The simplest way to ask this question is:
“Why do I seemingly get better at billiards/darts/shuffleboard/etc. when I’ve been drinking than I am while sober?”
The slightly more complicated way to ask this would be:
“Does alcohol play any kind of positive role in hand-eye coordination as it relates to games of chance? And, if so, is there a kind of ‘perfect drunk’ state in which the body can reasonably expect better performance?”
Those are games of skill, not games of chance.
Apologies – I obviously have had too many myself.
Okay, so let’s revise – does alcohol consumption play any kind of positive role in games of SKILL?
Mild alcohol consumption can cause you to play more aggressively and take riskier moves, which can pay off (or not). If you count the hits and ignore the misses as we are wont to do, you could easily get the impression that it makes you better.
My friend was a competitive pistol shooter who said that it wasn’t uncommon for a person to take a shot before an event. It had to do with steadying the nerves.
I don’t know if it actually did make a difference or if the effect was only in he and his team mates minds.
I think it’s more likely that alcohol makes you THINK you get better.
Not the way I play darts! 
I kid, I kid … .
Anyway, alcohol may also help the player avoid overthinking his actions. If you pay too much conscious attention to what you’re doing in a game, sometimes your awareness of the situation can interfere with your ability to perform the task itself. We call this “choking”. A small bit of alcohol (or other substances) may help some people get into the zone, and the benefits of being in the zone can outweigh the minor motor and cognitive deficits the alcohol creates.
I was going to mention this. A small amount of alcohol helps me play pool, by simply relaxing me. Without it I will tend to overthink, and end up focusing on everything but a smooth stroke and follow through, which is where most of my problems come from. Unfortunately a little more alcohol reduces my coordination, so maintaining that level is very difficult.
Note to self: Always stand well behind **Hamster King **when he plays darts.
There’s a fine line between “relaxing” and “impairing.” If you can ride that line, alcohol can improve your performance at certain skill games. But it’s a thin line, and hard to maintain. Long-term it would be better to use some other form of relaxation technique.
Of course, that would take a lot of the fun out of it!
Meatros partially beat me to it, but I came in to say that I learned during the last Olympics that alcohol was considered performance-enhancing in shooting events. It’s also banned in professional snooker I believe, and in nine- and tenpin bowling. So it seems that it can be used to relax and steady your hand in games/sports similar to those found in most pubs. However, I think that 90% of the casual darts/billiards players who drunkenly swear “I’m actually better after a few drinks!” are in fact worse while intoxicated, they are merely suffering from alcohol’s extraordinary ability to make you feel more good-looking, intelligent, funny, and talented than you really are. You don’t notice that you actually sink 10% fewer shots while hammered because every shot that you do make is accompanied by your alcohol-ridden ego saying “Fuck yeah! I’m awesome! I’m the best pool player in this joint! And the best looking! And the most interesting and funny and insightful! Yeah!”
There can also be a practice effect. If most of your experience at, say, darts was when you were mildly buzzed, that might make it harder to apply that experience when sober.
There certainly is a fine line. I reckon it’s somewhere between one and two-thirds of a pint and one and seven-eighths of a pint. 
It worked for Dr. Johnny Fever.
You’ll probably enjoy this Mitchell and Webb sketch. It’s a few minutes long but totally worth it.
It worked for Kid Shelleen.
It works for writing code, too.
I would suggest a different “practice effect.”
My first game of darts always sucks. I only play a few times a year, so I start off pretty rusty. Each game gets better because I’m refreshing my skills. So… if I drank while I played darts (I don’t), you’d see a sort of bell curve, where the results improved until the amount of alcohol started to impair me. I could easily see that as thinking that a beer or two made me better.
I’ve only ever seen David Mitchell on QI; very different to see him in character.
If you have Netflix, check out Peep Show (also staring David Mitchell and Robert Webb as two odd-couple friends and flatmates). Hi-freaking-larious.
Beta blockers have been on the Olympics banned substances list for years, due to their ability to help shooters - and perhaps archers too - calm their nerves and eliminate jitters.
To the extent that booze can do the same thing, it might help target oriented bar games.