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Subject: Statistical analysis by ReiXou rss

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John Farrell
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Olivier Reix posted a series of Beowulf articles on his blog "Les Ptits jeux du Mercredi". These articles are similar to my review, but go into much more depth. For the enlightenment of BGGers who don't read French, I have translated them here with permission.

Here are some notes and probabilities for your consideration, in particular on risks.

1. Division of cards

In the 100 cards, there are 18 cards of each type (Travel, Triendship, Wit, Courage, Fighting) and 10 Beowulf cards (jokers). In each type there are 15 cards of value 1 and 3 cards of value 2 (totalling 21 symbols) except for the combat cards where there are 13 of value 1 and 5 of value 2 (totalling 23 symbols). The Beowulf cards are all of value 1.

2. Probability of succeeding at a risk

Assume the starting conditions:
* 5 player game
* in hand I receive a Beowulf card and a combat card and the 5 other cards are evenly distributed (I receive therefore 1 more Travel, Triendship, Wit, Courage and Fighting).
* I'm first player

Question: what chance do I have of succeeding at the first risk in the game (Sailing to Denmark)?

Of the 100 cards, there are 15 cards I know: the 7 cards in my hand, plus 4 jokers and 4 combat cards which are in the hands of the other players. Of the remaining 85 cards, there are 17 + 17 + 5 (jokers) which permit me to succeed at the risk - 33 with one symbol and 6 with 2 symbols. My chances are then:

* ((85-39)/85 x (84-39)/84) = 29% chance to take a scratch

* 33/85x46/84 + 46/85x33/84 = 42.5% to win exactly one symbol

* 33/85x32/84 + 6/85x46/84 + 46/85x6x84 = 22.5% to win exactly 2 symbols

* 6/85x33/84 + 33/85x6/84 = 5.5% to win exactly 3 symbols

* 6/85x5/84 = 0.4% (let's call it 0.5) to win exactly 4 symbols

Each time I take a risk, I hope to win:

* 0.29 scratches (therefore with 10 risks you can expect to take 3 wounds)

* 0.425x1 + 0.225x2 + 0.055x3 + 0.005x4 = 1.06 symbols

You can use those numbers to roughly assess your chances when taking a risk, but be aware that:
* the numbers change dramatically as cards are played and seen (and so you know what remains in the deck)
* they change slightly depending on the number of players and the initial cards you have in your hand
* they are also slightly different if combat cards are taken in the risk, because they have different numbers of symbols (and they are initially distributed differently).

3. Distribution of alliance tokens

The 36 alliance tokens are divided like this:
* 12 gold tokens (4x1, 4x2, 4x3)(24 gold in all)
* 24 fame tokens (6x1, 6x2, 12x3) (54 fame points in all)

4. What can you hope for taking an alliance token?

Each time you take an alliance token you can hope to gain:

* 24/36 = 0.66 gold and
* 54/36 = 1.5 fame points

Or in other words:

* 1 time in 3 : 2 gold on average
* 2 times in 3 : 2.25 fame points on average

It follows then that using the rules of the basic game (which I don't recommend, after all, the game is simple enough), it is generally better to take an alliance token (expecting 2.16 victory points) than 2 gold or 2 fame points (= 2 victory points). With the advanced rules it is less clear but if you have no particular need for gold or fame, it's more interesting to draw an alliance token.

5. Finding the values of things

There are multiple occasions to win resources during a game, but the major differences between players occur during the major episodes and the treasure episodes. Let us concentrate on the former.

Of the 36 episodes, 12 are major episodes which are comprised of:
* 7 clockwise auctions
* 5 simultaneous auctions (that is 4 plus the final auction, which isn't really an auction at all)

One intuitively feels that some auctions are more crucial than others, in particular (after saying that the author doesn't pay attention to theme) those corresponding to the great combats of Beowulf - against Grendel, against the Sea Hag, against the Dragon, and during the raid on Friesland. I will assess the importance of each episode with the help of 2 values:
* Amplitude, which estimates the difference in victory points (VPs) between the winner and the loser of the episode
* Sum, whic estimates the number of VPs to win or lose on the episode, to see if the episode is positive or dangerous.

To do that, it's necessary to estimate the value of each thing in VPs:
* fame points: simple, they are directly VPs
* infamy tokens: also simple, they take off 2VPs, so they count as -2
* gold: not simple to estimate precisely but because we need to calculate I have take 1 gold equal to 1 VP which agrees with the basic rules and the choices proposed in the Selection episodes
* alliance tokens: we saw earlier that an alliance token is with 0.66 gold and 1.5 VPs. I have therefore added those two values together and estimated each alliance token at 2.16VPs.
* cards: in all honesty it's very hard to evaluate the value of card which changes greatly according to the situation. Based on the Selection episodes, I have made a dubious estimate of 1VP (very dubious in fact). I must say that cards are worth more at the beginning of the game than at the end. All the same, because they all have good times to be used, I will stick with 1VP. Note that the 2 card symbols must be I think valued at 2VPs and those with 3 or 4 (special cards) at 3 or 4 VP or else one makes a very very big mistake. Overall, assume then that 1 symbol = 1VP and 1 card = 1VP, that's approximate but it will do well enough.
* scratches and wounds: these also aren't simple as their influence is not linear. We know the first wound (= 3 scratches) loses 5VPs, add a second and a third and you lose another 15VPs (making 20VPs total). The fourth also loses 5VPs (25 in total) and others as well, but I doubt you can achieve that playing seriously. Scratches are still irregular, the first are without direct consequence but the 3rd costs you 5VP and the 9th costs you 15VP. I have estimated wounds to lose on average a bit more than 5VPs each (5 for the first, 20 for the third, 25 for the 4th) and I have rounded off to -6VP but in my opinion they cost a bit more than that. That cost makes a scratch worth -2VP which seems to me to be about right (remembering that scratches function in 3s, so the first is always less critical than the second which is less grave than the 3rd which costs you a wound).
Note that when you take a wound you can also draw 2 cards, which makes the approximate penalty -4VP.

Note: according to this, when one takes a risk one can get on average 0.29 scratches at 1.06 card symbols. Following our evaluation, 1 card = 1 VP, therefore our expected gain from a risk is 2x0.29+1.06 = about 0.5VP, therefore generally positive. This result seems to encourage us to take risks in the Risk episodes (during the auctions there are different stakes which enter into the calculation), as long as we don't already have 2 scratches.

* the special cards: for the 5 special cards with symbols, I value them according to their number of symbols (1 symbol = 1VP) which gives us 4VP for the iron shield and 3VP for the others.

King Hrothgar: he allows you to retrieve one card to your hand. Combined with a special symbol card he is very strong, and you can at least use him with a 2 symbol card. Estimated at 2.5VP.

Queen Wealhtheow: she lets you refuse a scratch, a wound, a misfortune (-2VP) or even the double wound of the dragon, which is naturally her best use (if you want to avoid the loss of 15 or 20 VPs in that case). Generally, using her for something other than a wound seems inefficient. I estimate her therefore worth a minimum of 6VP (a wound) knowing that you even better use her with the dragon.

Beowulf's Daring: I admit I haven't done the precise calculations but I that a 5 card risk must be worth on average 3 symbols (maybe 4), therefor let's say 3VPs.

Beowulf's Resolve: you ignore a failed risk (and therefore a scratch = 2VP) and you can even continue to play in the case of an auction. 3VPs for the road.

Warrior's Rest: -2 scratches = 4VPs.

6. Major Episodes: Amplitudes and Sums

We then get the following results for the major episodes (I have rounded to the closest number):

* the simultaneous auctions:

3. King Hrothgar's Hall (Friendship) : Amplitude 3 / Sum 11
10. Sea Hag's Attack (Navigation) : A : 8 / S : 3 (*)
17. King Hygelac (Friendship / Wit) : A : 3 / S : 12 (*)
22. Beowulf King (Friendship / Courage): A : 2 / S : 11 (*)
36. Death of Beowulf (All) : A : 4 / S : 15

* the clockwise auctions

5. Grendel's Attack (Courage / Fighting) : A : 9 / S : 4 (*)
8. Celebration (Friendship / Wit) : A : 3 / S : 9 (+)
13. Hunting the Sea Hag (Courage / Navigation) : A : 6 / S : 8
14. Encounter the Sea Hag (Wit / Fighting) : A : 9 / S : 4 (+)
20. Raid against Friesland (Fighting / Navigation) : A : 9 / S : 2 (+)
31. Dragon's Rampage (Wit / Navigation) : A : 8 / S : 1
34. Dragon Battle (Courage / Fighting) : A : 17/ S : 1 (+)

(*) designates the episodes where the last player will be in position to choose during the following Recovery episode: they can reasonably expect to take a 2 symbol card
(+) designates the episodes where the last player will be in a favourable position during the following simultaneous auction (in case of ties).

Sorting by decreasing order of Amplitude (from most crucial to least important in terms of VPs), we get:

17 VP (1 episode): 34. Dragon Battle (Courage / Fighting)
9 VP (3): 5. Grendel's Attack (Courage / Fighting) / 14. Encounter the Sea Hag (Wit / Fighting) / 20. Raid against Friesland (Fighting / Navigation)
8 VP (2): 10. Sea Hag's Attack (Navigation) / 31. Dragon's Rampage (Wit / Navigation)
6 VP (1): 13. Hunting the Sea Hag (Fighting / Navigation)
4/3/2 VP (5): the others.

Sorting in increasing order of Sum, we get:

1/2 VP (3): 20. Raid against Friesland (Fighting / Navigation) / 31.Dragon's Rampage (Wit / Navigation) / 34. Dragon Battle (Courage / Fighting)
3/4 VP (3): 5. Grendel's Attack (Courage / Fighting) / 10. Sea Hag's Attack (Navigation) / 14. Encounter the Sea Hag (Wit / Fighting)
8/9 VP (2): 8. Celebration (Friendship / Wit) / 13. Hunting the Sea Hag (Courage / Navigation)
11/15 VP (4): 4 simultaneous auctions

We see that this is similar to the Amplitude ordering, but the first seems to mean more to me.

Conclusions

Let's see what we can learn from the classification by amplitude.

* the attentive reader will remark that the fighting cards are used in the 4 most crucial auctions. It's well for us that we're stocked with 2 symbols at the beginning of the game. It's necessary to manage them well throughout the game.

* Friendship never matters at important moments!

* Navigation in 4 of the 7 most crucial auctions, and one time exclusively (Sea Hag's Attack) followed by Courage (3 times) and Wit (twice)

* the simultaneous auctions are of less consequence than the clockwise auctions, with the notable exception of event 10 Sea Hag's Attack (Navigation)

* 6 auctions out of 12 create a large divergence in VPs (Episodes 5/10/14/20/31/34). You need to be ready for those, in particular for the battle against the dragon which can be very very bad except eventually for the owner of Queen Wealhtheow who can pass with impunity through this episode.

Note: the gold episodes create equally large divergences in VPs but are not analysed here (later maybe?)

(bravo if you have made it this far, but all the same I fear for your mental health )

7. Evaluation of the symbols (Fighting, Courage, Wit, Navigation and Friendship)

Here is a small summary of the characteristics of the 5 symbols available in the game. The Beowulf cards count as a joker and aren't assessed here. For each symbol I have listed:
* the number of symbols available in the deck
* the episodes in which that symbol appears: risks, auctions, others (RR = Recovery, where you choose cards)
* for the significant episodes, I have indicated the Aplitude of the event in the form of the available returns
* finally I have summed the Amplitudes concerning the symbol in order to try to measure their importance to the game

Symbol: Friendship

Availability: 15x1+3x2=21

2. Risk (with Courage)
3. Simultaneous auction (seul) *** (4/2,5/2/2/1)
6. RR
8. Clockwise auction (with Wit) *** (4/4/3/1/1)
11. RR
16. Risk (with Navigation)
17. Simultaneous auction (with Wit) - *** - (4/3/3/2/1)
19. RR
22. Simultaneous auction (with Courage) *** (4/3/3/2/1)
26. RR
28. 2 Friendship => 1 alliance token **
36. ****

Total amplitude: 18*


Symbol: Wit

Availability: 15x1+3x2=21

4. Risk (with Courage)
6. RR
8. Clockwise auction (with Friendship) *** (4/4/3/1/1)
11. RR
14. Clockwise auction (with Fighting) ********* (5/3/2/-2/-4)
17. Simultaneous auction (with Friendship) *** (4/3/3/2/1)
19. RR
26. RR
30. Risk (with Fighting)
31. Clockwise auction (with Navigation) ******** (4/2/1/-2/-4)
36. ****

Total Amplitude: 27*


Symbol: Navigation

Availability: 15x1+3x2+3=24

6. RR
8. Clockwise auction: 3 Navigations to win
10. Simultaneous auction (alone) ******** (4/3/1/-2/-4)
11. RR
13. Clockwise auction (with Courage) ****** (4/3/2/1/-2)
16. Risk (with Friendship)
19. RR
20. Clockwise auction (with Fighting) ********* (5/2/1/-2/-4)
26. RR
31. Clockwise auction (with Courage) ******** (4/2/1/-2/-4)
36. ****

Total Amplitude: 35*


Symbol: Courage

Availability: 15x1+3x2+3=24

2. Risk (with Friendship)
4. Risk (with Wit)
5. Clockwise auction (with Friendship) ********* (5/3/2/-2/-4)
6. RR
11. RR
12. Risk (with Fighting)
13. Clockwise auction (with Navigation) ****** (4/3/2/1/-2)
14. Clockwise auction: 3 Courages to win
19. RR
22. Simultaneous auction (with Friendship) *** (4/3/3/2/1)
26. RR
34. Clockwise auction (with Amitié) ***************** (7/2/2/-2/-10)
36. ****

Total Amplitude: 39*


Symbol: Fighting

Availability: 13x1+5x2+3+4=30

5. Clockwise auction (with Courage) ********* (5/3/2/-2/-4)
6. RR
7. 2 fighting => 3 gold *
11. RR
12. Risk (with Courage)
13. Clockwise auction: 3 Fighting to win
14. Clockwise auction (with Wit) ********* (5/3/2/-2/-4)
19. RR
20. Clockwise auction (with Navigation********* (5/2/1/-2/-4)
26. RR
30. Risk (with Wit)
32. Gold auction: 4 Fighting to win
34. Clockwise auction (with Courage) ***************** (7/2/2/-2/-10)
36. ****

Total Amplitude : 48*

Analysis

The spread of amplitudes for the cards is incredible, as we suspected earlier. The distribution will be about Fighting 48 / Courage 39 / Navigation 35 / Wit 27 / Friendship 18. Of course the numbers are only indicative but the spread is significant and seems to mean that at the beginning of the game:

1 Fighting = 1.2 Courage = 1.4 Navigation = 1.8 Wit = 2.7 Friendship!!!!

On reflection, it seems to me that it is necessary to measure the strength of each symbol against the number available. That tends to equalise things and one gets these numbers if distribution is considered:

1 Fighting = 1 Courage = 1.1 Navigation = 1.25 Wit = 1.9 Friendship

8. What to choose from the recovery episodes?

Here is at least a guide to your choices during the recovery episodes (RR). Evaluating the symbols for each of the 4 recovery episodes, we deduce for each an order of choice:

Episode 6 Fighting 39 / Navigation 35 / Courage 30 / Wit 27 / Friendship 15

Order of choice:
2 Fighting / 2 Navigation / 2 Courage / 2 Wit / 1 Beowulf / 1 Fighting / 1 Navigation / 2 Friendship / 1 Courage / 1 Wit / 1 Friendship

Episode 11 Fighting 39 / Courage 30 / Navigation 27 / Wit 24 / Friendship 12

Order of choice:
2 Fighting / 2 Courage / 2 Navigation / 2 Wit / 1 Beowulf / 1 Fighting / 1 Courage / 1 Navigation / 2 Friendship / 1 Wit / 1 Friendship

Episode 19 Fighting 30 / Courage 24 / Navigation 21 / Wit 12 / Friendship 9

Order of choice:
2 Fighting / 2 Courage / 2 Navigation / 1 Beowulf / 1 Fighting / 2 Wit / 1 Courage / 1 Navigation / 2 Friendship / 1 Wit / 1 Friendship

Episode 26 Fighting 21 / Courage 21 / Navigation 12 / Wit 12 / Friendship 6

Order of choice:
2 Fighting / 2 Courage / 2 Navigation / 2 Wit / 1 Beowulf / 1 Fighting / 1 Courage/ 2 Friendship /1 Navigation / 1 Wit / 1 Friendship

In summary:

- Fighting / Courage / Navigation are your priorities, in that order

- Wit is not bad in the first two choices, but weak thereafter

- it's almost always better to take a single other symbol than a double Friendship (that's less clear in the first choice).

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Olivier REIX
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Thank you for the translation, John.

Maybe this should have been posted in the "strategy article" section.
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John Farrell
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ReiXou wrote:
Maybe this should have been posted in the "strategy article" section.


Yes, probably. However ynnen suggested I post my analysis as a review, so I figured yours went in the same place.
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Jay Little
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For the amount of time, effort and analysis put into these, I think they provide valuable content and are valid uses for the review system... I hope others agree, as well. It may focus on a specific aspect of the game, but diversity in reviewing a game's mechanics is pretty darn useful.
 
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Matthew M
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ynnen wrote:
For the amount of time, effort and analysis put into these, I think they provide valuable content and are valid uses for the review system... I hope others agree, as well. It may focus on a specific aspect of the game, but diversity in reviewing a game's mechanics is pretty darn useful.


I've moved them to the Strategy section because that is a more appropriate category. Users will go to the Review folder to find opinions and summaries of a game. Users will go to the Strategy folder for tips on play and for this kind of hard analysis.

-MMM
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