Information theory and player archetype choice in Hearthstone
IInformation theory and player archetype choice in Hearthstone
Mathew Zuparic a , Duy Khuu b , Tzachi Zach c a Defence Science and Technology Group, Canberra, ACT 2600, Australia b Australian National University, Canberra ACT 2601 Australia c Ohio State University, Columbus OH 43210 United States
Abstract
Using three years of game data of the online collectible card game
Hearthstone , we analysethe evolution of the game’s system over the period 2016–2019. By considering the frequenciesthat archetypes are played, and their corresponding win-rates, we are able to provide narrativesof the system-wide changes that have occurred over time, and player reactions to them. Ap-plying the archetype frequencies to analyse the system’s Shannon entropy, we characterise thesalient features of the time series of player choice. Paying particular attention to how entropyis affected during periods of both small and large-scale change, we are able to demonstrate theeffects of increased player experimentation before popular decks and tactics emerge. Further-more, constructing conditional probabilities that simulate understandable player behaviour, weanalyse the system’s information storage and test the explain-ability of current player choicebased on previous decision-making.
1. Introduction
Player choice in most adversarial games (
Chess , Go , etc. ) is limited to on-board actions,where the distinguishing factor of success is typically experience enabling players to makebetter choices over the course of a match. Collectible card games (CCGs) such as Magic: theGathering , Yu-Gi-Oh! and
Hearthstone require players construct their specific deck of cards before they engage other players. Decks consist of a limited number of cards (approximatelythirty) from a potential pool of thousands. The act of constructing a deck is arguably the mostmeaningful choice a CCG player will make, usually determining the tactics that will be pursuedduring a match. For CCGs with a significant number of players, popular decks (also referred toas archetypes ) inevitably emerge over time.Online CCGs exist within a substantial system.
Hearthstone for example currently boastsapproximately one hundred million players world-wide. Figure 1 presents an abstraction ofthis system, divided into three layers: tactical , meta and authority . Gameplay occurs at thetactical layer, where players attempt to make the best moves given the cards in their hand. Deckconstruction also occurs at this layer, where players choose cards based on their preferred styleof play, relying on past experiences to refine their choice of cards. The meta for an onlineCCG can be articulated as the emergence of archetypes and corresponding tactics over time.As explained by Carter et al. [7], though the meta is peripheral to the rules and mechanicsof the game itself, it describes the environment that players will experience. The authority Preprint submitted to Elsevier September 30, 2020 a r X i v : . [ n li n . AO ] S e p igure 1: Abstraction of the online CCG system. layer in Figure 1 is the remit of the publisher of the CCG. At various times, the publisher willinitiate change into the tactical layer of the CCG, either by changing cards, adding new cards orchanging rules and/or play mechanics. Due to the underlying difficulty involved in deck construction, the majority of past researchin CCGs has been focused on the discovery of optimal decks and/or determining the best game-play options. Demonstrating this difficulty, in September 2018 standard players of
Hearthstone could construct approximately 4 . × unique decks. To efficiently explore this samplespace, Garc´ıa-S´anchez et al. [16] applied evolutionary algorithms to continuously evolve initialdecks and showed a noticeable improvement in win-rates against the majority of other deckarchetypes. In [17] the authors used evolutionary algorithms to create decks which outper-formed archetypes within the meta when played by an artificial intelligence (AI). Most recentlythe authors in [18] tested their methods in an international Hearthstone
AI competition, placingin the top 6% of entrants. Bursztein [5] demonstrated an algorithm which predicted opponentmoves in
Hearthstone , boasting over 95% accuracy after the second turn. Remarkably, thepublisher requested that the author not publicly release the algorithm as it was game breaking.Stiegler et al. [39] applied a utility system to automatically construct decks which consideredmetrics relating to gameplay (cost effectiveness, card synergies, etc. ), as well as popularity ofcards in the meta. The authors found that their algorithm was able to complete deck skeletonsinto currently popular archetypes. Fontaine et al. [15] applied an algorithm which incentivisedboth novelty and performance to imitate player decision-making and determine a set of populardecks. Once decks were established the authors negatively affected ( nerfed ) some commonlyselected cards to limit their viability. Counter-intuitively, the authors found that the algorithmincluded some cards more frequently after they were nerfed, suggesting that even an objectivelynegative change may have a positive impact on the perception of a card’s viability. Bhatt etal. [4] found that decks constructed by their algorithm possessed some degree of generality,performing well against decks not in the current meta. Focusing on the meta layer of CCGs, deMesentier Silva et al. [11] applied an evolutionary algorithm to understand the impacts brought2n the meta by improving and/or nerfing various cards. While it was possible to balance themeta after initiating change, the authors noted that too much change was difficult to resolve.
This work focuses on the CCG system itself as presented in Figure 1 — to the best ofour knowledge a topic not actively studied thus far.
At a system-wide level we seek to under-stand how changes enacted to the CCG by the publishers influence player decision-making.
Wedemonstrate how small and large-scale changes initiated in the tactical layer influence the evo-lution of the CCG’s meta, and by extension the decisions players make in deck construction. Toenable this understanding we use three years of gameplay data from the CCG
Hearthstone , con-sidering both the frequencies that deck archetypes are played, and their corresponding win-ratesover the 2016–2019 period. To understand the effects that changes enacted by the publishershave on the
Hearthstone system, we focus on various information-theoretic measures as theyare a method to quantify the amount of surprise, randomness and complexity in an entire sys-tem. Analysing the
Hearthstone meta through the lens of information entropy, we understandand characterise the evolution of complexity and uncertainty in the meta at any given time. Fur-thermore, by considering the information storage exhibited within
Hearthstone’s meta, we areable to estimate how much previous player decision-making explains the underlying structureseen in the current state of the meta.
Though we focus on
Hearthstone , many of the elements discussed in this section apply tothe majority of CCGs. During a match, two adversaries take turns selecting and playing cardsfrom their hand . At the start of each turn a specific number of cards are randomly dealt tothe player’s hand, drawn from the deck which has been constructed by the player beforehand.During a player’s turn cards are activated by spending a predefined amount of in-game resource( mana ) which is replenished at the end of the player’s turn. Cards that have been activated arethen sent to the player’s discard-pile , out of play for the remainder of the match. The ultimategoal for each player is to reduce their adversary’s health to zero.Cards generally fall under two categories: minions and spells . Minions give the playercontrollable characters which enable a range of defence and offence options. Spell cards rangefrom single-use damage-dealing cards, to cards which perform sustained effects over multipleturns. Specific to
Hearthstone [5, 11] are also weapon cards, equippable by the player, secret cards, similar to spells but triggered once specific conditions are met, and hero cards, changingthe properties of the player’s in-game avatar. The synergies between card properties also affectplayer choice during deck construction. For instance, the activation of many high value cardsmay require complex conditions, which can only be satisfied if other specific cards are present.For more information regarding
Hearthstone card properties we refer the reader to [46].
Archetypes in CCGs generally fit into the following three categories:3
Aggro archetypes rely on aggressive tactics to achieve victory. They typically focus onlow cost cards with the intent of overwhelming the adversary in the early stages of thematch. Aggro players that cannot maintain significant tempo in the early-to-mid stagesof the match typically lose.•
Control archetypes rely on relatively high-cost and high-value cards to win in the laterstages of a game. The moniker control comes from the archetype’s strategy in the early-to-mid game of countering a variety of play-styles, thus granting player the time neededto initiate the intended late-game finishing tactics.•
Combo archetypes generally rely on cards which contain synergies, with the intent toknock out the opponent by playing a number of cards in conjunction with each other togenerate devastating effects. Much like control, combo archetypes must have some formof counter for aggressive early-game play-styles, but mirroring aggro archetypes, theyalso rely on knocking out control archetypes before their high-cost high-value cards areactivated.Specific to
Hearthstone in the 2016–2019 period, players must additionally choose one ofnine character classes:
Druid , Hunter , Mage , Paladin , Priest , Rogue , Shaman , Warlock and
Warrior . Characters provide unique abilities, and grant the player character-specific cards.Some characters generally favour specific archetypes due to these exclusive cards. For in-stance
Mages gravitate towards control archetypes due to the considerable range of spell cardsresulting in their ability to deal with a large number of adversary minions.In a recent survey of online CCG players, Turkay and Adinolf [40] established that playermotivations fell under 4 categories: immersion , competition , socialisation and strategy devel-opment . Thus, players can be motivated by more than simply winning, including finding com-binations of cards particularly fun to play. A Hearthstone specific example of this is the card
Marin the Fox , which when summoned creates a treasure chest that, once destroyed, grants theplayer with one of a number of extraordinarily powerful cards. It was recognised that success-ful implementation of this tactic posed many risks as a number of archetypes possessed abilitieswhich would allow the opponent to steal the resulting powerful cards for themselves. Neverthe-less, despite considerable risks this card did appear in a number of decks due to how satisfyingthe chest’s rewards were if obtained.
Information-theoretic measures such as Shannon entropy quantify the amount of surprise or randomness exhibited in a system [2, 27]. Hearthstone’s meta is a complex system wherearchetypes emerge and disappear, and a range of behaviours can be exhibited as time progresses.Measuring the amount of randomness displayed in the meta over time enables appreciation ofhow balanced active archetypes are, how effective recent changes were, and ultimately helpcharacterise the state of the
Hearthstone system. Past examples of Shannon entropy offeringinsights into complex systems include: Miranskyy et al. [28] who used entropy measures to un-derstand and compare rare events in defective software; Cao et al. [6] who developed Shannonentropy-based measures on graphs to understand the underlying complexity of graph families;4nd Aggarwal [1] who recently applied generalised Shannon entropy measures to provide deci-sion support in the face of multiple criteria that were often conflicting.Associated with information entropy is the concept of system criticality [21], sometimesreferred to as the edge of chaos [35]. In mathematical [22], physical [31], biological [30]and computational [23] systems, amongst others, criticality refers to the system being ableto respond and adapt to a rapidly changing environment. Intuitively, it can be viewed as adynamical system cycling through periods of relatively low and high entropy, spending themajority of its time in intermediate entropy values. For an introduction to this topic refer to[10, 32].We additionally apply the concepts of distributed information storage [29], closely related toinformation transfer [19]. The systematic explanation of how information is stored, processedand transferred in distributed systems began in earnest with Schreiber’s [36] landmark work on information transfer entropy , mathematically defining how information is transferred betweendistinct processes in distributed systems. Information transfer has since been applied to greateffect in a wide range of applications, including neuroscience [41], multi-agent dynamics [25],and social media [38]. A decade after Schreiber’s result, Lizier et al. [26] introduced localactive information storage (LAIS) to distinguish the dynamics displayed in cellular automata.LAIS has since been applied to understand how information storage properties affect networkstructure in biological and artificial networks [24], and characterise normal and diseased statesin cardiovascular and cerebrovascular regulation [13]. Wu et al. [47] demonstrated the util-ity of localising other information-theoretic measures as a means of overcoming their knownweaknesses regarding image recognition.This work applies LAIS to appreciate how much the past state of the
Hearthstone metacontributes to its current state. This is motivated by a number of studies which apply LAIS toexplore similarly themed questions on a number of complex systems. These include Wibral etal. [45] who measured the local time and space voltage neurologically generated by stimulatingthe visual cortex of an anaesthetised cat. The spatio-temporal structure of the correspondingLAIS data characterised how the onset of visual stimulus led to spatio-temporal surprise (ormisinformation) about the proceeding visual outcomes. Wang et al. [44] explored collectivememory/storage via an information-theoretic characterisation of cascades within the dynamicsof simulated swarms. Using the interpretation that the LAIS of a system component charac-terises the amount of past data used to predict the component’s next state, the authors calculatedthe system-wide active information storage (AIS) by taking the expectation value over all com-ponent states at any time period. They verified a long-held conjecture that information, usedfor computation by the swarm, cascaded via waves rippling through the swarm, and found thathigher values of storage generally correlate with greater dynamic coordination. Cliff et al. [8]explored the AIS within a multi-agent team by analysing implicit team interactions. The authorsnoted that when an agent’s AIS values were high its movements were largely predictable.5 .6. Mathematical preliminaries
For a set of K + { X T , X T − , . . . , X T − K } , the LAIS of the state X T attime T , based on its past K states, is given by a K ( X T ) = log P ( X T | X T − , . . . , X T − K ) P ( X T ) . (1)Positive values of Eq.(1) imply that the past states of the variable provide information andpositively correlate with the current state. Conversely, negative values of LAIS indicate that thevariable’s past history does not correlate with its next state and is synonymous with surprise.The expectation value of the LAIS (the AIS) A TK ≡ (cid:104) a K ( X T ) (cid:105) = (cid:34) K ∏ n = ∑ X T − n ∈ X T − n (cid:35) P ( X T , X T − , . . . , X T − K ) a K ( X T ) (2)is the explain-ability [26] of the information in the system. That is, when compared to thecorresponding Shannon entropy, the AIS gives the amount of information in the current systemthat is explainable by the results of the previous time step(s). To further clarify this concept ofexplain-ability, complementary to AIS is the entropy rate E TK , given by E TK = −(cid:104) log P ( X T | X T − , . . . , X T − K ) (cid:105) . (3)When compared to the corresponding Shannon entropy, the entropy rate gives the amount ofinformation in the current system which is not explainable by the results of the previous timestep(s).Following Lizier et al. [26] and Crutchfield and Feldman [9], the contrast between what isexplainable and what isn’t in the system is made clear by the following duality relation betweenShannon entropy — labeled H ( X T ) — of the current state, AIS, and the entropy rate via H ( X T ) = A TK + E TK . (4)Thus by Eq.(4) the percentage of information within the system which is explainable by pastresults is given by A TK / H ( X T ) , and the remaining E TK / H ( X T ) being the percentage ofinformation not explained by past results. In the next section we detail the
Vicious Syndicate website which is the source of the
Hearth-stone data considered in this work. Using this data, we then construct sample timelines of somedeck archetypes, demonstrating the dynamic evolution of the meta over time. We then lookat the data through the lens of Shannon entropy. In Section 3 we construct conditional prob-abilities which simulate relatively simple, but nonetheless understandable, player deck choicebehaviour. These conditional probabilities are used to define the system-wide AIS values pertime period, ultimately applying Eq.(4) to understand how much of
Hearthstone’s past state ofdeck frequencies and win-rates contributes to its current state. In Section 4 we offer furtherdiscussion and detail potential future work. 6 - TokenMid - TokenBigAggroMiracleTauntQuestSpiteful Data Reaper Report A c t i ve D r u i d A r c h e t y p es Figure 2: Timeline of the
Druid -based archetypes present in the meta. Vertical lines represent system changesby the publisher — solid lines signify release of an expansion (new cards and game mechanics) in addition to arotation of a number of older cards out of the standard mode; dot-dashed lines signify the release of an expansion;and, dotted lines signify release of balance patches (changes to existing cards). The horizontal axis corresponds tothe
Data Reaper Report from which the data is drawn. The data for the first entry, report 30, was collected over theperiod 14-20 December 2016. The data for the final entry, report 80, was collected over the period 6-13 February2018.
2. Data explanation and exploration has been collecting
Hearthstone game data systematically since May2016. The data was used to produce weekly
Data Reaper Reports about the state of the
Hearth-stone meta-game [42]. Breaks in reporting occur near the release of new content by
Hearth-stone’s publishers. To contribute game data, players are asked to install a small plugin thatrecords their game play. That data is transmitted to the
Vicious Syndicate team to be processed.During any week between 2000 to 5000 players contributed game data, with tens of thou-sands of games being processed to produce reports. Specific numbers of contributing playersand processed games can be found in each of the corresponding
Data Reaper Reports [42]. Onlygames of rank 15 and above are included for reporting purposes. Only opponent archetypesare included for frequency reporting so as to avoid potential over-representation of archetypesfavoured by players who contributed data [43]. Deck identification algorithms are applied toclassify archetypes based on the cards played during a match. Though not every game provideda definitive identification, algorithms achieved a high success rate ( > all other archetypespresent in the meta at least twenty times per reporting period. To illustrate an example Figure 2 depicts the timeline (from December 2016 to February2018) of the
Druid -based archetypes present in the meta. Each archetype (14 in total) is notedon the vertical axis as they appear in chronological order. The horizontal axis corresponds to the7 ata Reaper Report from which the data is drawn. Each black horizontal bar designates the ap-pearance of that particular archetype in the meta over the appropriate time period. Each verticalline represents a specific change to the system: solid lines indicate the release of an expansion(with new cards and game mechanics) in addition to a rotation of a significant proportion ofolder cards out of the standard mode; dot-dashed lines signify the release of an expansion; and dotted lines signify release of balance patches (changes to a number of existing cards).The change that occurred after report 43 in Figure 2 was due to the release of the
Journeyto Un’Goro expansion which introduced 135 new cards (some with new play mechanics) to thegame. Additionally, a card rotation occurred during this time, making 208 cards released priorto 2016 unusable in the standard play format. Such rotations, which happen yearly around April,are designed to prevent certain powerful cards and tactics from dominating the meta for too long,and allowing new content to be released without requiring to account for all previously releasedcards when testing for overpowered tactics. Two of the rotated cards,
Emperor Thaurissan and
Aviana , greatly improved the viability of
Malygos Druid . Thus the extinction of this archetypeafter T =
43 in Figure 2 could be anticipated. On the other hand, player experimentation alsooccurred due to the release of new content, with two new archetypes seeing significant play —
Ramp Druid and
Token Druid . While
Ramp Druid lost popularity with players soon-after,
TokenDruid continued as a popular
Druid archetype until a patch released after report 55 nerfed thecard
The Crystal Core . This patch greatly affected the archetype
Crystal Rogue , causing it to fallout of the meta.
Crystal Rogue was one of
Jade Druid ’s worst match-ups, in addition to beinga very favourable one for
Token Druid . This flow-on effect led to the eventual disappearance of
Token Druid , and further cemented
Jade Druid ’s popularity in the meta.
In this work we label the set of all active archetypes in the
Hearthstone meta for a particularreporting period T as X T = (cid:8) X T , X T , . . . , X TN (cid:9) , T ∈ { , } , (5)where N ≡ | X T | . T ∈ { , } corresponds to the Data Reaper Report which was the sourceof the data [42]. This spans approximately three years, being collected over the period 14December 2016 to 27 December 2019.For each archetype X Ti , the frequency that it was played in time period T is labeled as P ( X Ti ) . For all X Ti ∈ X T the complete set of P forms a discrete probability distribution withthe property | X T | ∑ i = P (cid:0) X Ti (cid:1) = . (6)Figure 3 depicts the frequencies of active archetypes played over the period 10–18 April 2017,representing T =
44 in Eq.(5). In this figure all character classes are represented in the 26 activearchetypes, with the most frequently played archetype being
Midrange Hunter .This work also considers the win-rates between archetypes, with P ( W | P X Ti , A X Tj ) denotingthe conditional probability of winning , given that player ( P ) chose archetype X i and faced ad-versary ( A ) using archetype X j , at time period T . Win-rates and their transpose are equal to8 ade D r u i d R a m p D r u i d T o k en D r u i d M i d r ange H un t e r B u r n M age E l e m en t a l M age E x od i a M age F r ee z e M age S e c r e t M age C on t r o l P a l ad i n E l e m en t a l P a l ad i n M i d r ange P a l ad i n M u r l o c P a l ad i n D ea t h r a tt l e P r i e s t D r agon P r i e s t M i r a c l e P r i e s t S il en c e P r i e s t C r ys t a l R ogue M i r a c l e R ogue A gg r o S ha m an E l e m en t a l S ha m an M u r l o c S ha m an H and W a r l o ck Z oo W a r l o ck P i r a t e W a rr i o r T aun t W a rr i o r Archetypes0.050.100.15Frequency
Figure 3: Archetype frequency data for T =
44, collected over 10–18 April 2017. unity, leading to the identity P ( W | P X Ti , A X Tj ) = − P ( W | P X Tj , A X Ti ) , ⇒ P ( W | P X Tj , A X Ti ) = P ( L | P X Ti , A X Tj ) , (7)for W = win and L = lose . Additionally mirror match-ups amongst the same archetype areequal to 0 . i.e. P ( W | P X Ti , A X Ti ) = P ( L | P X Ti , A X Ti ) = . . (8) Shannon information entropy is measured via H (cid:0) X T (cid:1) = − | X T | ∑ i = P (cid:0) X Ti (cid:1) log P (cid:0) X Ti (cid:1) . (9)One of the main benefits of Shannon entropy is its ability in characterising the underlying com-plexity in the system [37]. In general maximum entropy values are obtained by uniformly dis-tributed probabilities — i.e. P ( X i ) = / | X | ∀ X i ∈ X . Thus Eq.(9) offers insight into howevenly distributed the archetypes in the meta are over any given reporting period if comparedthe value of the theoretical maximum H max (cid:0) X T (cid:1) = log (cid:12)(cid:12) X T (cid:12)(cid:12) (10)which is the logarithm of the number of active deck archetypes in the meta for any given timeperiod T . 9 ( χ T ) ℋ max ( χ T )
32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 T3.54.04.55.05.5
Figure 4: Graph of the Shannon information entropy (solid line) and its theoretical maximum (dashed line) —defined in Eq.(9) and (10) respectively — derived from the frequencies that archetypes are played over each of thereporting periods. Refer to Figure 2 for the specific meanings of vertical lines representing system changes.
Figure 4 depicts the Shannon entropy (solid line) and the corresponding maximum entropy(dashed line) derived from archetype frequencies for each of the reporting periods. Vertical linesrepresent changes to the system as explained in Section 2.2. Both entropy values largely mirroreach other over the entire time period which is expected. Noticeable increases and decreasesoccur immediately after a system-wide change has been introduced. The most common of theseoccurrences is a sharp increase, followed by a decrease in entropy until the next change occurs.This particular behaviour in entropy indicates a marked escalation in archetype experimenta-tion immediately after changes to the
Hearthstone environment. System entropy decreasessoon-after due to players understanding and settling on popular decks and tactics, which haveemerged due to the changes. For the majority of Figure 4, this behaviour in entropy and theassumed player decision-making it stems from is repeated semi-consistently.There are instances in Figure 4 where change led to a marked decrease in entropy values.As previously mentioned in Section 2.2, after T =
55 in July 2017 a patch nerfed the card
TheCrystal Core , which led to the extinction of
Crystal Rogue from the meta, along with otherarchetypes. Only the new
Jade Rogue emerged during this period, significantly decreasing theamount of active archetypes. A similar situation occurred after T =
62 in September 2017when a patch nerfed five cards. This decreased the frequency that
Druid -based archetypes wereplayed.
Mid-Token Druid and
Ramp Druid (amongst others) were extinguished from the meta,with only the new
Tempo Rogue emerging during this period. Such events which correspondedto decreases in system entropy relate to relatively small changes. These changes targeted a few10 T ℋ ∼ ( χ ) Figure 5: Graph of the normalised Shannon information entropy of the frequencies that archetypes are playedfor each reporting period. As with Figure 4, vertical lines represent system changes occurring between reportingperiods. archetypes perceived to be overpowered. Nevertheless, marked decreases in entropy values didoccur for the release of the
Rastakhan’s Rumble (after T =
113 in December 2018) and
Riseof the Shadows (after T =
125 in April 2019) expansions which were major changes. Unlikeother expansions their effect led to a reduction of the system’s entropy as they both saw a dropin active archetypes present in the meta. For both of these cases, the large drops in entropyvalues were reversed due to the minor changes initiated after T =
114 (December 2018) and T =
130 (May 2019). Both of these patches greatly encouraged deck experimentation and sawthe emergence of new archetypes.Figure 5 depicts the normalised Shannon entropy, which is the Shannon entropy divided bythe theoretical maximum value ˜ H ( X T ) = H ( X T ) / H max ( X T ) . (11)For any time period, Eq.(11) varies between ( , ) , indicating how close the Shannon entropyis to the theoretical maximum. As with the values in Figure 4, this graph displays sharp varia-tions when system changes are introduced. We conjecture that the evolution of the normalisedShannon entropy over time in Figure 5 shows hallmark signs of a system at criticality [21, 35].The system responds and adapts to a rapidly changing environment, cycling through periodsof relatively low and high normalised entropy values. High values indicate that all active deckarchetypes are equally popular. If a deck’s popularity indicates its likelihood to obtain victory,cases of high normalised entropy correspond to player choice offering little significance. Thiscase correlates with all active archetypes being equally probable of obtaining victory, with a va-riety of equally viable tactics. Conversely, relatively low values indicate that only a small num-ber of active archetypes are likely to be consistently victorious. Player choice would be biasedtowards those few archetypes, with experimentation kept to a minimum. System-wide changesoccur to veer the Hearthstone system away from these extreme situations [5]. The trajectory inFigure 5 spends the majority of its time at intermediate values. This indicates a scenario wherethere are a range of viable archetypes, and player choice is not arbitrary as archetypes have11arying strengths and weaknesses against each other. This scenario is reminiscent of Crutch-field and Young’s [10] concept of the complexity spectrum where a system displays the mostcomplexity between its minimum and maximum normalised Shannon entropy values.
3. Information storage and understanding player choice
Following Eq.(1), we construct the set of conditional probabilities P ( X Ti | X T − j , . . . , X T − Kj K ) which denote archetype X Ti being chosen, given that the archetypes { X T − j , . . . , X T − Kj K } wereplayed in the past. Importantly, the conditional probabilities possess the property of simulatingunderstandable player behaviour. Additionally, the conditional probabilities must satisfy theconsistency condition P ( X Ti ) = K ∏ n = | X T − n | ∑ j n = P ( X T − nj n ) P ( X Ti | X T − j , . . . , X T − Kj K ) (12)following Bayes’ theorem [27]. Using this notation, the LAIS associated with deck X Ti , givenpast choices, is given via a K ( X Ti | X T − j , . . . , X T − Kj K ) = log P ( X Ti | X T − j , . . . , X T − Kj K ) P ( X Ti ) . (13)This work applies the convention that LAIS values are zero if P ( X Ti ) = i.e. the archetype X Ti is not active for time period T . Additionally the AIS associated with each archetype, labelled A ( arch ) K ( X Ti ) , is given as the LAIS expectation value over all past choices A ( arch ) K ( X Ti ) = K ∏ n − | X T − n | ∑ j n = P ( X T − nj n ) P ( X Ti | X T − j , . . . , X T − Kj K ) a K ( X Ti | X T − j , . . . , X T − Kj K ) . (14)Archetype-AIS values in Eq.(14) are equivalent to the definition of agent rigidity given in [8],used as a measure of agent predictability. Also used in this work is the total AIS for each timeperiod, labelled A TK , which is the LAIS expectation value over all past and current archetypechoices A TK = | X T | ∑ i = A ( arch ) K ( X Ti ) . (15) This work is guided by the assumption that player decision-making is solely influenced bycomparing past archetype frequencies and/or win-rates to adjust choices accordingly. Thus, ifa player chooses archetype X j in time period T −
1, then the probability that archetype X i is12hosen in the next time period T is weighted by a function of both archetypes in the previoustime period. This is expressed mathematically via P ( X Ti | X T − j ) = ε ( i ) P ( X Ti ) K T − ( i | j ) , where K T − ( i | j ) = f (cid:16) X T − i , X T − j (cid:17) . (16)The weighting K compares deck-frequencies, and/or win-rates that players experienced againstarchetype X i , given they played archetype X j in time period T −
1. The function f results ina larger weighting if archetype X i was played more frequently, and/or had a higher win-rateagainst X j . Additionally, the coefficient ε ( i ) ensures that the consistency condition given byEq.(12) is satisfied. In order to test the assumption over multiple K -time periods the corre-sponding conditional probabilities are given via P ( X Ti | X T − j , . . . , X T − Kj K ) = ε ( i ) P ( X Ti ) K ∏ n = K T − n ( i | j n ) , (17)with the weighting factors K defined in Eq.(16). Additionally, by ensuring that the consistencycondition given in Eq.(12) is adhered to, the coefficients ε ( i ) are given as the following ε ( i ) = K ∏ n = | X T − n | ∑ j n = P ( X T − nj n ) K T − n ( i | j n ) − . (18)Thus, over a general number of K time periods, the AIS associated with each archetype X Ti , is A ( arch ) K ( X Ti ) = K ∏ n − | X T − n | ∑ j n = P ( X T − nj n ) K T − n ( i | j n ) ε ( i ) P ( X Ti ) log ε ( i ) K ∏ n − K T − n ( i | j n ) , (19)with the AIS for the entire time period T given by Eq.(15).Eq.(20) presents the exact forms of the weighting functions applied in Eq.(17) when con-sidering past archetype frequencies f FR : f FR ( ∆ P ) = Char i j × e ( ∆ P ) | ∆ P | e sgn ( ∆ P ) | ∆ P . | e ( ∆ P ) | ∆ P | e ( ∆ P ) | ∆ P | where ∆ P = P (cid:0) X T − i (cid:1) − P (cid:16) X T − j (cid:17) i (cid:54) = j , P (cid:0) X T − i (cid:1) − ¯ P ( X T − ) i = j , (20)and ¯ P ( X T − ) = | X T − | ∑ k = P ( X T − k ) | X T − | (21)is the mean value of the archetype frequencies played over time-period T −
1. The term
Char i j in Eq.(20) is a multiplicative factor which checks the character class of both archetypes, andreturns a value greater than unity if the character classes are equal, or 1 otherwise. As explained13 ( Δ P ) Δ P ⅇ sgn ( Δ P ) Δ P ⅇ ( Δ P ) Δ P ⅇ ( Δ P ) Δ P - - ΔΡ f FR ( ΔΡ ) ⅇ ( P - ) P - ⅇ ( P - ) P - ⅇ ( P - ) P - ⅇ ( P - ) P - Ρ f WR ( Ρ ) Figure 6: Plots of the specific forms of the functional responses applied to the weightings in Eq.(16). Left panelshows archetype frequencies f FR detailed in Eq.(20), and the right panel shows win-rates f WR detailed in Eq.(22). in Section 1.4 each of the character classes gain access to class specific cards which requirea resource investment to both obtain, and learn how play effectively. Thus the term Char i j simulates the resource hurdle and/or unwillingness involved in changing character classes. Thiswork sets
Char ii =
2, assuming that players are doubly likely to choose a deck if it is the samecharacter class as the deck they played in the previous time period.Additionally, Eq.(22) gives the forms of the weighting functions applied in Eq.(17) whenconsidering past win-rates f W R : f W R ( P ) = e ( P − . ) | P − . | e ( P − . ) | P − . . | e ( P − . ) | P − . | e ( P − . ) | P − . | where P = P (cid:16) L (cid:12)(cid:12)(cid:12) P X T − j , A X T − i (cid:17) i (cid:54) = j , ¯ P (cid:16) L | P X T − , A X T − i (cid:17) i = j , (22)and ¯ P (cid:16) L | P X T − , A X T − i (cid:17) = | X T − | ∑ k = P ( X T − k ) P (cid:16) L | P X T − k , A X T − i (cid:17) (23)is the mean value of the win-rate for archetype X i over time-period T − X i rises linearly the more it was played (left panel) and the better it performed against X j in the previous time period (right panel). The remaining coloured trajectories present non-linear responses. In the left hand plot, the quadratic weighting in orange grows slowly initially,but then experiences the sharpest rise as ∆ P → .
27. This range is chosen due to the largestfrequency (occurring at T = Lackey Rogue played 26.5% of the time. In contrast tothis, the pink trajectory experiences its sharpest rise immediately after ∆ P =
0, with a steadyrise afterwards. An equivalent picture is presented with the four trajectories on the right handpanel of Figure 6. If the win-rate is less than 50% then the exponentials have a negative ar-gument, leading to minimal weighting. If the win-rate is greater than 50% the arguments are14 A T ( frequencies ) Figure 7: Plots of Eq.(15) for K = positive and the weightings grow non-linearly, for all but the black trajectories. Mirroring theleft hand panel, the quadratic weighting in red grows slowly initially, experiencing the sharpestrise as P →
1. This range is chosen due to the largest win-rate (occurring at T = Taunt Warrior winning against
Cube Rogue
Figure 7 depicts the total AIS defined in Eq.(15) for K =
1. The functional responses ofthe past frequencies correspond to Eq.(20), with the colours of each trajectory matching thecolours given to each functional response in the left panel of Figure 6. A major feature ofFigure 7 is the marked difference between the AIS values with different functional responses.The almost-linear (grey) and squared (orange) responses display similar AIS values, except at T ∈ ( , ) . The highest AIS values are obtained by the pink trajectory, whose functionalresponse rises the sharpest as the difference between the frequencies becomes greater than 0, aswitnessed in the left panel of Figure 6. Thus, the functional response which rises the sharpestimmediately after the archetype under consideration compares favourably best aligns with ac-tual player behaviour.The AIS values in Figure 7 also experience a significant decrease whenever they cross timeperiods where a major change is introduced into the system. Thus players base significantly lessof their decision-making on past outcomes immediately after such change. This is illustratedby considering the five changes which happened between T =
39 and T =
63. At the end ofFebruary 2017 ( T =
39) a patch was released which nerfed the cards
Small Time Buccaneer and15 A T ( win - rates ) Figure 8: Plots of Eq.(15) for K = Spirit Claws in order to break the dominance of
Aggro Shaman in the meta. This patch, as wellas the patches released after T =
55 (July 2017) and T =
62 (September 2017), only affected ahandful of cards and were intended to affect a small number of archetypes in the meta. Thoughsuch small changes substantially changed the meta and its corresponding Shannon entropy (asshown in Figures 3 and 4), these changes did not substantially change the decision-makingplayers employed to choose archetypes, having very little effect on the AIS values. Interest-ingly, AIS values actually increased immediately after T =
55, meaning this change actually reinforced past decision-making. This is in stark contrast to AIS values occurring immediatelyafter T =
43 (April 2017) and T =
58 (August 2017), with the release of the
Journey to Un’goro and
Knights of the Frozen Throne expansions, respectively. In fact, all of the major decreases inAIS values in Figure 7 occurred immediately after significant changes were introduced.Figure 8 presents the total AIS defined in Eq.(15) for K =
1. Functional responses of the pastwin-rates are given in Eq.(22). The colours of each trajectory in Figure 8 matches the coloursin the right panel of Figure 6. Figure 8 displays many of the same features already discussed inFigure 7. These include notable decreases in AIS values immediately after significant changeis introduced, and subdued responses (or slight increases) for minor change. Also, the highestAIS values are obtained by the green trajectory, which has a similar functional response to thepink trajectory which obtains the highest AIS values in Figure 7.16 A T ( combined ) Figure 9: Plots of Eq.(15) for K = Figure 9 presents the AIS values which combine the archetype frequency and win-rate func-tional responses given by K T − ( i | j ) = f FR ( ∆ P ) (cid:124) (cid:123)(cid:122) (cid:125) Eq.(20) × f W R ( P ) (cid:124) (cid:123)(cid:122) (cid:125) Eq.(22) . (24)Each trajectory in Figure 9 is composed of two colours. These signify which of the functionalresponses were combined on the left hand panel (archetype frequencies) and right hand panel(win-rates) of Figure 6. The trajectories in Figure 9 generally display similar features to thoseseen in Figures 7 and 8. The main difference however is the marked increase in AIS values inFigure 9, approximately doubling the values seen in Figures 7 and 8.Recalling the duality relation between Shannon entropy, AIS and the entropy rate in Eq.(4),we can compare the AIS values in Figure 9 with the Shannon entropy in Figure 4. Doing so en-ables us to appreciate how much of the current state of the Hearthstone meta is captured by thefunctional responses we have used to simulate player decision-making. Figure 10 plots the ex-act value of this explain-ability ( A T / H ( X T ) ) per time period, for the highest A T values takenfrom the green-pink trajectory in Figure 9. The value of A T / H ( X T ) varies between [ , ] forany system. A value close to zero signifies that the assumptions used to construct the AIS revealvery little about the current state of the system. Likewise, a value close to unity signifies that the17 A T / ℋ ( χ T ) Figure 10: Plot of the percentage of uncertainty (or surprise) within the
Hearthstone meta that is explained byconsidering past archetype frequencies and win-rates via the functional response given in Eq.(24). assumptions applied to construct the AIS offers a near-to-complete explanation of the currentstate of the system. The assumptions in Eq.(24) are designed to simulate understandable playerbehaviour, with more popular and better performing archetypes having greater probability ofbeing chosen. For most time periods, Figure 10 shows that this simple principle explains ap-proximately 20% of the
Hearthstone meta. Though our assumptions do not take into account thenuances of player motivations when faced with deck construction, the fact that AIS values dropso dramatically immediately after large changes validates our assumptions. As discussed inSections 2.2 and 2.4, deck construction and tactics experimentation generally increases imme-diately after such changes, leading to the emergence of new archetypes, and the correspondingShannon entropy. Hence, during these periods it would be incorrect to assume that relying onpast results to inform current decisions would lead to good outcomes. The dramatic decreasesin AIS values during these periods in Figures 7, 8 and 9 validates these assertions.
Figure 11 provides a heat-plot of the archetype AIS values per time period — A ( arch ) ( X Ti ) viaEq.(14) — which were used to generate the largest combined AIS values (pink-green) in Figure9. The horizontal axis of Figure 11 indicates each of the 166 deck archetypes considered in thisstudy, which are ordered alphabetically within each of the nine character classes. The verticalaxis indicates the data reaper report for the archetype AIS values. Horizontal lines in Figure 11indicate changes occurring between reporting periods, as per the convention detailed in Section2.2. White regions signify archetypes that did not contribute AIS values for that particular timeperiod. Non-zero AIS values signify that the frequency of play of these archetypes at time T correlates with the state of the meta at time T −
1. The darker the colour, the more pronouncedthe correlation.Figure 11 reveals in greater detail the impact that change has on the various archetypes,useful when comparing to the global picture given in Figures 7–10. Periods experiencing smallchanges generally have minimal effect on the AIS values of the majority of the active archetypesin the meta. Contrast to this, time periods experiencing large changes generally display disrup-18 ruid Hunter Mage Paladin Priest Rogue Shaman Warlock Warrior313539434751555963677175798387919599103107111115119123127131135139143147 Archetype D a t a R eape r R epo r t Figure 11: Plot of Eq.(14), giving the archetype-AIS values per time period which were used to generate thepink-green total AIS values in Figure 9. tive effects, with a sizeable proportion of archetypes receiving zero AIS values, and a notice-able change in the AIS values of the remaining active archetypes. Focusing again on the majorchange occurring immediately after T =
43, Figure 9 displayed a sizable drop in total AIS val-ues due to the
Journey to Un’goro expansion and card rotation. The impacts of these changesare made clearer in Figure 11, with a number of the active archetypes receiving no deck-AISvalues past T =
43. Nevertheless, archetypes which survive to T =
44 actually obtain a sizableincrease in AIS values, such as
Midrange Hunter and to a lesser extent
Miracle Rogue . Dueto these archetypes surviving the change and performing relatively well in the meta, we inter-pret the relatively large increase in AIS values as these archetypes offering players a means toreinforce their previous decision-making during a disruptive period.
4. Discussion and future work
In this work we applied a number of information-theoretic measures to characterise andunderstand three years of game data of the online CCG
Hearthstone . Producing the system’sShannon entropy using the frequencies that deck archetypes are played provided a unique anduseful characterisation of
Hearthstone’s meta. One striking trend which manifested across themajority of the time-period was that most of the variability in the entropy appeared immediatelyafter a system-wide change had occurred. Sharp increases in entropy values, usually followedby decreases immediately after, implied a marked escalation in deck construction experimenta-19ion after change had been initiated. Entropy decreasing soon-after can then be understood asplayers understanding and exploiting the strong decks and tactics which emerged due to thesechanges.Additionally, by constructing conditional probabilities that particular archetypes were cho-sen in the current time-period based on the past state of the system, we examined the infor-mation storage exhibited in
Hearthstone ’s meta. Importantly, the weightings used to constructthe conditional probabilities simulated simple player decision-making. An undeniable featurewhich emerged from the resulting AIS values were the significant decreases experienced duringperiods of major change, implying that players base significantly less of their decision-makingon past outcomes during disruptive periods. Furthermore, small system changes did not signif-icantly change the underlying decision-making players employed in their archetype choices. Insome instances AIS values actually increased, implying that such changes effectively reinforcedpast player decision-making.There are a number of avenues to further this work, both for CCGs and wider applicationareas. Similar to [8], it may be possible to combine exploration of information transfer en-tropy and AIS in an attempt to establish if the
Hearthstone system in Figure 1 displays theprimitives (storage and communications) of a universal computer [14]. One could also consider
Fisher entropy [34] in an attempt to uncover control parameter(s) which influences CCG-systemcriticality. An additional generalisation would include trying to algorithmically-optimise AISvalues by producing weights to replace the mathematical functions — Eqs.(20) and (22) —used in this work. This optimisation would come with the challenge of interpreting the resultsthrough the lens of understandable player behaviour [20]. It would also be meaningful to con-sider the impact of constructing conditional probabilities based on archetype choices beyond T − i.e. K > et al. [15] by including generalised entropies similar to those considered inProkopenko et al. [33] to maximise synchronisation/coordination in artificial systems with theintent of information-driven evolutionary design .Applying similar methods to other game-related application areas, we posit that it wouldbe possible to gain appreciation of the evolution of other games with online systems similarto Figure 1. An equivalent analysis of the real-time-strategy game Starcraft II , with its mixof human players and AI [3], may offer non-trivial insights on the impacts of AI interactingwith wider society. Finally we hope that the information-theoretic results obtained about thenature of decision-making behaviour in epochs of system-wide change will be used to examinerelevant data sets stemming from wider society. Such applications include: understanding theeconomical impacts of shifts in the international political landscape [48]; and awareness of thechanging nature of population behaviours [12].
Acknowledgements
This work was supported by a Research Fellowship under Defence Science and TechnologyGroup’s Modelling Complex Warfighting strategic research initiative. The authors are grateful20o the
Vicious Syndicate team for agreeing to share their data set for research purposes. Weadditionally acknowledge Ivan Garanovich, Scott Wheeler, Keeley McKinlay, Carlos Kuhn,Alexander Kalloniatis, Sean Franco and Daniela Schlesier for fruitful discussions, and MikhailProkopenko for helpful feedback on an early draft of this work.
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