Sub-optimal strategies and Donkeyspace

• Brampton Booster
• Feb 21, 2015
• Comment

Optimal.
Optimal.
Optimal.

Do you think being optimal means that strategy is the best way to play? Does it trounce suboptimal plays each and every time? And if you said no, in what way would a sub-optimal play be preferred?

What Is Optimal

I think the largest misconception about optimization it that is it is “the best way to play”. But that’s not exactly what optimization means. In a theoretical sense, it means you play with the least exploitably. It means that regardless of what the opponent does, you won’t be at any larger disadvantage then if you used a suboptimal tactic. The problem with people solely grinding optimal strategies is that they leave out much of the mortal factors. Much of the time¹ we are facing opponents with suboptimal deck builds and suboptimal strategies and when confronted with that, sometimes it is necessary to also play suboptimally in order to exploit that weakness.

It is important to understand that optimal plays are based only on the situation, ignoring the human bias. A scenario:

• you are certain to lose next turn
• your opponent has a infinite wall of cards
• they decide to 2-pass your vanguard
• and you pulled a trigger on the first check

The optimal solution is to put everything on the vanguard. The sub-optimal solution is to pass it to a rearguard. You are playing in a way that provides you the lowest chance of losing or “playing least exploitably”. Other situations call for different optimal and suboptimal plays taking into account all the mechanics and knowledge about the specific situation.

Now let’s turn this around for a moment. If you had an infinite wall of cards, why did you decide to 2-pass? Clearly that is a suboptimal strategy that introduces a way to be exploited. But what if your hand was actually an infinite wall of grade 3’s, barring enough to defend the remaining two unboosted rearguard lanes? It would then be your optimal play to land a two-to-pass. At the same time, would the optimal solution for the opponent be to pass it to the rearguard? No! Remember that each individual has a different set of information and that being optimal only accounts for the information they acquired. Should you have learned their hand was full of grade 3’s, it would then be reasonable to say that would be optimal. Optimal plays are very local and are heavily dependant on individual scenarios.

Looks good on paper

Let’s imagine Rock Paper Scissors. Now imagine the opponent, say his name is Frank, randomly throws each option at the same probability with no bias (1/3,1/3,1/3). How would you play in order to get the highest chance of winning?

Let’s try throwing out 100% paper. In this case, you have 1/3 chance of winning, 1/3 chance of losing, and 1/3 of a tie.
Can we get better? Let’s try to use paper 50% of the time, and switch to scissors for the other half. Again, you are confronted with a 33,33,33 spilt. In fact, regardless of what you try to do it’s impossible to get anything more then a third. So what’s the point of picking any strategy if whatever you do seems have the same outcome by Frank?

So you are fed up with thinking as there is no need to do anything other then throw paper over and over. But all of a sudden, Frank starts to play scissors! From this switch, he suddenly moves to a 100% win rate and causes you to lose all your money. This is called exploiting. Frank deviated from his optimal strategy in order to exploit and take advantage of your suboptimal plays. Knowing this you decide you want to make him taste a bit of his own medicine by exploiting his suboptimal tactic you decide to try rock. Your smirk turns into surprise as you realize he chose paper this time. Go home he says, you’re too dumb.

Donkey Spaces

Donkeyspace² refers to the space in which a suboptimal tactic is used to exploit another suboptimal tactic. An even slyer definition of donkeyspace refers to the space in which using a suboptimal tactics is used to encourage a suboptimal tactic to exploit the original tactic, only to use a suboptimal tactic to exploit that suboptimal tactic. Talk about inception.

Even in high level tournaments or matches, professionals constantly enter this donkeyspace and this is were the whole “mindgame” meta comes from. Before we get too far, I like to define mindgames as a more general definition of donkeyspace, which includes external interactions such as a players habits and psychological states. Donkeyspace is closer to a theoretical approach where you have thought of the suboptimal plays and knowing that in a clash of two specific plays will yield a positive pay off to you. Mindgames do not necessarily need to have a mechanical goal in which you end up with the edge, but rather incite and bait the opponent until you can exploit this weakness. While both donkeyspace and mindgames require you to understand the current state of an opponent, I prefer donkeyspace as using a players knowledge against them and mindgames as using their own psychology against them.

It is rare to see optimal players make it to the top in any tournament, and this is because these players are so entrenched in the game mechanics and theory they don’t see the human interaction and fail to take advantage of plays that would yield in victory. If you know a person always tends to paper, why are you falling on an optimal strategy when the “best” strategy is actually to play scissors? Is it because they know you know they prefer paper and will pick rock just for that one special time?³ Optimal players avoid risk and this is one of the points that sets them back from winning tournaments. They aren’t planning to win, they are concerned with not losing.

Extension to Vanguard

Theory and practice always have conflicting points. When playing against people and using donkeyspace keep this in mind:

Theory:

1. Always assume the opponent plays optimally.
2. Always assume your opponent is an expert in navigating donkeyspace.

Practice:

1. Your opponents are all idiots.
2. They are too dumb to be mindgamed.

And sadly, this is quite true in vanguard. Don’t try to do fancy stuff to lead the opponent into a maze of decision trees. More often then not it’ll go right over their heads and they just to the same thing they have always done. They lack both the knowledge and the adaptability to take what you give them. The key is to figure out your opponent skill level and judge whether or now he will react when presented with a suboptimal play. And vice versa don’t get too hung up when your opponent consistently makes poor choices, just take advantage of them in a straightforward manner until they learn or lose without worrying about potential “traps”.

I find that playing on a donkeyspace or mindgame level a really satisfying thing, as the constant baiting and provoking as well as out-thinking the opponent really plays into account here. You have all the tools at your disposal and you really are playing the game to the best it can be. You can predict the opponent and counter their counter to your plan. You aren’t blind to your choices and what results from it and you do feel that you have control over the game. But this can’t be reached unless your opponent also has the knowledge and experience to match up. While it’ll be great to have all players to play like in “theory”, the tournaments and attitudes voiced by the majority currently falls right into the “practice” ideology.

Absolutes or Nash Equilibrium

Much of the later part of this article focuses on using and abusing suboptimal plays in vanguard, but I need to point out a couple of things that can’t be played suboptimally because you gain no situational advantage for doing so. If I

All decks should have 12 Crits and 4 perfect guards⁴. There is not one set of triggers that play better against another set of triggers then 12 crit. It is strictly:

• Tier 1: 12C 4H
• Untiered: everything else

Same case for perfect guards. There are still no builds viable enough so far as to even consider quintets.

I guess other ones would be to ride to grade 3 as fast as possible and put crits on vanguard if they are at 4 damage and you can pass, and other small things.

Exploiting terrible players

The format will be divided into the particular suboptimal tactic your opponent has, what is “optimal”, what “suboptimal” play can be subjectively considered superior, and details as to why. I invite you to think of the optimal and suboptimal decisions before looking through it.

Scenario

Your opponent is not running the optimal lineup (Something like 8C/4D/4H). You are at 4 damage and have enough to guard all his attacks. You have a good chance of winning if you tank for a couple of turns, but if you decide to let his vanguard through you’ll have a better chance of winning with the cards you saved. Your opponent initiates an assault on your vanguard with no additional crits.
Your plan is to…

Optimally: Guard it

Sub-Optimally: Not guard it

I guess one can start by arguing I never specify how much better not guarding the attack actually is in the question. But we can look at winning plays based off of two distinct directions. The first is having the correct pieces for a extensive finisher such as Bad End+Ogre or Immortal+Samurai. But that shows a complete polarity, a 30# chance of losing and 70% chance of winning (since no deck can survive a full zerg rush). Had you wasted cards to block the opponent’s attack not only did you drop a 70% win rate but your proceeding rush may not be as soundly unstoppable as it was before. The second way are viable finishers, akin to restanding vanguards and giant pumps. They aren’t the all or nothing plays as in the first example, but they uniformly applicable to present and future. The success of these finishers is to attack at a point where the opponent does not have enough defense, and as we need to predict a 70% they pull non-crits (read, 5k bodies) that leaves an opponent where it could lead to your favour. Hence, based on the information you should know I’m dealing with the second case.

Your opponent has an astounding (1 - 41/49*40/48) 30% to get at least one crit. I mean, that’s almost one in every three games. Surely you wouldn’t risk it by letting it pass. Your hand looks decent enough to survive a couple of turns and you aren’t guaranteed to win next turn either.

But this is where randomness plays out. Neither you nor your opponent is going to know exactly what is going to happen in next three turns. Optimal takes into account the average of those three turns but not the precise number. So what if you start to pull below average and your opponent gets above average? It is absolutely in the realm of probability that you can lose.

But there is also something inherent in the way the trigger ratios are presented. The 4 draw trigger allows them to succeed in very long battles. The reason why 12 Crit is superior is due to the game favouring offense more than defense, and the mean number of turns it’s advantage starts to decay is sometime after natural game speed. But if you keep holding back with 12 crit then you’ll start to lose a the battle of attrition. Remember you have a 43% chance to break out the crits so the advantage is clear.

You might be asking why guarding it is still optimal given my above point. That’s because 30% is still a fair amount of risk, provided that tournaments are best of 1 or 3 (depending on the format). It would be on average more advantageous to you to keep going for a few more turns, as an opponent that lacks access full crits troubling in other departments. But playing suboptimally lowers the amount of chaos or “randomness” remaining and closer to a set of micro-scenarios where theory takes a greater hold.

Scenario

You are stocking a full field. Your opponent has all but a side booster, but uses his sole stage 1 rearguard to attack your grade 2 rearguard. You don’t have a replacement attack on hand but you still have a booster to guard it. Your plan is too…

Optimally: Guard it

Sub-Optimally: Not guard it

Let’s start with the net advantage of the two possible scenarios. If you get an attacker the turn after then you lost nothing (if you consider that the RG attacked the VG, then you would need \5/ or an interceptor amount of shield). But if you fail to pick up an attacker you lost an important 2-stage attack. Even if you have no \5/ to stop the rearguard attack, a \10/ would only cause you to lose 5k rather then a full 10k. When we add in the probabilities then the solution is clear; Assuming a grade ratio of 8:11:14:17 you have a 38% of the time to draw an attacker. If you happen to have a set of grade 1 units that can be substituted for then it could be 47%. Still, when compared to the higher 62% percentage of getting that -5. Summarized:

Not Guarding: (0)(19/49) + (-10)(30/49) = -6.122
Guarding (min): (-5)(19/49) + (-5)(30/49) = -5
Guarding (max): (-10)(19/49) + (-10)(30/49) = -10

The reason why you would consider a decision such as this is dependant on the deck. Or specifically what your ‘replacement’ could be. Unfortunately most of the good rearguard effects are only for grade 2’s so your opponent can take out two birds with one attack. So this only works when your opponent is attacking a weak or empty grade 2 so you can pull a legion mate or some other card to replace it. In contrast a legion mate can be sacrificed for re-legioning. If the legion mate has a raw advantage over a temporary attack advantage then swapping it for a beatstick can provide a solid rush.

So far I’ve been hammering you with what irrational plays you can make in order to secure some other sort of advantage. Yet I haven’t asked for any situation where you would force an opponent would make an irrational play. Now we flip the scenario around and propose the question of what you will do when you could send a 1-stage at the rearguard. You want your opponent to not guard the attack. To do so, you should target the grade 2 rearguard with the lower threat. It might seem irrational as well but you are opting to cause the opponent to be irrational as well. It’s a interesting gambit which requires you to understand an opponent’s reactions corresponding to what they have available in their hands.

Scenario

Take the previous scenario, and change one of the following: Either you have a grade 1/3 unit being attacked instead, or the opponent is throwing a 2-stage lane. Your plan is too…

Optimally: Not guard it

Sub-Optimally: Guard it

A bit of math churns out: 5 * 19/49-5 * 30/49 = -1.122

And against a full -5, it hurts to consider dropping it. The sub-optimal clause is described in the first example, where you aim to reduce entropy in order to secure a quicker game. I believe you can work out the kinks in the solution given the former examples.

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