>
Let's say I have 4 robots and a maze. The only choices the robots are able to make in the maze is to go left, right, forward, or back. 
The first robot has a program it will follow. The program is complicated enough that an observer would have a hard time figuring out the decision tree. 
The second robot will base its choices off of the random particle decay of a chunk of radioactive material inside the robot. 
The third operates with both. It follows a program, but the program is influenced by the particle decay. 
The fourth is a black box, with no way of knowing what is inside the robot or what factors influence the path the robot takes. 
These four robots are released into the maze without any markings showing which is which. After watching them operate for a certain amount of time, do you think you could reliably tell which robot is which? >

Of course.
That's how PacMan works. >

Your realization about determinism/randomness is correct and is a very old realization in philosophy.  
Your conclusion that this makes free will 'impossible' is malformed, in the same way that 'free will' as a concept is malformed.
The standard response to this realization is to say that our intuitive notion of free-will is incoherent and needs to be amended into something that makes sense, rather than to say that free will is 'impossible'.  
Basically, the intuitive notion of free will which this realization argues against isn't even well-formed enough to be possible/impossible, it's just underdefined and vague.  
The most popular coherent notion of free will is compatibilism, which basically says that free will is measured by how much an agent is able to act based on it's own internal impetuses and desires and intentions, versus how much its actions are constrained by outside forces.
This formulation of free will is independent of the determinism question - the agent can be guided deterministically by factors internal to itself, it can be constrained by random assaults from outside itself, whatever.  The definition instead just focuses around how much self-determination the agent has, how much it's behaviors and outcomes are self-motivated rather than externally-determined. >

The problem I have with compatibilism as you're describing it and as I generally understand it is indeed that all it's doing is redefining a term to mean something that most people don't seem to mean by the term. Although it's why I tend to like to clarify with say "libertarian free will" which rather obviously doesn't seem to exist.
However, the idea that free will can be determined seems rather to fail most intuitions on what we're trying to get at with free will. I think. And as such, I guess I need more information on how redefining free will in that way is useful - i.e. how do we expect people to use the term? Why not just stop at the idea that free will is an incoherent intuition. >

Free will is usually a poorly defined term. For many, it's an overly strict definition that, as you've described, becomes impossible. But take the definition "the ability to act at one's own discretion.". The fact that your discretion is predictable or random or both doesn't make it not your discretion. >

I don't think this addresses the problem, your just changing words. You've swapped "discretion" for what OP attributed to "free will". Whether or not you have control over your own discretion is inherent to the discussion. The question is whether your conscious experience of discretion (or decision making) is a happy delusion or the experience of yourself can actually make actionable change on the universe. >

The problem, as I mentioned, is that 'free will' is a poorly defined term. I've provided a definition of free will that is compatible with any of the universes you've proposed. Maybe you could say what definition you're operating under? >

I'll define free will as the ability of the experience of self to make actionable change in the universe.
I certainly agree that this is mostly a game of semantics, and maybe we're using different words to grasp at the same idea, but i guess I just don't understand your choice of definition.
It's more or less saying "whether or not choice is random or influenced its still choice" which sort of avoids the whole point. No one is saying we don't have the experience or perception of choice in our consciousness, I think the idea is whether or not that experience of choice or discretion is actually influenced by our conception of self or just a product of the material things that compose the universe. >

My experience of self is what is making these words be written now. Even if you assume that my experience of self is a 'clockwork self' inside a 'clockwork universe' it doesn't change the fact that my experience of self is what is making these words be written now. So it seems that free will, by your definition, is compatible with the clockwork universe. >

Hmm. I'll concede that my definition needs work. You're correct, my given definition is compatible with a clockwork universe.
I guess what I'm getting at is why do you think the thoughts you do? Why did you chose to write down the words you did? You didn't pick the clockwork self you are.
When did your free will start? 
You've given me some things to think about >


the universe is random: If the universe is random, then that means that all of our actions are random and we can not possibly have free will.

The Universe is Random.    I am not.    
I am very predictable.   
I will choose not to eat black liquorice every time it happens across my path, for  I have the free will to not eat that foul shit. >

Can you choose to like liquorice? >

Can you define "free will" first? >

I can choose - I have free will.
That's it. You can do mental gymnastics to retroactively justify anything, just don't get cause and effect confused. 
The idea that the universe is deterministic is naive and at worst a coping mechanism for people that can't comprehend chaos. >

How do you explain the FMRI studies that show people what they were going to choose about a third of a second before they're aware of the choice they are going to make? Either you have cause and effect backwards here, or it's not your consciousness or self in any normal sense of the words "making the choice".
I'm not sure what gives you the sense the universe isn't deterministic but much of physics seems to disagree with you at a macro scale at least. At extremely tiny scales there's quantum randomness, but like the OP said, that doesn't seem to concur with any conception of making a choice. >


If something is not predictable, then it is random. 

I think this is the leap you're making without proving it. Is random really the opposite of predictable, and is it truly a dichotomy? 
Isn't the Free Will Hypothesis generally a 3rd alternative to these two options, and you're writing it off out of hand? 
How would you object to this modification of your premise?
"If something is not predictable, then it is either random, or determined by the Free Will of some actor(s) who can influence it"? >

I wouldn't say that not predictable means random, I think you should consider a third alternative, where some things are influenced by something kind of external from the universe (our souls). It sounds a bit absurd, but that's what people basically believe.
I personally believe in case 1 (all deterministic). >

This kind of deep examination of free will is, frankly, useless.
We can only observe one outcome. It doesn't matter if it was because of precise deterministic laws or a random roll. >

Death and taxes are predictable.  Nothing else >

What I think is the root here is that strict materialism is incompatible with certain forms of free will.  That is, if all motion is purely caused by the physical interactions of particles and never by our thoughts, then we are not willing ourselves to move, we're just imagining that because we moved.
If we have libertarian free will, then our thoughts have to actually be causes in and of themselves.  This means rejecting strict materialism. >

I’m not sure if physicists have decided if the universe can be truly random or displays pseudo randomness. i.e. to what degree the universe is probabilistic. 
But, I don’t think anyone can claim to have “free will” absent complete knowledge. We can only have pseudo free will because none of our choices are truly informed - your range of choices will always be artificially constrained by your/societies past, and your understanding of that past; by your/societies present, and your understanding of that present; and an incomplete understanding of available options and an inability to properly calculate opportunity costs. 
So, yea… we may have “free will” but only along a fairly narrowly constrained array of choices. >

Free will is only a meaningful concept juxtaposed to force or coercion.
If someone had a gun to their head while they signed a contract, a court will throw it out because it wasn't signed of their own free will. 
Philosophically, people who believe in an all powerful God with rules like "don't murder" and "don't commit adultery" question why an all powerful god would have created a world where murder and adultery are possible but forbidden when he presumably could have created a world where they were not possible. Here people point to "free will" as an explanation.
Outside of that, we are agents that make decisions based on our experience and our biology. We don't choose our biology and we don't choose our experiences, which ultimately makes our decisions deterministic, but it's free will in the sense that we're not being coerced into a particular decision. >

I don't see what predictability has to do with free will. Free will implies being able to freely choose. Even if you choose the same thing over and over again, nothing is stopping you from choosing something else.
It is the absence of the ability to choose that removes the possibility of free will.
It is possible that there is some entity/being that knew that I would reply to this post and could even predict what I'd write, but it is my choice to do so. I am acting on a choice, and not on an instinct. >

Firstly, being "not predictable" and "random" are not opposites.  I can have no idea whether something will go left or right, but that left or right are the only options makes it not random. 
However, in your argument you miss out on what the very idea of freewill is.  It's not that the outcome is random, it's that your will determines it.  From an observer perspective it may appear random in that you couldn't predict the output of the will, but for the agent involve it was their choice.  That's the essence of freewill.  It posits an idea of self as in control.  It doesn't matter that the overall system of the universe is unpredictable at all because within that system there is a determining agent - you. >

Case 1:
A prediction is based on potentialities. I can never be 100% certain that if I drop my coffee cup, that it will fall on the ground and spill. It's just very very likely. On the same level, I don't know that I will be driving home after work today, it's just a safe prediction. 
What you are thinking of in philosophy is called determinism. The idea dates back to ancient Greece an you can read Emmanuel Kant for a much better explanation of what this means for the nature of humans than I could ever decide.  It's the idea that events are always determined by previous events. Like the chicken and the egg or the butterfly effect.
I would say that everything is both random and detirmined at the same time. Or at least, that is the best way to understand it and apply the consequences of it to life. The deterministic nature of the universe is so vast and complex that you might as well describe it as random. There isn't really a good reason to think of the universe in terms of this causality for the most part.
Lets start with reading this post and work backwards. I read this post avoiding my job. You posted it for some reason. You thought it for another reason. The thing that caused you to have that thought is another thing. If we had omnipotence we could trace it back to the big bang. The earthquake in Turkey and Harry Styles winning album of the year at the grammy's could also be directly tied to the big bang. 
Nobody in their right mind would or could ever comptemplate the connection between Harry Styles and the cause of the Earthquake in Turkey despite their connection. This is why it is safe to say the universe is random.
Case 2:
Why can't the randomness be caused by free will? If free will is real, it would cause chaotic randomness. Think of a freeway where all cars are controlled by the same computer network. It works perfectly. That would be perfect determinism. Now throw a random event in there. A drunk driver is randomly thrown into the mix going in the wrong direction. It would cause chaos.
It would be random and if the computer programming had never dealt with it before, it wouldn't be programmed to compensate. There would be collisions that would cause more collisions, and so on. 
This is essentially what happens and it is a good analogy for the course of life. When you think you have things figured out, something big happens and you realize you don't have any control at all, your parent dies, there's a deadly earthquake, you get cancer, etc. Your able to predict things gets shattered from time to time and you change. >

I agree with what you've laid out, and for that reason I've adopted a different definition of free will. I view free will as the means by which "you," by which I mean the deterministic structural/chemical system that is your body, make decisions independent of outside control. External phenomena dictate the conditions and events to which you must react, but your reaction isn't being controlled by something external to you. There are obviously some soft lines here - chemically-altered behavior still involves your own physiology, as an example - but I think that the basic principle is useful. Even if your decisions are dictated by your nature, that nature is who you are, and we can make judgements about you and your nature. >

Short of introducing something additional to your metaphysic, you've got the meat of the problem more or less, but haven't yet reached the final conclusion. 
Our Will is a product of our Minds, and our Minds are either an emergent property of their base components [brain tissue, neutral connections, laws of physics, and so on], or are reducible to these things. 
If our Minds are reducible to these things, and no emergent property exists, then in so far as the universe is deterministic or random, so too are our Minds, and thus, our Will. That is, our Will owes as much to the factors that create "us" as our Will would owe to "us ourselves." The quick conclusion of which is that we only perceive ownership of our Will, but cannot truly say to generate it ourselves. 
If our Minds are some emergent property, then perhaps there is grounds to claim that more is at work. Perhaps it could be that some element of our Mind is not beholden to cause and effect, that there exist a space in which we could be truly Free, and thus that our Will, not bound to the laws of physics but only to the laws of Logic, could also be free. That we would truly say "I might have done other," and mean it in the fullest. Perhaps, but we are reasoning backwards here, and searching for the answer, rather than reasoning from knowledge. It is one thing to search for Freedom, it is another thing to lift up a rock and discover it. 
So far, we reason as you have, and find it very difficult, perhaps almost magical, to suppose that freedom actually exists. How can we possibly say all the world obeys the laws of nature, save for the small spot between our eyes? That spot alone, us ourselves, is exempt. 
A strange thing to say. But we feel free, and ascribe meaning to that feeling, as if it were something discovered and thus true by the merit of its finding. 
Here is the conclusion; if you did a thing freely, then you did so freely. If you did a thing unfreely, then you did so unfreely. If you regret, or blame, or take pride in the act, then either the feeling is justified by virtue of the freedom which you exercised, or as unjustified as all other acts. Either you are free, and permitted to delight in your freedom, or you are not, and absolutely nothing changes. 
Some will say that without freedom, we have no grounds to blame or regret or judge, but if there were no freedom, we would not be free to do otherwise than we already are inclined to do. 
In so far as it is in your ability, freely or by the machinery of your Mind, do not fret. Take the stoic path, do what you might. Thankfully, you will convince yourself in the small moments of life that you are free, that you do things of your own accord, and will not feel burdened by the wondering. >

My mother calls me every night. I know she is going to call me. So her actions are not random. Does that mean she's not really doing it? Why can't determinism and free will both be correct? >

This is a false dichotomy fallacy. You are excluding some of the most widely believed assumptions. I.e...
Case 4: consciousness is neither predictable nor random, but rather self-determining. 
You may recognize this typically as a spiritual belief, but it doesn't have to be. It's not proven, but neither are any of your claims. >

Can you give me an example of free will that would be possible in any fictitious reality?