No replies to this topic

[Water Bear]

Howdy. So everyone assumes that if you play enough games your tier will eventually be one no matter what. That's mostly due to comments from this thread:

http://mwomercs.com/...-tiers-and-psr/

In particular people look at this graph:

http://i.imgur.com/TpRKr5n.jpg

and conclude that since there are more ways to go up, it must be that your tier climbs forever even if you win half the time.

I just wanted to say that this only may or may not be true. To demonstrate this I cooked up a simple example.

Let's start by saying your chance of winning is 50%. We need a way to denote the events "win and get a very high score," "win and get a low score," "lose and get a medium score" etc. To do that, let's denote the event 'very high score' by the symbol v, 'high score' by h, 'medium score' by m, and 'low score' by l. Let's denote winning by w and losing by l.

Then P(v|w) is the probability that you get a very high score and you won, P(l|l) is the probability that you got a low score and lost, so on and so forth.

What I am about to do is make up a reward function inspired by the graph in the second picture linked above. I am then going to make up two different probability distributions describing how a hypothetical player scores given that they won or lost, and show that your expected point gain is negative in one case and positive in the other.

Let's say that when you win, your rewards are as follows:

R(v|w)=3, R(h|w)=2, R(m|w)=1, R(l|w)=0,
R(v|l)=1, R(h|l)=0, R(m|l)=-1.5, R(l|l)=-3.

Now let's make up some probabilities for how we score based on whether we won or lost.

Say P(v|w)=.05, P(h|w)=.1, P(m|w)=.3, and P(l|w)=.05. Further let's say that P(v|l)=0.025, P(h|l)=.05, P(m|l)=.3, and P(l|l)=.125.

Now our expected score towards our tier is 3 times the probability P(v|w) plus 2 times the probability P(h|w) plus 1 times P(m|w) plus...so on and so on, where we add the possible rewards multiplied by their probabilities of occurrence.

In this particular example, that expected value is about -.15.

If we were to assume that P(v|l)=0.05, P(h|l)=.1, P(m|l)=.3, and P(l|l)=.05 (and leave the other probabilities unchanged) then this expected point change is about 0.1 (positive).

What is the purpose of this post? To illustrate that we cannot assume that just because there are "more ways to gain points" in the picture linked above that everyone will get tier 1 by just playing at a 50/50 win loss forever.

We would have to know more about the exact reward scheme, and about our own chances of scoring well when winning or losing.

Notes: (Some may be added in edits if need be)

The fact that you win 50% of the time enters into the above discussion in the following way. Note that the probability that you win is P(v|w)+P(h|w)+P(m|w)+P(l|w) = 0.5.