Getting highscores and probability....

SuperInfinity

I was idly trying to get highscores in arcade videogames at a forum recently and it struck me how useful the mathematics that come into this straightaway are.

Suppose Billy is playing a game where for every point following, he has X chance of surviving (surviving = not losing a life). Suppose this X remains the same throughout the game (which often happens after you reach a certain level of an arcade game). Suppose his average score per life is 15,000 over 1,000 games.

What is the chance that Billy will get 15,000 or above on his next go? Is the answer exact or is it an estimate and what does the estimate depend on? What is the standard deviation of his score? Can you identify the mathematical constant the probability of getting a certain score is similar to and try and explain why it is like this? (I'm not 100% clear on this myself yet!!)

Suppose Billy has to play Adam in a game with one life each. Adam's average score is 16,000 over 1,000 games. What are the chances that Adam will beat Billy?

What if both players have three lives, what are the chances Adam will win then?

Anyone who has played a videogame knows that if everything goes haywire near the start, it's better to just restart the level as that attempt is probably hopeless anyway. Restarting saves time overall.

Suppose Billy has 3 lives and is trying to make a high score but has a quota of 8 hours in which to do so. Assume Billy gets exactly 3,000 points each minute. Before what two times (if he loses his first life early on or his second life a little later on but not late enough to justify his continuing) is it appropriate for Billy to restart his game to maximize his highscore probability?

That last question is actually guesstimated all the time by people playing for a highscore, and similar things are guesstimated at all other times in life to see what's "worth" doing. It would be an extremely useful one for people who are serious about highscores to answer. You could also make X a function of time for when game difficulties vary during the game. Or X a function of where you are in the game, or both if you're really crazy.

I believe this is a pure form of mathematics where you are coming up with things yourself and can clearly see where everything is ocming from, rather than depending on esotheric formulae etc. Some of the ways people teach mathematics such as simple probability is ridiculous, they teach combinations as simply rules to be learned... what a joke. :rolleyes:

Have fun with your mathematics. :cool:

SuperInfinity

I'm watching Who Wants to be a Millionaire and it is head-banging-off-table stuff.

There's a guy who had a 50/50 and a phone a friend, and he knew the answer was either a or c. Now if there was ever a time for 50/50, this was it.

Usually 50/50 offers no help, but this there was a 2/3 chance that he would get the answer straightaway. What does he do? He phones a friend and wastes that life time.... ridiculous.

And another thing: if you have $8,000 and you know it's one of two but have no idea which it is.... FOR THE LOVE OF GOD PICK ONE OF THEM AND GO WITH IT!!!! The most you can lose is 7,000 while you have a 50/50 chance of winning 8,000 and a chance of winning 16,000, 32,000 etc.

This guy actually did BOTH of these things. He didn't use his 50/50 for the question where he knew it was either/or, then for the next question he used his 50/50 so he had at least a 50% of choosing whatever it was. But he decided to take the money and leave.

It doesn't make any sense, no matter what way you look at it. It's ridiculous.

Fremen

Some people can't afford to gamble money like that, paradoxical as it might seem. I was in a situation a few years ago where I was sitting on three grand in a poker game, and had the choice of gambling two thousand to potentially walk away with six (last three players in a tournament, about five blinds each and turning into a total crap shoot).
I was twenty two at the time, and two grand was an insane amount of money. I simply wasn't able to justify the risk.

There's a concept known as "risk aversion" which comes into play in a major way when dealing with individuals. it's built into just about every financial model out there. You're correct that on average it's better to guess at a 50/50, but people get very protective about wealth when they have it in hand. Even monkeys have been observed to display this behaviour.

The set-up you described in your first post is typical of what's known as mathematical modelling. This is where you make a bunch of reasonable assumptions to make a seemingly impossible problem analytically tractible. It's the kind of thing that typically wins the young scientist's competition.

There's a whole undergrad conpetition about mathematical modelling. You can see previous problems here:
http://www.comap.com/undergraduate/contests/mcm/previous-contests.php

SuperInfinity

Fremen wrote: »

Some people can't afford to gamble money like that, paradoxical as it might seem. I was in a situation a few years ago where I was sitting on three grand in a poker game, and had the choice of gambling two thousand to potentially walk away with six (last three players in a tournament, about five blinds each and turning into a total crap shoot).

Yeah if I'm understanding this correcly your decision was mathematically sound, you could have done either. These people clearly lost out, even if they had zero chance of getting the following question they lost out. But they also would have had a good chance of knowing or somehow getting the question for 32,000, where they would have at least a quarter chance of getting 64,000. If it were just lose 7,000/win 8,000 I might have sympathy for taking it then. I think as well as risk aversion, maybe they don't want to feel "humiliated" by losing money, ironically they really humiliate themselves by this.

The woman even actually had a good theory as to what it was and was right but took the money anyway. As it turns out, the one that the guy guessed it might have been was wrong.... he probably thought: "I made the right decision so" :rolleyes: I accept there can be a non-linearity at the higher levels, for example I wouldn't go 50/50 with 10 million to make 30 million. But with 8,000 euro it's kind of ridiculous.

Thanks for the rest of your post, interesting.

equivariant

SuperInfinity wrote: »

Usually 50/50 offers no help, but this there was a 2/3 chance that he would get the answer straightaway.

That's assuming that the 50/50 eliminates combinations with a uniform distribution, which I don't believe to be true.

SuperInfinity

equivariant wrote: »

That's assuming that the 50/50 eliminates combinations with a uniform distribution, which I don't believe to be true.

It does, as Chris Tarrant constantly says it does and then he says: "would the computer please take away two completely random wrong answers and leave us with the correct answer and the one remaining wrong answer".

Fremen

SuperInfinity wrote: »

It does, as Chris Tarrant constantly says it does and then he says: "would the computer please take away two completely random wrong answers and leave us with the correct answer and the one remaining wrong answer".

Depends on your definition of "completely random". If you mean maximum entropy, then yeah, it's uniform. If you mean each event has some probability strictly between zero and one, then you can pick whatever distribution you want.

I know for sure the pub game doesn't pick uniformly.

SuperInfinity

Fremen wrote: »

Depends on your definition of "completely random". If you mean maximum entropy, then yeah, it's uniform. If you mean each event has some probability strictly between zero and one, then you can pick whatever distribution you want.

I know for sure the pub game doesn't pick uniformly.

Not so helpful this time Fremen. We know nothing is ever completely random, dur. I recommend not replying to this.

Fremen

I doubt you understood my comment. I doubt you know what a random variable is either.

SuperInfinity

Fremen wrote: »

I doubt you understood my comment. I doubt you know what a random variable is either.

Then you are mistaken.

I don't care about your issues, please take your pseudointellectual nonsense elsewhere. Throwing out terms like that in an attempt to pretend you know something about them when you obviously have no idea what they mean yourself.

Once again: I don't care about your issues. A 6 year old child can understand what Chris Tarrant means there, and here you are trying to obfuscate it. I already asked you not to reply back.

LeixlipRed

Ladies, anymore of this jazz and the ban hammer comes out. Grow up.