*By MannySkull*

**About the author: ***MannySkull is a casual Hearthstone player that has experience competing online (playoffs, open tourneys, and leagues like THL) and in LAN events (Dreamhacks). He has been a Legend player since 2016 but never the type of top 100 player. He has been fortunate enough to have several Pro Hearthstone friends (some currently Grandmasters (GMs), some formerly GMs, and some aspiring to be GMs) and had the opportunity to learn from them and also helped them with lineups, ban decisions, and queuing decisions. He is a full professor in Statistic and Economics at a top US university, has published articles in top journals, and teaches graduate level classes on these topics.*

Hearthstone players often criticize queuing decisions from competitive players in tournaments; most notably Grandmasters (GM) players. From arguing that there was a better queuing decision to claiming that queuing order does not matter in a conquest format, if you participate in discussion forums you will see it all. Motivated by these debates, my friend Pasca asked me to write an article explaining why queuing order in conquest matters and what are reasonable thought processes you can expect from rational players. These are two different questions, so I decided to start with a description on how players should think about queuing and will leave the “how it affects the odds of winning” for another article. If you are looking for a three line explanation in a TLDR, then stop here and go do something else.

I will illustrate the concepts using a win rate (payoff) matrix for a match (Bo5 setting) between two of my GM friends: Muzzy and Justsaiyan. Here we do not discuss bans so we assume each player starts with three decks in their lineup (after bans already happened) and have to decide what deck to queue first. The first important aspect to note is that we do not question the win rate probabilities (take them as given and assume these are the relevant win rates for the players to make their decision). The second aspect is that margins in Hearthstone (HS) are often small, so we assume that players care about small margins (that is, they would play a deck with a win rate of 51.2 over another one with a win rate of 50.8 without hesitating). The third aspect is that both of these players are smart, they know each other super well, so we will not consider situations in which one takes advantage of the other by being “smarter”. The final aspect is that in the subject of Game Theory there are multiple notions of solutions, optimal behavior, and equilibrium. Here I will use the most basic ones exclusively.

The main concepts you should hopefully understand after reading this article are:

- Deciding what to queue requires the player to understand what a
**best response**is, whether there is a pure equilibrium or not, and also make an assumption on the level of rationality of the opponent. - Understand that when there is an
**equilibrium**, players cannot make profitable deviations (that is, they cannot be better off by playing off equilibrium). - Concepts like
**dominant**and**dominated strategies**may simplify the analysis significantly in some cases. - Finally, in many cases the player will have no choice but to
**randomize**the starting deck to avoid being predictable in repeated iterations of the game (like, for example, the GM circuit).

## Case 1: Dominant Strategy

Sometimes, although not often, players may have a **dominant strategy**. A **strategy** here is simply a deck to play, and dominant means that the player wants to play that deck *regardless* of what his opponent could potentially play. This makes the queuing quite simple, as illustrated in the following example (Muzzy is the row player and Saiyan is the column player. The probabilities are from the point of view of the row player (Muzzy)).

**Best Responses**: I highlighted in green Muzzy’s best responses to Saiyan’s deck. For example, if Saiyan plays Malygos Druid, Muzzy’s best response to that decision is to play Libram Paladin (60.2% win rate). I marked in purple bold Saiyan’s best responses to Muzzy’s deck. For example, if Muzzy plays Libram Paladin, his best response is to play Highlander priest (which is unfavored with a 47.2% win rate from Saiyan’s side but it does better than either Druid or Warrior).

**Dominant Strategy**: Libram paladin is the best response on Muzzy’s side *for all the three* decks that Saiyan can possibly play. This means that Muzzy is better off playing Paladin regardless of what Saiyan plays, and so Muzzy should queue paladin first.

**Equilibrium**: Saiyan is smart and realizes that Muzzy will queue Paladin, so Saiyan rightfully decides to queue Priest (his best response to Paladin). But this does not affect muzzy’s choice: muzzy wants to queue Paladin when Saiyan queues Priest. This is a notation of “equilibrium”: when no player has incentives to deviate from their strategy if you give them the option to do so ex-post. The outcome of this game is that Muzzy queues Paladin and Saiyan queues Priest and no player can be better off deviating from this strategy.

Most often there are no dominant strategies and so the analysis becomes more intricate. However, before getting into more sophisticated analysis it is good practice to first check whether there are dominant strategies for one or both players.

## Case 2: Equilibrium in pure strategies

Even when there are no dominant strategies there could be equilibria in pure strategies, as the following example illustrates. The table shows the win rates where I marked their best responses using the same coloring as before (highlighted green for Muzzy and bold purple for Saiyan).

**Best Responses**: Paladin is Muzzy’s best response to Druid and Mage is his best response to Warlock and Mage. Saiyan’s best response to Druid and Mage is Zoo Warlock, and his best response to Paladin is Mage. There are no dominant strategies in this situation.

**Equilibrium**: We can see there is an equilibrium in pure strategies: Muzzy plays Mage and Saiyan plays Warlock. Even after each player finds out what the other one is queuing, they still want to stick with their choices as there are no profitable deviations.

**Thought Process**: The fact that there is an equilibrium, does not mean that players will play it. However, it’s the only outcome where you can tell a story that is consistent with behavior of two players that are smart. Suppose Saiyan thinks “I’ll queue Mage!”. What does he need to think about Muzzy’s behavior to justify such a choice? Saiyan must be thinking that Muzzy will queue Paladin (Saiyan wants to be able to get that sweet 67.8% win rate). But then, what would Muzzy have to think to queue Paladin? Muzzy has to believe that Saiyan is queuing Druid. Why would Saiyan do that? Well, it turns out Saiyan **never** queues Druid first (this is called a **dominated strategy**). He is always better off queuing Zoo or Mage. This means Muzzy never plays Paladin first. Ok, What if Muzzy plays Druid? Well, Muzzy **never** queues Druid first because he is always better off playing some other deck (Druid is a **dominated strategy** on Muzzy’s side). Saiyan, understanding that Muzzy will figure out that his Druid is dominated, notices that Muzzy is always better off playing Mage and so Saiyan responds with Warlock. Muzzy, understanding this, realizes that his best response to Warlock is Mage. And we go back to (Mage, Warlock) as the outcome of this game.

## Case 3: No equilibria in pure strategies

We get to the last case I want to illustrate (there are other situations that I will not cover here). In this case there are no equilibria in pure strategies (I will not talk about mixed strategies in this article). The table shows the win rates where I marked their best responses using the same coloring as before (highlighted green for Muzzy and bold purple for Saiyan).

**Equilibrium**: No equilibrium here. If Muzzy plays Demon Hunter, Saiyan wants to queue Rogue. If Saiyan queues Rogue, Muzzy wants to queue Warlock. If Muzzy queues Warlock, then Saiyan wants to queue Warrior, in which case Muzzy wants to queue Demon Hunter.

**Thought Process**: Saiyan is smart and realizes that Muzzy is never queuing Paladin First (dominated strategy) and Muzzy is smart and realizes that Saiyan is never queuing Warlock first (dominated strategy). So, Muzzy must be playing Demon Hunter or Warlock and Saiyan must be playing Warrior or Rogue. At that point, the only rational choice for them is to randomize which deck they choose (not necessarily by flipping a coin, but I will not get into the right odds of randomization). You may think, “Well, muzzy’s Demon Hunter gives him better odds on average”. True, but if Muzzy plays his best **average win rate deck** and Saiyan anticipates that, then Saiyan would queue Rogue and get an edge. Same, you could say Saiyan has better average win rate queuing Rogue. But if Muzzy anticipates this, he would queue Warlock and get an edge. And again, although I am not formally considering the repeated aspect here, remember that these players know that they will play each other often and so they care about not being predictable.

HS players when facing situations like this one talk about “mind games”. However, I do not think this is a good terminology because it implicitly assumes that players have different levels of rationality. Let’s call “level 0” the player that plays his highest average win rate without contemplating what the opponent could do. Let’s assume for a second that Saiyan is a level 0 player, so his queues Rogue. If Muzzy is also a level 0 player, he queues Demon Hunter. But, if Muzzy is a “level 1” player (a level 1 player is a player that knows his opponent is a level 0 player) Muzzy would queue Warlock. Sweet. But now, if Saiyan is a “level 2” player (a player that knows his opponent is a level 1 player) then he would queue Warrior to counter Muzzy’s Warlock. And you can keep going. So, taking advantage of your opponent because you think you are smarter than them, only works if you are effectively smarter than them or if you get lucky. Then, call it mind games.

Remember that decisions taken under uncertainty should be judged with the information that the player had at the time of making the decision. Make sure you keep this in mind next time you want to criticize a queuing decision.