During the release event, after seeing me attempt to multitable (e.g. play two or more drafts simultaneously) two different drafts, my wife mused what—if any—were the benefits from playing with multiple accounts in the same draft queue (henceforth, multiqueuing). Her question, as she is another academic, was entirely about the cost and benefits as opposed to the normative implications (here is a long discussion of that topic, though I have not slogged through it myself) of engaging in such behavior. I immediately rejected the notion of multiqueuing being beneficial based on some inherent, passed-through-the-halls gamer logic that it is generally negative expected value (EV) for friends to be in the same draft and it is likely a similar situation if you controlled more than one account in a draft. My follow-up reaction to my immediate dismissal of the proposal (which included a few hours of work later) was to calculate the math behind doing such a thing and see if my intuition held (this is a time-consuming habit.)

To be clear, it is against Hex policy to use multiple accounts in the same queue and collusion between two players is cheating. Whether economics favored multiple entries into a single tournament does not outweigh the cost of losing your account(s). The risk and cost of detection will trump any marginal gains from this behavior. That being said, it is still not efficient for winning players to be in the same draft nor is it wise for a player to be in the same draft with multiple accounts. This holds even when we increase the value for first place substantially.

To calculate the benefits of entering a draft with multiple accounts, we need to consider the opportunity cost. For the sake of comparison and ease of calculation, we will assume the choice is to enter the same queue twice or to enter two different queues; by doing this, we allow the costs and the benefits to mirror each other. The cost to enter a queue is roughly 550p where we will assume that, on average, you will get 150p of value back in cards from the packs that you open. To make calculation simplistic, we are assuming that the value of a given pack is 200p for both entering into queue and winning packs (we can always modify this to account for auction house prices and the chances of receiving a primal). To ascertain the benefit, we will assume both accounts have the same probability of winning any match for each account and this win percentage is static for any round they are in. Again, these assumptions are for simplicity and we can treat them as an average winrate with some understanding that it may be harder to win a later round than a normal round; we can include this, but we are going to make the math simple for now.

To create the expected value of entering a draft, you take the probability of each event multiplied the reward for that event. To know the probability of an event, we take the likelihood of you winning a given round based on your match win percentage. So, for example, your likelihood of winning round 1 is x and your likelihood of losing round 1 is 1-x. The likelihood that you win a tournament is x*x*x or $x^3$ and the result of winning the full tournament is worth 1000p (5 packs). Taking all possible outcomes into consideration, we get the following calculation:

$800x-800x^2+1200x^2-1200x^3+2000x^3-1100$

We calculate the player’s win-percentage multiplied by their expected payoff for each outcome minus their cost of entry. We double this as they are playing two different drafts, so the outcome reduces to:

$800x+400x^2+800x^3-1100$

This is a pretty straightforward utility calculation and, given our basic calculation for the value of winning packs, a player gains the ability to go infinite above a 72.69% winrate (checking my own winrate, I am at 73.41%, but that number is a bit misleading as the data is right-censored; you never see what happens in round 2 and 3 if you lose round 1).

To answer the initial proposition, we need to make a much more difficult calculation. To really get at the value of entering in the same queue twice, we need to figure out every end-state of a draft that involves two accounts, when those accounts play against each other, when those do not play against each other, and the probability of the player winning/losing each round. This makes a very messy calculation that is the following:

$\frac{1}{7}(400-400x+600x-600x^2+1000x^2)+\frac{6}{7}(x^2-2x-1)*0+\frac{6}{7}*\frac{1}{3}x^2(1000+800x)+\frac{6}{7}*\frac{2}{3}*x^2(1600x^2+2*(x*(1-x)*1400x)+2000(x*(1-x)*(1-x)+800(1-x)(1-x)+\frac{6}{7}*2x(1-x)*(400(1-x)+600x(1-x)+1000x^2)-1100$

There is quite a bit going into the above utility calculation: In addition to needing to know a player’s winrate, you need to figure out the likelihood that they get paired up against themselves in round 1, 2, or 3 after calculating if either account wins/loses each of those rounds. This mess of an equation is certainly reducible and we can pinpoint what winrate you need above to be a winning player (above 78.78%), but our real question is: When is the above equation for multiqueuing more profitable than multitabling? To do that, we can set the equations equal to each other or, in what is a bit more exciting, we can graph it:

The solid line is the EV for multitabling, the dashed line is for multiqueuing, and the dash-dotted line is the difference in the expected value between the two options. Notably, multiqueuing has some benefit for losing players (<50% winrate), but that benefit is tiny. The threshold is at 50% and winning players (>50% winrate) are hurt by multiqueuing at an increasing rate. Thus, if you are a winning player on average, you should enter as many drafts as possible, not enter with as many accounts as possible.

In the introduction of this article, I mentioned that I initially had this conversation and did this calculation during the release weekend which had a unique prize for winning a draft: an alternate art Filk Ape. Does it matter if there is a lopsided prize pool for first place?

Here, the relationship becomes more stark. I put the value of winning first at an extra 1000p for simplicity. Again, the threshold for value is at the 50% mark suggesting that winning players should not multiqueue, but should multitable. The penalty for multiqueuing is even more steep as, when you enter in the same queue twice, you are losing out on an opportunity to win another Filk Ape. Dramatically, if you had a 100% winrate; you could only win one Filk Ape in a single queue, but would be guarantee two in two separate queues. As such, the better player you are, the more, distinct drafts you should enter.

Importantly, this result also holds for players of roughly equal skill. If you and your friend are nearly equally skilled in Hex, you are both better off being in other queues unless you are both losing players.

In this discussion, there are a few things that I assume away for simplicity. The most overt simplifications are the cost and benefits utility values, but changing those will not dramatically alter our results in a way that will lead to large changes in our predictions. However, there are a few things that may alter our conclusions. First, there is a time component: A player may be unable to enter multiple tournaments, but is able to enter into the same queue; however, this is an unpersuasive objection as queues fire consistently and if you can play in the same tournament on two accounts you are nearly as able to play in different tournaments.

Second, some people may argue there is a benefit to collusion that adjusts a player skill over a player that is multitabling. However, even this must be taken with grain of salt as for every percentage you gain in increasing an account’s likelihood of winning a draft, you are decreasing the likelihood that the other account is going to win. This makes it a bit hard to do a comparative winrate assessment as you would still have to find an average between the two accounts that may just even out.

Third, and this only applies to a single player with multiple accounts, we also have the opposite effect of objection number two; that is, if you are playing multiple games at once, your winrate is likely to decrease even as a skilled player. Juggling two games is difficult, you are likely to miss nuance in a given game, and you may not remember everything you need to for both games (such as deck composition, troop threats in the deck, removal in the deck, etc.). This is a common issue in poker where players will multitable in two to twenty-plus tournaments or cash games; players will miss much of the nuance of their opponents and their betting habits. However, if you are still a winning player after taking an EV hit of multitabling, then multitabling can increase your overall winnings (quantity comes to trump quality).

The final thing worth mentioning is the thing that I always neglect. It is feasible that two winning players may enter the same queue neither for collusion nor for expected utility, but for the fun of being in the same draft and playing against each other? I do not know how to compute fun, so there is that.

### Zubrin

Michael Allen is a competitive HexTCG player, co-host of the 2 Turns Ahead podcast, and founder and moderator of the Hex Subreddit.