# Thinking about fine-tuning of explicit probability

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Masacroso
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Joined: 05/05/2014

Hi. I hope you can help me to think together about this topic: fine-tuning of explicit probability, random vs deterministic success.

1) Deterministic events are defined by 0% or 100% probability
2) Pure randomness is defined by the homogeneous distribution of probability between all possible results, e.g. 50% probability for 2 possibilities as throwing a coin (100%/2=50% -> homogeneity of probabilities). For 3 possibilities 33% (1/3) is pure randomness, etc.

So for a event with 2 possible ends, as the throw of a coin, a 25% probability for some of the results is the average between pure randomness (50%) and pure determination (0%).

Of course the complementary is the same i.e. 75% and 25% are the average points between pure randomness and completely deterministic event with 2 possible ends.

Closer to 50% means closer to absolute random event (I repeat: in a 2 possible ends event like the throw of a coin) and closer to 0% or 100% means closer to deterministic event.

So 25% (or 75%) means 50% randomness in the event. And e.g. 10% (or 90%) means that the 2-end event have a 20% of randomness and so on.

For a event with more than 2 ends seems a bit more complicated. A event with n possible ends have a homogeneous probability of 100/n i.e. when probabilities of all success are 100/n then the event is completely random. Lose randomness of a event is going out of homogeneity of probabilities so going >100/n or <100/n.

On these case of n possible cases on a event the loss of randomness is more obvious when one case have a big probability of success i.e. when some case is more probable that the sum of all others, when probability is >50%.

Anyway the mean of deterministic event is attached to one case so to determine the real loss of randomness is not enough taking into account the homogeneous probability of 100/n, we need to have an ability to do a prediction reasonably determined over all the possible cases/ends of the event.

So, in the end, the mean of deterministic success is the same in the case of a event with 2 possibilities, as the throw of a coin, and the case with n possibilities, i.e., determinability (=predictability) of a event is measured by the probability of one event. If a event have a probability of 75% then, as in the case of the throw of a coin, the event have a 50% of randomness/predictability.

Well this is all by the moment. I hope that my terrible english-ability don't ruin too much the legibility of the text :D

P.S.: I said "explicit" randomness to differentiate of the implicit randomness that come from ignorance as when you deal cards.

X3M
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Joined: 10/28/2013
You did a good job. Did you

You did a good job.

Did you know?
There is also more deterministic behaviour when you combine 2 random events of each 50%.

Whereas throwing 2 coins. Both need to end up with side A and B for a certain event.

Thus we have 4 possibilities.
AA, AB, BA and BB.

Yet only 3 outcomes.
AB and BA are the same.

Where this one outcome has a 50% of chance. The other 2 have 25% chance. The average chance is 33%.
This means that AB/BA is more deterministic than AA or BB.

But how to call AA and BB?
Well, I suggest they are also more deterministic as well than an absolute randomness. Because the absolute randomness here is 33%. While they are both only 25%.

***

The fun part starts when you have deterministic outcomes combined with outcomes that appear to be completely random.

When rolling a dice, each chance is 1/6th.
However, when there are only 3 possible outcomes, the absolute randomness is 2/6th.

Now some games have 3 possibilities when rolling a dice with 3/6th, 2/6th and 1/6th chances.
The 2/6th is a random event. While 3/6th will be the deterministic one. Of course 1/6th is also deterministic, since this one will appear less than average.

I noticed that in my game, and many others, the total casualities becomes more and more deterministic when the number of attacking soldiers increases.

The most known game in this is Risk. There is no risk when you have 2 equal armies fighting each other with 1000 soldiers each side. This because both sides will almost certainly loose the same ammount than predicted. Each roll might appear random. But the chance never changes. And you roll more often than there is a random.

So you could say that having more rolls than possibilities in a game is also deterministic.
A game can be very deterministic while it appears to be completely random.

Zag24
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Joined: 03/02/2014
Ummm...

I started out wondering if you knew something I didn't, or you were making stuff up. I'm leaning towards the latter.

This term "explicit probability" -- that's something you've made up, right?

In serious discussion of probability, you should drop the percents and just use fractions. So if you have an action (i.e. rolling a die) with n possible results with equal chance for each, you would say that any one of those results has 1/n chance of happening. (100/n is just wrong. You were trying to say 100%/n, which is the same as saying 1/n.)

This sentence: "If a event have a probability of 75% then, as in the case of the throw of a coin, the event have a 50% of randomness/predictability." I just don't get what you're trying to say, at all.

If I understand you, you're trying somehow to say that an event might have results skewed from equal probability of each result, and trying to say that this is somehow different from randomness, having an element of determinism. I don't believe that this way of thinking about it is helpful at all, neither intuitively nor mathematically. An Event (i.e. rolling a 6 in one try on a fair die) has some chance of happening (in this case, 1/6). It sounds like you're trying to say that this event with a binary result (it happens or it doesn't) but not a 1/2 probability, is therefore partly random and partly deterministic. If you can find some utility in thinking about it this way, then I'll be surprised and impressed; but I doubt you can.

X3M
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Joined: 10/28/2013
I don't know if you replied

I don't know if you replied to only his post. Or to mine as well.

To put it simply.
A game runs on random options decided by any tool nessesary.

Flipping a coin.
With a chance of 1/2nd side A and 1/2nd side B, it is completely random. Each option has just as much chance to be choosen than any other option.

Rolling a die.
The same goes for dice. Where each number has a chance of 1/6th to be choosen.

Deterministic means that the outcome is most likely to be the same over and over. There is a smaller chance that it is going to be another outcome.

With flipping one coin, you cannot have this.

But with flipping several coins, there are outcomes that simply happen more often.

With dice you can also say that 5 or lower means a hit. Than the hit is deterministic. While with 3 or lower, the chance has reached a maximum of randomness.

If a game runs on the outcomes of random events. Then the deterministic events decide the thread through the entire game.

Zag24
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Joined: 03/02/2014

Yeah, I had not seen your post when I made mine. (I spent a while typing, because I was called away to a meeting.)

So I understand your point, from a gamer's point of view. (My background in probabilities is mathematical.) So my question is: what value do you get out of thinking about random events as having some deterministic aspect to them?

Statistics already has a measure of "randomness" of an event, which is standard deviation. I can tell you exactly how this measurement is valuable, what sort of predictions and calculations the knowledge of this value will make possible. I'm looking for the same sort of thing for this value.

Masacroso
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Joined: 05/05/2014
Zag24 wrote:I started out

Zag24 wrote:
I started out wondering if you knew something I didn't, or you were making stuff up. I'm leaning towards the latter.
[...]

If you can find some utility in thinking about it this way, then I'll be surprised and impressed; but I doubt you can.

I understand what you said. I just tried to talk not too much technically.

Yes, of course 1/n is 100%/n, I preferred use % for avoid misunderstanding... maybe a bad decision of me.

And you are right about "deterministic", it is just a way of talk non serious. Im talking about the predictability, the chances of predict a event with success.

You can predict a event with a 90% of probability more easily than one of 50%.

When I said that the 75% of a event represent just a 50% of unpredictability/predictability its because a event with a 50% of probability is just pure randomness... you cant predict absolutely nothing about it.

When you move out of 50% of probability your chances of success in prediction increases to the absolute predictable points of 0% or 100%.

I know that the things changes when you repeat a random success or you do multiple random events at once (like throw 2-3 or more dice). I know binomial distribution, permutations and the combinatorial basics.

Bur what I was thinking is about the meaning of the numbers. If I have some meaning on them I can fine-tune a game more easily.

Zag24
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Joined: 03/02/2014
Fair enough.

Masacroso wrote:
Bur what I was thinking is about the meaning of the numbers. If I have some meaning on them I can fine-tune a game more easily.

Fair enough. I apologize for coming off a bit harsh, earlier. Coming up with a new way to think about something like probability is really a value in itself. All great new ideas started out as an idea that somebody else probably said was a thought not worth exploring.

Masacroso
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Zag24 wrote:Masacroso

Zag24 wrote:
Masacroso wrote:
Bur what I was thinking is about the meaning of the numbers. If I have some meaning on them I can fine-tune a game more easily.

Fair enough. I apologize for coming off a bit harsh, earlier. Coming up with a new way to think about something like probability is really a value in itself. All great new ideas started out as an idea that somebody else probably said was a thought not worth exploring.

No problem man, you are kind. Your point have value, Im not the best on probability mathematics and I like A LOT maths.

I think the core of maths are in probability... from the theory of groups, theory of numbers... I really dislike when maths of probability are taken as a "not as important part of maths", you know that these maths are more related to economics and less (at least in a classic point of view) to physics or engineering.

Anyway: my knowledge is poor. The approach to maths searching for "meaning" is "wrong" (because maths objects are pure abstract "geometry" and the "meaning" is some perspective ever, philosophy of language can explain this very well... I dont have the words on english to continue a text about that and anyway this is off-topic) but useful and valid (in the history of maths it have some important). It can create some perspective.

X3M
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Joined: 10/28/2013
I find this discussion rather

I find this discussion rather interesting.
Since you want to have an equal chance for all the players.
In that case you need to get to the fine line of randomness in the long run. If you have to much determinism. Than it is decided on before hand who is going to win.

This determinism can be caused by 2 factors in the game.
1. The player shows his/her skill (planning/experience/swiftness etc.) and there for wins.
2. Certain situations give a sure to win outcome in most events.

Where 1 is generally accepted.
2 however needs to be avoided as much as possible. Because it transforms the game into something predictable.
No one wants to play with the "almost always" losing side.

Quote:

In physics we have quantum mechanics. They are also all about statistics. The position of an electron within an atom is decided by statistics, there is actually a probability that the electron is there. Electrons simply move very very fast within the atom, thus creating a shell of this probability.

In chemistry we have calibrations. These too are all about statistics. After all, how well can a machine measure? Some analytical machines count particles, and you only get a good analysis when you count enough particles.

2 cases of determinism in nature through a lot of randomness. ;)

Masacroso
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X3M wrote:I find this

X3M wrote:
I find this discussion rather interesting. Since you want to have an equal chance for all the players. In that case you need to get to the fine line of randomness in the long run. If you have to much determinism. Than it is decided on before hand who is going to win.

This determinism can be caused by 2 factors in the game. 1. The player shows his/her skill (planning/experience/swiftness etc.) and there for wins. 2. Certain situations give a sure to win outcome in most events.

Where 1 is generally accepted. 2 however needs to be avoided as much as possible. Because it transforms the game into something predictable. No one wants to play with the "almost always" losing side.

Quote:
In physics we have quantum mechanics. They are also all about statistics. The position of an electron within an atom is decided by statistics, there is actually a probability that the electron is there. Electrons simply move very very fast within the atom, thus creating a shell of this probability.

In chemistry we have calibrations. These too are all about statistics. After all, how well can a machine measure? Some analytical machines count particles, and you only get a good analysis when you count enough particles.

2 cases of determinism in nature through a lot of randomness. ;)

Yes, I wrote a very similar commentary here.

I understand what you says X3M... We need to avoid, as much as possible, a deterministic nature of a game to evade boredom or annoyance but, at the same time, we need to let that a player get stronger by success in his decisions.

This type of "contradiction" is what make a game interesting. You can see, by example, that a bad move in a chess championship early can cause the player to abandon because he knows the advantage of the other player is unbeatable.

And if you put too much random on the game or just deny value to success plays of players then the game lose his competitive dimension. For a social game its ok but for a competitive game of course not.

So a important point of analysis in a game is how important a decision is in the stage of the game: in the early, mid or late stage of any game, in others words: how the weight of a decision evolves (weight of a "move" or "play" related to the probability of win the match or the round).

Generally in classic games early decisions have way more weight than mid or late decisions. And because this the game on mid and late stages it is generally determined more by mistakes than success of the player.

Anyway this is very arguable, I just says here some of my perceptions of some classic-board games as chess and baduk.

On poker or other games maybe different because the weight of a decision is not strongly determined by the time is played because the strength of a bid is chosen by the players circumstantially. Anyway some degree of weight-determination exist on poker due to accumulative nature of the pass of time than generally mean accumulation of some kind of points or something similar.

And, of course, in board or cards games probabilities narrows when the time pass (less positions, less cards, less valid strategies cause difference on score... less degree of freedom in general).

In goofspiel this is an explicit core of the game itself: the narrowing of probabilities when the turns/time pass.

A balance for the narrowing nature of many games i.e. the evolution to deterministic state of many games is the concept of round. When you play a game divided in several rounds you are forcing the players to play from zero (reset the state of game) over and over. So the best player is who accumulate more successful rounds.

This concept is used in all sports: chess, football, tennis, poker, etc.

EDIT: I must add a last rumination about the contradictory nature of a game: any game have a end but, at the same time, the funny of a game is being played.

So the contradiction is that a game consist of a determinate final state (the end) and a undetermined state of properly game. If the final of a game is determined in some point but it continues then there is a no-game, just pseudo-game, at least in a competitive vision about what a game is.

So the point is that a competitive game must not be a pseudo-game or, at least, must not be a pseudo-game the greater amount of time of play.

Anyway in some point the game that is being played must be quasi-determined, i.e., must be a potential (very high probable) pseudo-game.

If the game is a real game all the time/turns (undetermined final state beyond a reasonable probability) but the final turn/move/play then the game loses the competitive nature or maybe very random. In others words: if the qualitative final state of a game (qualitative: major winners/losers) is determined just in the last move/turn then the game may be excessively random.

What I am trying to says is that a game, to be competitive, must have a portion of pseudo-game on itself because a game represent a evolution of some kind... evolution to the final state, from a pure undetermined state from a pure determined state.

This transition from undetermined to determined is present in all competitive games. If there isnt an evolution then the game is extremely random (or extremely complex) so it isnt competitive. So, in some point, the game that is being played is completely determined or, at least, in a high degree of probability it is determined (=pseudo-game).

EDIT 2: Another strategy to avoid excessive complexity/randomness of a game is the concept of points. The points itself, if there isnt a big difference in score, arent the direct representation of a state of win/loss.

This is why the combination of scoring by points and set up a game in rounds is very useful to control the depth and evolution of a game.

Masacroso
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Adding more info: it seems that the way I was thinking is basically the same idea of entropy.

X3M
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Joined: 10/28/2013
More faces of determinism / A deterministic score

To add a third one to my last post of deterministic games. Some games follow a certain story line, forced by the higher certainty of "dice rolls". In this the game makes sure that both players roughly suffer the same concequences. And the last couple of rounds are the real random aspect of the game where the effects of a dice roll can be a disaster.
(Monopoly)

To add the fourth one. The game has a constant randomness. But the very next round, there is an event that will return the balance again.
As example: if a soldier receives damage, he/she starts to panic and will show a higher chance on throwing grenades instead of just shooting. These grenades do more damage. So if this soldier will have -10% in health, it will also have +10% in damage.

*****

Could it be possible to assign deterministic scores to games?
The higher the score, the more deterministic the game is?

For this we could calculate the highest chance and divide it by the average/random chance by looking at all possible outcomes. This is the score for a single roll sequence of dices.

It could also be asigned to a number of rounds that a game could take. For example, "A game of Goose". Where the average time is 18 rolls/player. The chance that this game has 18 rounds/player is the highest.

If it comes to war games, you could even calculate the deterministic factor of even groups of units.

pelle
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Joined: 08/11/2008
I don't know about all the

I don't know about all the words used in this thread. I have a quite good grasp of the maths behind it though, and also designing some games have made me think a lot about this (some stuff I wish I thought of earlier).

The two main things I think are (and very similar to what others mentioned above, but in different words, I think):

- If you use dice, make sure that there are not a few rolls that are very important. Make sure there are lots of rolls. But also that players can affect the probability of success. If we play a big wargame and each of us makes 500 attacks, if my attacks are always made with 60 % of success and your attacks have 50 % of success I am almost guaranteed to win the game. There is randomness, but in reality it will not mean that anyone is going to win by "luck".

- In a game with few rolls, which is unavoidable in a small game unfortunately, like my Trenches of Valor game, as a player I don't think it works very well to have 1/6 probability rolls that can instantly win the game for one player. My game has that problem. Because as a player you look at the odds and when something is only 1/6 you can't be blamed for not planning for that. I think the probabilities should mostly be kept around 50 %. Wish I thought about that earlier. If I know that positioning my unit in one location will give my opponent a 40 % chance of hitting it, it is something I can expect and it will not upset me (or my plans) much if he scores a hit. If my opponents score a 16 % hit (or I miss a 84 % hit) that can be really annoying. I know this is very subjective, not mathematical, but I think it makes sense when designing a game, so I try to think about that now. If I designed a game using just 1d6 much now I would keep almost all rolls as hits on 3 or 4, a few rolls hits on 2 or 5, but never or almost never hits on 1 (unless it is for something not so important just for flavor that will not win the game for either player).

Masacroso
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pelle wrote:I think the

pelle wrote:
I think the probabilities should mostly be kept around 50 %. Wish I thought about that earlier. If I know that positioning my unit in one location will give my opponent a 40 % chance of hitting it, it is something I can expect and it will not upset me (or my plans) much if he scores a hit. If my opponents score a 16 % hit (or I miss a 84 % hit) that can be really annoying.

It depends on the effect of the roll. You can setup the roll to any probability for something, 10%, 50% or 70%. But the effect of the roll (snowball) it is the thing that make you put some probability or another.

By example: you can setup to a 1/6 of probability something that make change your strategy for this turn or the next, not something that change the strength of something if not that change your options.

If you setup something at 50% it will be more to add some atmospheric effect or maybe something not directly related to a player, maybe just some variable to create a mao or something (something imitating brownian movement or random walk by example).

These are just some examples... games are a thing to wide. All depends about the effect of the roll and what you want to feel/make with a game. I was just discussing the possible meaning of a probability but anyway all depends on the context of the roll and meaning of the roll to see if is what you want to do with the game or not.