# Using evolutionary game theory to explain and predict the meta of Yugioh Duel Links

TL;DR: Evolutionary game theory, initially developed for biology, has been successfully applied to other domains such as economics, sociology, and anthropology. This post will use it to explain the various meta of Yugioh Duel Links and predict the nonexistence of the kind of meta which is simultaneously diverse, accessible, and without being Rock–paper–scissors.

• Introduction
• Multi-species Competition
• Predator-Prey
• Rock–Paper–Scissors
• Conclusion and discussion

## Introduction

Fur Hire is an archetype released in the mini box Clash of Wings. Although powerful, they were generally not considered as overwhelming as Sylvan or pre-nerf Cyber Angel. However, contrary to people’s expectation, it recently became a Tier 0 archetype and left Amazoness and Masked Heroes far behind.

These cute creatures have since infested the competitive play. More than 50% of decks in tournaments feature Fur Hire, and tons of players are complaining about it on Reddit daily, just like what they did to Sylvan and Cyber Angel. This unexpected situation raises the question: What can we do to create a healthy meta?

This question seems to be too big to answer. In this post, I will, instead, analyze the formation and properties of various kinds of meta in an attempt to understand this game better. With this understanding, players could have more reasonable expectation about this game and find their suitable ways of playing.

For this purpose, I will describe three types of meta with the help of evolutionary game theory: Multi-species Competition (MC), Predator-Prey (PP), and Rock–Paper–Scissors (RPS). All of them can find examples in the Duel Links history.

Name Evolutionary Model Drawback Epoch
MC Malthus not diverse Fur Hire
PP Lokta–Volterra inaccessible Cyber Angel
RPS Replicator equation cyclic domination Spellbook

Furthermore, I list here three properties that players may expect the meta to have: diversity, accessibility, and without being cyclic domination.

• Diversity: No Tier 0 decks. The strongest deck does not have more than 50% share in a competitive environment.
• Accessibility: Free-2-play. With reasonable effort, the players can play any deck they want.
• Cyclic Domination: Rock–Paper–Scissors. Deck A wins against Deck B, which wins against Deck C, which wins against Deck A. This chain, of course, can be longer (e.g., Naruto), and each component of the chain can contain multiple decks.

I will show you that the three aforementioned meta types each have their own drawback and that none of them can have the three properties simultaneously. This conclusion works not only for Duel Links but also for other card games, such as Magic the Gathering, Hearthstone, and Shadowverse. To my best knowledge, I am the first person to use evolutionary game theory to analyze card game meta.

## Multi-species Competition

In this section, I assume the existence of several strong decks and that none of them has a significant advantage against another (i.e., when two of them battle, both have a winning rate around 50%). Under this assumption, I will show that one of those decks will be predominant and eventually eliminate the other decks.

Without loss of generality, let us consider an environment with only two strong decks. Let $x_t$ and $y_t$ be the population of these two decks at Moment $t$. The population dynamics can be represented by the following differential equations [ \frac{dx_t}{dt} = a x_t , ] [ \frac{dy_t}{dt} = b y_t . ] The population increase is proportional to the current population, which is justified by the fact that the more players playing this deck, the more likely it ranks high, and the more players try to copy this deck. The parameters $a$ and $b$ depend on two factors: it increases with the intrinsic strength of the decks and decreases with the cost of the deck.

The solution of the above equation is given by [ x_t = e^{at} , ] [ y_t = e^{bt} , ] Thus, the proportions of both decks are given by [ \frac{e^{at}}{e^{at} + e^{bt}} \text{ and } \frac{e^{bt}}{e^{at} + e^{bt}} . ] If $a>b$, the 1st quantity will tend to $1$ and the 2nd quantity will tend to $0$ when $t \rightarrow \infty$, which means that, given enough time, one deck will eliminate the other.

If both decks are equally strong, the less expensive deck will have a large parameter (i.e., $a$ or $b$), which means the less expensive deck will eliminate the relatively more costly deck. This conclusion also works with the case of more than two decks: Given $n$ equally strong decks, the cheapest deck will eliminate all others.

This model perfectly explains the situation of the Fur Hire meta right now. Masked Heroes needs to go through a main box three times; Amazoness needs to go through a main box, level Mai to Level 45, and get three UR tickets for the Princess; Fur Hire needs only to go through a mini box three times, which is the cheapest of the three. Fur Hire is strong and cheap; this is why they appear everywhere in the meta.

## Predator-Prey

In this section, I assume the existence of a F2P deck and a P2W deck. The F2P deck is cheap and strong, and the P2W deck is nearly inaccessible but works excellently against that F2P deck in question. The F2P deck is essentially a prey, and the P2W deck is essentially a predator. Under this assumption, I will show that the meta behaves like a cycle. That is, the predator and the prey prosper and decline periodically and in turn have their peaks (as shown at the end of the section).

The population dynamics is given by the Lokta–Volterra equations. [ \frac {dx}{dt} = \alpha x - \beta xy , ] [ \frac {dy}{dt} = \delta xy - \gamma y , ] where $x$ and $y$ are the numbers of the F2P and P2W decks respectively. $\alpha$, $\beta$, $\delta$, and $\gamma$ are all positive parameters.

In particular, $\alpha$ represents the intrinsic strength of the F2P deck, $\beta$ and $\delta$ represents the relative strength of the P2W deck against the F2P deck, and $\gamma$ represents the extra cost of P2W deck against other random decks.

The Lotka–Volterra equations do not have analytic solutions. It can be solved numerically though, and the solution is represented in the following figure. It looks like that the F2P deck prospers first, and then gets hit by the P2W deck and declines (the P2W deck thrives instead); with the fall of the F2P deck, the P2W deck is less time-efficient against other random decks and thus is switched to the F2P decks (the F2P deck rises again). This behavior repeats periodically.

This meta has more than one Tier 1 deck, but the P2W deck is not accessible to all. A famous example is the meta of the pre-nerf Cyber Angel and the pre-nerf Three-star Ninja, with Cyber Angel being F2P and Ninja being P2W (three copies of Ninja Art of Transformation).

## Rock–Paper–Scissors

I have claimed in another post that Pokémon GO was essentially a Rock–paper–scissors game. Surprisingly, it seems that a diverse and free-2-play meta in Duel Links also has to be Rock–paper–scissors. The intuition is that, in a F2P environment, to keep a deck meta relevant, it has to be strong against some meta decks; and to avoid being predominant, it has to be weak against some other meta decks.

Nevertheless, the existence of cyclic domination does not automatically means that Rock, Paper, and Scissors will live in harmony. Instead, they can take their turn to be predominant and nearly extinct, which is not a diverse meta and is definitely not what we want to see. We want to know whether there needs some extra condition for Rock, Paper, and Scissors to coexist.

The best tool to analyze this problem is the replicator equation. I will restrain myself from writing excessively complex maths here. The idea of replicator equation is that a species’ increasing rate is proportional to the difference between this species’ fitness and average population fitness (Darwinism).

For example, if the majority of the environment at this moment is Rock, the fitness of Paper will be above average, and the fitness of Scissors will be below average; therefore, the population of Paper will increase, and the population of Scissors will decrease.

When we apply the replicator equation to the game Rock–paper–scissors, we can obtain some interesting insight. Let us write down the payoff matrix of Rock–paper–scissors.

Rock Paper Scissors
Rock 0 -1 $\mu$
Paper $\mu$ 0 -1
Scissors -1 $\mu$ 0

Here, the -1 stands for the loss of time and effort and psychological dissatisfaction per loss, and $\mu>0$ stands for the net gain (with the loss of time and effort deducted) per win. If $\mu=1$, we call it a zero-sum game.

It has been shown that

• if $\mu>1$ (positive sum), then the game will converge to the stable state where each of Rock, Paper, and Scissors takes 1/3 of the total population;
• if $\mu<1$ (negative sum), then Rock, Paper, and Scissors will in turn take over the whole population and then almost go extinct;
• if $\mu=1$ (zero-sum), then the game behaves like a cycle, just like the predator-prey case.

Therefore, in order to diversify the meta, we need to make $\mu>1$, which means that the prize of one win should cover more than two losses. In this sense, Konami is doing a smart job by giving rewards of accumulated wins in ranked duels.

In Duel Links, there was a short epoch where we observe Rock-paper-scissors. With the release of the Spellbook archetype, Sylvan was countered by Spellbook, which was countered by Amazoness, which was in turn countered by Sylvan.

## Conclusion and discussion

This post analyzes the three types of meta and predicts the nonexistence of the kind of meta which is simultaneously diverse, accessible, and without being Rock–paper–scissors. Among these three types of meta, the diverse, F2P Rock–paper–scissors seems to be the most promising, especially if we can create some long-chain Rock–paper–scissors.

It should be stated that the design of Rock–paper–scissors meta is tricky. Although Yugioh has tons of archetypes, finding three archetypes generating cyclic domination is still a hard task. Moreover, the frequent release of boxes makes it even more Herculean: in order for a box to be meta relevant (for more sales), it has to either fit in or destroy the current meta.

Another alternative strategy is to create F2P multi-species competition meta for short periods. Given enough time, the meta will naturally converge to a single deck. But when the box has just been released, and the meta has not been stabilized, we can observe a diverse meta. Therefore, if we can release the box frequently enough and do not give players enough time to overly optimize the build, we can achieve a diverse meta. The drawback of this strategy is that players will complain about power creep.

Konami also seems to have realized this problem, but they adopted a different (but equivalent) solution by making the game less F2P. They have reduced the gem income and increased the packs of mini boxes, which has been the players’ main complaint over the past weeks. In fact, this solution is, nonetheless, equivalent to accelerating the box release and attempts to achieve the same effect: diversifying the meta.

The best we can hope in this game is the long-chain cyclic domination, but since it is hard to design and even harder to maintain and Konami has essentially no financial motivation to do so, we players may have to make a compromise between the diversity and the F2P level.

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Written on July 31, 2018