GMP#4: R1 Hand, Polarization & Finding Cards
Table of Contents
Introduction
This article is devoted to the shape of Round 1 hands in Gwent: The Witcher Card Game. We will start from analyzing the space of possible R1 hands assuming natural mulligan strategies. Then we will define a ‘Gold Rush’ (GR) mulligan algorithm, resembling a player trying to get as many gold cards as possible. This algorithm would be used to calculate the probability of various hands after mulligan phase. We will act with GR on the old model deck introduced in GM&P1 and see how much mulligans impact the hand quality with respect to initial draws.
After various analyses and comparing blue/red coin hand strength we will move to the topic of polarization, gently touched in GM&P1. We will learn why it is necessary to polarize your deck and how exactly good polarization improves R1 hand quality.
Finally we will look into the most hot probability topic: what are the exact chances of finding cards in R1. Let’s stop babbling and start the math then…
1. R1 Hand
To avoid any possible confusion, R1 Hand is simply the hand we play in Round 1. While R1 mulligan phase discussion was more about play technique (GM&P3) – knowing crucial probabilities in order to make the correct decisions – the topic of R1 hand is closely related to deckbuilding.
1.1. The Space of R1 Hands
The space of R1 hands is the space of initial hands transformed by the mulligan phase. The exact way of transforming is dependent on player’s decision and it is impossible to make any analysis without assuming a mulligan strategy. While the strategy could have various depth and degree of adaptability to opponent’s deck, we would consider only the simplest ones.
Unless the strategy is very sophisticated, mulligan will start from removing bricks. There could be a hierarchy of bricks, harder ones removed first and softer having lower priority. We would assume no such hierarchy for the moment being. How much of the Initial Hand Space is cut out due to bricks?
Cutting the bricked hands out
It is the weakest and most justified assumption. Player tries to remove every brick from hand if possible. Let’s analyze how it impacts R1 hand space depending on the number of bricks in the deck build.
If only 1 brick in hand is possible out of K in the deck, then obviously all combinations with 2…K bricks are illegal as R1 hands. Summing the number of such illegal combinations with 2..K bricks we get how much shrunk the space is indeed. For ‘k’ bricks in hand we have (10-k) slots free for combinations of (25-K) elements (excluding additional bricks in hand prevents double counting). Also all the possible choices of particular bricks will be described by combinations of K bricks in k-slots and the final result is a product of these terms.
- Red coin
- Blue coin
Cutting out weakest cards
When bricks are mulled out, then the weakest cards (in R1 or overall) would typically have the greatest priority. We would not go into math here – only signaling the effect. Weakest cards will obviously interfere with bricks; the space would be reduced for 1(2) bricks on red(blue) only.
1.2. GR Mulligan Algorithm
Cutting the space does not really achieve much for computation purposes. The reason is that R1 combinations are not equally probable. The probabilities have to be deduced starting from initial hand anyway, one by one. Whenever there is no defined purpose (like finding a card etc.) we would assume a ‘Gold Rush’ algorithm defined below for computing R1 hand space.
Gold Rush mulligan algorithm
- If there are bricks in hand – mull out a brick
- If there is no brick – mull out the card with the lowest provision (at random if multiple choices)
- Continue mulligans unless highest total provision hand is already achieved
Gold Rush algorithm is defined in the provision space only. The name refers to searching for gold cards – algorithm guarantees finding as much of golds as possible (neglecting bricks) – while completely ignoring any matchup/R1 strategy wisdom.
1.3. Bricks and Provision Space
Working in provision space with bricks is tricky though, basically having to associate provision value with each possible brick (such value could even be negative!). To remain consistent with GM&P1, we would not include bricks yet.
I plan to do it in the future along with mapping provisions into real points.
2. Acting with GR on SK Witchers
Although Price of Power expansion happened meanwhile, we would keep work with the same SK Witchers deck as the main model example (have a look at GM&P1).
The complexity of the problem is linear with the number of mulligans. As we remember, the provision hand space is pretty small: 1136 inequally probable combinations. Direct acting with GR on each combination poses no difficulty.
(The model deck has two bricks, but as stated in subsection ‘Bricks and Provision Space’ we have to ignore this fact at the moment)
2.1 Total Provision
As could be seen, each subsequent mulligan has visible impact on provision distribution. Vertical dashed lines denote the mean value of respective distributions. Continuous curves are Gaussian fits. Main descriptors are juxtaposed in table below.
Average provision value of hand (mean) increases almost linearly with the number of mulligans used. On the other hand, there is no visible trend in standard deviation (std). Mulligans would not even out hand power disparity between games (or players) at R1 stage! Of course, weakest and bricked hands have a good chance of getting improved, but after all the average disparity (std) will remain about the same.
The difference between ‘initial’ and ‘blue’ mean values is higher than the standard deviation of initial. In human language it translates to ‘new quality’. Maximal total provision of 10 cards in our model deck is 96. We reach 68.75% of this value on average when drawing initial hand, while after blue coin mulligans the average quality reaches 76%. On average each mulligan is worth roughly 2.5 provision points for hand quality.
After numerical comparison, let’s turn to qualitative one. As you may remember from GM&P1, bins with 6 provision width were defined. We would keep the same bins now and see how much hand quality improves above ‘Normal’ level (TLG naming convention is used).
Particular numbers are not important here, but you could easily see how extra mulligans establish new quality standards. Hand viewed as ‘Good’ (or even ‘OP’) at the initial stage is standard after mulligan phase. Variance is still high, but the disastrous scenario of ‘Absolute ass’ hand becomes almost impossible and hands weaker than ‘Normal’ goes from ~33% to <15% on red (<10% on blue).
2.2. Provision Discrepancy
Playing a game of Gwent, obviously red-coin and blue-coin players hand qualities are independent events. Then according to probability theory, mean provision advantage of blue is simply equal to difference of mean provision values, which is ~2.5p.
Nevertheless, the shape of the discrepancy curve is still an interesting object to study.
It appears that ~60% of the time the total provision value of blue-coin player’s hand is greater than red-coin player. That’s a significant power gap, increasing chances of blue coin player whenever contesting Round 1 is crucial for a given matchup.
The |Δ| provision curve is very similar to one known from initial hand study.
Compared with initial hand power disparity only a slight increase is observed. Blue coin hand quality advantage does not have too much impact here.
3. Polarization
In GM&P1 we touched a bit the topic of polarization. In simple words, polarization describes how much a deck is biased towards high-end gold cards. We defined a parameter to measure polarization of a deck as a fraction of provision variance and maximal possible variance for a given provision cap.
We have shown that polarization has negative impact on the consistency of initial hand power. Also we ‘played’ the game of ‘TurboGwent’, where player drawing more provisions wins. This way we learned that less polarized decks are slightly favoured due to asymmetry of provision distribution.
To sum up, at the stage of initial draws polarization had neutral or maybe even slightly negative impact. On the other hand, if you compete in Gwent or follow main content creators, you probably know that high enough polarization is essential for a well-built deck. Round 1 mulligans is the first stage at which properly polarized decks starts to be favoured. Let’s see exactly how it happens.
3.1. Intuitive View on R1 Mulligans
We have computed a lot of numbers so far. But do we understand them? Time to stop being exact and develop an intuitive view on what happens in R1 mulligans with respect to provisions. Let me tell you some useful lies.
R1 Mulligan phase means just throwing off the weakest cards from initial hand. This way in N mulligans you could get rid of N weakest cards. Those cards would be replaced by random cards from the deck. We know average value of cards in the deck – it is just ProvisionCap/25. For example in the model SK Witchers deck average provision quality of cards drawn would be 6.6p.
It is easy to see that when 4p cards are mulled out, the net gain of value is equal to 2.6p. This fact is in very good accordance with the results of numerical computation (‘Acting with GR on SK Witchers’). Expected gain is slightly lower there because 4p cards are not always available; it also explains why each next mulligan improves the hand a bit less.
3.2. Trash is good
Exchanging 4p cards obviously raises hand quality the most. Therefore in order to get maximum provision benefit from mulligans there has to be a common access to 4p cards in the number equal at least to the number of mulligans.
Imagine that you play a deck full of 6 and 7 provision cards instead and the mean value is still equal to 6.6 as in the model deck. Then the net gain from every mulligan would be around +0.6p. Summing numbers up, the average power advantage of a polarized deck on blue coin would be already equal to 6p, which is roughly equivalent to playing with one gold card more.
That’s the way how Polarization comes into play at the R1 stage. Nothing else is important but the number of 4s in the decks. No matter if your high-end golds are 10 or 13p. It would come into meaning in the subsequent game phases only.
One final remark. Everything has a price. Provision gain in hand is at the expense of lowering average deck power. It possibly leads to slightly worse topdecks in later rounds.
4. Finding Cards
When it comes to finding cards, hypergeometric probabilities are the ultimate answer. If N=25 is the number of cards in the deck, n=12(13 on blue) is the sample size, ‘K’ is the packet size (no. of cards we want to find) then {‘k’=number of desired card in hand} could be read from cheatsheets below (all numbers are in % and rounded to 0.1 for better visibility).
4.1. Using R1 Probability Cheatsheets
(you could also find cheatsheets as a Google Sheet here)
Unlike GM&P1, probabilities shown in spreadsheets above are ‘>=’, which means the odds to find at least ‘k’ cards from the packet in hand. Let’s ask few typical questions probability questions and answer them using tables.
- Drawing a single card
It doesn’t really require hypergeometric, but ‘>=1 of 1’ row in back-to-back table answers the question. On red coin there is 48% chance of finding desired single card, on blue coin 52%.
Gwent v9.0 examples: finding Kurt in Devotion Jackpot to purify Defender/ finding Cupbearer in Ball Imprisonment/ finding Oneiromancy in any meta deck
- Drawing at least one of two cards
Another fundamental probability. We use ‘>=1 of 2’ row in back-to-back table. R: 74%, B: 78%. This problem typically arises in decks playing a R1 card with a single tutor. Also thinning pairs are common examples
Examples: Amphibious Assault w/Jan Natalis, She-Who-Knows w/Oneiromancy, Portal w/Avallac’h: Sage, Archespores in Force of Nature Koshchey
- Completing Poison Package
Nilfgaard Imprisonment Ball and Syndicate Midrange Jackpot were the meta decks playing Poison packages in Gwent v9.0. NG package was made of 2xFangs and Cupbearer (tutoring via Roderick ignored), SY package of 2xFisstech and 2xFisstech Trafficker (there was also Salamandra Hideout for offensive poison, but we will neglect this part). Usually at least a poison pair is called for to get the removal value. We use then respectively >=2 of 3 and >=2 of 4 in ‘back to back’ table to get 50% and 70.3%. Syndicate had +20% more chances to get poison working in R1!
- Drawing at least ‘k’ out of K golds in R1 hand
Golds are too big of a package to include in back-to-back table, so we have to use Blue and Red coin sheets. For example in the model SK Witchers deck we have defined 10 cards as golds (GM&P1). What is the chance to draw at least half of them? We search for k ‘>=5’ and K=10. The answer is R: 59.6%, B: 71.6%.
4.2. Finding Golds
In the case of Golds the exact numbers are perhaps more interesting than ‘>=’. Let’s see what are the odds for various number of golds in the initial hand.
,where exact values are put in the table below.
The chance to miss all your golds is less than 1/10000 on both coins. It means that such an event would occur maybe 1-2 times in whole career of most Pro Gwent players. Wait… did something raise your eyebrow? The numbers…
4.3. Don't think so.
If you look closely into values in the table and bars in the chart, it becomes clear that Red(%) and Blue(%) are mirror images of each other. What is the cause? Try to figure it out yourself before moving on to the answer below…
On red coin you draw 12/25 cards with no real impact on what you draw – it is completely random. Imgaine you play a game of Gwent with friend, but using only one 25 cards deck. You draw 12 cards from the top (red) and your friend 13 cards (blue) from the bottom.
It becomes clear that your chances to find ‘k’ golds are exactly equal to your friend’s chances of finding ‘K-k’ golds! If your chances to draw 5 golds are 31.2%, then your friends chances are too 31.2% for (10-5) = 5 golds.
Closure
Thanks for reading!
After the most nerd article in GM&P cycle (GM&P3), this one is in my opinion really rich in useful information, especially when it comes to deckbuilding. I hope you feel some insufficiency after lecture – there was no time for various conclusions and case studies making use of the probability tables. If so, then let me know what R1 case studies would you like me to investigate in the future. Feel free to contact me on Twitter or Discord (there is actually a Twitter thread for this purpose having 9 likes and zero suggestions KEKW).
I predict to make a short follow-up article in two weeks. I hope it would be a fancy one as I invited some of the top Pro Ladder players to collab and write some numbers. Then maybe two monographic articles at the start of the next Gwent season.
Stay tuned!
Written by:
lerio2
