I actually liked FFXIII a lot. The characters were pretty solid in my opinion and here's why I think so.
Lightning: Her character was pretty defined and shows what happens when you make a child take responsibility too early in their life. Lightning and Sarah's parents died when they were younger and Lightning had to take the role of supporting the both of them. She enlisted in the military for that financial reason and stayed motivated in that profession to ensure their lifestyle.
She continued to rise through the ranks due to her hard work and dedication. However she never took the time out to stop and take in situations. She had always been in full throttle since their parents passing and deal with a problem head on without thinking too deeply on it. Focus on what's ahead and press through. That makes for a great soldier, but one that has a short life span. And this shows in the story.
For the majority of the time, she suggests taking PsiCom head on, rather than running away or finding somewhere to hole up and plan. Why? Because that's what she had to do to survive when she was taking care of herself and Sarah. To deal with the problem at hand, rather than waiting for someone else to do it for her, or help. It keeps her from thinking about how bad the situation is, and she can proform better without having to worry about the small details.
Around the time everyone starts meeting up together again, she sees what a mistake that is because not only is she simply reacting to the situation when it came to her, her actions are putting a child at danger. And even worse, pushing him towards revenge. One of her most defining points in the story is when Hope questions everything she's been doing and she does one thing that most characters in stories--hell. Most people in life never do.
She says: "I was wrong." She admits fault to her decision to try to rush in without thinking and attempts to make up for what she exposed Hope to. While playing, you can see her change from someone who runs in head first to someone who considers the well being of everyone around her and sees that her actions affect those around her. Also the fact that she treated Hope like a son/student was some pretty good character development.
Snow: I personally liked Snow. But I can see why people dislike him. His constant need to "Be the hero" and to say it out loud could probably get annoying to some. But for me, it tells a lot about his personality. He's a guy who's trying to do right, but can't seem to. So he sticks his chest out and puts himself in harms way anyways for the sake of others. However, there's a problem with how he carries it. He uses it as a mask for his own guilt.
After his attempt to free people from being purged at the start of the game, failing, and losing hundreds of people to that slaughter of a battle (Including Hope's mother) he tries to brush it off at first, knowing that's what happens when you fight. People die. People have to die for their to be change. However, he has a harder time dealing with that for the people he has come in contact with directly. Sarah and Hope's mother are examples of this.
With Sarah, he blames himself for not finding a way to keep her safe. And for that, he takes responsibility in trying to find a way to bring her back. Needless to say, he was grasping at straws. Even when her sister had given up on her, Snow kept trying to find a way to rescue her and make up for his own failing. This shows a sign of actual love in a story to me. Rather than just "I have to help this girl for some reason...I don't know why..." reasoning that other stories in games give, he was engaged to her. He took the aspect of "Til death do us part" seriously and like anyone else who would, he stayed true to it the way he knew how.
His guilt over Hope's mother was the most ground breaking though. When Hope questioned him about being responsible for the death of others, he at first tries to word his way out of it. Ignore it in a sense. But after pressed about the issue of how do you make up for a life loss, that you caused he admits that there is no price to pay for a life. That even if he begged and said he was sorry, that wouldn't be enough. Even when Hope tried to kill him, Snow told him that he deserves whatever Hope thinks he has to dish. But his mother asked Snow to see to it that Hope stays safe. Her last words were for her childs own safety and security.
And he does carry through with that promise and continued to try to. That shows dedication in a character but also eventual acceptance towards ones faults and the costs others had to pay.
Sazh: I can't help but think that Sazh was the best character in the game. And a statement that Extra credits made about throwing a black guy in the game for the sake of it....I felt was incorrect. (For one, if you want it to stop happening, stop talking about it and stop seeing the aspect of race altogether. But that's another topic altogether.) Sazh fits the mold for the aged adult that gives the reality of the situation due to his experience in life. However he shows how conservative things can become when you age.
Sazh admitted this himself. When you get older, you don't think outside of the box. You don't see things from a different perspective and it's hard to accept that what you've thought to be true was false. And that understanding the position of others is harder as you age as well. And it's true. For example: How many people have actually stopped and thought "I wonder how people who are lower classes get through their life without insurance, a house or a career." or "I wonder how people on death row feel when they didn't commit that crime."
Sazh expresses his concern with that kind of mind set and somewhat hints towards how it bothers him that so many people--even himself tend to fall within that problem. There's also the interesting aspect that he is a father. When Hope is angry at the fact that his father had ceased spending any time with him and thinks that he doesn't care, Sazh takes it personally in the sense that he is a single father and that nothing could replace his son.
However he doesn't lash out at Hope. He simply says "Fathers always care." While that may not be true for many people, it does hit home for people who want to be fathers, or are good fathers and can understand that sentiment. His endless devotion towards his son in the story is one of the most fleshing parts of Sazh. His interaction with Vanille shows a man who is confused as to why childen are dragged into situations like theirs at all. You can see from the way talks to the younger cast, he thinks that the younger generation should let the older generation handle dangerous or tough paths while the younger enjoys their youth.
Also there is something to say for a man who is on the run from a law he doesn't understand, but he is not willing to continue do to his position. He's lost his wife and his son has been taken away from him. At that point he feels he has nothing more to lose, and nothing more to continue for. There's some more things, but I should stop there with him. Seriously, he is a good character.
Vanille: ... This is a tricky one because I don't even like her. But I can't deny the fact that she shows a character that is trying too hard to mask her failings by not letting others know what was her fault. Or even a more sensitive aspect, that she's vulnerable. She's someone who is constantly running from her own problems, afraid to face them and tries to not have anyone point that finger and tell her that she is in trouble.
I'm sure we can all understand that feeling. After all, we have all done something that has wrecked us with guilt and even caused flat out pain, misery or disappointment to those around her. What I DON'T like about her is the exact act she puts on to hide the fact that she's caused everything to happen. I hate the bubble-gum happy happy girl personality. But honestly, a carefree cheerful girl is one that most people smile at on the surface. Yet on the inside something is wrong and people feel that.
Fang: This is also a tricky one. I couldn't pick up all too much character development past "the big sister" vibe from Fang. She more or less shows dedicated friendship on the level of being non-related family. But with the limited content given on how they were before hand, its hard to give much detail into it. I'll have to think about it.
Hope: I do not understand why people dislike him. The majority of the reason I hear is this: "He's a whinny *****! I can't stand him! All he does is complain!" Well, put it like this. If you had no emotional bond with a man who is your father and the majority of direct love you receive in your family comes from your mother, and she was killed directly in front of your eyes, how would you react at the age of 14? For the majority it would take a cold, unfeeling person to not carry that heavily.
Yes, he did whine. However, the only one who was giving him love in his life was torn away from him at an age where we start to wonder just why are we here, who do we need in our life, and what defines us. On average, family tends to be a big part of all three of those, be it negative or positive. And what positive he had was literally killed off right in front of his eyes. A situation that wasn't someone he knew direct fault, he chose to blame it on Snow. The guy who couldn't save his mother as she was dangilling from his finger tips.
Now, normally we can understand that Snow tried to save her. He did. However Hope seeing her fall put horse blinders on. All he could see was the man who didn't pull her up and the only thing keeping him motivated was revenge on that person, even driving him to want to murder Snow over his loss. It makes sense in an odd way. I know I'm the type of person who will die before harm comes to those I love. However, if someone is taken from me, by the actions of another person--be they small or large--my reaction will not be rational.
All I would see is my loss and how unfair it was to lose someone for what I would think is stupid, or even that they did not deserve it. I'm sure many other people have said the same or felt the same along those lines anywhere from "If you hurt my wife/husband...I'll kill you." or "If you hurt my daughter/sister/granddaughter/son/brother/etc. I'll bury you." So his reaction is not unbelievable. The fact that Hope saw that his mother wouldn't want him to sink to that level and forgiving Snow for what happened and he never went through with it shows maturity in the character.
But again, I'd like to go back to his age of 14. It's a confusing time in all of our lives and stacking the death of direct family on top of that would be devastating. And they showed that in his character. Not only the negative aspect but the positive. Instead of using it as a crutch forever, he used it as a chance to grow, and mature.
Now, for the most interesting character of them all in that game. And if you've read this tex wall then...well...you can guess who I think this is.
Hope's Mother: I cannot help but think that she was the most interesting character despite her low screen time, and I wish Extra Credits would have given this one props in their female character episode. I'm making this a point because of a conversation I've had with my mother. Despite troubles that she's had with one of my brother's she had said this to me:
"You know... A mother's love is something that is always there. There is a difference between liking your children and loving them though, but that love very rarely leaves. Despite everything he's said, done, or put me down about the worst aspects of my personality...I still love him. And if anyone were to ever try to do something to him, you had better believe that outside of all that harsh treatment I received from him, I would plunge a knife into the heart of a person who would try to take his life. I would go to the courts, tell the judge my crime, confess of it and if I were to be placed in prison for taking a life, I would not regret it.
There is no substitute for the child you carried with you for those nine months and there never will be. And I would fry in hell before I would see someone steal that short time away." When considering a mother's will to protect their child and keeping any threat as far away from their child as possible, no matter what the cost... it would seem that Hope's mother wins the cake for this. While some may argue "Well, if she wanted to protect him, she should have just stayed next to him the entire time and never left his side."
That would not have been enough. Her goal was to keep them as far away from Hope as possible and if picking up a gun, that she's had no experience with and never taken a life before would -possibly- even stall them from reaching him she would do it in a heart beat. And she did. When asked by Snow if she was sure about her decision, she looked him in the eye and said "Yeah, moms are tough." In that one sentence she more or less stated that if she had to, she would plow through an entire army to keep her son safe from harm.
The best aspect of her character was in two parts before her death. The first being during an explosion she dove in front of Snow---a very large and hardy man who looks like he can be smacked with a 2x4 and laugh--to shield him. This has a lot to say about the role of a mother in general. In dangerous situations, a mother would think "What if this was my child in trouble?" and act before thinking. By attempting to shield his body from harm, she showed her devotion to protect. And right before her death, she told Snow to take care of Hope. To make sure he gets home.
Rather than worrying about her own well being, she is worried about her son over all. I think that says a lot for her character, given how little she was around.
Whew. That was a lot. But honestly, the characters in this game were fleshed out well, went through development and had flaws and insecurities. In general I tend to see a trend that "It's cool to hate this game" or "YOU HAVE TO LIKE THIS GAME!" I call it Sheepfluenza if you give weak reasons like "He whines too much.

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