GMP#2: Mulligans, Accessibility & Thinning Paradox
Table of Contents
This article was supposed to be an in-depth analysis of R1 probabilities in Gwent, yet when I looked right into the first mulligan phase, the general topic of mulligans and determinism appeared to be too rich and important. I had many misconceptions myself and I hope this article will clarify a lot of things.
We will start step by step by defining mulligan (what exactly happens when you view it through math glasses?) and mulligan phases. Then we will move to the main part: accessibility problem. How many cards could you access during a game of Gwent? What is the probability of finding a single card? We will start from finding accessibility groups, then do some math and visualizations for the most simple cases. This part will conclude with some cpu assisted (yet exact) computation results.
Next, based on observations made in the main part, we will try to answer the question: ‘How many games of Gwent are written?’. Is the combined number of possible mulligan shuffles and deck orderings processible by CPU or not really?
The last chapter would be my first real DISCOVERY (it is peculiar to discover something universal in a computer game, I know…). Ladies and Gwentlemen, let me introduce you to THE THINNING PARADOX.
Have a good read!
1. What is a Mulligan?
The general meaning of ‘mulligan’ term is a second chance to perform an action, usually when the first attempt went in a wrong (most often also unlucky) way (wiki). According to this definition, mulligans in Gwent are additional chances for improving your hand quality and playability, while bad draws are analogy of ‘failure’ in the first attempt. Mulligans as an element of gameplay are supposed to alleviate the card drawing RNG. Especially improvement of very weak or bricked hands is enabled by mulligans, so that worst case scenarios are most often cut out from the space of possible events.
2. Mulligans in Gwent
How does mulligans exactly work in Gwent? Let’s look into it step-by-step.
A single mulligan in Gwent is an act of choosing one card from your initial hand, moving it into a temporary stack, and replacing with the top card from your deck. Card drawn is immediately visible in hand (unlike in some other card games, e.g. Hearthstone) .
Mulligan phases arise after initial draws in every round and consists of a strict maximal number of mulligans, never less than 2. Player going first (blue coin) has 3 mulligans available in R1, while player going second (red coin) – 2 mulligans. Also each draw exceeding 10 cards limit at the round start grants an extra mulligan.
When the mulligan phase ends, all cards from the temporary stack are shuffled back to the deck in a random way. Mulligan phase lasts about 40 seconds.
3. Determinism and Accessibility
Determinism is the opposite of randomness. While draws are obviously random in Gwent, the order of cards in the deck at the very start is kept during the game (unless changed by specific cards like Maxii etc.). This kind of determinism have to be acquainted for – some cards could be just inaccessible whole game by the ruling of Gwent gods.
It leads to a question: how many cards are accessible via draws and mulligans? Let’s try to consider different boundary scenarios, which will lead us to three accessibility groups: Guaranteed, Accessible, Inaccessible.
3.1. Accessibility Groups
- Best Case (Inaccessible Group)
Let’s start from the most simple, best case scenario, where each card you mulligan out lands straight at the bottom of the deck. Each mulligan and each draw means accessing a new card in such case. No thinning, no tutors, no other deck manipulation tools. How many cards are you going to see? Let’s count for blue and red and assume no extra mulligans.
Without any deck manipulation, 2 cards are always inaccessible on blue coin and 3 on red coin. There is instant low bound of 8%(12%) to completely miss a card. Let’s call this group ‘Inaccessible’.
- Worst Case (Guaranteed Group)
In the worst case, each card mulliganed out lands back on the top of the deck. Out of card drawn in R2/R3 only a fraction is new.
Blue: 10+3+2(3 card mulled out are back in hand and 2 mulls get 2 new ones; one old left in hand)+3 = 18
Extra mulligan on the blue coin does not even improve low-bound accessibility when you are terribly unlucky. On both coins top 18 cards are guaranteed to be found in a full 3 rounds game.
- Usual Case (Accessible Group)
From cases above we deduce that 19th to 23rd (22nd) cards are accessible, but not guaranteed; everything depends on draws.
- Event Space
Let’s now wonder how big is the space of possible events. Deck order is determined, so we may just use 1 to 25 numbers to distinguish cards.
The differences between various scenarios come only from RNG of shuffling back mulliganed cards. For the red coin there would be 3 mulligan phases with 2 mulligans. The order of these two cards in deck doesn’t matter as both were already accessed, so we look for combinations of slots indexes on two places. For example in Round 1:
R1: 15!/(13!*2!) = 7*15 =105
and in Round 2:
R2: 12!/(10!*2!) = 66
Multiplying these numbers gives all accessibility scenarios with respect to mulligans.
Total: 105 x 66 = 6930
Just as in the case of initial hand combinations, the number is too big to be processed by human, but very small for a casual CPU computation task. It is possible then to analyze all mulligan events strictly without using any shortcuts provided by math.
3.2. Finding Cards from the Accessible Group
Let’s calculate the most simple cases of accessing the first/last accessible card.
- Accessing the last card (22, red)
In the picture above we see the order of cards in deck before R3 which leads to drawing 22 in the last mulligan. All 4 card mulliganed out are after 22 and their order is irrelevant. One of the methods to evaluate probability of such event is going step by step along with mulligans. There are 4 slots behind 22 available in the 1st mulligan. The probability is 4/14. Each card shuffled in below 22 increases the number of slots below 22. Consequently, in 2nd we have 5/15. In R2 3 cards are drawn and again: 3rd mulligan: 6/11, 4th: 7/12 (notice that 7 is exactly the final number of cards behind 22).
All events stated above must occur in one game, so the total probability of accessing 22 is equal to product of all probabilities, which is 1/33 ~ 3%. At the same time it is the chance that not a single card you mulled out during the game is found back (1/143 ~ 0.7% on blue)!
Comparison of blue coin and red coin results show that the accessibility difference is not so huge when it comes to maximal reach; finding additional card (23) thanks to blue coin is very improbable.
- Drawing the first accessible card (19, red)
There are 7 slots preceeding ‘19’ available in the 1st mulligan. The probability is 7/14. Each card shuffled in before ‘19’ increases the number of preceeding slots . Consequently, in 2nd we have 8/15. In R2 3 cards are drawn and again: 3rd mulligan: 4/11, 4th: 5/12 (so that there are 5 cards before ‘19’ when R3 starts and it could not be accessed).
‘19’ will be accessed in ~96% (95/99) of games on red. On blue the negative scenario is multiplied by 6/13, so we get ~98.1% odds of finding ‘19’.
- Accessible Group – final results
To compute the probabilities of accessing all cards from ‘19’-‘23’ range I used aforementioned combinations of possible shuffle-back slots. This method gives exact and very nice numbers, which confirmed the maths presented above (by very nice I mean fractions of integers; for example Twitch favourite 23/33 is here)
The impact of extra mulligan on the blue coin is clearly visible. While the possibility of accessing extra card (‘23’) is rather irrelevant, and the ‘19’-th is pretty much guaranteed anyway, the difference in %odds is considerable in ‘20’ to ‘22’ range. Extra mulligan transfers into +17% chances of finding ‘21’st card!
4. Determinism and Finding a Single Card
Knowing that each card position is equally probable after initial draws and computing the exact odds of finding cards from ‘accessible’ group we could easily get total probability of finding a single card in a game of Gwent. Before we do it though, let’s estimate the bounds.
As 18 cards are always accessible, the probability of finding a card during 3 rounds game could not be less than 18/25 = 72%.
From the other side, 3 cards are (in the case of blue – almost) inaccessible – the probability of finding a card could not be higher than 88%.
To sum up, if we denote the probability of finding a card by ‘p’, then 72% < p < 88% and the lucky guess would be p=80%. Let’s move to exact results.
The chances to find a single card are very close to 80% on both coins, and this number could be memorized for practical purposes. The raw difference in accessibility between coins is 1.55%, which means in 3 games per 200 (on red) you would miss a crucial card just because of playing on red rather than blue coin.
Now let’s see if determinism impacted the %chances indeed. Calculating the odds of finding a card when the deck is shuffled after every round is very simple: 1 – chance to miss the card completely = 1-(chance to miss R1)*(chance to miss R2)…
Red: 1-(1-12/25)*(1-5/15)*(1-5/12) = 79.78%
Blue: : 1-(1-13/25)*(1-5/15)*(1-5/12) = 81.33%
Results are exactly the same! Determinism does not have impact on finding a single card during the game!
5. How many games of Gwent are there written?
Discussing mulligans and determinism is the perfect time to investigate how many different scenarios are written for both players during a game of Gwent. By ‘written’ I mean independent of players actions and card effects (for which each possible RNG outcome could be another path).
Let’s assume all mulligans are used by default. In GM&P1 we have shown that the order of cards in initial hand is rather irrelevant – let’s keep this assumption.
In this article on the other hand we have shown that the order in the deck really matters. There is 15! possible orderings of the deck and while it is possible to cut sth here and there from this number with another assumptions, we will keep it general.
Order also has to be included when shuffling cards back into the deck now, so the number of combinations on every stage has to be multiplied by n!, where n! is the number of cards shuffled back. Also last mulligan has to be taken into account. We are outside of the simplified world of accessibility problem!
Red: 3268760*15!*Comb(15,2)*2!* Comb(12,2)*2! Comb(9,2)*2! = 97.858.829.376.000
You know what? Gwent is a game for two where coinflip determines who starts. Let’s multiply these number together (and by 2 for the coinflip variable) for total the number of ‘written’ games between 2 players.
Games: 248.985.112.657.868.094.283.776.000.000 = 2.49E+29
Ocean of fate. And we didn’t even consider any possible choices made by both players. Gwent is far too big to be analyzed directly with all possible orders and mulligans, let alone optimizing gameplay with these. Smart, not brute is the way.
6. Thinning Paradox
Thinning is always good. It lets you scroll through the deck more efficiently… ~ Abraham Lincoln
Well, if you think that the statement above is true, then… you are generally right, but thinning has a non-zero chance to backfire with THE THINNING PARADOX!
Imagine you managed to thin one card from your deck in R1. Let’s assume you play one win-con card in the given matchup, but it is not in the hand yet: you have to find it. The card is at the position N in the deck, let’s say N=22, so that you live on the edge. Then there are two cases to consider. First one: the card thinned is before ‘22’, second one: after ‘22’.
The first scenario is obviously very good, the second one at the first glance does no harm, but… Mulligans comes in. Thinning a card lying under ‘N’ means less slots available for cards shuffled back into the deck. Let’s go back to ‘Accessing the last card’ study.
With one less slot behind ‘22’ we have to update 3rd mulligan: 5/10 and 4th: 6/11. Finally we got 2.6% chance of finding ‘22’ (-0.4% with respect to no-thinning example).
If you are unlucky, even thinning is your enemy!
Thanks for reading!
Hopefully you feel some insufficiency after the lecture. Lot of questions arise in my head either just by looking more deeply into something as simple as mulligan act. The topic of thinning deserves deeper discussion – I plan to write some monothematic article on thinning in the future. However, promised R1 probs has priority for now. The topic of mulligans itself will be continued in articles dedicated to particular game stages.