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Merir
05-21-2013, 01:30 PM
I'm sure we've all seen the Head Games card spoiled in today's The Guild Experience article. For those of you who don't remember the mechanic:

Secretly choose 5 or 10. Target opposing champion guesses which number you chose. If they guess incorrectly, deal damage to them equal to the number you chose. If they guess correctly, deal damage to you equal to the number you chose.

When I saw this I thought that it's a pretty cool idea for a card, adds a bit of randomness, risk vs. reward and all that, but something troubling occurred to me after a few moments. I would like to preface this by saying I'm not very good at math, and if this turns out to be a silly mistake on my part I'll be only happy to be proven wrong.

Now let's imagine a situation where you play the Head Games card, and your opponent defends based on a coin-toss. Heads, he picks 5, tails, 10. So he has a 50/50 chance of picking either one, but leaves no recognisable pattern for you to exploit.

What is your expected value in terms of damage dealt (positive) and damage taken (negative) when you secretly choose 5 against this opponent?

0,5 * 5 [you deal 5 damage 50% of the time when he chooses 10] + 0,5 * (-5) [take 5 damage when they choose 5] = 2,5 - 2,5 = 0

When choosing 5 against an opponent who is making their choice based on a coin-toss, you can expect to deal exactly zero damage over the long run - or, the damage you deal will be exactly offset by the damage you take. The same is true when choosing 10 against this opponent:

0,5 * 10 [deal 10 damage when he picks 5] + 0,5 * (-10) [take 10 damage when he picks 10] = 5 - 5 = 0

What ever you choose, as long as your opponent defends 50 / 50 without a repeating pattern your expected value for playing Head Games is exactly zero. This leads me to the conclusion that Head Games is a waste of resources to play, and a waste of space in your deck. Even worse, having a Head Games in your deck actually benefits your opponent: every time you play it, you use four resources for zero gain, and having this card in your deck makes it less likely for you to draw cards that actually benefit you.

As I said before, I'm not good at maths and might have made a mistake in thinking about this scenario. Please prove me wrong, since the concept is interesting and it would be nice to be able to play this card without it hurting you.

Chance
05-21-2013, 01:34 PM
Yeah simple logic dictates that this card would never see competitive play but coupled with other cards in PvE it could be interesting. 4 drop for a novel idea is just to much to afford in any PvP format.

Karnegal
05-21-2013, 01:36 PM
This isn't a card that's designed for tournament players who are trying to grind out every advantage they can. It's a card designed to appeal to casual players who enjoy the guessing element and a chance to "go big." Not every card is designed to be a tournament staple.

Xenavire
05-21-2013, 01:36 PM
Except when they call wrong and you kill them in a few hits.

It is a gamble, and a big one, but there will be gimmicks (and there is already a combo) built entirely around this deck.

It isn't inherently bad - it just requires set-ups or support.

Turtlewing
05-21-2013, 01:36 PM
Well dealing damage to yourself isn't the same thing as dealing negative damage to an opponent.

As such the expected value is not as easy to calculate as you make it out to be.

But in general yes, this card looks like a bad play unless you have some synergy with it and some unspecified other card(s).

houjix
05-21-2013, 01:38 PM
The only way it become remotely constructed playable is if you have a reliable way to mitigate the damage to yourself with some sort of redirect back to them. Then it no longer matters what they choose as it become always 10 damage to them.

Digital_Aether
05-21-2013, 01:39 PM
You effectively proved that with a card that's got 50/50 odds, it has equal odds of being both beneficial or harmful. That's what's evening out. That's the whole point of the card though. It's a card for gamblers, because you CAN do a bunch of damage in a single instance. It doesn't do you good long run, unless you're "lucky". Now, strategically, a lot of players might always say 10 on defense, to prevent a worst case scenario. That means a guess of 5 I would assume would end up empirically testing beneficial. Who knows what the AI would do though.

Xenavire
05-21-2013, 01:42 PM
The only way it become remotely constructed playable is if you have a reliable way to mitigate the damage to yourself with some sort of redirect back to them. Then it no longer matters what they choose as it become always 10 damage to them.

http://hextcg.com/wp-content/uploads/2013/01/BlindingLight_Gear_Web.jpg
No redirect, but you are safe - making it a much smaller gamble.

Turtlewing
05-21-2013, 01:42 PM
The only way it become remotely constructed playable is if you have a reliable way to mitigate the damage to yourself with some sort of redirect back to them. Then it no longer matters what they choose as it become always 10 damage to them.

Or if you have a good ability that triggers when you take damage (like when you take damage draw one card for evert damage taken, or whenever you take damage put an X/X troop into play where x is the damage taken).

There are lots of ways it could be useful, but they all rely on something other than Head Games, setting up the beneficial environment.

Karnegal
05-21-2013, 01:42 PM
The interesting mental game here is what happens when you're at 10 or less life. Then you really have to try to figure out what sort of play your opponent is making.

houjix
05-21-2013, 01:43 PM
http://hextcg.com/wp-content/uploads/2013/01/BlindingLight_Gear_Web.jpg

I was talking more about redirect than prevention, because paying 7 to do maybe 0,5, or 10 damage seems really bad.

dogmod
05-21-2013, 01:46 PM
If I am guaranteed to lose the next turn than this card doesn't have any negative value just positive potential... Limited situation of course

FireSummoner
05-21-2013, 01:46 PM
As I see this card as a defending player, you should always guess 10. If the active player chooses 5 that is 4 resource cost for 5 damage. If the active player choose 10, you did 10 damage at no cost plus they lost a card. By always choosing 10 you guarantee to cut the damage in half. As a result, I don't think you should play this card unless it combos with other effects.

MatWith1T
05-21-2013, 01:46 PM
Game theory principle of Loss Aversion predicts that most players would secretly choose five, and if their opponent plays it, they would guess 10.

This card would be a lot more interesting if they changed the damages so the expected value didn't work out to zero.

Xenavire
05-21-2013, 01:46 PM
It depends on the situation, and if your deck is already based on prevention and dealing direct damage, you wouldn't be that bad off. It isn't a perfect combo, sure, but it could be playable in the right setting. PROBABLY in limited/extended formats.

Vengus
05-21-2013, 01:48 PM
As I see this card as a defending player, you should always guess 10. If the active player chooses 5 that is 4 resource cost for 5 damage. If the active player choose 10, you did 10 damage at no cost plus they lost a card. By always choosing 10 you guarantee to cut the damage in half. As a result, I don't think you should play this card unless it combos with other effects.
Exactly. The only time you should think about it is when you have 5 or less lifepoints left. Otherwise, choose 10.

FireSummoner
05-21-2013, 01:50 PM
It might be worth playing if there were different effects on the card but as a defending player why would you allow your opponent to hit you for 10? It is the one playing the card that is really taking the gamble. Spending 4 resources hoping to do damage to your opponent seems like a last ditch move to me.

Boojum
05-21-2013, 01:55 PM
Yeah, with default loss-avoidance choices (player playing the card chooses 5, opponent chooses 10), it's a 4-cost, 5 direct damage that only targets players. That's (http://gatherer.wizards.com/Pages/Card/Details.aspx?multiverseid=30572) nothing (http://gatherer.wizards.com/Pages/Card/Details.aspx?multiverseid=179440) special (http://gatherer.wizards.com/Pages/Card/Details.aspx?multiverseid=139477). Obviously there's a chance for more if the opponent cooperates, but I don't see it being especially viable in constructed decks.

Xenavire
05-21-2013, 01:56 PM
Well, there is the chance they pull a fast one on you and call 5, causing you to get blasted in the face - you could counter with 10 and blast them.

It is called mind games after all.

FireSummoner
05-21-2013, 01:59 PM
At that point I think the player deserves to take 10 damage in the face.

Turtlewing
05-21-2013, 01:59 PM
It might be worth playing if there were different effects on the card but as a defending player why would you allow your opponent to hit you for 10?

Hypothetical:

"I have 9 life. since we both know 10 damage will kill me he expects me to guess 10 and will choose 5 in order to get me down to 4. I choose 5".

Alternatively:
"she has 11 life and I have 9, but I also has a 6/6 she can't block. If I choose 5 and she falls for it I win the game, if not I still survive to try finishing her next turn. So I choose 5". Assuming she's as smart as I am she'll guess 5 because she knows 10 is too big a risk for me but 5 has the potential to end her just as well as 10.

Xenavire
05-21-2013, 02:00 PM
Well, the point of the card is either take the safe bet, or try to be a smartass - it will be interesting to see if someone ever tops a tournament using these (at least a draft).

FireSummoner
05-21-2013, 02:23 PM
Hypothetical:

"I have 9 life. since we both know 10 damage will kill me he expects me to guess 10 and will choose 5 in order to get me down to 4. I choose 5".

Alternatively:
"she has 11 life and I have 9, but I also has a 6/6 she can't block. If I choose 5 and she falls for it I win the game, if not I still survive to try finishing her next turn. So I choose 5". Assuming she's as smart as I am she'll guess 5 because she knows 10 is too big a risk for me but 5 has the potential to end her just as well as 10.

Option 1 Again I don't see how 4 resources for 5 damage is a good trade off. Also, why risk losing the game? In the end it is the one using the card that is risking the resources to do damage. If the defending player does damage why have the card in your deck.

Option 2 you already have the advantage with a 6/6 that she can't block. That is the card that is winning the game for you not the Head Games. If anything she should hope you are greedy or arrogant and hope to kill you with the card backfiring, lol.

Again just my thoughts but I think there are better things to spend 4 resources on, unless it combos with something.

But it might be worth trying out if you have the second equipment on it. It allows you to reveal it and discard your cards from your opening hand to draw the same amount of cards. That I think is a better "mind game"

Edit: It took me awhile to find the cards but Rage Fire and Burn do 2 damage for 2 plus escalation and 2 damage for 1.

houjix
05-21-2013, 02:27 PM
But it might be worth trying out if you have the second equipment on it. It allows you to reveal it and discard your cards from your opening hand to draw the same amount of cards. That I think is a better "mind game"
Unfortunately that's only PVE and who knows how they will program AI to respond to a card like Head Games.

Clawdius
05-21-2013, 02:32 PM
http://hextcg.com/wp-content/uploads/2013/05/HeadGames_Gear_Web.png It is a nice card, if only when combined with Tendrils of Memory. The specifics of actually using the thing on the other hand, is tricky. Would an unknown human opponent tend to pick 10, just because if they were right it would do 10 damage to their enemy? Or might they more often choose five, because if they have to take damage it may as well be as little as possible? It's tough to say, but as I say more for the chance of having it in your starting hand.

FireSummoner
05-21-2013, 02:34 PM
Side question: what rarity is it?

houjix
05-21-2013, 02:35 PM
white = common
green = uncommon
blue = rare
red = legendary
orange = promo

FireSummoner
05-21-2013, 02:39 PM
white = common
green = uncommon
blue = rare
red = legendary
orange = promo

Thanks stupid question on my part matches the color on the Theoretical Wraps which is rare.

Turtlewing
05-21-2013, 02:40 PM
Again just my thoughts but I think there are better things to spend 4 resources on, unless it combos with something.


Well duh. No one is actually saying the card is good absent some synergy with another card or equipment. My post was just illustrating two cases where the defender would be better off choosing 5, than 10. I completely ignored the strategic question of "why include this card" as the point was to address the tactical question of "my opponent played this card how do I best respond".

The card is designed to be a gamble, and to make your opponent try to "out-guess you". Those sort of cards are rarely competitive, but there are people who play casually that like that style. And again, it just takes one other card with a crazy synergy to make this card viable. Or failing that a limited format like a draft, where you might not have the option of including a better burn card in it's place.

Jugan
05-21-2013, 02:41 PM
I'll make it simple for you guys.

Guess 10 every time. It becomes a bad lava axe.

Also, the black go stones in the picture should be on intersections, not between them.

FireSummoner
05-21-2013, 02:46 PM
I'll make it simple for you guys.

Guess 10 every time. It becomes a bad lava axe.



I pretty much already stated that.

Jugan
05-21-2013, 02:46 PM
I pretty much already stated that.

I guess I owe you a cookie.

FireSummoner
05-21-2013, 02:47 PM
I guess I owe you a cookie.

I am going to hold you to that one.

MatWith1T
05-21-2013, 02:51 PM
Also, the black go stones in the picture should be on intersections, not between them.

Ha. Good catch.

Qorsair
05-21-2013, 02:51 PM
Also, the black go stones in the picture should be on intersections, not between them.

Well, well... look who never learned to play Go using Void Society rules.

Clawdius
05-21-2013, 03:29 PM
Also, the black go stones in the picture should be on intersections, not between them.

Ahhh, but the white are only mostly on the intersections, so it could be another game. Or it's a mix to baffle or annoy people, hence highlighting the name head games.

Erebus
05-21-2013, 03:38 PM
I wonder how this card would work out if it was a choice between Drawing 3 (or 5) cards and dealing 10 damage.

Showsni
05-21-2013, 03:49 PM
I'll make it simple for you guys.

Guess 10 every time. It becomes a bad lava axe.

Also, the black go stones in the picture should be on intersections, not between them.

If defensively everyone guesses 10 every time, it becomes a better Lava Axe; Lava Axe costs 5.

Xenavire
05-21-2013, 04:13 PM
If defensively everyone guesses 10 every time, it becomes a better Lava Axe; Lava Axe costs 5.

Except when the opponent guesses you out and you take 5 in the face.

I am loving this card, you just can't win an argument about it.

Willzyx
05-21-2013, 04:32 PM
The equipment, Tendrils of Memory (click Head Games to see it) also makes Head Games into a free mulligan if you draw it in your opening hand. That makes it a little more worth playing.

Clawdius
05-21-2013, 04:41 PM
The equipment, Tendrils of Memory (click Head Games to see it) also makes Head Games into a free mulligan if you draw it in your opening hand. That makes it a little more worth playing.

Given that if you are say, playing a multi-mana flavor hand and draw 1 of each but not great cards, you can trade in for 5 more including potentially land and at the very least close the distance from the top of your deck to the next mana... Combine with Void Marauder and Head Games can actually be a very effective start to a game, or Monkey of the Nine Tails in PVE will allow you to potentially discard enough actions in the first turn to play Void Marauder on your very first turn for free, an extraordinary defensive start and a card with devastating potential.

Tyrfang
05-21-2013, 04:50 PM
At the expense of having an equipment slot dedicated to it...and running the card...

Willzyx
05-21-2013, 05:53 PM
At the expense of having an equipment slot dedicated to it...

I was under the impression that equipment was on a per-card basis. Is this not true?

Gwaer
05-21-2013, 06:15 PM
Nope, your "character" has equipment slots like a normal mmo. The equipment you wear modifies whatever cards. You have to be kind of picky with your equipment.

Clawdius
05-21-2013, 06:44 PM
At the expense of having an equipment slot dedicated to it...and running the card...

Yeah, you wouldn't be able to fully gear to take advantage of the Void Marauder in my example, but it would be worthwhile IMO with the right deck setup. It certainly might not be the best move in PVP since you won't have the equipment, but in PVE I imagine it will be useful.

Tyrfang
05-21-2013, 06:48 PM
PvP doesn't utilize equipment bonuses except for the anything-goes variant.

KingBlackstone
05-21-2013, 07:57 PM
Clever idea, terrible card.
It'd be more interesting/viable if it used two coin tosses.
75% of the time you'd come out with a net gain (only 5 damage) and 25% of the time you'd come out with the maximum loss (10 damage). The numbers are pretty much the same but it's a lot more exciting.

Grissnap
05-21-2013, 08:23 PM
I have a feeling there are going to be a few cards like this in the first set that are basically, "Look at this novel gameplay mechanic. This card isn't so great, but the effect is neat, right!?"

Anyway, the General's Tent uses the same mechanic and is much better.

KingBlackstone
05-21-2013, 08:24 PM
I have a feeling there are going to be a few cards like this in the first set that are basically, "Look at this novel gameplay mechanic. This card isn't so great, but the effect is neat, right!?"

Anyway, the General's Tent uses the same mechanic and is much better.
That's true. I can't say I terribly mind it, either, with the lack of vanilla cards. A few bad effects is better than no effects. :)

yovalord
05-21-2013, 09:14 PM
All im going to say is, this card would make for some pretty intense play when both players are at 5HP or lower.

Tyrfang
05-21-2013, 09:19 PM
All im going to say is, this card would make for some pretty intense play when both players are at 5HP or lower.

So would a Fireball type card.

Kalius
05-21-2013, 09:38 PM
I'm just waiting for them to reveal some card that gives you something when you take damage. Hell, I'd love this card for MtG because I have a deck where, when I take damage, I get that many creatures and gain that much life. so for that deck, it would be "do I hit the opponent for 10, or gain 10 creatures?"

Even if there isn't something in the first set that gives you an effect for taking damage, there will probably be something eventually. Just follow the same rule that MtG follows- you really can't rule out most cards as being amazing or crap on a preview until you know the full set. Then you can get a better idea of what each card can do in every format.

Grissnap
05-21-2013, 09:42 PM
Point taken. I do agree though that it would be a really fun card to play casually (headgames is a great card name).

funktion
05-21-2013, 10:27 PM
Over the course of a large magnitude of games it's a net wash, but over the course of a small sample size it has very high variance. In general, extremely high variance cards are better in the hands of bad players. It gives them one more out in a game they are likely already losing, whereas a stronger player can win solely through their mechanics more reliably.

Grissnap
05-21-2013, 10:58 PM
Of course the mind always wanders to constructed first, but would you want to pick this in draft, or put it in your deck if you picked it?

Considering this card if both champions are at 20/20, I think the smart player playing this card would choose 5 damage, and the smart defensive player would choose 10. Why? Because the consequences of guessing wrong for both players is massive damage, especially if the turn player has board presence. So lets say now that player using Head Games is at 10 life, and his oppenent is at 10 life. At this stage, the defending player has much more incentive to choose 10, because if he guesses wrong with 5 he loses, and if he guesses right with 10, he wins.

When the defending player is at 5 health is when things become a head game, and this card would be exceptionally powerful as a finisher if the opponent playing it has way more health than his opponent - but in those instances Head Games is more of a win-more card.

So with that analysis, and excluding the risk taking / randomness factor out of it, the card ends up being a 4 cost, 5 direct damage to hero card. As a wow player, that would be fairly underpowered rare - but Hex is more like magic, and I'm not a magic player, so I don't know how powerful something like this would be. Still, I think it would be an okay pick if you were already going Ruby, especially if you also went Blood or Diamond with lifedrain/healing. First pick first pack? Almost certainly not.

Kalius
05-21-2013, 11:38 PM
So with that analysis, and excluding the risk taking / randomness factor out of it, the card ends up being a 4 cost, 5 direct damage to hero card. As a wow player, that would be fairly underpowered rare - but Hex is more like magic, and I'm not a magic player, so I don't know how powerful something like this would be.

As a magic player, 4 mana for 5 damage is your average burn spell( 1 mana for 2 damage, 2 mana for 3, etc...). Of course, your average MtG player also wouldn't gamble on it without a way to negate the penalty. On the other hand in a draft, a magic player might grab the card just to remove the risk involved of someone actually playing it against them, because while it is a gamble, it's a potentially game ending gamble.

SacrificialToast
05-21-2013, 11:41 PM
So with that analysis, and excluding the risk taking / randomness factor out of it, the card ends up being a 4 cost, 5 direct damage to hero card. As a wow player, that would be fairly underpowered rare - but Hex is more like magic, and I'm not a magic player, so I don't know how powerful something like this would be. Still, I think it would be an okay pick if you were already going Ruby, especially if you also went Blood or Diamond with lifedrain/healing. First pick first pack? Almost certainly not.

Well, Lava Axe (http://gatherer.wizards.com/pages/card/Details.aspx?multiverseid=205117) is generally regarded as a pretty terrible card, and this is only 1 cheaper than that. Burns are always better when you can kill creatures with them, unless you can actually finish off your opponent. Really, just including burns that can target either is better.

Deathfog
05-21-2013, 11:50 PM
Just gotta hedge your bets with both cards that have an effect on damage dealt to you and damage dealt to your opponent. By default its a poor choice, but build part of a deck around it and other effects and it could be devastating.

Merir
05-22-2013, 12:11 AM
I would like to make a point about the game theory perspective some have mentioned - defender always picking 10, attacker always choosing 5. While this is true from a loss aversion point of view, if your opponent knows you play either choice 100% of the time, you're always going to lose. Your only option as an attacker is to mix up your 5s and 10s and force the defender to guess (otherwise you will take the damage every time). There is only one 5:10 ratio that cannot be exploited by the defender - 66/33

This time imagine a scenario from the defenders point of view, playing versus an attacker that chooses 5 66% of the time, and 10 33% of the time:

[value of defending with 5] = 0,66 * 5 + 0,33 * (-10) = 3,33 - 3,33 = 0

[value of defending with 10] = 0,33 * 10 + 0,66 * (-5) = 3,33 - 3,33 = 0

Any other ratio of 5:10 by the attacker can be exploited for a net gain by choosing either 5 or 10 every time. If attacker picks 5 more than 66%, defender wins by always picking 5. If attacker picks 5 less than 66%, defender wins by always picking 10. The more the attacker deviates from the optimal 66/33 split, the higher the net gain for the defender.

The best way to play this card, it seems to me, is to use a 66/33 split when attacking, and 50/50 while defending, resulting in a wash. The exceptions would be times when 'your health < 5 =< 10', when the loss aversion principle comes into play.

LargoLaGrande
05-22-2013, 12:23 AM
The problem with looking at it from that perspective is that the outcomes aren't as simple as just 5 or 10 damage to you or your opponent because head games costs a card and 4 mana to cast. With a 66/33 split you're trading a card and 4 mana for an expected value of 0 damage.

Merir
05-22-2013, 12:26 AM
The problem with looking at it from that perspective is that the outcomes aren't as simple as just 5 or 10 damage to you or your opponent because head games costs a card and 4 mana to cast. With a 66/33 split you're trading a card and 4 mana for an expected value of 0 damage.


Correct, I already made this point in the original post. 66/33 still is the 'best' way to play this card, and even then it's a loss for the attacker.

Grissnap
05-22-2013, 12:49 AM
So at best, in certain situations, it can deal 5 for 4, otherwise statistically its a 4 cost deal 0 card.

That is pretty bad... but its like the lottery. Put it in your deck and you might just get lucky! :p
But the way it plays still seems more fun of a mechanic than if it were a paper card with rules like:
Choose one at random, "deal 10 damage to opposing champion", "deal 5 damage to opposing champion","deal 5 damage to your champion","deal 10 damage to your champion".

Note to self... do not draft this card :p.

Rapkannibale
05-22-2013, 01:25 AM
IMO like others have said this card gets much more interesting web the opponent is at 10 or 5 life.

If he/she is at 10 they could guess that you chose 10 to kill them. However maybe you chose 5 because you assumed the opponent would think you would chose 10 to kill them. But maybe you are double or triple bluffingn. :)

Of course if the opponent choses completely randomly then yes it is still just a 50/50 chance. This card is not for everyone but there are players that like these kind of elements and they should get some cards too. :)

Jacklau89
05-22-2013, 03:22 AM
This card is probably interesting, just not enough. Warning for boring economic game theory below.

---------------------------

To put it simply, the card yields strictly negative payoff for anyone who plays this card (player 1), conditional on the opponent (player 2, or P2) using the idea of mixed strategy nash equilibrium, i.e. putting 1/2 probability on playing 10 while 1/2 on the act 5. Hence P1's expected net payoff is negative since you need to take into account the resource cost of playing. (EDIT: my previous claim on 33/66 probability is incorrect as this is a sequential game, not normal form game. If P1 expect P2 to put anything but 1/2 on his strategies, P1 can always exploits this by choosing the strategy that P2 plays less often. Then there will be a net gain for P1.)

But this question becomes more interesting when we get to the point when both players have less than 10 healths: this is the situation where choosing 10 yields strictly higher payoff then 5, hence the act 10 is more salient. Since it is impossible for P2 to commit using the mixed-strategy ex-ante (such commitment is time inconsistent as the rewards is now 10+εi, where ε is some intangible value that could not be measured quantitatively for all player i), it now comes down to a one-shot non asymmetric game with uncertain payoff.

Behaviour game theory suggest an Lk model of strategic thinking, where player with Lk always best response to other Lk-1's strategy. In this case, we denote L0 as naive individuals who will always choose 10. Then a L1 player 1 will best respond to it by choosing 5, while L1 P2 will continue to choose 10 (since L0 P1 chooses 10). Further a L2 P1 will choose 5 while L2 P2 chooses 5, etc. Hence depending on your information about the opponent's type (L0, L1, L2,...), this initial response one-shot zero sum game has a different prediction relative to the mixed strategy nash equilibrium, where a player could strictly gain from outguessing others by utilizing its (potentially larger) information set.

---------------------------
In short, it seems to be a fun and at least a marginally useful card :p

Merir
05-22-2013, 04:55 AM
I have no idea what the poster above me is saying, but I was going to post about scenarios where one or both players have between 6 and 10 health so here I go. Hopefully I'm not just repeating Jacklau89, and if I'm talking out of my ass I'm sure he or somebody else can set me straight.

As a basis for these scenarios, I'm assuming the following to be true: If you risk a loss without an equal chance of winning, when you have the option to always stay in the game, that play has a value of negative infinity. If this assumption by me is incorrect, the following will likely not hold.

All scenarios have you as the attacking player.

Scenario #1: Your hp is between 6 and 10, opponents hp is higher than 10

In this case you can never, ever choose 10, since you'd risk losing instantly but you can't win since 10 damage won't kill your opponent. You will always have to play 5, and your opponent, knowing this, can always play 5 against you. In theory he has to play 10 some of the time to force you into this strategy, but in practice it should be unnecessary - the mere threat of him possibly playing 10 will force you to always play 5.

Result: you take five damage.

Scenario #2: Your hp is higher than 10, your opponent's hp is between 6 and 10

Your opponent will always play 10 because otherwise he's risking a loss without a chance to win. You can play 5 every time (again, in theory you'd need to balance your range by playing 10 occasionally, but in practice I'm not sure if that's necessary).

Result: you deal five damage.

Scenario #3: Both your and your opponent's hp is between 6 and 10

If you play 10, the game is over - one of you dies.

If the opponent is using a 50/50 split, the result will be a wash (or a net loss of four resources for you). Your choice doesn't matter in this scenario, since the end result is always a cointoss, though you might want to stick to the 66/33 split to avoid being taken advantage of.

If the opponent deviates from the 50/50 split, you can exploit that freely. If he picks 10 over half of the time, you benefit by always going 5. If the chooses 5 more than 50% of the time, you'll kill him outright more often than not by always picking 10.

Conclusion: Scenario #2 seems to be the only time when playing a naked Head Games is clearly beneficial.

Jacklau89
05-22-2013, 05:48 AM
Merir: I know its all sorts of boring economic theory but hea, at least its not a bad model for behaviour. And of course I forgot to mention the analysis only applies when both you & your opponent have 6~10 healths, otherwise it just reverts to mixed strategy case :)

In your Scenario #1, if you expect your opponent to understand the logic you posted and always play 5, then your dominant strategy would be playing 10 instead (EDIT: the threat of your opponent playing 10 is not "credible", as there are no ways for your opponent to commit playing 10 under ALL conditions). The revert is also true in Secenario 2. This illustrates its impossible to have a case where playing a single strategy is "always the best".(in short no pure strategy nash equilibrium)

Regarding whether to use 50/50 or 33/66 split, it seems to depends on how you treat the payoff of that card. Let me think on that a bit. (EDIT: I think the optimal mixed strategy should be 50/50 for both players.)

Qorsair
05-22-2013, 06:10 AM
I'm glad we finally got someone who can get into some of the game theory behind the card.
And based on everyone else's responses, this card is going to be a ton of fun in non-tournament play.

Jacklau89
05-22-2013, 06:25 AM
I'm glad we finally got someone who can get into some of the game theory behind the card.
And based on everyone else's responses, this card is going to be a ton of fun in non-tournament play.

Still game theory did not told us whether spending 4 resource to deal 5 damage is a good choice or not (sounds a little bit meh if you ask me...though I am no expert in mtg). If it turns out to be a sucker for mere 5 damage, this card gets EVEN MORE interesting now: the probability distribution (prior) of a player choosing 10 is going to be biased (towards 10), conditional on you put this into your deck.

Alternatively speaking, unless you plan to go for 10 most of the time, you would not have choosen this card in the first place. Hence, if both players taken this into account, the choice of 5/10 is going to be even more uncertain than one might have thought.

Merir
05-22-2013, 06:28 AM
Merir: I know its all sorts of boring economic theory but hea, at least its not a bad model for behaviour.

It sure isn't, I just wish I understood it properly :)

I'm not sure if I understand what you mean by the threat of your opponent playing 10 is not "credible", as there are no ways for your opponent to commit playing 10 under ALL conditions. Are you just saying that there might be situations where the opponent is unwilling to play a 10, such as him being at 11 hp and not wanting to risk dropping down to 1 and possibly losing next turn? If so, that's a valid point. I didn't take it into consideration.

You sound like you know what you're talking about, so good thing you chimed in. Obviously the play in these situations isn't as cut-and-dried as I made it out to be.

ramseytheory
05-22-2013, 06:51 AM
OK, so Jacklau89 is one to watch for in tournaments. ;)

The combo with Blinding Light seems really bad to me. Blinding Light isn't free - it's a 3 mana card in its own right. So at the 50/50 equilibrium, which is the best-case scenario, you're spending an average of 5.5 mana and 1.5 cards to do a whopping 3.75 damage. And you need the full 7 mana available for the strategy to even be feasible. And the extra card you might be spending could be used to mitigate damage elsewhere. And most other direct-damage cards would, on top of giving you better efficiency and probably better average damage, let you target troops instead for extra flexibility. It's not worth it.

I hope a viable combo does turn up, though, ideally involving an instant that can be hidden until it's used. Someone plays Head Games on you - do they have the relevant card in their hand? Do they even have it in their deck, or are they just using the threat of it to make you guess low so they can hit you for 10? How valuable is the effect going to be to them? Everything would get a lot more interesting.

Jacklau89
05-22-2013, 06:55 AM
Merir: I cannot say for sure those theories (or even myself) are correct 100% of the time. After all, theory can be refuted at any time:)

When we refer some strategy (or threat) as "credible", in economics, we usually means it is ALWAYS in a player's best interest to do so. Consider the case where you play the "Head Games" card against your friend in the same room, and he told you "Oh don't you dare to choose 10, since I am always going to choose 10 whatever you do".

This "threat" is clearly not credible, as his statement of " ...always going to choose 10.." is not true. If he sneak peak at your monitor and find out you choose 5, do you think he will still commit himself to choose 10?

Nevertheless, if it is the case where a player plays chicken (hence play 10 whatever you do) because he only got 11 health, and 10 points of damage means a sure defeat for him, you are introducing the concept of "AMBIGUITY AVERSION" (belongs to the field of Subject Expected Utility), not risk aversion as the odds are unknown. And I hate discussing SEU because it sucks so bad:)

Jacklau89
05-22-2013, 07:05 AM
I hope a viable combo does turn up, though, ideally involving an instant that can be hidden until it's used. Someone plays Head Games on you - do they have the relevant card in their hand? Do they even have it in their deck, or are they just using the threat of it to make you guess low so they can hit you for 10? How valuable is the effect going to be to them? Everything would get a lot more interesting.

Yap I do wish there are better combo to enhance this card as it seems to be the essence of why digital TCG is better than physical. It would be a shame if it goes to waste because of imbalance.:(

Merir
05-22-2013, 07:12 AM
Merir: I cannot say for sure those theories (or even myself) are correct 100% of the time. After all, theory can be refuted at any time:)

When we refer some strategy (or threat) as "credible", in economics, we usually means it is ALWAYS in a player's best interest to do so. Consider the case where you play the "Head Games" card against your friend in the same room, and he told you "Oh don't you dare to choose 10, since I am always going to choose 10 whatever you do".

This "threat" is clearly not credible, as his statement of " ...always going to choose 10.." is not true. If he sneak peak at your monitor and find out you choose 5, do you think he will still commit himself to choose 10?

Nevertheless, if it is the case where a player plays chicken (hence play 10 whatever you do) because he only got 11 health, and 10 points of damage means a sure defeat for him, you are introducing the concept of "AMBIGUITY AVERSION" (belongs to the field of Subject Expected Utility), not risk aversion as the odds are unknown. And I hate discussing SEU because it sucks so bad:)

All right, so my logic fails because situations exist where it's not always in the opponent's best interest do choose one option 100% of the time. We don't have perfect information about the opponents situation nor his thought process. This creates ambiguity about our own best play, so there cannot be a situation where choosing one or the other all the time is always the best play. I guess I get it. Makes sense.

TheWrathofShane
05-26-2013, 03:55 PM
I don't like how your tapping out @ sorcery speed and spending a card to theoretically have a 50/50 chance of hurting yourself...

Now if the risk was this card did nothing, then it would be more interesting. The concept is neat, but unless involving some kind of efficient synergy where you don't mind blasting yourself, it wont see much casual play.

opaopa13
06-12-2013, 09:03 AM
e: Oh, look at that, the thread is more than one page long. This is in response to just the first page. Whoops.

Your math is fine. You can actually take it one step further by looking at mixed strategies.

Basically, while there doesn't exist any pure strategy Nash equilibrium (that is, a set of choices such that no player can benefit themselves by changing their choice) and neither player has a dominant strategy (that is, a choice that is superior regardless of what their opponent chooses), there does exist a mixed strategy Nash equilibrium: a set of probabilities such that neither player can benefit by deviating from it.

In this case, if you assume that "opponent takes X damage" is exactly as good as "I take X damage" is bad (which certainly isn't always true, but it's a reasonable place to start), then you can calculate that:
* If the caster chooses "5" two-thirds of the time, the target will be indifferent between choosing "5" or "10"
* If the target chooses "5" one-half of the time, the caster will be indifferent beween choosing "5" or "10"

At that point, neither the caster nor the target can improve their expected value by deviating from their strategy.

Imagine you're a caster against a target who chooses "5" or "10" randomly. It's pretty obvious that, no matter how you make your decision, choosing "5" will give you a 50/50 chance of taking or dealing 5 damage, while choosing "10" will give you a 50/50 chance of taking or dealing 10. Either way, your expected value is 0. There's no strategy which will let you get away from that.

Likewise, if you're a target against a caster who chooses "5" 2/3rds of the time, then you can choose "5" and take 5 damage twice as often as your opponent takes 10 damage (which is "fair"), or you can choose "10" and take 10 damage half as often as your opponent takes 5 (also "fair"). Again, no strategy will let you get away from that.

That said, this analysis only holds up against two robots playing Mind Games against each other an infinite number of times in a void. Other cards, remaining life and the potential to analyze your opponent's playstyle (a risk-adverse player might always choose "10", preferring a guaranteed "small" hit to a potential "big" one) completely ruins the "payoff is simply damage dealt" assumption. Even so, the caster has to pay a card and 4 resources just to play a "fair" game. Mind Games NEEDS support just to be better for the caster than the target, which is a shame. And what if your opponent has a Blinding Light of his own?

But forget everything else I've written: if the target decides to simply always play "10" no matter what, then Mind Games becomes 4 resources to deal 5 damage to target champion, making it a slightly more efficient but way less flexible Burn to the Ground. For a card like this to be really be exciting, the punishment for being predictable has to be higher.