Falsification

It sounds like your friend is using 50-50 to mean "we have no evidence either way". In itself, that's problematic; but probably the bigger issue is violation of Conservation of Expected Evidence. In other words, there are things he could see which would make him think it more likely that this is the Matrix (e.g. unmistakable glitches in ordinary physics), so the fact that he hasn't seen any of them should make him think it less likely that he's in the Matrix.

The more common name is the simulation hypothesis.

I don't know if it's been discussed in the sequences, but this paper argues, "Unless we are now living in a simulation, our descendants will almost certainly never run an ancestor-simulation."

As far as I'm concerned, the simulation hypothesis is firmly in the class of propositions which probably do not matter on the time scale of my expected lifespan.

Occam's razor is in your favor here, although there are more compelling arguments than the one your friend is making (see paper-machine's comment).

Matrix is a nice metaphor, but what exactly does it mean?

If it only means that there is some lower level below us, something like ones and zeroes in a computer memory, I agree -- we are constructed from quarks and similar particles, which we don't see in normal life. By scientific theories and experiments, we can discover these elementary particles, just like people living in a computer simulations could experimentally discoves some basic properties of their universe. Are we just using "computer" as a metaphor for "physics"?

There is more in the movie Matrix metaphor. It says that there are intelligent beings outside of this universe, and that this universe was designed and is manipulated to fulfill their goals. It also says that there are bugs in this program, which allow people to do magic, win against these intelligent creators and escape outside.

Note that the manipulation is necessary for the metaphor, because if this universe is just a fair simulation of physical laws, there is no difference from being a real universe. We can discuss whether a simulated universe has the same "qualia" as the original one, but the point is that if the simulation is exact, then the same things will happen in both universes, therefore your suspicion of being in a simulation is useless, and you cannot break out of the simulation any more than "2+2=4" could break out of your calculator. And psychologically speaking, the manipulation and possiblity of escape is what makes all this Matrix discussion interesting, isn't it? If you say "let's assume that some aliens outside of our universe made a detailed movie of it, and are watching it now, so in some sense we both exist in the reality and as a movie stars" that's boring. It would not motivate you to do anything differently.

The motivation in the Matrix movie did not make sense -- even if the basic idea wouldn't contradict thermodynamics, the machines would have made a smarter choice by using pigs instead of humans. Let's say that instead of energy they would harvest human thoughts. (We could assume machines that don't understand their own design, can't build a machine-friendly AI, so making a simulation for humans and letting them solve problems is useful for them.) Even so, it would be completely stupid to make a simulation where people can break the rules simply by the power of their imagination or will or whatever. How exactly is the gravity simulation routine programmed that when Neo simply thinks "I believe I can fly" strongly enough, the simulator will change the gravity vector of Neo's body? And by the way, if the Agents can posess anyone in the Matrix, why can't they simply possess Neo (instead of running after him) or why can't they simply kill him? Why can't agents do the same kind of magic and make Neo levitate helplessly in the air? Why shoot at Neo when levitating any object could kill him just as well? Simply, why the simulation constrains the machines to some rules, but not Neo? It does not make sense, except as the classical paranoid setting -- an enemy that is both omnipotent and omniscient, but at the same time kind of stupid and helpless against an adversary who knows.

The difference between "Matrix hypothesis" and "simulation hypothesis" is that the former is a pseudo-scientific justification for whatever, while the latter adds up to normality. This world may be real or simulated, but the rules remain the same, so you should not care; what difference does it make whether the quarks are simulated by computer, if the simulation if flawless? If the simulation has flaws, they may appear as new physical laws and maybe we can use them to extract some energy -- so in this sense it is useful to look for them, but it falls under the umbrella of particle physics. (A simulation hypothesis is not helpful here, unless it contains details about how exactly are we simulated and what kind of bugs can we expect in the simulation.)

In other words: If you believe that we live in Matrix, what testable predictions do you make that you wouldn't have made otherwise?

Privileging the hypothesis: here

The concept of 'living in the matrix' is only meaningful if there is some difference with not living in the matrix. Thus, there should be something, at least in principle, that we can observe. And based on this observation (or lack thereof) we can determine probabilities.

Regarding the 50%, suppose (for the sake of argument) that there really is some property of the universe that is either true or false, but we have no evidence whatsoever in either direction. Then the probability is simply unknown, not 50/50. Of course, as a Bayesian prior starting point, you can use 50%, but that is only a rule-of-thumb, and shouldn't import all the connotations of a "50%" in cases where we do have facts.

You can't assign all hypotheses 50% prior probability, "Occam's razor" or otherwise.

If you assign H1 and H2 50% probability, then you shouldn't assign H1 & H2 50% probability. If you assign "The world is a simulation" 50% probability, you shouldn't assign "we live in the matrix" 50% probability. If you assign "There is an objective state of affairs, which my senses don't reflect in the expected way" 50% probability, you shouldn't assign "The world is a simulation" 50% probability.

To get around this, you either need to privilege the matrix/simulation hypothesis aggressively, or give up on your beliefs being consistent under even the simplest applications of modus ponens. Neither seems defensible.

First of all, that we are "in the Matrix" is one of a zillion zillion functionally equivalent and indistinguishable theories of "simulated reality". While we're at it, maybe we're in the Matrix_1 in the Matrix_2. Maybe in the Matrix_3. Maybe in the Matrix_4. Etc.

So your friend is just incorrect. Once you admit "The Matrix" as a possibility, you're committed to admitting an infinite number of theories into your universe of discussion, so the odds of The Matrix being true are 1 out of infinity.

Bur as long as we're not contesting the statistical regularities we observe where ever we are, I don't care which of the zillion we're in. Until your friend comes up with observable and falsifiable consequences to his Meta Reality, I will treat it as a superfluous complication to my model.

Until "we're in the Matrix" can pay rent, it's not living in my head.

As mentioned by others, it's important to draw a distinction between having no information about the relative probability of two events, and saying they're equally probable.

If you aren't going to apply occam's razor here to eliminate the matrix possibility, then don't you also have to give equal weight to all other alternate possibilities which are mutually exclusive with these two, no matter how complex? Each time you think of a new idea, it will have to share probability with the other ideas and the probability of each isolated scenario will approach zero.

For example, what if our brains are controlling artificial bodies that exist outside of any matrix, but our sensory input is being edited to alter specific details we observe (such as making us unable to notice clues that this is happening)?

Check out Occam's Razor. The Simulation Hypothesis requires that a real, physical universe exists, and that someone is simulating another universe within that "real" universe. P(our universe is a simulation within a "top level" universe) < P(ours is the "top level" universe), given no further evidence of simulation. The God hypothesis (typically) assumes the existence of a complex, sentient being -- not really a simple explanation when known physical laws can describe our observations.

By Solomonoff induction, we should suspect we are in the matrix if we can find a mathematical description of a simpler universe that simulates ours even when taking into account the loss of simplicity that results from having to precisely identify our own universe simulation in that other universe (having to identify the simulation prevents all universes from being reduced to the simplest universe which simulates all possible universes).

Although, in the terms of Solomonoff induction it only even makes sense to ask the question if being in the simulation makes a testable difference (i.e. interference from outside entities or we are fated to break out).

First, an obligatory SMBC reference. Second, the probabilities might be guessed as 50/50, but the confidence intervals on those numbers are so large, they make the numbers themselves meaningless. And third, would you do anything differently if someone convinced you that you are simulated? If not, what does it matter? If yes, why aren't you doing it already?

His argument is based on two things:

Being in the matrix and not being in it each have a prior probability of 50%.

Our world is equally likely both in the matrix and out.

I vehemently disagree with this one. For one, we have never been told that we're in the matrix. This would most likely happen if we are in the matrix, and almost definitely not happen if we are not. Also, some people live lives much worse than to be expected in a world with a remotely benevolent behavior. There is no reason for so many people to suffer. It's possible that I'm wrong about that, in which case the problem is that there is no reason for so many other people not to suffer.