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The Mathematics Behind Throws

Discussion in 'Wolf' started by Zass, Feb 5, 2013.

By Zass on Feb 5, 2013 at 9:41 PM
  1. Zass

    Zass Well-Known Member

    Have you ever wondered what the best throw to use is? Should you do your most damaging throw? Should you mix up your throws? Should you do the weakest throw because no one breaks it?

    Fighting games are based on two players making decisions simultaneously. Different choices give different payoffs.

    Throws are an especially interesting situation in VF5:FS, because the attacker and the defender have to simultaneously choose from three options. The defender is trying to predict what the attacker will do, while the attacker is trying to do the most damage possible while not being predicted by the defender. It almost seems like a game of cats going in circles -- as soon as you develop a pattern, your opponent can develop a counter pattern to exploit your pattern. Is there any way to win the mind games?

    Nash equilibrium is a strategy that cannot be exploited. In other words, you could tell your opponent what your strategy is, and he could do nothing to change your payoff. The Nash equilibrium strategy for the attacker means that you are playing optimally to maximize your damage, no matter what your opponent does. For the defender, it means your are playing optimally to minimize your average damage.

    Either player can deviate from this strategy, but doing so exposes that player to being out-predicted. So you may find that your opponent is weak, and that you can always do your most damaging throw. This increases your damage, but your opponent may learn from this and always escape your throw. Then you modify your strategy, and so the game of cats begins again... Nash equilibrium is an escape from the mind games.

    In this video, I outline the concept of Nash equilibrium, and go through a case study of deriving the optimal probabilities of Wolf's standing neck clutch throw stance. This video is lengthy and mathematical in nature:

    For those of you that want to skip the math, this video shows the optimal throw percentages for each of Wolf's setups:

    Some of the conclusions presented may be unintuitive.


Discussion in 'Wolf' started by Zass, Feb 5, 2013.

    1. Genzen
      I watched both videos, and they're done quite well, since I, a relative idiot in the realm of mathematics, was able to follow what you were saying. However, I was hoping to use your video to allow me to apply this to my characters' throws, but at the point where you 'solve the equation', you don't tell morons like me how - you just sort of say 'go and do it'.

      Maybe you could edit the video or add a commentary or something, to show how one would go about solving this equation (or if you're really good, design a spreadsheet that just allows someone to plug data into it and have it do the maths for them).

      Seems like a very useful case study, and gives a lot more weight to using neutral throws. I've always felt like I should use neutral more because it's almost never guarded, but I've been finding it hard to give up the extra damage on the directional throws. I'll definitely use neutral more now - I'd just like to know the weights for my two characters so I know what kind of ratio I should be trying to apply.

      Also, I liked the study on guaranteed damage vs variable damage. The idea of gauging one's skill against their opponent makes a lot of sense. I play poker in a amateur fashion, and I always try to play risk-averse (tight), because I want my results to be based on skill and 'out-playing' my opponent rather than just blind luck. That being said, if I were to play against an obviously better player than I am, I'd be more tempted to risk it on a weaker hand and hope it holds up, because I'm unlikely to win the more subtle areas of the game (blind-stealing, check-raising etc). Applying that to VF is very interesting - when I play my brother, I'm the weaker player, and often find myself 'throwing hail marys' that I wouldn't use against a more evenly-matched opponent.
    2. SDS_Overfiend1
      ^^ says the guy who has a answer for every answer...
    3. capt_catalina
      Very "Numb3rs" of you to bring it up! A class act.
    4. Zass
      Thanks for the feedback. To solve these kinds of systems, Wikipedia has a good explanation. From http://en.wikipedia.org/wiki/System_of_linear_equations

      Doing this for a system of four variables, as we have in the video, would take some time to go through. I encourage you to try the above method by hand to check for it.

      Online solvers do in fact exist, and I used one myself to check the answers. Here is the one I used:

      You should be able to enter the equations from the video and get the same answers.

      I would encourage you to use the above system to solve for your characters. I would be happy to double check your work.

      It might be interesting to solve the weights for each character, and possible add that to do the wiki?

      Another situation where risk aversion/seeking behavior comes into play is when there is a large life discrepancy. Players that are very far behind tend to exhibit risk seeking behavior, as they throw "hail marys". It's very common for a player that is far behind on life to move towards the edge of the ring, inviting the opponent to play in the very risky ring out game. This is risk seeking.

      On the other hand, a player that has a large life lead will often play conservatively, staying in the center of the ring and avoiding risky moves that can be punished. This is risk aversion.

      Thank you for your feedback on the videos!
      VFnumbers and Tricky like this.
    5. Genzen
      Okay, so I tried to copy what you did for Eileen.

      I started with your little grid thing:

      -----------Back (q1)-----Neutral (Q2)----Towards (q3)

      Back -------0--------------60-----------------60

      Neutral ---40--------------0------------------40


      Then I wrote out the equation things like you did:

      P1*0 + p2*40 + P3*70 = E

      P1*60 + p2*0 + p3*55 = E

      P1*60 + P2*40 + p3*0 = E

      P1 + P2 + P3 = 1

      When I copy and paste the equations into the solver you linked, I'm given this:

      { e = 600/17, p1 = 28/85, p2 = 33/85, p3 = 24/85 }.

      600/17 = 34.29
      28/85 = 0.33
      33/85 = 0.39
      24/85 = 0.28

      So, if I've done this right, then the expected average damage (E) is 34.29. Backwards should be done 33% of the time, neutral 39% of the time, and forwards 28% of the time.

      Is that right? Seems like it makes sense to me - neutral is done the most because it's the least escaped direction; backwards is second because it's the most damaging; forwards is last because... well, it's last.

      Next, I wanted to work out the defender's choices against Kage (my main opponent).

      I started with the table again:

      -----------Back (q1)-----Neutral (Q2)----Towards (q3)

      Back -------0--------------60-----------------60


      Neutral ---40--------------0------------------40




      Then did the equations:

      Q1*0 + Q2*60 + Q3*60 = E

      Q1*40 + Q2*0 + Q3*40 = E

      Q1*86 + Q2*86 + Q3*0 = E

      Q1 + Q2 + Q3 = 1

      Pasted them into the solver:

      { e = 2064/55, q1 = 103/275, q2 = 17/275, q3 = 31/55 }.

      2064/55 = 37.52
      103/275 = 0.37
      17/275 = 0.06
      31/55 = 0.56

      Meaning expected damage (E) = 37.52. Backwards should be escaped 37.52% of the time, Neutral 6% of the time, and Forwards 56% of the time.

      Forwards is TFT, and a lot more damage than the others, so it makes sense it would be preferred (though over 50% is surprising). Neutral is weak, but I'm unsure if I did something wrongly because 6% seems very low.

      When you consider these numbers (assuming they're accurate), and the other values of TFT (ring-out, wall-splats etc), it seems like TFT should be broken a very large amount of the time, which is annoying, because it makes his 60 damage backwards throw close to being guaranteed, else you risk eating the massive 86+ nonsense.
      Chanchai and Tricky like this.
    6. SDS_Overfiend1
      And You are?
    7. Zass
      You have a 70 in your first equation. It should be:

      (assuming defender always picks back)
      p1*0 + p2*40 + p3*55 = E
      ^------------ we deal 55 damage if the defender picks back, and the attacker picks towards, not 70 damage.​

      The other three equations I get the same as you. When I compute the final answer, I get an expected damage of 33.42

      The numbers are off because of the error in your first equation. Try with the correct numbers and I think you'll find a different relative weighting.

      I'm not sure where the 86 comes from (I don't play Kage, but I assume it's a combo you land after his towards throw?). Using the numbers you provided, I come up with the same answer as you.

      I understand your surprise that 6% is very low, but it is correct. The number is very low because doing a neutral throw (40) is close the same value of mixing up between the TFT (86) and back throw (60). I had the same surprise myself when computing some of Wolf's setups.If you look at 4:51 in my second video, you see that the defender percentage for neutral is only 1.6%. That's even lower!

      One way to see why this is, is to imagine the following. What if Kage's directional throws (towards and back) both did 80 damage? Well in that case, alternating between the two directional throws would deal 40 damage on average. In this case the defender should completely ignore the neutral throw (in other words, break it 0% of the time), and only focus on breaking left and right. Kage's back throw only does 60, but it's enough to make the big mixup between towards and back, so the defender is forced to defend against these most of the time.

      If you compute Kage's attack percentages, you'll find that Kage should use neutral 47% of the time, to maximally exploit this.

      In general, you'll find that as the damage for directional throws increases, the probability that the defender should pick neutral gets smaller and smaller. If the directional throws get very damaging, then the probability eventually becomes zero, which means the defender should just never even use that option (and neither should the attacker, since he will get more damage from just mixing up the directional throws. This is exactly what happens with Wolf's Burning Hammer and F5 at the end of the video. Those throws do so much damage that doing neutral isn't worth it for either party.

      Kage's most often used throw is not the back throw, but the neutral throw, which he will use close to 50% of the time. This is because the defender is forced to defend between the TFT and back throw most of the time.

      Excellent job computing these results!
      Chanchai and Tricky like this.
    8. Mister
      Zass i've seen your videos on youtube before you posted them here and when you posted here i was kinda happy to discuss this topic with you. To simplify your work and the meaning of nash equilibrium how would you translate into practice? Should we just do more neutral throws?
      Thanks for any tips.
    9. Genzen
      Thanks for the follow-up, and for catching that error - was a copy and paste thing from yours and I forgot to change the values!

      What I meant about backwards being 'almost guaranteed' is that, because it's so risky not to try to break TFT, the defender is pretty much forced into always breaking that direction, meaning that Kage can use that and throw out backwards and it will land a large portion of the time, rather than having to use neutral as his 'guaranteed' throw, like most do. I guess it's just a case of conditioning yourself as a defender to stick to the percentages and keep working in that back throw-escape once every three or four attempts, to keep it somewhat varied. Can't really say I'd ever go with breaking neutral against Kage, unless, like you mentioned at the end of your video, my health was low enough that neutral could finish me as well.
    10. Zass
      Great question. One thing I'd like to clear up right away is that Nash equilibrium isn't necessarily the best move. For example, let's say you are playing against an opponent who never breaks any throws. In this case your best response is to just do your most damaging throw every single time.

      Okay, so what does it mean then?

      I think of all the different mind games in throws as a field, like a soccer field. Every strategy corresponds to a position on the field. So there is the "always do your most damaging throw" strategy. That's a position on the field. I like to think of the red dots on the field as the attacker's strategies, and the blue dots as the defender's strategies.


      Some red strategies are really bad, some are good. Some blue strategies are bad, some are good. A common red strategy would be "Always attack with your most damaging throw". That's a popular one, and I use it a lot in random online matches. Another common red strategy is "Always mixup your two best throws".

      A common blue strategy is "do nothing". That's a very bad strategy. It's also very common in online play :). There's also the blue strategy of "break towards all the time". I see that a lot in tournaments against Wolf.

      So there's many possible strategies. But the point is, each red strategy can be exploited by a blue strategy, and each blue strategy can be beaten by a red strategy. Sometimes there's many ways to beat a strategy. The blue strategy of "do nothing" loses to almost every red strategy!

      So this dance of strategies is what I think of as "the field". I think of it as a soccer field. And you and your opponent are dancing on the field.

      Nash equilibrium in green is a special place on the field. It's like "base". If either player goes on base, then the expected damage is locked in, and there's no way to force any more, or any less damage. It's a way to avoid the dance of strategy picking and counter picking.

      Why would you want to go on base?

      For the attacker, base is the maximum guaranteed average damage.

      If you're the attacker, maybe you want to go on base because the defender is in your head. He's escaping every throw you try, no matter how much you guess. Or maybe he's just escaping a lot of them. Another reason is, maybe you are risk averse. You don't want to risk going on the field and dancing back and forth. You just want to deal your damage and not have to think about it. Maybe your opponent is at half life, and you don't want to mess around with fancy tricks, you just want to deal what you can. Maybe you are in the finals at Sega Cup, and you know your opponent is smart, and you don't want to risk being outplayed.

      For the defender, base is the minimum guaranteed average player.

      If you're the defender, you might want to go on base because you keep getting out predicted. Maybe your opponent made you eat three giant swings in a row, and just when you finally started to defend it, he switched to F5. You're tired of eating these throws. Or perhaps you don't want to get on the field in the first place. Maybe you are in a tournament against an unknown opponent and you just want to play it safe and solid.

      If either player goes to base, then the expected damage listed will be dealt on average, and there's nothing the other person can do about it. It's a powerful concept, because it cannot be exploited.

      So, how do you play against someone who is refusing to come off base? What if he won't play with you in the field? You can try to taunt him off. For example, if I was Wolf as the attacker (red), and the defender was just play nash equilibrium all the time (on base), then I might try to trick him into leaving base. I might start attacking with F5 every single time. Every time, no matter what. I'm trying to lure my opponent into thinking "okay, this guy is a moron. He's not mixing up his throws, and I'm going to start defending backwards all the time. That way I won't take any damage!". If I can trick him into doing that, then success, I've got him off base, and I can start doing a Giant Swing to get him. This is, as I'm sure you all know, a very common tactic for Wolf players. Just do F5 all day, and once they are convinced you are never going to do a giant swing, that's the time to do it :). That's Wolf 101.

      But if my opponent refuses to leave base, if he never deviates from Nash equilibrium despite my taunting and trickery then there's really nothing I can do.

      Of course, the same applies if I'm the one on base too. If I'm playing against the world VF champion, I'm probably going to use Nash equilibrium when I'm doing red strategies, because I don't want to get into a guessing game with this guy. Some of you guys on VFDC I felt this way against when I played online. You broke every giant swing. You broke every F5. You even broke neutral when I was trying to be tricky. I remember a game I played against Dennis0201. He broke every single throw correctly every time. I felt like someone was giving him hints of what I was going to do. Against a player like that, I'm going to run to base and be happy with my damage. I'm not going to get on the field and dance with you!

      I hope that analogy is helpful. I know that analogies can sometimes make things worse, but I wanted to share with you how I think about it. I really do think about it as players dancing on a field playing games, and running to base. Whether you choose to use base or not, it's helpful to at least know it's there, and how much it is worth.

      So if you're doing your own strategy and it's working for you, you should keep doing it. Like in most online matches with random bad players, just do what you normally do. But if you're playing against someone really good, who is owning you on the field, then you might want to consider switching to Nash equilibrium.
      Chanchai, Shouta, R_Panda and 2 others like this.
    11. Tricky
      You had me at equation

      I love you! You're my fav.
      phanatik, R_Panda and Zass like this.
    12. daRockReaper
      I have to admit I kinda enjoy thinking of VF as mathematic equations. Thanks for this outlook. I would like to think about Blaze's RD options this way but the discrepancy in frames and situation kinda makes this harder to visualize in this form
    13. Xzyx987X
      This is basically good information, but I can foresee a few problems applying it in real world situations. First of all, if you are trying to maintain an even statistical distribution for the throws you are using, you can't just do what the probabilities tell you to all the time. For example, if you had over the course of a match, used both Wolf's forward and back throw, you may by inclined to use Wolf's neutral throw next to maintain a good, even distribution. However, if you always insisted on keeping your distributions as close to matching the suggested probabilities as possible, you'd become predictable in cases where you'd already used several throws, which would lead to suboptimal results. This isn't an issue if your distribution is a true weighted random distribution, but maintaining such a distribution in practice without revealing any biases to your opponent would be fairly difficult to do.

      The other problem is that your formula doesn't apply in situations where the amount of damage needed to finish an opponent off is less than or equal to the damage of a throw. For example, if your opponent knows that a forward throw will deal enough damage to finish them, but a back or neutral throws will not, they will be much more heavily biased to escaping forward throw. Or for another example, if all three throws are enough to finish them off, any advantage of weighting throw frequency disappears, and the ideal probability for doing all throws becomes 1/3.

      Still, overall, good stuff. This confirms what I've more or less believed all along, which is for optimal damage you should be somewhat biased toward doing neutral throws.
    14. DomAug
      the engineer in me, which is also the real me, is geeking out over this. this is a work of art. :D
      Zass likes this.
    15. Zass
      I believe the video addresses these two points at 12:35 and 19:36 respectively. I agree on both counts: You need to pick randomly, as opposed to trying to fill an even distribution, which is wrong. You also need to readjust weights if life totals are low, going to 1/3 on all throws once life totals are 40 or lower.
    16. erdraug
      So... does that mean everyone is now convinced throw escaping is a complete guessing game? ;)
      DomAug and Feck like this.
    17. Dennis0201
      About wolf's catch throw, defender's priority is always to avoid the worst scenario especially near wall or ring out situation. When against the players who know how to deal with wolf's catch throw, the suboptimal option becomes very powerful. Overall wolf still takes the advantage at this situation, nothing costs when they break your throw. If they do dare to break your suboptimal option with no fear and put themselves at risk to gamble, all I can do is admire them and accept the consequence.;)
    18. nou

      it always has been.
    19. ToyDingo
      I was unaware that anyone was confused about this...

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