Opportunity cost is everywhere

[D_acolyte]

I am not the best at expressing my thoughts but I figured I would do a large post about how I think around Malifaux. This is not new or unique, or at least not to me. Malifaux has several very interesting thoughts and ideas hidden in it in such a way that might go unnoticed if you do not examine the game and actions, such as opportunity cost, activation power, “true” cost and secondary synergy.

 

Opportunity cost:

I will start with opportunity cost because it really is the item at the center of it all. To help I will use examples of models and cards. Also be aware that what you are will to pay in a opportunity cost will be different from me and probably others, this is to be expected.

Opportunity cost originated in economics and can broadly be defined as the difference in return of a chosen action and one that is necessarily not taken. Lets say that you have a 1, 5, 7, 9, 13, and 13 in your hand and you wanted to summon a model that takes a 13. This is not a bad use of a card but what does that 13 represent, it could be a hit that you really needed later, a tactile action that you wanted to have, or a defense in a dire situation. With two 13 the cost of using one for a high summoning or a decisive blow is not a hard choice because you have another one to fall back on. Where as if your hand is a 1, 5, 7, 9, 10, and 13 this choice gets more difficult because the use of that 13 has a high cost then before and may be needed till later. We all have probably played games where holding onto a high card late into a turn has helped for some reason or another.

Though I have only talked about it with the numeric values there is a opportunity cost associated with the suits of the cards as well. The holding onto the 1 of mask for 3 turns just so Tara can be safe from a possibly deadly bullet is in fact banking your opportunity cost on protection. The numeric number of the cards has no value but the suit is extremely important and is where the cost resides but in holding onto that card instead of discarding it your lowing number of chances of drawing a high card into your hand by 1.

 

Some models such as summoning masters have ways of dealing with the opportunity cost of there summoning. I will discuss with the use of Nicodem, Ramos, Kirai, Dreamer as examples.

Collodi has no way of lowering/lessening the opportunity cost of summoning other than soul stones.

So thanks to Stryder, I actually found out that Nicodem has a way of lessening the cost of summoning models at the expense of corpses.

Ramos on the other hand can set the numbers of steam arachnids he seeks to summon which adjust the card value and suit he needs to summon a model. All you need to gain two models is a __ of tomb in you hand.

Kirai may sacrifice a friendly spirit to gain a +2 to her casting skill. This will let her summon a hanged on a 11 of crows instead of a 13 of crow. That is a change from a .04 percent of the time to .07.

The Dreamer on the other hand can have a daydream activate and sacrifice its self to gain lucid which gives him a + mask to his casting till the end of his next turn. This almost quadruple the possibility of summoning.

These all effect the opportunity cost of the action and the value of the cards.

 

Activation power:

I was helping explain to a new player this weekend about the power of activation numbers and the cost of activating models with companion or accomplice. This to can be thought of an opportunity cost in the game but has a lot of effect that is more abstract then cards. Very simply the activation power at any given time is the number of activation one side still has left minus the number of activation the other side has left. This means that killing a model that has already activated causes no change in the activation of the other side, though it may be worth it to prevent it use next turn. This also means that a model that gains reactivate adds activation power to there side.

The value of an activation is sometimes just the value of letting the board progress and seeing how your opponent acts. At other times it is the ability to act before another action happens. The cost of a chain activation is that by lowering your total alternating activation count you gain a more powerful moment in one activation count. This is all rather simple but what does it cost you. If your opponent opens up with a double defensive action on a model then they are trying to advance the field in an extremely safe manner. Similar actions are often taken, such as the double walking of small models. This is a use that we all do and think little of. The more interesting use of activation power is with how it runs into chain activation and sequential activation planing.

Chain activation can have a lot of power and can be a dangerous trap. The core of it is the player is exchanging 2 activation power instead of 1, the models that will activate, in order to gain a stronger activation. Is this always worth doing, no but like many things in the game it has its moments. I most often do this with summoning masters as the the generation and lost of the activation power are the same, excluding concern about other costs. One of my friends love to activate Ophelia and Francois in a chain activation, the problem he started running into with this very powerful activation is that he did it to early he was using actions to get into range of the targets he wanted to kill with this.

There is a strange concept from World War 2 humorously called the tank equations by some. The concept is the fire power and damage caused by one tank is that of one tank where the fire power and damage of 2 tanks is that of 3 tanks. This had to do with the time it took to destroy the second tank. In the game of Malifaux anything with armor 2 or high tends to take a significant amount of resources to finally kill, such as Izamu, Joss, or the Gracie. This is heightened when there are 2 such model near each other, such as Joss and Gracie. Luckily most of the time one of these models can be separated through just the difference in activation timing. Another use of the same idea of the tank equation can be found in the gremlins as they can work amazingly well at maximizing the use of there actions and there activation. Lenny can toss Ophelia and Francois 8 inches down the field and then be pushed into base when Ophelia activates. Though this is not so much a matter of armor it is a way of moving 2 to 3 dangerous models across the board quickly and will soak up the resources of some crews to a disproportionate amount.

Sequential activation planing is when to optimize a model for its role you will have another model activate to modify it in some way. An example with the Dreamer summoning and the use of daydreams to give him Lucid before he activates. This in effect means that to ramp up the Dreamer you have to think about 2 separate independent activation and what the board may look like when you take the empowered Dreamers actions. Another master that is very involved with this is McCabe and his handing/taking of upgrades, his ideal cycle is the model with the upgrades activates, McCabe activates and pics up the upgrades to hand to another new models, that newly upgraded model activates. For the Gremlins Lenny is a great model for sequential activation planing with his toss and his giving of rams.

 

“True” cost:

What is the worth of a soul stone when your actually playing? That is a hard questions and for me I figure it is 2 wounds because I mostly use them to prevent damage. By contrast I find what is the “true” cost of a model much easier, even if that model uses souls stones. The true cost of a models in stone is a combination of what you pay for the model and upgrades as well as what stones you spent on the model through out a game. This number is one that has the ability to change but a close approximation can be calculated. I want you to think of your favorite henchman and/or master, now how many stones do they use in a game on average? Now that you have the number, that is what I use for a close approximation of what a model will used in a game to help figure out the average true cost. I tried to explain this to someone once they, there response is that it sounded like a tax. This is not a tax but rather a way of assessing how I personally use soul stones and is that worth the opportunity cost of the use and do I have enough to do what I normally do with my soul stones.

 

Secondary synergy:

This is synergy that is important to the model but does not stand out in any great way given something else. For instance Yan Lo being a spirit master because of Fury of Yomi or the combination of Ototo with Sidir to make use of Empty the Magazine giving slow. One of my favorite recently, this last week, was with Anious and a few talking about how great he is with Tara. I told them I would run him with Lady Justice and got a few puzzled looks to which I responded well not exactly her but with her Death Marshals because they bury thing in a guild force and he can make them fast. This is a great example of people glossing over a models synergy or confusing it. Anious and Tara have a synergy through burying models and through action point control but I find that Tara tend not to bury models herself so there is a go between in this synergy that people gloss over. Alternatively, if you identify that go between, in this case the bury and models that do it, you open up another option you might miss. Anios with Death Marshals and the Brutal Emissary can really leverage his effect on buried models possibly as much or more then Tara. Same goes with Anios ability to move markers and Illuminated, Lilith, the Carrion Emissary.

 

I hoped you enjoyed this. As I said I do not think there is anything new or revolutionary here. Just a little economics, probability, opinions and concepts.

Edited December 8, 2015 by D_acolyte
Found out something on a model mentioned.

[Myyrä]

Your thoughts are very good, but the probabilities are wrong. Here's a link explaining how the probability of drawing exactly one high card to your hand from the deck is calculated. ...or you can send me a pm if you want me to calculate it for you.

 

Edited to not look like a personal attack.

Edited December 6, 2015 by Myyrä

[Hateful Darkblack]

<MOD HAT>

Myyra, let's please steer away from things that could sound like personal attacks. It may be more useful to provide your own math instead.

</MOD HAT>

Edited December 6, 2015 by Hateful Darkblack

[D_acolyte]

Your thoughts are very good, but the probabilities are wrong. Here's a link explaining how the probability of drawing exactly one high card to your hand from the deck is calculated. ...or you can send me a pm if you want me to calculate it for you.

 

Edited to not look like a personal attack.

I am well aware of binomial coefficient and the also the way it works, I chose to use a simple approximation and even stated that. What is did was a simple probability for the purpose of understanding, I even stated that this is with replacement to keep the math simple.

Lay out of the question: what is the probability of drawing a high card on any independent draw? that is simply the number of high cards/the cards in the deck. which in a full deck is .24.

Now that i know that, I can ask what is the very simple probability of drawing at least 1 high card. There are several ways of doing this, but I chose the simplest for easy of understand because I am posting it on the forum. Saying having at least 1 high card can be turned around as having only 1 high card for simplicity. which then means it is the number of time I need the draw of the high card times the probability of that draw times (1-the probability of that draw) times the rest of the draws to get a very simple number which though not a 100% accurate is probably good enough for illustration and making the point.

I hope my reasoning on why I kept my math simple works for you, especially because there are people that when they start seeing complicated mathematics get turned off or scared.

<MOD HAT>

Myyra, let's please steer away from things that could sound like personal attacks. It may be more useful to provide your own math instead.

</MOD HAT>

Thank you Hateful Darkblack, I did not see the original post but as this is a text base medium I try hard to read thing in the lease offensive manner possible and exclude what I believe the tone is. Thank you again.

[solkan]

Your math isn't simple, it's wrong.

 

[Myyrä]

Your thoughts are very good, but the probabilities are wrong. Here's a link explaining how the probability of drawing exactly one high card to your hand from the deck is calculated. ...or you can send me a pm if you want me to calculate it for you.

 

Edited to not look like a personal attack.

I am well aware of binomial coefficient and the also the way it works, I chose to use a simple approximation and even stated that. What is did was a simple probability for the purpose of understanding, I even stated that this is with replacement to keep the math simple.

Lay out of the question: what is the probability of drawing a high card on any independent draw? that is simply the number of high cards/the cards in the deck. which in a full deck is .24.

Now that i know that, I can ask what is the very simple probability of drawing at least 1 high card. There are several ways of doing this, but I chose the simplest for easy of understand because I am posting it on the forum. Saying having at least 1 high card can be turned around as having only 1 high card for simplicity. which then means it is the number of time I need the draw of the high card times the probability of that draw times (1-the probability of that draw) times the rest of the draws to get a very simple number which though not a 100% accurate is probably good enough for illustration and making the point.

I hope my reasoning on why I kept my math simple works for you, especially because there are people that when they start seeing complicated mathematics get turned off or scared.

Maybe I shouldn't have posted a link to the binomial coefficient as you did try to make a simplifying assumption.

The real problem I had with your math was that it was wrong even if you assume the probability to draw high cards remains constant. The right formula to calculate the probability of getting exactly one high card when drawing six cards would be:

6 * p * (1-p)^5, which would be about 0.37 or 37% for p = 0.24

The probability to draw at least one high card would be:

1 - (1-p)^6, which would be about 0.81 or 81%.

[D_acolyte]

Good point there, shows I should not right things at 4 am and post them the next morning with out more strenuous proffing. Sorry about that.

[Myyrä]

In case you were wondering, the real probabilities without any simplifying assumptions when drawing from a full deck would be 37.7% for exactly one high card and 82.6% for at least one high card.

[D_acolyte]

Math aside the important thing is the thought process. This is why I like having data more then just calculating probabilities.

I actually do not really see a point in putting in the math into the main post because of all the discussion there is already from it.

[Myyrä]

I actually think that it is pretty much necessary to have at least basic understanding of probabilities involved to be able to make informed decisions. You don't really need to understand the math behind it all, but you should have some basic understanding how much having a , +1 stat or built in suits increases your probability of succeeding.

[D_acolyte]

I actually think that it is pretty much necessary to have at least basic understanding of probabilities involved to be able to make informed decisions. You don't really need to understand the math behind it all, but you should have some basic understanding how much having a , +1 stat or built in suits increases your probability of succeeding.

Unless you are implying that I do not, I do not really see the point to this statement. As for me, I have a working understanding even though it escapes me at 4 AM. I have a degree in statistics and math trough differential equations. Hence why i like data more then numeric probability calculation, been a long time from when I had to do it last.

As for others, I sort of assume that they have an understanding of that. Might not be good or might not be as obvious to them.

Edited December 6, 2015 by D_acolyte

[solkan]

The thing is, the reasoning behind "What are the odds I'm going to get X cards of one suit? (13/54)^X*(1-13/54)^Y" leads to telling someone "You just discarded the red joker.  Since there are fifty four cards in the deck, your odds of drawing the red joker on your next card is 1 in 54."  If you're going to make the assumptions necessary to make that reasonable statement, you really should warn people.

 

[Myyrä]

I actually think that it is pretty much necessary to have at least basic understanding of probabilities involved to be able to make informed decisions. You don't really need to understand the math behind it all, but you should have some basic understanding how much having a , +1 stat or built in suits increases your probability of succeeding.

Unless you are implying that I do not, I do not really see the point to this statement. As for me, I have a working understanding even though it escapes me at 4 AM. I have a degree in statistics and math trough differential equations. Hence why i like data more then numeric probability calculation, been a long time from when I had to do it last.

As for others, I sort of assume that they have an understanding of that. Might not be good or might not be as obvious to them.

I was referring to your earlier post where you stated that you don't see the point in including the math. I was trying to say that it is fine to drop the math but probabilities aren't irrelevant to decision making in Malifaux, and you weren't my primary audience as much as everyone else possibly reading this.

I know several people who don't understand the significance of probabilities for the decision making and/or aren't able to form even a rough estimate of a probability of succeeding in an action. (And I'm not trying to point any fingers here.)

[Hateful Darkblack]

Your thoughts are very good, but the probabilities are wrong. Here's a link explaining how the probability of drawing exactly one high card to your hand from the deck is calculated. ...or you can send me a pm if you want me to calculate it for you.

 

Edited to not look like a personal attack.

Thanks for your edit!

[D_acolyte]

The thing is, the reasoning behind "What are the odds I'm going to get X cards of one suit? (13/54)^X*(1-13/54)^Y" leads to telling someone "You just discarded the red joker.  Since there are fifty four cards in the deck, your odds of drawing the red joker on your next card is 1 in 54."  If you're going to make the assumptions necessary to make that reasonable statement, you really should warn people.

 

I did not touch on one card of one suit, the closes I came to that was the was listing a .04 for summoning a hanged. This number is simply 2/54 which is admittedly rounding to the hundredth, for this I figured that is an accurate degree of precision. Otherwise when I did have my math I mentioned that a high card was counted as an 11+ and did not go into the combination of card and suit other in detail other than a discussion point.

The point of the post was to try and generate a conversation or though around how resources are used in the game.

For the most part I find modeling draws in Malifaux to be futile event because it takes a lot of assumptions mostly around human behavior. I meal look at this:

Assumptions to account for humans:

1. High cards will be replaced with card of note starting at a 10 (change this number as you wish) and go through the red joker, alternatively you could also include the black joker because getting it out of the deck is very helpful.

2. There are 4 level of hand use: low, mid, high, all

   1. On turns of low hand use you will keep all cards of not and discard all others at the start of next turn.

   2. You will use half your cards of not and discard half of all the other cards at the start of next turn. Assume the other half of the cards not of note are used during the turn.

   3. High turns you will all but 1 card of note. No cards will be discarded at the start of next turn.

   4. All cards in your hands are used during the turn, next turns number of cards in the deck is the full 54 cards.

3. The soul stone pool of the player is 6 stones.

4. If there are not 3 cards of note and there is at least 4 stones in the soul stone pool assume a stone is spent and draw 2 more cards on any turn not marked as low.

5. This the number of cards in hand and discarded are not replaced into the deck, the starting number at the beginning of the turns will be different each turn.

6. Different factions, and possibly different masters, will have a different priority level to their hand according to turn.

7. Drawing card abilities of models that change the hand will not be looked at in this global level overview.

Now under these assumptions I can make either a mathematical model or generate a series of samples.

 

PS this is still going to wrong.

For the record I hate Wyrd UI on the forum sometimes, I had to change the font in the word doc to white and paste it in here because the change font color in the forum UI did not work with the formatting from word for some reason. I think it has to deal with all the bullet points.

Edited December 7, 2015 by D_acolyte
changing font color

[Stryder]

Nicodem has no way of lowering/lessening the opportunity cost of summoning other than soul stones.

Kirai may sacrifice a friendly spirit to gain a +2 to her casting skill. This will let her summon a hanged on a 11 of crows instead of a 13 of crow. That is a change from a .04 percent of the time to .07.

Nicodem can sacrifice corpses to increase his CA by +1 for each corpse at the start of his activation. Unlike Kirai, there is no limit and it lasts for his full activation. Therefore, Nicodem has a potentially unlimited CA and can summon a Hanged by discarding a Soulstone and flipping a Black Joker (so long as an entire army died around him recently...try against Hamelin)

[D_acolyte]

Nicodem has no way of lowering/lessening the opportunity cost of summoning other than soul stones.

Kirai may sacrifice a friendly spirit to gain a +2 to her casting skill. This will let her summon a hanged on a 11 of crows instead of a 13 of crow. That is a change from a .04 percent of the time to .07.

Nicodem can sacrifice corpses to increase his CA by +1 for each corpse at the start of his activation. Unlike Kirai, there is no limit and it lasts for his full activation. Therefore, Nicodem has a potentially unlimited CA and can summon a Hanged by discarding a Soulstone and flipping a Black Joker (so long as an entire army died around him recently...try against Hamelin)

Hmm, never knew that also never saw a person do it. I shall correct my main post when I am not busy. Thank you for the information.

[Ludvig]

Nice writeup! I think this will be helpful to a lot of players. I'm not touching the math discussion since I don't know much about probabilities, just that my master attacks have close to a 25% chance of getting the black joker  

I was one of the players you discussed "true cost" with here who didn't agree. You brought it up in a context of directly comparing SS cost and ability between models and made it sound like every henchman was actually balanced pointswise to a model matching it's true cost and not it's printed one. I argued that a henchman without stones should match a similarly priced enforcer in ability if you didn't use stones. In this context I agree with your construct true cost as a valuable instrument for choosing a list where not every henchman needs 5 soulstones to function efficiently and true cost as something to consider for getting the best result out of your soulstone users. Some SS hungry models just don't play well together and identifying which those are is necessary for getting the best result from those models.

 

[Gnomezilla]

Holy Calculus for the masses! but this is the kind of discussion I was hoping to find. With numbers. Thank you both very much for flagging the mistakes I could not spot.

[D_acolyte]

Nice writeup! I think this will be helpful to a lot of players. I'm not touching the math discussion since I don't know much about probabilities, just that my master attacks have close to a 25% chance of getting the black joker  

 

And that is Bayesian statistics lol.

I was one of the players you discussed "true cost" with here who didn't agree. You brought it up in a context of directly comparing SS cost and ability between models and made it sound like every henchman was actually balanced pointswise to a model matching it's true cost and not it's printed one. I argued that a henchman without stones should match a similarly priced enforcer in ability if you didn't use stones. In this context I agree with your construct true cost as a valuable instrument for choosing a list where not every henchman needs 5 soulstones to function efficiently and true cost as something to consider for getting the best result out of your soulstone users. Some SS hungry models just don't play well together and identifying which those are is necessary for getting the best result from those models.

 

I am not the best at getting my thoughts across often. The concept of true cost makes it hard for me to compare henchman to anything else because "well they are both x stones but I spend y stones on this guy to keep him there how much of a drain are the to models and which is more efficient in the end". Which then trickles into crew choices. I am crazy like that and can count the number of masters I play with no henchmen on one hand so I think they are worth it, it is just if I bring Nekima she need all the stones factored into how I plan my crew and if there is a lot of competition I instead bring a mature nephilim or Barbaros. If I do not know a henchmen I general put the stones they should use at 3 when I am planning things.

Holy Calculus for the masses! but this is the kind of discussion I was hoping to find. With numbers. Thank you both very much for flagging the mistakes I could not spot.

Calculus, lol. For a more exact model we have moved pass that and now are working on sample generation.

I am glad you to liked it. Next time I will do the draft of it during daylight hours.

Feel free to P.M. me if you want my take on anything else, the expansion of anything here or have an idea you want me to bounce around in my head till it makes sense.

 

[Myyrä]

Nice writeup! I think this will be helpful to a lot of players. I'm not touching the math discussion since I don't know much about probabilities, just that my master attacks have close to a 25% chance of getting the black joker  

 

And that is Bayesian statistics lol.

And here I thought it was called pessimism bias...

[D_acolyte]

And here I thought it was called pessimism bias...

ROFL! Normally I would agree but after playing a game and seeing the same sniper eat the black joker every turn I am willing to believe a lot.

Edited December 9, 2015 by D_acolyte

[JoeJones]

All I know is that if I have a plus flip to damage, I'm likely to flip a severe and a black joker. Correspondingly, if I'm on a double negative flip, I have a reasonable chance to flip a red joker, especially immediately after the above. 

In summary, bad things happen.

[Ludvig]

Nice writeup! I think this will be helpful to a lot of players. I'm not touching the math discussion since I don't know much about probabilities, just that my master attacks have close to a 25% chance of getting the black joker  

 

And that is Bayesian statistics lol.

And here I thought it was called pessimism bias...

I liked d_acolyte's term for it better   Truth be told I am a pessimist at heart but Sonnia is just a black joker magnet for me. I eould definitely pay ss for the upgrade Rasputina has. Other masters usually do ok in regards to the joker.