Identical games of billiards with one double the speed of the other The situation: two friends play billiards and we should all hurry to the plane already but the game is still yet not finished. Time is about to run out and players decide to play the rest of the match double speed. The question follows:
Suppose we play a game of billiards (we'll call it the game A) and have information about every ball position and every momenta at every instance of time. We have all the input forces and all masses and a known friction. Could we create a game identical to the game A but twice as fast (the game B)? What parameters do we possibly have to change? Maybe something like a game table half the measures, balls half the size, and something...? Is there need to change any fundamental physical constants?
Basically the question is: can we double the speed of a dynamical system of billiards by changing universally certain parameters?
 A: 
Is there need to change any fundamental physical constants?

No, I don't think so. You are still dealing with a classical system, so if you were able to scale everything correctly, as you have already mentioned, I don't see anything obvious that would prevent you from finishing the game. Two players playing billiards at 95 percent of the speed of light should not notice any difference to their Earthbound friends, playing the same game. I think that is equivalent to what you are asking here. Apologies if I have misunderstood you.
But I would be interested in alternative answers that I might learn from.
A: In the ideal case, if you have full information, yes, it's easy to scale as you suggest.
But the system is chaotic, so you'll need knowledge of an awful precision the reproduce the game even at the same speed.
And in the non-ideal case, especially if you need a higher speed-up, then the corresponding scaling-down will at some point make air resistance important, so you might need to reduce pressure, and you'll also need different materials to keep friction properly scaled.
A: Our answer assumes the following for the billiard table:


*

*Ball-ball and ball-cushion collisions preserve momentum and energy.

*You are not using spin (aka English), which also assumes that the balls are not rolling. Notice that this is not true in a game of pool, but corresponds to an idealized billiard table.

*All balls have equal mass.

*You cannot/will not perform the next shot until all balls are at rest.


Your problem is reduced to the following: Given an initial configuration of a billiard table, can I achieve the same final position in time $T$ AND time $T/2$? You simply repeat the question for each individual pool shot.
Since all balls have equal mass, (and also assuming your billiard is not relativistic) momentum and velocity are equivalent. We are now going to examine 3 distinct cases based on the type of friction:
Case 1: No friction: Possible but not Physical
In this case, simply doubling the momentum would have the game play at double speed, since all the balls would cover double the distance at the same time. There is a problem however; the balls would never stop to allow for a second strike. So this case is a bit useless.
Case 2: Constant Friction: Physical but not Possible
In the case of constant friction, the velocity of a particle travelling for time $t$ is simply $v = v_0 - at$ (assuming $a>0$). However, in the billiard table you are measuring distances, so you probably want to use the formula $v_0 = \sqrt{2da - v_f^2}$ where you mainly substitute the final velocity as $v_f = 0$. 
You can immediately see the issue here: the first equation gives you the time you need to go from some $v_0$ to 0 velocity (stop). The second equation tells you how much distance you will cover in that time. As you can see, there is no way that you can half the time while keeping the same distance. This is important, as you want to play out an identical pool game.
Case 3: Complex Friction: Physical and Conditionally Possible
This is the most common non-constant case about friction. It depends on the ball velocity (i.e. it is stronger the faster the particle goes). In Newtonian dynamics, you would write this as a drag force of $F_d = -\gamma \dot{x}$. Notice that $\gamma$ introduces a time-scale, because it has units of inverse time. Since this drag needs theoretically infinite time to make velocity be exactly zero, we assume that after 5 timescales $\gamma$ (and you can use any number here) the balls have stopped.
The equation for the velocity is simply $v(t) = v_0*e^{-\gamma t}$. This gives you the equation for the distance: $x(t) = \frac{u_0}{\gamma}\left(1-e^{-\gamma t}\right)$. You want to have $x(t) = d = const.$ AND $t \ge 5/\gamma$ (5 is an arbitrary number here). Thus, you solve:
$$ v_0 = d\gamma \frac{1}{1-e^{-\gamma t}}$$
Notice that here we do have 2 parameters that we can interchange: both $v_0$ and $t$ are free. So the whole problem boils down to: given that a $v_0$ gives you $t$, is there a $v_0'$ that gives you $t/2$? Fortunately, this is always possible as long as $t/2$ is also $\ge 5/\gamma$. Specifically, if your game A is performed in time $t$ then the solution of the initial velocity in game B is:
$$u_0' = \frac{d\gamma}{1 - e^{0.5\gamma t}}$$
which is is always solvable since $e^{0.5\gamma t} < 1 \; \forall \; t \neq 0$. The angles are not discussed because they are not affected by the friction type (assuming no spin).
There is a catch here: We have not used the case of both constant and non-constant friction together, which is the more likely way for a real pool game. Also, rolling is a huge factor since after the first couple of instances that your cue stick contacts the cue ball, the ball starts to roll. These are complications that well exceed the scope of this answer, but we hope that we have given you satisfactory insight to the question.
Final Answer
Whether it is possible to have a pool game play out in double the speed depends on the type of friction. You do not need to change parameters, you only need to adjust initial velocity.
In the case of including spin, the answer is very complicated. My intuition would say that in general no, you would not be able to play the exact same game in half the time simply by adjusting the initial velocity. 
