooms yet. Therefore we first analyze the other 4 rooms to get a rough craft point value and use that to evaluate room 5.

Normalized room craft point value for each room:

* 1: 0.31
* 2: 0.31
* 3: 0.22
* 4: 0.24
* 5: 0.25 / 0.20 (rounds 1-4 / rounds 5-6)

In order to get the value for the bonus token of room 5, we took the average of 3 cloth crafting points and 3 room crafting points, which is why the value of that room drops slightly on rounds 5-6. Again, we take average of these 5 rooms to get the room craft point value of 0.27 for rounds 1-4 and 0.26 for rounds 5-6.

Now, the last thing we still need to calculate is the value of golden tokens, which is simply 1.93 for the jewel on rounds 1-4 and 1.56 on rounds 5-6. For the golden room the value is always 1.31.

Summary of the analysis

* Money: 0.3
* Red quill: 0.31 (x2 on rounds 1-4)
* Yellow quill: 0.37 (x2 on rounds 1-4)
* Blue quill: 0.44 (x2 on rounds 1-4)
* White quill: 0.44 (x2 on rounds 1-4)
* Black quill: 0.31 (x2 on rounds 1-4)
* Average quill: 0.37 (x2 on rounds 1-4)
* Happiness: 0.33
* Cloth craft point on rounds 1-4: 0.35
* Cloth craft point on rounds 5-6: 0.18
* Room craft point on rounds 1-4: 0.27
* Room craft point on rounds 5-6: 0.26
* Golden jewel on rounds 1-4: 1.93
* Golden jewel on rounds 5-6: 1.56
* Golden room: 1.31

Many simplifications were made in the process, but there are also some calculations behind those numbers. All the results match quite well with my intuition, but the drastic drop of the cloth craft point value on rounds 5-6 was surprising. However, it may not be surprising if you consider that you get quills twice from the cloth craft point points you obtain on rounds 1-4 and the quills you get on round 4 are included in both scorings.

Now we can proceed to analyze the cards one by one. In the below analysis AP means action power in victory points, TB means total benefit when recruited (this calculation includes using the action every other round, getting the quills from income, paying the money at the end of the game, but it does not include removal of exhaustion) and EB means exhaustion benefit, which is the benefit you get for removing exhaustion from this person instead of one of the persons which are available at the start. You can get exhaustion benefit only once per round and you don't get it from round 6.

***

Cards sorted by total benefit on rounds 1-2:

Cloth 5 + white quill (actor): 5.97
Cloth maker 8: 5.53
Room 4 + white quill (actor): 4.98
Builder 8: 4.79
Goldsmith: 4.52
Cloth maker 6: 4.37
Builder 6: 3.82
3 cloth craft points + white quill (actor): 3.75
3 room craft points + white quill (actor): 3.67
Helper boy: 3.64
Happiness + blue quill (actor): 3.14
2 room craft points + yellow quill (actor): 3.11
3 quill income (actor): 2.96
2 cloth craft points + red quill (actor): 2.78
Red quill + yellow quill (actor): 2.69
Money income (actor): 2.60
Happiness + white quill (actor): 2.54
4 craft points: 2.43
Unhappiness + yellow quill (actor): 2.23

Cards sorted by total benefit on rounds 3-4:

Cloth 5 + white quill (actor): 3.59
Room 4 + white quill (actor): 3.12
Cloth maker 8: 2.73
Builder 8: 2.66
Goldsmith: 2.59
Helper boy: 2.41
Cloth maker 6: 2.27
Builder 6: 2.22
3 quill income (actor): 2.07
3 room craft points + white quill (actor): 1.99
Happiness + blue quill (actor): 1.91
2 room craft points + yellow quill (actor): 1.84
3 cloth craft points + white quill (actor): 1.82
Money income (actor): 1.71
2 cloth craft points + red quill (actor): 1.47
Red quill + yellow quill (actor): 1.34
Happiness + white quill (actor): 1.31
4 craft points: 1.20
Unhappiness + yellow quill (actor): 1.16

Cards sorted by total benefit on rounds 5-6:

Goldsmith: 0.66
Builder 6: 0.63
2 room craft points + yellow quill (actor): 0.58
Happiness + blue quill (actor): 0.55
Builder 8: 0.54
Cloth 5 + white quill (actor): 0.45
Money income (actor): 0.37
Room 4 + white quill (actor): 0.36
3 room craft points + white quill (actor): 0.31
2 cloth craft points + red quill (actor): 0.30
Cloth maker 6: 0.17
Helper boy: 0.13
3 cloth craft points + white quill (actor): 0.01
Cloth maker 8: -
3 quill income (actor): -
Red quill + yellow quill (actor): -
Happiness + white quill (actor): -
4 craft points: -
Unhappiness + yellow quill (actor): -

Cards sorted by exhaustion benefit on rounds 1-3:

Cloth maker 8: 1.02
Cloth 5 + white quill (actor): 0.61
Builder 8: 0.35
Cloth maker 6: 0.32
3 cloth craft points + white quill (actor): 0.16
Goldsmith: 0.15
Room 4 + white quill (actor): 0.09

Cards sorted by exhaustion benefit on rounds 4-5:

Builder 8: 0.84
Cloth 5 + white quill (actor): 0.37
Goldsmith: 0.36
Builder 6: 0.33
Cloth maker 8: 0.23
Room 4 + white quill (actor): 0.22
3 room craft points + white quill (actor): 0.01

Notes

A bit below I'll analyze a bit more carefully the cards which are better than average. But here's the list of very poor cards, don't pick these:

* Red quill + yellow quill (actor)
* Happiness + white quill (actor)
* 4 craft points
* Unhappiness + yellow quill (actor)

Individual analysis

Cloth maker 8:

AP1-4: 2.80
AP5-6: 1.43
TB1-2: 5.53
TB3-4: 2.73
TB5-6: -0.07
EB1-3: 1.02
EB4-5: 0.23

Clearly the best card in the game. You get insane 2.8 victory points per action and also the highest benefit for removing exhaustion. The action is also quite robust so you don't have to take it as your first action very often. So get this card early if you can and remove exhaustion on turns 1-3 to make the card really shine.

Builder 8:

AP1-4: 2.12
AP5-6: 2.04
TB1-2: 4.79
TB3-4: 2.66
TB5-6: 0.54
EB1-3: 0.35
EB4-5: 0.84

A good card, and special in the sense that this card remains good even on the later rounds. On rounds 4-5 you get the highest exhaustion benefit out of this card, so recruiting him on rounds 1-4 and using him 3 times on last 3 rounds seems like a very solid strategy. If you recruit builder 8 it's usually everything you need to get all the rooms you want, especially if you manage to pick all the small rooms, which seems to be clearly the optimal strategy with rooms.

Goldsmith:

AP1-4: 1.93
AP5-6: 1.56
TB1-2: 4.52
TB3-4: 2.59
TB5-6: 0.66
EB1-3: 0.15
EB4-5: 0.36

A very solid card and one nice thing with this card is that you can usually pick the golden token with your last action because other players cannot take it. You should always prefer jewels instead of golden rooms, even on rounds 5-6. Pick the golden rooms only when you have to as a fallback. I was slightly surprised that the goldsmith seems to be only slightly better than Shakespeare on rounds 1-4, so he doesn't have higher exhaustion benefit than 0.15.

Cloth maker 6:

AP1-4: 2.10
AP5-6: 1.07
TB1-2: 4.37
TB3-4: 2.27
TB5-6: 0.17
EB1-3: 0.32
EB4-5: -

A good card, if somebody gets to pick builder 8 or goldsmith in front of you, but you manage to take cloth maker 6 on the same round, you're still very much in the game. Just like with cloth maker 8, the action strength drops drastically on rounds 5-6, so don't remove exhaustion token from this worker on rounds 4-5.

Builder 6:

AP1-4: 1.59
AP5-6: 1.53
TB1-2: 3.82
TB3-4: 2.22
TB5-6: 0.63
EB1-3: -
EB4-5: 0.33

An average card. Usually isn't enough to get full points from rooms alone and doesn't offer any exhaustion benefit early. He remains useful for the whole game though unless your rooms are full, so if you don't get anything from above, he might be the best pick for you.

Cloth 5 + white quill (actor):

AP1-4: 2.39
AP5-6: 1.57
TB1-2: 5.97
TB3-4: 3.59
TB5-6: 0.45
EB1-3: 0.61
EB4-5: 0.37

On paper, this card looks awesome and almost as good as cloth maker 8. Total benefit is actually higher because you do get 2 quills from the quill income, which is taken into account in total benefit calculations. Cloth maker 8 is better only because of its larger exhaust benefit. Another sweet thing with this card is the fact that you often get good initiative with it because it's an actor, but the problem is that you'll always have to fight for the initiative and spend your first action to use this actor, because otherwise the green tokens may be stolen. And with a bit of bad luck, green tokens may not appear at all. Each round there's roughly 21% chance for that event, so it's very likely to happen at least once during the game.

If you miss the action once, this actor becomes average or slightly below average card and if you miss it twice then it already becomes poor. So if you're good at outguessing and/or feeling lucky, pick this actor. You may win, but you may also lose horribly. This actor is much better when cloth makers do not appear early, because then you don't have to bid so low every round to get the initiative.

Room 4 + white quill (actor):

AP1-4: 1.87
AP5-6: 1.42
TB1-2: 4.98
TB3-4: 3.12
TB5-6: 0.36
EB1-3: 0.09
EB4-5: 0.22

Similar to the actor above. Very solid card on paper, but has the same problems. The likelihood of room 4's not appearing is even lower than for cloth 5, but there are more craftsmen who can pick those tiles and depending on the groupthink, you have to be very careful when playing with this actor. If you can get cloth maker 8, cloth maker 6, builder 8 or goldsmith, then definitely pick them instead of this actor, because they are more robust, but this is a good card, if it works.

3 cloth craft points + white quill (actor):

AP1-4: 1.94
AP5-6: 0.98
TB1-2: 3.75
TB3-4: 1.82
TB5-6: 0.01
EB1-3: 0.16
EB4-5: -

Quite robust, offers a good action for first 4 rounds and is an actor, so helps with initiative. Not one of the priority picks, but if there's nothing good left, this may still be available and it's quite beneficial to take.

3 room craft points + white quill (actor):

AP1-4: 1.69
AP5-6: 1.21
TB1-2: 3.67
TB3-4: 1.99
TB5-6: 0.31
EB1-3: -
EB4-5: 0.01

Another robust actor card, which has an action which is not great, but remains good for the whole game. Maybe a good choice on rounds 2-4 when all the good cards are already taken.

Helper boy:

AP1-4: -
AP5-6: -
TB1-2: 3.64
TB3-4: 2.41
TB5-6: 0.13
EB1-3: -
EB4-5: -

I calculated the values by assuming that you will get 1 extra cloth crafting point and 1 extra room crafting point for each round. If you don't get that much out of the him, total benefit should be lower. One good thing about him is that you can bid lower and still get a lot of stuff done, which helps in getting good initiative and bonus points for being 1st.
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Petri Savola
Finland
Espoo
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designer
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Please note that there may be mistakes, typos or just bad assumptions or simplifications in my analysis. I thought somebody may appreciate it, so I spent some time to post it here. If you want to continue the analysis or change something to get even more accurate results, please feel free to do so. And comment below if you observe any obvious mistakes.
 
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