Jeph Loeb has a tendency to depict Batman villains in a strange way. In The Long Halloween and Dark Victory, we see the bulk of his rogues gallery actually working together to achieve a common goal. In fact, the latter has Two-Face actually conducting a sort of mock trail in his lair with all of the villains (including the Joker) in attendance, watching and participating as Two-Face prosecutes witnesses as part of his deranged scheme.Of course, this is ludicrous. One of the distinguishing features about most of the notable Batman villains is that they all have distinct neuroses and pathologies that render most of them utterly incapable of working together. It is not a rational decision, rather that most of these rogues have deep-rooted psychological afflictions, many of which mirror some aspect of the Batman. As such, they have different motivations and goals, different means to achieve those goals, and different reasons to kill Batman. In fact, it's been argued ad nauseum that the Joker may not even want to kill Batman, for this act would extinguish his very nature of being. The one truth is that, barring certain less-insane villains like the Penguin, most Batman villains have a burning desire to find, identify and kill the Batman on their own.
Most people would intuitively argue that it would make more sense for these villains to pool their skills and cooperate in order to finally rid the world of Batman. However, the decision to work alone is not entirely irrational. In fact, we can use very basic tools of utility and game theory in order to work out such a decision for a Batman villain (let's say the Joker, being the most insane and distinguished) and show that cooperation is not necessarily the optimal choice.
First, we have to make certain assumptions. Specifically, we need to assign probabilities of capturing Batman and figure out how much these probabilities increase due to the addition of a new cooperating villain. We also need to assign utility values for the Joker for each scenario. Let's start with utilities.
For not killing Batman, we can obviously assign the Joker a utility of 0.
For capturing Batman on his own, let's assign the Joker a utility of 10.
For capturing Batman with the help of x other villains, the utility would be 10/x.
The last one is sort of tricky. This means that if the Joker cooperates with one other villain (say Two-Face) and together they manage to kill Batman, then the utility for each would be 5. In effect, this means that the villains "split" the utility of 10.
Many of you might be wondering why it is that the more villains there are, the less utility one of them receives from killing Batman. Well, consider the pathology argument above. Obviously, if we factor in the Joker pathology, his utility for killing Batman with cooperation would be less than that of capturing Batman on his own, but greater than 0 as Batman would still be out of the picture. The thing is, as more and more villains enter the party, the Joker will feel less and less accomplished if they wind up killing Batman. Again, it is his very essence of being. He wants nothing more than to kill the Batman on his own, so it should make sense that the satisfaction he derives diminishes as more rogues are brought on board for the mission.
Aside from pathology, there are other reasons why the Joker's utility would diminish as such. One is that the villain who finally achieves victory would be held in the highest esteem among the criminal underworld and feared the most by the Gotham elite. Victory over Batman is largely a symbolic projection of status to the rest of Gotham City--and this is something the villains all desire. Therefore, if the Joker were to finally kill Batman, he would effectively "rule" Gotham.
Consider an even more tangible reason. The villain who kills Batman would gain access to his identity. He could therefore do with this identity anything he pleases. Assuming Nightwing and Robin won't be a problem to neutralize, the Joker could gain access to the batcave: a goldmine of wealth and technology. He could further auction off Batman's identity to the highest bidder. Even though he would be dead, I have no doubt that most of the villains would want to purchase this information to exact revenge on Alfred, Dick, Tim, etc.
For all the reasons then, it makes sense for the Joker's utility to diminish as more villains are added. The more rogues that take party to the death of Batman, the less the Joker will feel satisfied, the less influence he will have over Gotham City, and the less actual benefits he will reap after his death. For simplicity's sake, I assume that the villains "split" the utility of 10.
Now, let's assign the probabilities. I'm going to assume that each Batman rogue has a 2% chance of killing Batman alone (and this is being very, very generous and neglecting the individual skills of each rogue for simplicity). You would then think that adding villains to the scheme would increase the probability of killing Batman by 2% with each new rogue. Except, this ignores the economics law of diminishing returns, which states that as you increase the factors of production, the marginal benefit of those factors decreases. Usually, this applies to outcomes which are continuous (such as production of goods) rather than binary (to kill or not to kill Batman), but we can apply diminishing returns in this case to the probabilities. The theory is that as you add villains, working together will prove more difficult and planning more arduous. Therefore, the probability of getting Batman will increase, but by a marginally smaller amount with each villain added.
Thinking of probability as output, let's assume that in each state,
p = 2*y^0.9, where
p = probability of killing batman and
y = number of villains involved in the scheme.
Hence, we have a diminishing returns function. If there is only one villain involved in the scheme, the probability of killing Batman is 2%.
If there are 2 villains involved in the scheme, the probability becomes:
p = 2*(2)^0.9 = 3.73% (the probability increased by 1.73 percentage points)
If there are three villains involved, then:
p = 2*(3)^0.9 = 5.38% (probability increased by 1.65 percentage points)
And so on and so forth. Now armed with the knowledge of probabilities and utilities, let's conduct an analysis of whether it makes sense for the Joker to team up with Two-Face and the Scarecrow. We must analyze the expected utility of each scenario (teaming up and working alone).
First let's calculate the expected utility of working alone for the Joker. The equation is:
EU = p * (Uk) + (1-p)*(Unk) where
EU = expected utility
p = probability of killing Batman
Uk = utility of killing Batman.
Unk = utility of not killing Batman
We know that for the Joker, the utility of killing Batman alone is 10 and the probability of killing Batman by himself is 0.02. Hence:
EU = 0.02*(10) + (0.98)*0 = 0.2
Hence the expected utility of the Joker killing Batman on his own is 0.2.
Now, we analyze the expected utility of the team-up. We know that the probability of the Joker, Two-Face and the Scarecrow killing Batman is 0.0538. The utility would be 3.33 each. Hence:
EU = 0.0538*(3.33) + (0.9462)*0 = 0.179
Hence the expected utility for the Joker of the trio killing Batman is 0.179.
Since the expected utility of the trio killing Batman is less than the expected utility of the Joker doing it by himself, the Joker should prefer to work alone. Hence using simple economics, we have shown that it makes perfect sense for the Joker not to cooperate with other villains. Of course, this is incredibly simple and there are many other issues to consider. One of these issues is whether it would make sense for the Joker to cooperate, but then backstab the other villains. This issue will be considered in a subsequent post.
You assumed that working as a group makes the three villains less likely to kill Batman. (If the three of them made separate attempts they would have a total chance of roughly 6%; for the three together you assumed 5.38%.)
ReplyDeleteThen you made some questionable utility assumptions, hand-waved about the Joker's feelings of personal accomplishment, invoked game theory, simplified your analysis to just one case, did some math, and concluded that the villains shouldn't work together.
I can make your argument more simply: "We assume that the villains are less likely to kill Batman if they work together. Therefore they shouldn't do that."
Minor point: If x is "other villains", then it should be 10/(x + 1), shouldn't it?
ReplyDeleteI think it's a safe bet that the Joker isn't likely to work well with anyone to accomplish his goals. He isn't going to "put his cards on the table" and would no doubt attempt to come out on top not just vs. Batman but anyone he is working with.
ReplyDeleteHow does this same analysis play out if the villain isn't the Joker?
Batman's villains can't play well with others long enough to play a game of poker, let alone kill Batman.
ReplyDeleteUnlike the Flash's villains -- the Rogues work well together and that's why they're a serious threat to the Flash in spite of their goofy Silver Age origins and gimmicks...
An EXCELLENT analysis, sir! I tremble to think of how much more complex the formula would be if you figured in Batman bringing his bat friends along. I suspect that Batman's bringing along a Robin or Batgirl would decrease his effectiveness as it would divide his attentions between defeating foe and rescuing friend, plus Joker would have no scruples about killing Batman's buds, thus increasing his foe's utility.
ReplyDeleteDoesn't the Joker get value from the team-ups beyond the scant possibility of killing Batman? In "Hush" (yes, I know, but bear with me), the Riddler claims that it was easy to get the Joker to go along with the latest Loebfest because he loved the dead Robin/Clayface gag. If we accept Morrison's "supersanity" conceit, then his motivations for joining/not joining the Arkham All-Stars would change every day (not to mention the questionable logic in using algebra to try to predict Mista J's behavior).
ReplyDeleteThis post is amazing
ReplyDeleteDan -- Good point. Except, the decision I was laying out was one villain by himself vs. three combined, rather than three separate vs. three combined. The Joker has to decide whether to work alone or to work with others. Yes the probability of killing Batman if the three of them were working separately would be close to 6%, but the Joker cannot count on other villains simultaneously hatching plots. Even if they did, the issue here is the probability that he does it himself. So it's like flipping a coin three times and asking what the probability is of getting heads on the third flip.
ReplyDeleteCisco -- Oops. That was a typo. I meant for x to be the total numbers of villains. Good eye!
Mikey -- Some villains would be more cooperative for sure. The Penguin would likely cooperate well. The Riddler would too (although he is now reformed). But I think most of the A-list rogues (Joker, Two-Face, Scarecrow) are less likely.
Gatsby -- Oh boy, the classic "to save the girl or catch the villain decision." One day I'll have to write this up.
Glen - Nice use of Morrison's supersanity concept. Obviously there are different interpretations of the Joker and some of them might take an odd pleasure in teaming up. I happen to disagree with Loeb's interpretation, but to each his own.
Craig -- Thanks!
Good stuff. Actually, separate villain attacks would reduce the Joker's chance of success since someone else might kill Batman first--which would prevent Joker from doing so.
ReplyDeleteThe problem with using diminishing returns is that we aren't talking about one Joker vs 3 Jokers, we are talking about 3 villains with a variety of skills, ideas and resources. This can be made up for with Batman's incredible ability to start fights between people who have captured him.
Your probability equation is all wrong and your statement that probability is based in economics and diminishing returns is backwards - economics uses probability.
ReplyDeleteThe way to calculate this sort of probability, x = 0.02% and y = number of additives is:
p = 1 - (1 - x)^y
So three villains working together means the probability of killing batman is p = 1 - (0.98)^3 = 5.88%
This analysis leads me to wonder about the economic advantage to any one villain, the Joker perhaps, in killing off the other villains, so as to protect the Batman for his own villainous purposes. (Either to be able to kill or unmask him, or preserve the Joker/Bats dichotomy that drives him.) I think the Joker would find that an amusing project, and it might make for a dramatic story.
ReplyDeleteFollow me? If there is only one Batman-worthy villain left, his odds of eventually killing Batman would seem to rise, at least in contrast with these hypothetical cooperation schemes.
Maybe we introduce a new villain for the purpose. He starts offing B- and C-list enemies of the Bat, the Joker says, "Fantastic, why didn't I think of that?" and then the Joker hunts the hunter to keep the game for himself. Meanwhile, of course, Batman begins investigating the trail of bodies that is leading right to him.
There you go. Top-selling graphic novel rooted in economic theory. All rights reserved.
A good idea, but your analysis is incomplete, as well as contingent on the functional form for the probability you specified and the division of the payoff for killing Batman as part of a coalition.
ReplyDeleteIt is incomplete in that it ignores in the calculation of the expected utilities the value to the villain of *someone else* killing Batman ("Unk" in your terminology). So, for example, if the Joker doesn't join the scheme, a coalition of two other villains might kill Batman. Or, one of the two other villains acting alone might kill Batman. If the Joker does join a coalition, then (a) at least one other player has to join as well -- otherwise, it's not a coalition, and (b) someone else may or may not join, in which he/she will try to kill Batman on his/her own, and might succeed while the coalition fails.
Presumably, the value to a villain of Batman dying, but not by the villain's hand, is between zero and ten, but of course, as you note, this depends heavily on the individual villain's psychological makeup.
With the correctly specified expected utilities in hand (which may, of course, differ by villain, and must be specified for each possible choice by each villain given 0, 1, or 2 other villains wants to join a scheme), you're going to have a 2x2x2 game. (Not, as you analyze it, a 2x1 decision problem). Look for Nash equilibria in pure strategies if they exist; if not, look for mixed strategies (i.e., villains cooperating with some nonzero probability). I think the game as written satisfies Nash's conditions, so at least one Nash equilibrium should exist.
Regarding functional form, we could tweak the (non-sacrosanct) 2*y^0.9 and 10/x formulae and get a completely different answer.
Decreasing returns to scale? Please. Didn't your momma teach you anything about cooperation? Two heads are better than one.
ReplyDeleteEconomic assumptions should be carefully considered before being applied. Consider combat, for example. If you can gang up on someone, your chance of beating him is clearly larger than the sum of the individual probabilities. Economics is as its worst when people use math such as expected utility calculations as a substitute to careful thinking rather than as a complement to it.
--annoyed game theorist (who won't waste any further time replying to your rebuttals because he has actual research to do)
Dan said...
ReplyDelete"If the three of them made separate attempts they would have a total chance of roughly 6%"
Anonymous said...
"So three villains working together means the probability of killing batman is p = 1 - (0.98)^3 = 5.88%"
Dan is laughably wrong, and Anonymous is correct to treat the problem as a binomial distribution with n=3, k>0 http://en.wikipedia.org/wiki/Binomial_distribution
However, "Anonymous," if I may nit pick, that is NOT the probability of success of all three rouges working *together*. It is merely the probability that one or more of the three will have success if they each try to kill the batman in *independent* incidences. Binomial probability can only be applied to independent events.
The probability of success with co-operation is much harder to quantify because of the various ways in which the rouges can aide and/or hinder each others' actions.
I agree with you.
ReplyDeleteThe best example of your analysis would be the Reaper in Batman: Year Two.
I got a question though.
Who is the flying white hooded villain in that picture?
You're missing a major factor in your calculations - the villains not only get points for actually killing Batman and sharing his loot, they also _lose_ points every day Batman's alive, because he interferes with their nefarious plans. Even though joining the other criminals in ganging up on Batman decreases their share of the loot and glory, it increases the chances that they have a high future income stream from a Batman-free Gotham. It's probably a big win for them, in spite of diminishing returns, though of course they also have some incentive to stab each other in the back to reduce the amount of sharing they'll have to do in the future.
ReplyDeleteIt sounds like this econoblogger has never been in a fight.
ReplyDeleteNeither have I. I mean, not a real one. But if you put one fighter against two opponents, each the fighter's equal, the odds of him winning don't drop from 50% to 25%. They drop to more like 10%. Add a third opponent, the odds of him winning are probably 1%.
If, as your model supposes, good cooperation is possible, the utility function of adding more villains should be exponential, not logarithmic.
Now, if you want to complicate the model some, there is some negative utility to killing associated with killing Batman. If you consider that the death of Batman represents the creation of a power vacuum that the dregs of Gotham will be rushing to fill, then the villain who has expended great resources to kill Batman may end up without the resources needed to capitalize on the opportunity.
I believe the Inverse Law of Nina should also be taken into account. 1 ninja or villian, badass; a bunch of ninja, lame.
ReplyDeleteI have serious reservations about this argument. I express them here:
ReplyDeletehttp://www.youtube.com/watch?v=Xko-glcr1rI
Ugh. This is a sad misapplication of economics. The factors involved in the decisions of these characters (fictional though they may be) are not nearly as simple as some 'utility value' to an objective like killing Batman. It takes a complex human motivation and over-simplifies it to the point that a discussion about the math involved does nothing to actually describe the real (fictional) act of trying to kill Batman.
ReplyDeleteEcon is about commerce, and does not accurately describe how people really make choices outside of that realm.
Thelvyn: The mystery villain in question is A-Mortal of the Hyperclan. Of "I KNOW YOUR SECRET!" fame from Morrison's first arc on JLA.
ReplyDeleteI like the gist of this analysis. Of course, it seems like you understand that the variables become much more intricate, as well as hard-to-predict with the use of different specific villians, and different specific scenarios.
ReplyDeleteThe one flaw that I see in the logic, however, is this: The value of teamwork should have a positive modifier, not a negative one.
First, three people attacking one person generally have the advantage of striking while their foe's attention is elsewhere.
Second, is the complement of each Batman foe's derangedness. For all that the foes of Batman have distinct and occassionally crippling neuroses, each of those neurotic compulsions is offset by a complementing form of genius.
Let me say it simply: There are no stupid (or even average-intelligence) characters in Batman.
The Riddler is a stellar example. He views Batman (a noted genius) as the only person worthy of opposing him. He creates games, riddles, and puzzles that require immense IQ to solve.
The benefit to any given villian for working with another villian is that they gain a skillset they do not possess. For example, the Scarecrow, whose genius for hallucinagenic drugs, coupled with the Riddler's skill with the creation of puzzles and traps.
What about the Penguin's fairly vast access to criminal resources (thugs, weapons, plans, illicit information) combined with the unmatched talent for scheming possessed by the Joker?
~Jot
This is a great starting point on how villains do not work well together. However, there still are many issues that were not covered in this article that greatly affect the way the Batman villains work.
ReplyDeleteFirst each has their own way they want to kill or toy with Batman. Villains like the Riddler and the Joker want more to play with his mind while villains like Bane and Clayface want to physically destroy him. This difference in style automatically causes a riff in the teamwork of any villain team.
Secondly, each villain wants the credit of being the one to either kill or unmask Batman and does not want to share the credit of this accomplishment of this with anyone else.
We know that the chances of killing batman are zero because the writers would never let him die. The number that matters for this discussion is the Joker's perceived chance of killing batman, as that is what he would need to weigh. Ignoring stories such as Dark Knight in which he has no intention of killing the batman, he usually perceives his chance of killing batman with his latest plan as quite high (higher than 2%, I'm sure). If you're going to get inside of the criminal mind, then try to approximate the numbers inside it instead of making up a number that we all know is zero. You know when you pick up a Batman comic that he's going to survive.
ReplyDelete@CalebGT
ReplyDelete*SPOILER*
Apparently you haven't been reading your Batman comics lately (I actually haven't either, but I have my ways), as the good Bat was just killed... Er... Sort of. Darkseid has a messy way of sort of killing people to a semi-permanent degree. I doubt it'll stick (if it hasn't already been reversed somehow).
*/END SPOILER*
@Jot
You are forgetting to take into account (at least to a large enough degree) the weaknesses of each villain, which is also added alongside each strength. While Riddler may be able to cover for a certain degree of Two-Face's coin induced antics, there will be a point where it becomes debilitating to the overall effort (and thus beneficial for him to be cut from the team).
@Original Post
Nice job, I found it very interesting. I also must laugh at the rift caused between mathematicians and psychologists in these comments. While I'm no math wiz (more of a psychologist actually), I think you've pretty much nailed the probabilities minus the infinite array of variables that would be impossible to account for.
-DuLake
Jeeze people.
ReplyDeleteScrap all the economics. Scrap all the probabilities.
Send Batman on vacation for a month and just let Superman clean up gotham.
This example ignores the added probability of being able to multitask, occupy separate places simultaneously, etc., all things Batman cannot do.
ReplyDelete"...NOT the probability of success of all three rouges working *together*."
ReplyDeleteI suspect the eyebrow pencils would be more dangerous to Batman than are the rouges. Mascara applicators could also pose some problems, as might lipsticks. Rouges, it seems, are quite a ways down on the list, unless their colors clash to such a degree as to cause lethal injury.
It is not always true that two people attacking one have an advantage greater than simply arithmetic. As in this article, a diminishing probability is also possible. Trained fighters are frequently able to use the poor coordination of fighters not used to working with each other against them.
ReplyDeleteI got stuck wondering if Two-Face should be counted as one villain or two.
ReplyDelete@Dan
ReplyDeleteWorking as a group is not less likely to kill Batman. It's just that the probability to kill Batman as a group is not proportional to the number of the villains.
For instance, Batman is more likely to be killed by a group of 3 (5,38%) than by one villain (2%)
Why propose new arbitrary formulas when this one isn't even adequately analyzed? (And yes, any formula describing the effectiveness of comic villains is going to be an arbitrary approximation unless Brainiac provided it himself.)
ReplyDeleteWhat I found interesting about the math is that according to the formula provided here, it will take 78 (well, 77.223) cooperating villains to have a 100% chance of killing Batman. That's a certainty that no number of independent villains will ever achieve.
Since having to split the utility pie for working together is apparently exactly the same expected value as randomly choosing which villain gets full credit, the Joker and his buddies should select the method maximizing the probability of Batman's demise. Independent attempts are better for small numbers of villains, but in this case, that means transitioning from independent attempts to cooperation when there are at least 41 (40.317) bad guys.
So, as long as there are are least 41 villains gunning for Batman, they're better off working together. Otherwise, not. How many were in that picture again? ... It looks like only 22. Unless there are 19 more in hiding, the ~36% chance of success they'll have on their own is better than the ~32% chance they'll have together.
This feels just a little over my head. But that might be the whole Just getting out of bed feeling.
ReplyDeleteNow hypothetically, The Joker utilizes two to three others to kill Batman and succeeds. Yay, the Batman is dead, hurrah! now let's celebrate. The Joker's going to be like "Who said anything about WE?" and likely kill his cohorts. He doesn't strike me as someone who would share the glory. Thus, the victory would likely not be so diminished in his delusional mind.
Bryce -
ReplyDeleteYour scenario works only for untrained fighters. This is because at an untrained (or minimally trained) skill level, the chaos of a fight is amplified and an untrained mind is pretty quickly overwhelmed.
However, as the solo fighter's skill increases - regardless of his opponents' skill level(s) - his odds of reaching a favorable outcome decrease only slightly, UNLESS his opposition is specifically trained to fight as a team.
I have been on all sides of this scenario, losing and winning when alone and when outnumbering my opponent. Multiple uncoordinated attackers completely lose the advantage of maneuver, and simply become movable obstacles for their teammates.
Even under your supposition of all having equal levels of training, without the additional team training to delegate zones of control and task handling, one versus 3 is not a 90% loss rate for the individual (not even close, in fact).
My envisioned scenarios, of course, are based on a favorable outcome for the individual, which is to emerge from the confrontation intact and free. I also assume adequate room for maneuver and the existence of an escape route. YMMV, of course.
There are actually 3 stages to a battle, and several levels to an engagement. I'l simplify:
ReplyDeleteLogistical: Ensuring all necessary personnel and materiel are in the proper places and accessible. This is most important.
Strategic: Holding the proper ground for initiating the battle.
Tactical: Maneuver once the battle is begun.
The "more is better" argument only really applies to the tactical stage, and only assuming they are willing to cooperate. If someone winds up with the garotte around Bats' neck, the others might want to interfere a bit.
I'd guess they'd cooperate in the logistical stage, backstab each other at the strategic level, and infight in the end stage of the tactical engagement.
A clandestine event would follow this progression:
Reconnaissance of possible engagement areas.
Planning.
Placement of resources.
Approach.
ENGAGEMENT.
Reinforcement (if needed).
Conclusion of engagement.
Departure from the area, in this case, with proof of Bats' death/with the captured Bats, OR, strategic withdrawal in event of failure.
You'll need to calculate probabilities at each stage.
@Jot: There are no stupid (or even average-intelligence) characters in Batman.
ReplyDeleteWasn't there the Killer Croc, not to mention countless muggers?
@Jot: There are no stupid (or even average-intelligence) characters in Batman.
ReplyDeleteWasn't there the Killer Croc, not to mention countless muggers?
My only comment is your equation for diminishing returns in regards to the probability of villains defeating Batman.
ReplyDeleteYou can set a value for the exponent such that you would continue to get diminishing returns but that the expected utility could either be close enough the expected utility 0.2 that it the difference may not be noticed by the joker.
Example,
If you set your equation to
p = 2*y^0.978
then
p = 2*(3)^0.978 = 5.85672163
plugging that solution into the equation for expected utility you find
EU = 0.0585672163*(3.33) + (0.9462)*0 = 0.195
The expected utility is lower then if the Joker killed Batman by himself, but it is unknown that if the difference in utility would be noticeable.
nice post.
ReplyDeleteit's very useful for me.
thanks. :)
seriously men, you believe that all villains can make a good team, please don't make me laugh, just think about it all that insane character together, there's no way to victory.
ReplyDeleteNot many people would say this. Youve got some guts. I will say this, though. If you dont want to alienate any readers, youre gonna have to stop generalising so much. Maybe you should try seeing both sides of this issue instead of assuming that yours is the only valid opinion. Id still read it, I like the way you write. But I can see some people getting upset.
ReplyDeleteSuch kind of articl is our need , it can give us help from different aspect for different people . after reading this article , I learn a mount of knowkedge increase my power , my families and I all like you article ,reading you article is our best love. May be you are Sports fans? Do you like NFL jerseys and NBA jerseys or Puma Shoes ,Puma sneakers and Ecco shoes and Clarks shoes , Nike Sneakers is Cheap Nike Shoes ,these have the high quality,low price,professional service,Just For You if you like.Thank you again for writing this article!
ReplyDeleteHello.. Firstly I would like to send greetings to all readers. After this, I recognize the content so interesting about this article. For me personally I liked all the information. I would like to know of cases like this more often. In my personal experience I might mention a book called Green Parks Costa Rica in this book that I mentioned have very interesting topics, and also you have much to do with the main theme of this article.
ReplyDeleteThe market in which shares are issued and traded either through exchanges or over-the-counter markets. Also known as the equity market, it is one of the most vital areas of a market economy as it provides companies with access to capital and investors with a slice of ownership in the company and the potential of gains based on the company's future performance.
ReplyDeleteThanks of Buy Generic Levitra |
Hello Dear,
ReplyDeleteReally your blog is very interesting.... it contains great and unique information. I enjoyed to visiting your blog. It's just amazing.... Thanks very much for the share & keep posting such an informative articles. I'm looking forward to your blog. Thanks!
This information is very interesting, I really enjoyed, I would like get more information about this, because is very beautiful, thanks for sharing!
ReplyDeleteno nonsense muscle building program
Great Contribution no words for description thanks a lot sharing this nice information.
ReplyDeleteSmith Alan
You are so generous! I just love finding beautiful and artistic freebies like this. They really brighten up my day, thank you!
ReplyDeleteThis is a very different analysis I ever see..I think this is equal to google analysis...hehe
ReplyDeleteI am thoroughly convinced in this said post. I am currently searching for ways in which I could enhance my knowledge in this said topic you have posted here. It does help me a lot knowing that you have shared this information here freely. I love the way the people here interact and shared their opinions too. I would love to track your future posts pertaining to the said topic we are able to read.
ReplyDeleteHello,
ReplyDeleteReally this is very interesting blog post here.... your blog is so interesting for reading. It contain very unique and very helpful information. Thanks very much for sharing such an interesting information. Keep posting....
Wow! It's awesome blog post here.... really very interesting for reading..... I enjoyed it. Thanks for the share.... just keep posting such an informative articles, I want to know more about this topic.
ReplyDeleteRegards,
Sam Woods, NY
thanks a lot for the informative article. i am really interested to read more from you.
ReplyDeleteToday, most of the businesses have set up their web presence. Hence, to concentrate your endeavors on running your business online you need to seek the pertinent help of a SEO solutions expert.
ReplyDeleteThanks for sharing this post with us. It get amazing experience at time of reading.
ReplyDeleteInteresting Post.....
ReplyDeleteReally great post, Thank you for sharing This knowledge.Thank you very much for sharing this knowledge.this graph really conveyed the part which i was looking for.
ReplyDeleteI love to read this post
ReplyDeleteI was reading something similar on another website that i was researching. I will be sure to look around more.
ReplyDeleteI found so many interesting stuff in your blog especially its discussion. From the tons of comments on your articles, I guess I am not the only one having all the enjoyment here! keep up the good work
ReplyDeleteI have to say, I dont know if its the clashing colours or the bad grammar, but this blog is hideous! I mean, I dont want to sound like a know-it-all or anything, but could you have possibly put a little bit more effort into this subject.
ReplyDeleteYou've got an excellent point to say about this subject, and so a lot knowledge. I believe that you know how to make people listen to what you've to say, particularly with an issue thats so essential. Im glad to learn this weblog. Great work, man!
ReplyDeleteI was questioning what is up with that bizarre gravatar??? I do know 5am is early and I'm not wanting my finest at that hour, but I hope I don't seem like this! I would nonetheless make that face if I am asked to do a hundred pushups. lol Anyway, in my language, there aren't much good supply like this.
ReplyDeleteDo you have issues with batman
ReplyDeleteHi, thank you for sharing this great info. Was just browsing through the net in my office and happened upon your blog. It is really very well written and quit comprehensive in explaining with a very simple language.
ReplyDeleteI love your site! did you generate this your self or did you outsource it? Im searching for a blog style thats similar so thats the only reason I m asking. Either way keep up the nice work I was impressed with your content really..
ReplyDeleteBuy Vermox
Its amazing to see this post.
ReplyDeleteThanks for posting.
This is the best ever site I see. The articles are very informative.
ReplyDeleteThis is an interesting information..... Keep posting such an amazing blog post.
ReplyDeleteGreat post thank sfor th enic eshare!!
ReplyDeleteGreat post thanks for the read!!
ReplyDeleteProblemet med at bruge aftagende afkast er, at vi ikke taler om en Joker vs 3 jokere, taler vi om 3 skurke med en bred vifte af kompetencer, ideer og ressourcer.
ReplyDeleteJeg synes, det er en sikker satsning, at Jokeren ikke er sandsynligt, at arbejde godt sammen med nogen til at nå sine mål.
ReplyDeleteI would like to thank you for the efforts you have made in writing this article
ReplyDeleteComputer Repair Manchester
exell.articl.
ReplyDeleteI certainly enjoyed the way you explore your experience and knowledge of the subject! Keep up on it. Thanks for sharing the infobobble head doll
ReplyDeleteIt was a beneficial workout for me to go through your webpage. It definitely stretches the limits with the mind when you go through very good info and make an effort to interpret it properly. I am going to glance up this web site usually on my PC. Thanks for sharingengraved gifts
ReplyDeletevery informative post, thanks for sharing it with usseo sheffield
ReplyDeleteThese kind of post are always inspiring and I prefer to read quality content so I happy to find many good point here in the postAfrican Mango
ReplyDeleteI tremble to think of how much more complex .
ReplyDeleteThanks for the posting. Loads of excellent writing here. I wish I had found this site soonerengraved gifts
ReplyDeleteHappy to see your blog as it is just what I’ve looking for and excited to read all the posts. I am looking forward to another great article from you. fast vpn
ReplyDeleteIt was really nice to study your post. I collect some good points here. I would like to be appreciative you with the hard work you have made in skill this is great article. Meratol Reviews
ReplyDeleteI certainly enjoyed the way you explore your experience and knowledge of the subject! Keep up on it. Thanks for sharing the info
ReplyDeletewooden kitchen worktops
I’ve seen progression in every post. Your newer posts are simply wonderful compared to your posts in the past. Keep up the good work
ReplyDeleteiPhone 5
I recently came across your blog and have been reading along. I thought I would leave my first comment. I don't know what to say except that I have enjoyed reading. Nice blog. I will keep visiting.
ReplyDeleteYou really did a good job, computing Batman's fall if the Villains work together.
ReplyDeleteBut for my assumption, if all Batman's work together. It will only lead to the downfall of each other. Since they have there own personality and is full of themselves. I rather suggest they must work individually.
Great article, I love reading it since I'm also a Batman Joker fan.
Learn to Play Guitar
Polo-Tops.com is a personal online shop that offers a wide range selection of affordable and fashionable men and women’s designer polo, polo shirt, ralph Lauren polo, lacoste t-shirts and lacoste polo. Visit us online to avail of some discount and promotion. Orders more than £50 could be free shipped.
ReplyDeleteIt's great to see a blog of this quality. I learned a lot of new things and I'm looking forward to see more like this. Thank you.
ReplyDeleteWe all eat healthy food to gain energy but it is seen that some people want more energy then what an ordinary food meal provides. Well! Special energy drinks become an integral part of the diet chart of such people who become dependent on these drinks.
ReplyDeletehome products
energy drinks
vitamin supplements
I really like the math involved here. I will also say that Batman's enemies were generally far more interesting than many of the others.
ReplyDeleteYou please post more article on such topic ? I really appreciate it. Thanks man!
ReplyDeleteIt's my first visit on this website and i think this website is perfectly one of the useful website for sharing.
ReplyDeleteSmead labels
hello, i think that this post is the best that i have read.
ReplyDeleteGreat posts thanks!
ReplyDeleteYou made some good points there. I did a search on the topic and found most people will agree with your blog.
ReplyDeleteGreat post thanks for the read
ReplyDeleteI am really thankful to you for the information you have provided. You are helping others to grow their knowledge by sharing Efudex Cream such a valuable information you have. This post is amazing & I'm glad for it.
ReplyDeletehello, i would like to read more information about this topic.
ReplyDeleteGreat stuff from this post. Ive read your stuff before and you're just too awesome. I love what you've got here. Thanks
ReplyDeleteinteresting! although I have no expert, but I want have to know more and more, on your blog just interesting and useful information. Keep it up!
ReplyDeleteThe way you explained it is really awesome and makes every one to read till the end. Keep posting. This is a great job.
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteYou raise a lot of questions in my head; you wrote an excellent post, but this post is also mind provoking, and I will have to think about it a bit more; I will return soon.
ReplyDeletebuy fertility drugs online | buy soma online
Thank you for the efforts you made in writing and sharing your points of view. Looking forward to learn some more from you. Keep it up.
ReplyDeleteYour game theoretic concept is very good and knowledgeable.That is very funny about joker pathology.also batman is an excellent game.Reverse Cell Phone Lookup
ReplyDeleteBatman is a very interesting game.Your post will help those gamer who like this game.
ReplyDeleteYou have a great mathematical knowledge. Why not try to Be An Expert Trader by visiting some Automated Forex Trading and download Forex Trading Software.
ReplyDeleteThis is a great inspiring article. You put really very helpful information. Keep it up.
ReplyDeleteI just love the way you work. Thanks for sharing this great and interesting stuff. Fabulous post! I really enjoyed that.
ReplyDeletebuy from world best online pharmacy com
ReplyDeleteThanks for sharing, much appreciated and useful post, congrat and keep on track!
ReplyDeleteWow! It's awesome blog post here. really very interesting for reading.Like those pictures.. I enjoyed it.
ReplyDeleteI used to have the original comic that image was in.
ReplyDeleteDallas Website Designer
Merchant Services
Nice post! Thanks for sharing!
ReplyDeleteA very nice informational blog.Keep on making such important blog post.Your work is really being appreciated.
ReplyDeleteHello, I am very happy to read your blog. This is another fantastic blog post what I have read. There are so many tools available that it becomes difficult to know which will work. And how best to use Buy Atorvastatin Online.
ReplyDeleteI really enjoy reading the post, thanks for sharing I really like it, I already bookmarked it, thank you guys.
ReplyDeleteGreat article post. I appreciate your writing skills. They are excellent. The knowledge of the subject is pretty good.
ReplyDeleteThis analysis is spot on in my opinion. Great rundown and very interesting indeed. My FirstYears Wave Stroller site is not as in dept when it comes to reviews but only in comparison to this post. Great work.
ReplyDeleteNice post. This post is different from what I read on most blog. And it have so many valuable things to learn.Thank you for your sharing!
ReplyDeleteGreat article, by the way. I agree that this material represents a little-seen about Batman, and deserves greater attention. webwatcher
ReplyDeleteim student of economics,game theory and algebra is too difficult.thank you for sharing about game theory.
ReplyDeleteAll villains think they are the best, and dont want to cooperate to one another. Why, simply because they are too high of for themselves. A single villain has a high rate to defeat a hero than working together.
ReplyDeleteForex Trade System
FX Trading Strategies
There are may person searching about that now they will find enough resources by your post. credit repair attornies
ReplyDeleteThis is my first to visit on your site and so far I really like the way you explained it is really awesome and makes every one to read till the end. sample letter to dispute
ReplyDeleteAs such, they have different motivations and goals, different means to achieve those goals, and different reasons to kill Batman.
ReplyDeleteI have viewed so many blog post but yours are different. I like to ask how you composed your articles for it really leaves an excellent impression on me. Just keep on posting interesting facts. I'm already a fan! Liability insurance
ReplyDeleteDear admin can you please tell me what is this Zahnarzt Berlin about?
ReplyDeleteDear admin can you please tell me what is this SEO about?
ReplyDeleteI'm really inspire to read on your post. It show a lot of useful stuff. credit charge offs
ReplyDeleteThe most perfect villain I've ever seen of Batman is Joker. I find it very interesting. I love Dark Knight.
ReplyDeletephentermine no prescription
It’s encouraging to see writing like this online. I respect the work you do and the effort you have to put into getting this kind of material ready to write. uniblue scam
ReplyDeletewow great i have read many articles about this topic and everytime i learn something new i dont think it will ever stop always new info , Thanks for all of your hard work!
ReplyDeleteReally great post, Thank you for sharing This knowledge.Excellently written article, if only all bloggers offered the same level of content as you, the internet would be a much better place. Please keep it up!
ReplyDeleteI ACTUALLY was happy to discover this web-site.I want to to thanks for your effort for the great study!! I surely loving every single little bit of this and I had you saved as a favorite to look at fresh things you text. And this my website DENTAL IMPLANTS. About DENTAL IMPLANTS ifomation
ReplyDeleteDENTAL IMPLANTS REVIEWS
I am fully agree with your post.. Its really brilliant mind thought and superb for new generation.. please keep it.. I hope every one in future will adopt your information. thanks.
ReplyDeleteI am fully agree with your post.. Its really brilliant mind thought and superb for new generation.. please keep it.. I hope every one in future will adopt your information. thanks.
ReplyDeletevery nice and great informational blog.Keep up it on making such important blog post.Your work is really being appreciated by some one.thankd
ReplyDeleteExcellent entry!!
ReplyDeleteAnd nice blog!!
Thanks for sharing!
I hope you have a nice day! Very good article, well written and very thought out. I am looking forward to reading more of your posts in the future.
ReplyDeletePosh limo services in memphis tn takes great pride in servicing our clients. Best limousine
ReplyDeletecompany in Memphis.Posh limo services in Memphis tn has never had any complaint's since we
opened in 2000. We are very proud of this.
the presentation method is very good because first they explain the intitial things and secong phase it is start to describe the main things whick is necessary to explain it.
ReplyDeleteHi! Your article rocks and is really a very good understand!…
ReplyDeleteHmmmm!!! Interesting topic and i love to read it completely... its a relevant of my business and i also something share new about it please read it and ask me some important tips...Web agency perth
ReplyDeleteThe story is horrific and disgusting.
ReplyDeleteGreat information in you post. keep it up.
ReplyDeletelater,
Casa Grande plumbers
nice comic picss !!
ReplyDeleteVery nice post. I just stumbled upon your blog and wanted to say that I’ve truly enjoyed browsing your blog posts.
ReplyDeleteI was looking for cleaning products and came into this blog thanks for sharing it. credit repair reviews
ReplyDeleteThanks for sharing!!! The things have been put in a smart manner and at the same time are also entertaining to read out.
ReplyDeleteInteresting suggestion regarding Batman Villains.
ReplyDeleteBut doesn't the nature of being a villain create a lack of trust making working together less likely?
Wonderful content along with cheers pertaining to expressing this information.
ReplyDeleteThe blog was absolutely fantastic! Lots of great information and inspiration,
ReplyDeleteboth of which we all need!
I thought this was an excellent analysis overall, never mind what the other haters have said here, great post.
ReplyDeletePhillip Morris
Liverpool University
boilx @ treatment for boils
i actually thought this was an excellent post, being a super-hero freak myself i can appreciate your angle on this. I write about Natural remedies for hot flashes but as a hero i don't need such things lol.
ReplyDeletehmmm nice blog.. its really great.. keep posting.. i like the way you have written this article and you have made it very interesting a fun to read.
ReplyDeleteThanks for giving us such quality knowledge. Waiting for your next interesting posts.
ReplyDeleteIn fact, I thought it was great place, super-hero phenomenon that I had the corner values. I write about natural remedies for hot flashes and a hero, but I did not dry
ReplyDeleteliking the mavlin theme
ReplyDeleteThis is a great article. I am pretty much impressed with your good work.You put really very helpful information. Keep it up.
ReplyDeletewould you mind updating your blog with more information? It is extremely helpful for me.
ReplyDeleteThis is great post, I love the way it was written the whole review.
ReplyDeletecool article
ReplyDeletei read all article its very good and informative i agree with your ideas its very helpful for us keep it up good job
ReplyDeletevery vivid and useful.. i think you can give more information for me for reference!
ReplyDeleteI like your website. Thank you for great information. I will come back to your website again.
ReplyDeletebest regards
Great informative post and i really likes your information, most of the peoples are likes your blog because its having the good knowledge. thanks for your good informative post.
ReplyDeletei just wanna thank you for sharing your information and your site or
ReplyDeleteblog this is simple but nice article I've ever seen like it i learn something today...
kwiaciarnia
ReplyDeletelekomania blog
narkomania blog
terapia blog
uzależnienie blog
help blog
leczenie blog
greener grass
wojciech blog
stop drinking
perkins restaurant coupon
tires On sale
countertop
lampy nice
lampy old
lamps
lamp ikea
lampy salonowe
lampy wiszace
lampy luxusowe
lampy dekora
lampy ladne
lampy nowe
pozycjonowanie
kwiaciarnia
los angeles seo
Took me time to read all the comments, but I really love the article. It proved to be very helpful to me and I am sure to all the commenters here! It’s always nice when you can not only be informed, but also engaged! I’m sure you had joy writing this article.
ReplyDeleteI absolutely match with your post.
ReplyDeletewow . . .really nice calculation . .I really love watching batman series. . thanks for sharing a post. .
ReplyDeletephone spy software for the safety of my phone.Android phone locator for the tracking of my location.
android gps apps locator and tracking my route.
android tracker also a tracker for my gps.
phone tapping software
free spy software
cell phone spying
cell phone spyware
cell phone tapping software
android gps
spy phone android
android phone tracking
mobile phone spyware
cell phone monitoring
How long have you been blogging for?
ReplyDeleteGreat post. i like it. feeling great when reading your post. The content of the articles there will be a lot of attractive people to appreciate
ReplyDeleteYou have made some questionable utility Analysis..
ReplyDeleteNice post, very thoughtful, but I think you might have too much time on your hands to go into this much detail about these villains.
ReplyDeleteThis can't truly have success, I suppose so.
ReplyDeleteHello, Such a wonderful blog i ever read. Please keep posting good blogs. Thank you very much.
ReplyDeleteGreat post I must say. Simple but yet entertaining and engaging... Keep up the good work!
ReplyDeleteYour post is great but the answers to the post are even better. All the theories about the villains cooperating with each other are pretty great!
ReplyDeleteThis is a really good read for me, Must admit that you are one of the best bloggers I ever saw.Thanks for posting this informative article.
ReplyDeleteIts really a nice and wonderful post. Keep it up.
ReplyDeleteThanks for this blog post. I think it's fantastic.
ReplyDeleteReally it's amazing to read this post.
ReplyDeleteflyer printing miami
vehicle wrap miami
Good stuff, I have a hard time believing villains would work together.
ReplyDeleteThe post is very nicely written and it contains many useful facts. I am happy to find your distinguished way of writing the post. Now you make it easy for me to understand and implement. Thanks for sharing with us.
ReplyDeleteI have read your post thoroughly and got mind blowing information.... Please send me mail if your are going to improve something advance..
ReplyDeleteThanks..
Flood damage repair Cleveland
What does it say about me that I find mathematics applied to cult culture phenomena to be irresistible?
ReplyDeleteawesome one.thanks a lot for sharing.
ReplyDeleteKamagra Online
This is such a crazy concept, but I really like it - you are very creative!
ReplyDeleteI find this post very interresting. My english is not very good, but I really love this blog. thank you for sharing..
ReplyDeletebusiness checks
I very much enjoyed reading your article. I know you put a lot of work into this and that deserves recognition. Very high quality article. Please keep a great job!
ReplyDeleteHi , i got this Article , i was searching some thing relevant to this, And i am feeling lucky, as its the perfect one for what i am looking for. I will share this link on face book.
ReplyDeleteHi , i got this Article , i was searching some thing relevant to this, And i am feeling lucky, as its the perfect one for what i am looking for. I will share this link on face book.
ReplyDeleteInteresting post !!! i love to see it, when people need such type of site. which has great information about this topic... Great for new generation..
ReplyDeleteThanks..
It won't truly have success, I consider this way.
ReplyDeletesteel building
I saw a great deal of helpful information above!
ReplyDeletemetal buildings
Good post, good information - I like the equations on whether the villains can work together.
ReplyDeleteWonderful work! That is the kind of information that should be shared around the internet. I appreciate your writing skills. The knowledge of the subject is pretty good.
ReplyDelete