Solution to pascal’s wager

For those who don’t know, Pascal’s wager is an attempt to make a logical argument for believing in god. The idea is that if you agree that heaven has infinite value, it follows that if your belief is that the probability god exists is greater than 0 (and even Richard Dawkins would only rate himself at 6 out of 7 in the strength of his belief that god does not exist) then the value of believing in god is infinite, because any non zero number multiplied by infinity is infinity – if you assign a probability of 0.01 of god existing then the value of believing in him to reach heaven is 0.01 * infinity = infinity. For more information, check out the wikipedia article.

Allowing the possibilities of states that have infinite value (such as heaven) implies that any decisions that can both potentially lead to reaching these infinitely valuable states necessarily have equal value. For example, exercising for 50 minutes and punching one’s self in the face are both of equal value to someone who believes that the probability of reaching heaven after doing either of these things is not exactly 0.

Assigning a future state the value of infinity in decision systems has a very powerful destructive effect. Because a future state of infinite value has a non-zero probability of being reached, all the parent nodes have infinite value too and that will ultimately lead to the current node also having infinite value, meaning that there will be indifference to all current decisions. Thereby making it not much of a decision system anymore because it has no way to discern between any choices.

Those who do believe in states of infinite value but don’t believe that all decisions that don’t rule out the possibility of reaching all values of infinite states have equal value are forced into resolving the contradiction. Either they must accept that essentially all decisions have equal value to them, or they must deny the existence of the possibility of states that have infinite value – it’s not consistent to hold both beliefs. And it’s my belief that denying the existence of states of infinite value is the most intuitive decision that most people already agree with. I don’t have any proof that this is the correct assumption, but I think most people would agree with me that anything that results in almost all decisions having exactly the same value is not valid.

While adding in a new rule for decisions systems is a heavy handed approach to solving what seems to be a small problem in Pascal’s Wager, it actually solves the problems of the destructive forces infinities have on decision systems.

At this point you may be thinking “but wait, if heaven does last infinity long and each year has a value greater than 0, it’s value must be infinite”. In that case I would like to outline how it is possible that even though heaven can last infinitely long, it does not necessarily have infinite value even though each year has a positive value. Consider the series 1/2 + 1/4 + 1/8 + … Does it add up to infinity? It doesn’t, here’s a visual proof

Things that have an infinitely long existence in the temporal realm do not necessarily have infinite value even though each instance of that point of time has a value greater than 0.

Similarly, you can assign a probability distribution of the value of an unknown entity such that it never adds up to infinity. E.g. the probability it is worth 1 is 1/2, the probability it’s worth 2 is 1/4 and so on. This is another way to assign a positive probability to all possible values of an unknown entity without requiring you to be absolutely certain that it is not worth more than a given amount (i.e. not needing to assign a probability that it has a value above x to be exactly 0).

In summary, this post is meant to outline the fact that you must either accept that almost all decisions have the same value or that states that have infinite value do not exist.

The Nature of Computation

If you were building an AI for a game of chess it would be strange if you did anything other than modelling the game. But take a look at the universe we live in and the best equations to explain the reality. Some physicists will tell you that time does not exist. That time and space are really made of the same thing. It is my conception that if we put all the greatest minds together working on a chess AI, they would come to the conclusion that there is something more fundamental that the pieces and the squares. There is some more fundamental substance that they are both constructed from that would serve us better when creating an AI. Just as space-time  is named, we’ll call this fundamental matter square-piece.

This square-piece would allow a much more effective AI. An argument against this might be that we don’t know what reality is defined as, but chess *is* defined as pieces on a board obeying various rules therefore it can’t be anything else. This is wrong however. Just because a problem can be defined in a certain way does not mean it cannot be reduced to a simpler problem. Reducing the game so that the AI can make computations on the fundamental matter would be more effective.

I enjoy making wild statements that are difficult to verify (this seems to ruffle some feathers at times but I don’t care), so here’s another: our brains are actually operating on this fundamental square-piece when playing chess. This is how we can still compete current day chess AI that just iterates through potential scenarios on a weak processor that has a ‘mere’ 100 million transistors on it.

I think to exploit this square-piece a cellular automaton is needed. The space of algorithms that can be designed by humans is infinite in size but that does not mean it explores the entirety of the space of algorithms. There are many algorithms our brains could never comprehend that are essential to building intelligent systems. Instead of designing a CA, we need to search for one that does what we need. Play chess well.

A Novel Method to Determine the Best Player in a Team

This method can be used in teams games where the first to a fixed number of points wins and where players gain points that are the result of a specific individual on the opposite team playing poorly. This type of analysis would work for a game like Halo or any first person shooter – by killing a specific enemy player you gain a point. The basic idea is that if you die a lot you’re bad because the enemy gains points from you dying (because they killed you) and killing a lot is good.

I’ve been in countless arguments about what the best method is to determine the best player using only the information of kills and deaths in the game. Some players argue that the kill death ratio (kills divided by deaths) is the best measure of skill. I used to take the stance that kills minus deaths is the best measure. But I’ve created a novel measure that takes account of all available information to determine the bets player.

Hypothetical scenario: you’re going to play in a game of first 50 kills in a 2v2 against the strongest opponents in the world. You know that in any given encounter the probability you kill is 0.3 and the probability you die is 0.7. You may choose either of the following team mates.

xXx1337sNiPeRXxX who is guaranteed to have 47 kills 48 deaths.
DogrA_aRMA who is guaranteed to have 2 kills and 1 death.

Using either the kill death ration or kills minus deaths would lead you to believe that DogrA_aRMA was the best player to choose. But in fact your chances of winning are dramatically higher with xXx1337sNiPeRXxX.

With xXx1337sNiPeRXxX you would need 3 kills to win the game. The enemy needs 2 kills. There can be at most 3 + 2 – 1 = 4 encounters, this is because the closest the kills at the end can be is 50 to 49. It can therefore be assumed that there will be exactly 4 encounters and that you need to have 3 or more in these encounters to win. It doesn’t matter that some of these kills would result in games where one team has over 50 kills – it still is equivalent to enumerating the permutations of wining when there are no kills beyond the limit.

The probability of winning takes a binomial distribution. The chances of you winning with xXx1337sNiPeRXxX is

With DogrA_aRMA there are at most 97 events. You need at least 48 of these to be kills. The probability of winning is therefore

As you can see, this is dramatically lower than with the other player, even though our measures of his kill death ration and kills minus deaths was higher. Strange indeed. It suggests that these statistics are flawed.  In the kill death ratio the information of the number of encounters is completely lost. With kills minus deaths the number of encounters is also lost but you can determine a minimum number of encounters from it. But then how should we figure out who the best player is?A new method is needed. One in which no information is lost… a method which will put and to the debate. One statistic to rule them all. It works as follows.

The basic idea to evaluate the strength of a player is to figure out the team’s probability of winning without player by replacing him with a player who has the skill level of the average of the rest of the team. This value is then subtracted from the probability of the team winning a rematch with no replacements. This gives us the score.

Not listed is the probability of Team 1 and 2 winning which is 0.159661361932217 and 0.840338638067789.

The formula for column H is

This one minus the probability that you lose and don’t get the kills required to win. It’s done like this because of how the BIDOMDIST function in spreedsheets works. 119 is the maximum number of encounters and 59 is the least number of deaths you need to lose.

The higher the score, the more that player did towards making his team win the game. Here, it shows player K made the biggest contribution to the game, player L came in second and D in third. The scores do not represent skills when comparing between teams. But within the teams they do. So we know that within team 2 the best to worst is K, L, J, G, H and for team 1 is D, E, F, B, A, D. It would be wrong to say that player D is better than player G due to his score being higher.

Download the open source spreedsheet here if you’re interested.

Computers and Optimizing Schedules

When will we begin to use computers for useful things such as scheduling? There are many ways we could vastly improve our life by taking advantage of the computational power we have available. One big area in in scheduling. Schools in particular could take advantage of this. Right now the system is:

Lump people born in a 365 day period together.

Put them in a fixed schedule.

Put them through the school at the same rate regardless of their ability.

In a strong system which took advantage of computation, the following properties would be true:

No groups of people would be  fixed – everyone would be treated as an individual

There would be no fixed schedule. The computer would determine on the fly what your schedule is (for the next x days).

Smarter students would progress through the system faster. This is the whole reason we use it. If all students have to take the same classes then this whole algorithm is pointless.

I’m not an expect on scheduling algorithms but this is a feasible problem. Here’s how the solution would look:

It would need to be run on a very powerful computer. Actually, the more power the computer had the better the scheduling algorithm would work. This is because the bottleneck to how good of a schedule you can design is the amount of computing you can do (as it should be – a sign that you’ve found the solution to a problem is when the bottleneck is computational power and not designing better algorithms).

The only real problem with system is that at the start and end of the scheduling there would be bunching just like in the normal way we do it. Classes can’t run if there’s just one student. It has to wait until there are enough students to make it worthwhile. A student in the middle of his school lifetime would have the most flexibility whereas those at the end would have the least. It’s like how a binomial distribution is. The intervals at the start are quite jerky and jumpy, but the middle is smooth.

The strength of the algorithm would improve as the number of students improved because it allows for greater flexibility. This is why it is ideal to have a larger group.

I think some people would sturggle to imagine what an optimized timetable would look like. It’s pretty simple. The better students progress faster through the system because they have to take fewer classes. The computer would figure out who the best students were by looking at their tests results and so on. There’s actually an episode of Star Trek where a super computer decides everything for a civilization including all moral decisions, and I’m reminded some what of that but this is an ideal calculation for computers. Humans are terrible at optimizing schedules.

Zero.

Zero was discovered by an Indian mathematician some time around the 9th century AD. It seems odd that it took so long for someone to realize we could use 0 in mathematics. But even today people still struggle with the concept of 0. We haven’t fully integrated it into how we contemplate 0. We have words for things where 0 is there.

Bald – a person with 0 hairs on their head

Virginity – A person who has had sex 0 times

We also have words for people who do things greater than 0 times:

Murderer – A person who has murdered more than 0 people

Car owner – A person who owns more than 0 cars

We shouldn’t use language like this. It gives people a poor understanding of nature and how zero plays a role in it.