# Gwent Math & Probability 6: Tutors

### Introduction

A card capable of pulling out another card/set of cards from the deck  is called a tutor. Also cards/abilities helping to find chosen cards in a direct way (e.g. Pincer Maneuver, Fisher King…) are commonly called tutors or tutoring tools. On the other hand, non-direct means increasing the probability to access a card are referred to as thinning (you could find the detailed thinning discussion in the preceeding GM&P5 – The Basics Of Thinning article).

The ‘tutoring’ verb doesn’t stem from any English semantics, but originates in one Magic: The Gathering card – Demonic Tutor released back in 1993.

As we can see, finding one card from the deck and reshuffling it, Demonic Tutor was the most classical tutor possible 😉

### Tutors In Gwent

In Gwent there are multiple ways to directly find a card from the deck in every faction. Let’s explore some possibilities before moving on to strict analysis (Gwent 10.3) .

• Classical Tutor&Play (Royal Decree, Call of the Forest)

Cards playing a target from the deck, usually with a small restriction – Royal Decree works on units only, Call of the Forest on Scoia’tael units only and so on.

• Classical Tutor&Play Semi-Determinant (Alzur Double Cross, Marching Orders…)

Cards which rely on some conditions which prioritize which exact card would be tutored. Pulling out a random card in case of multiple targets being on the same priority level. Usually require specifically crafted decks. Lower provision cost of these tutors compared with more flexible ones (classical) is a form of payoff for smart deckbuilding. Recently ADC seen play in Magpie’s Alchemy deck in Top16 qualis of the Season of the Wolf

• Thematic Tutor&Play (Jan Natalis, Whispess:Tribute, Ge’els…)

Tutors restricted to specific categories and tags only. These tutors usually have few targets and consequently some bricking risk outside thematic decks.

• Echo Tutors (Oneiromancy, Amphibious Assault, Blood Eagle…)

Echo effect enables tutoring desired cards twice, but in different rounds. Echo tutors preferably should be found early on. Oneiromancy stands out here as a classical tutor&play with no added value, while especially AA is a value card with tutoring capabilities.

• Classical Tutor&Draw (Matta Hu’uri, Iris’ Companions, Octavia Hale…)

Apparently there is no card in Gwent working in exact analogy to Demonic Tutor – putting a card from the deck in hand. Matta Hu’uri draws the highest provision card into hand, but also tutors lowest provision cost card for opponent and elongates the round by 1. Iris’ Companions (Demon btw.) draw a card of choice, but a card from hand gets discarded. Octavia Hale works in the closest analogy to Demonic Tutor, but is restricted to specific Bounty archetype cards only.

• Postponed Tutor&Draw (Dol Dhu Lokke location order, Naglfar secondary effect, Fisher King…)

Placing a card at the top of the deck, so that it is guaranteed in the next round. These tutors could be combined with cards like Sunset Wanderers or discard package for immediate tutoring.

• Cursed Scroll

The stratagem tutoring a card of choice and putting a card from hand at the bottom of the deck.

• Leader Ability Tutoring (Imperial Formation, Pincer Maneuver…)

Leader ability tutoring is always at hands, but using it means some commitment at the same time.

• Complex Tutors (Jan Calveit…)

There are some unique tutoring cards with special effects which has to be studied separately.

• Combined tutors (Naglfar + Sunset Wanderers, Albrich + Tactical Decision…)

Sometimes tutoring is improved or realized by a combination of few cards rather than one.

### Tutor Value

Let’s start from having a look at the original Demonic Tutor once again. If you have MtG or other mana system card games experience it is clear that tutoring is done at some expense. It would be 2 mana in the case of Demonic Tutor. Consequently, consistency is gained, but tempo is lost. Opponent may put some value cards on the board in the same turn and get tempo advantage, which needs to be neutralized.

Gwent uses Provision instead of Mana system. In the case of immediate tutors, provision is the cost. Such kind of expense is more complex and harder to see than mana usage. It is a model case of “what you see and what you don’t” in economics. To estimate the value of a tutor in each game we would have to compare him with an imaginary replacement worth same provision. Or with one of other cards upgraded to a better one, probably a strong gold.

Playing a tutor obviously lowers deck peak potential (let’s call it: firepower), but could improve average performance. Tutors are especially useful when particular card is needed at particular game stage.

Also tutors have added value of flexibility – sometimes your best pick would not be the only gold card you missed, but a bronze card playing for great value in particular matchup.

For the most generic math analysis, let’s forget gameplan and crucial cards at the moment. We live in Provision and Power world only. What conditions have to be satisfied for a tutor to play for a better value than a direct replacement card?

### Terminal Tutor Value

It was shown in GM&P5 that without thinning the chances to find a single card are equal roughly to 80% (81.3% on blue/79.8% on red). Imagine that a set of N cards with value higher than tutor replacement is played in the deck. Then whenever one of N cards is missed, tutor plays for better value than the replacement.

##### Rough Estimate Of Missing Chances And Why It Is Wrong

Let’s define $$P_{full}(N)$$ as the probability to draw full target package of N cards. The chance $$P_{miss}(N)$$ that at least one card from the package will be missing is obviously equal to:

$P_{miss}(N) = 1 – P_{full}(N)$

When evaluating probabilities in real time, it is hard to perform complex math. Then it is tempting to calculate $$P_{full}(N)$$ as a simple product:

$P_{full}(N) = P_{full}(1)^N$

Such expression is not correct, but still could be a useful rule of thumb estimate. What is exactly wrong?

Imagine an infinite deck made only of Raging and Elder Bears. We want to find peaceful Elder Bears (EB) and avoid dangerous Raging Bears (RB). Let’s assume we draw 2 cards from such deck and the proportion between number of EB and RB is 2:1.

As the deck is infinite, both drawn cards have 2/3 chance to be EB and there is (2/3)^2 chance to avoid RB.

Now let’s thin the deck to just 3 cards. Does (2/3)^2 still obey? No, it doesn’t. After first card is drawn (2/3 for getting EB), the card pool becomes drastically changed. It is 50:50 between EB and RB in the second card. The overall chance to avoid RB is then equal to (2/3)*(1/2) = (1/3). It is (1/9) less than global estimate.

When drawing full packet is considered the estimate is always higher than real value. The smaller the packet with respect to deck size, the smaller the error. The real value is achieved, as always, with hypergeometric probabilities.

The results in the table above show how often whole package of N target cards is found. A staple to remember: a package of 3 cards is completed roughly 50% of time in a game of Gwent without tutoring. Simple product estimates are compared with strict hypergeometric results. As could be seen % differences grow with package size, but aren’t generally big enough to matter in practice for a Gwent player.

##### Shelf Method

If cards in the tutored package are of similar power and there is no clear hierarchy between them, then package probability results could be used with a common point value for whole package.

In other words, not drawing full package means that tutor is worth U points, where U is equal to average point value of cards in the package (U – upper shelf)

On the other hand, if all cards from the upper shelf are found, then one from lower shelf has to be played. These would typically consist of low-end golds and strongest bronzes. Analogously we define L as mean point value on the lower shelf.

Consequently, the expression for tutor value would be:

$V = (1-P_{full}(N))*U + (P_{full}(N))*L$

, where V means average point value of the tutor and $$P_{full}$$ is the probability of drawing full upper shelf package.

### Tutor Value In R3 - General Case

Shelf method relies on the assumption that all target cards are of similar value and consequently are tutored with same frequency. It isn’t true in the general case.

If package is incomplete, then tutor value is equal to best card missed, not to average value of package cards. Instead of arithmetic mean, various weights has to be associated with each card according to tutoring hierarchy.

This time the results we need for further analysis is the probability to find exactly one (last in hierarchy) card from N cards package. Then it would be possible to climb step-by-step with package size and determine coefficients for each card in hierarchy.

Each column in the table is simply equal to coefficients (weights) which should multiply cards value in respective rounds. Row ‘1’ refers to best card in the targets hierarchy, ‘2’ to 2nd best and so on.

Let’s see how it works on example of following deck from Gwent 10.2 playing Royal Decree (let’s neglect The Flying Redanian thinning for the sake of simplicity).

There are 6 especially worthy targets for Decree in this deck. Let’s consider R3 tutor value and assume following (arbitrary and matchup dependent of course) value for targets:

Savolla – 20p

Professor – 15p

Sigi Reuven – 15p

Philippa – 12p

Caesar – 12p

Whoreson’s Freak Show – 10p

Other targets on average – 8p

According to full packets calculations, there is 23.9% chance to complete K=6 cards target package in R3 (one extra decimal place would be used there with respect to tables). If you look closely, coefficients always sum to 1-full_packet odds for a given K. Therefore, the obtained value doesn’t need any further adjustments and rescaling.

V = 20p*19.4% + 15p*16.2% + 15p*13.4% + 12p*11.0% + 12p*8.9% + 10p*7.2% + 8p*23.9% = 13.3p

Estimated R3 value of Royal Decree in the demo Jackpot deck is equal to 13.3 points for 9 provision, which is very good meta level, but not broken. It is hard to think of any replacement for this card making better average value in this slot.

If you want to be even more precise it is possible to include Kurt and Salamandra Mages in the package and move ‘other targets’ category to just the weakest bronzes.

### When Not To Play A Tutor?

Going back to shelf method:

$V = (1-P_{full}(N))*U + (P_{full}(N))*L$

, we could see that average tutor value depends on 3 macro parameters

• packet size – the bigger the target packet, the lower $$P_{full}$$ and higher tutoring value
• U – average value of tutor targets
• L – average value of backup targets in case whole packet got completed

Let’s once again take Royal Decree as an example. Neglecting other synergies, a 9 provision cost card should play at least as raw 12 points to be at the meta level.

To be consistent with the example in the paragraph above, let’s keep L value as 8 (it depends on particular deck greatly). What minimal value of U is needed for various package sizes in order to achieve 12 points?

Results for small packet sizes may seem odd after Jackpot+Royal Decree deck demo, but not always tutors have ~20 targets from upper and lower shelf combined.

Especially in the case of thematic tutors (John Natalis, Whispess:Tribute…) the total number of targets is very low, and upper shelf is very tight.

In fact, in Gwent 10.2 meta the Alumni deck played Jan Natalis with just one upper shelf target – Amphibious Assault, and few lower shelf ones, like Winch or Boiling Oil . Apparently effective AA value wasn’t far from 28.6 points in an Alumni deck! It used to be rated as ~18 points in Midrange Zeal NR times (power curve ratings), but obviously the Alumni/PMP synergies elevate effective AA power by a lot in this deck (also Natalis itself is +2 points, sometimes +4 with extra synergies like Raffard’s Vengeance).

Unless there is a card comparable in effective power with AA in Alumni in your deck, never play a tutor just for it.

If there is one really good card at the high end, which you would like to tutor, try to add other targets from the same category (if only present on the meta level) to your deck.

### Flexibility in R3 (Tutor Toolbox)

Due to polarization of Gwent decks, tutor flexibility value in R3 is most often niche. It would rely on choosing a card getting good situational value instead of simply playing the first missed best card.

To understand flexibility let’s imagine that we play a solitare NG deck relying on Illusionist spam. The only mean of control would be 6 provision card – Peter Saar Gwynleve, and our only tutor for whole deck is Royal Decree.

We queue on ladder against a Devo Pirates deck. During R1 and R2 opponent plays normal cards from a standard Pirates list. Going to R3 with last say we cannot expect any green power, so Peter gets mulled out because his expected value is too low. The play goes on and with 4 cards left opponent suddenly plays Dagur Two Blades… We would probably get cheesed, lose by 40 points and uninstall Gwent, but if Royal Decree is still saved in hand, we could tutor Peter on last say for the rescue.

Sometimes situational card use may arise not only due to surprise cards hidden in opponent’s hand, but also opponent playing in an unexpected way. Let’s say that we queue Syndicate Bounty deck. Peter value could hardly be expected. During R3 though opponent greeds by spending with Sea Jackal everytime on 9 and 7 coins. Once again, Peter could be summoned from Decree if his value exceeds best missed card.

Tutor could be then a kind of toolbox, enabling reaction to various unexpected plays. The effect would only really matter though when lower shelf card value exceeds upper shelf, which is hardly achievable due to power gap between golds and bronzes.

Perhaps the most important period in Gwent when tutor flexibility mattered was during Bomb Heaver/Scenarios binarity. Bomb Heaver used to be 4 power card destroying artifacts at 5 provision cost. Sometimes it was hard to judge if opponent plays Scenario or not and Bomb Heaver could lost the game if kept in hand. Effective value of Bomb Heavers was then equal to no less than 16 points for 5 provision.

### Package Consistency (Combos)

While we already assessed tutor power, the question of how the consistency of finding N targets package does exactly improve hasn’t been explicitly addressed yet.

The presence of tutor for all cards from the package means that N+1 package could be considered of N (including tutor), where missing one card still means completeness of N.

There is only 3% chance to miss the crucial single card in R3 if playing a tutor! Also, if a combo deck with tutor is supposed to be competitive in R3 and all elements from the package are essential, its size should never exceed 4 cards (if only a single tutor is used for consistency).

Also 2-card combos could be launched consistently with a tutor (91%) in R3, and 3-card/4-card ones would still be competitive whenever good enough for a win-con (84%/76%). Comparing R2 and R3 probabilities we could see, that a combo deck would seldomly be able to unlaunch its power before final round. It means that combo decks would struggle both when pushing and being pushed in R1/R2.

Finally, tutors are absolutely necessary for full combo-reliant decks to be competitive! Even a 2-card combo could not be played consistently enough in R3 without tutor! (64%)

### Tutors in R1 and R2

While finding crucial cards for R3 is the most common application of tutors in Gwent, sometimes they are used for powerful carryover / abuse plays in R1. Also when under pressure or forced to bleed opponent in R2, a tutor has to be used earlier.

Let’s have a look back at Drawing Full Packages table, this time summary for R1/R2/R3

As the probability to miss crucial cards in R1/R2 is higher, the relative power of tutor in the round is also higher. Let’s move back to the model Jackpot deck and forget nuances. Applying same ‘General Case’ math to R1 and R2 we obtain:

R1: V= 17.1

R2: V= 15.5

Does it mean that tutors shall be used R1? HELL NO! But they can help in emergency situations, when forcing R1/R2 outcome may be advantageous, or simply tutoring a card (like Melusine) is needed in your gameplan. Other than these, using tutor, especially on a gold card is a double commitment.

### Closure

Proper understanding of tutor cards is essential for sound Gwent deckbuilding. I hope that presented numbers helped to clarify various concepts associated with tutors and inspired you to build own deck with good tutoring!