Please note: This is not about an isolated problem at all, but rather a way of quantifying and visualizing an obfuscated and confusing game theory related model in an easy to grasp manner. For that I require the assistance of math experts in probability, which I am not. If you are interested in helping ,please show how to calculate the probability of results for the following game scenario.
I hope someone finds this interesting. I appreciate your help, Thank you!
Edit: Forum members have enlightened me to the complexity of the scenario and I appreciate all of the comments. I have made some edits in response to the comments..thank you.
Problem: A Battle Royale/last person standing elimination match tournament. Matches are comprised of 64 players per match, randomly populated. Players compete in 1 on 1 encounters throughout the match. 1 player is eliminated in each encounter until there is 1 player left in the match- the last person standing. A forum member asked how the match ups are decided. maybe it should be 64 players per match for simplicity sake with random matching of pairs for 5?? rounds.
At the end of each match points are awarded to the top 3 finishers as follows: 1st place =15 points, 2nd place = 8 points, 3rd thru 4th place = 4 points, and 5th thru 64th place =zero points.
There are two types of player: Player type L and Player type H. Player type L has no advantage over any other Player type L in an encounter and the outcome is 50/50 or a coin toss as to who wins. However, Player L has a disadvantage in an encounter with Player H. Player type H has a 20x advantage in winning an encounter with Player L.
Player H has no advantage in an encounter with another Player type H, where the outcome is a 50/50 coin toss as to who wins. However, Player type H has a 20x advantage versus any Player type L in an encounter.
Each Player will play exactly 150 matches in the season. A leader board reports every player’s accumulated points total score for the season. There are 4,700,000 type L players There are 300,000 type H players.
Without adding any other variables or complexity to the scenario above, what is the probability of the following expressed in 1 out of X and or other terms:
That a type L player would finish the season in the top 500 ranks of points accumulated (out of 5,000,000)? That the number 1 rank will be a type H? That the number 500 rank will be a type H? That the number 1 rank will be a type L?
Thank you very much in advance for your help.
Edit: This discussion has been productive and I realized that I needed to tweak the scenario a bit to better reflect the key elements of the real world situation. Comparing independent L and H distributions is interesting but as you noted L and H types are competing together in the same matches. In my tweaked scenario H players make up about 6 percent of the total population. Also the advantage was way to low, so I adjusted it to 20x.
For the sake of simplifying a ballpark estimate, maybe the starting point is to calculate the probability of H and L obtaining points through either a first or second place win in a single typical (“mean”) match. And then find a way to adjust for the probable deviations from the typical match in 150 matches.
What causes the results of a match to deviate from the “mean” other than the occurrence of no H player present in the match? How likely is it to have a match with no H player present? This seems like a way to come up with some ballpark probabilities for H or L appearing on the Leaderboard in the top 500, or the probability for H or L taking 1st place.
This is not a full answer due to lack of time on my part to work out the details (so don't accept it), but this is an outline on how to get one.
First step
First, we only observe the outcome of a fixed player. Say that there are $N\in\mathbb N$ players (in your case, $N=30000000$). Number the players in arbitrary order from $1$ to $N$ and fix some $k\in\{1,\dots ,N\}$. The probability of winning in a given game for player $k$ depends only whether he himself is H or L, and on the number of H or L players in that game. Say that the player plays $g\in\mathbb N$ games in total (in your case, $g=150$). Let $B_1^{(k)},\dots, B_g^{(k)}$ denote the points this player receives in games $1$ through $g$, respectively. The total number of points that player $k$ has at the end is $T^{(k)}_g \overset{\text{Def.}}=B_1^{(k)}+\dots+B_g^{(k)}$. From the modeling assumptions, the $B_1,\dots, B_g$ are independent and identically distributed. Furthermore, they only take finitely many values (in your case, for example, there are $4$ possible outcomes: One can get $0$, $4$, $8$ or $15$ points.)
From the Berry-Esseen estimates (see for example [1; Satz 15.51] for the formal statement, you may also find references using a search engine), if $(B_j^{(k)})_{j\in\mathbb N}$ is a sequence of iid real random variables satisfying $B_1^{(k)}\in L^3$ and $\mathsf V(B_1^{(k)})>0$, we have, with $T^{(k)}_g \overset{\text{Def.}}=B_1^{(k)}+\dots+B_g^{(k)}$, (and $\Phi(x)\overset{\text{Def.}}=\frac1{\sqrt{2\pi}}\int_{-\infty}^x\exp(-y^2/2)\,\mathrm dy$ denoting the CDF of the standard normal distribution)
$$\sup_{x\in\mathbb R}\left\lvert\mathsf P\left(\frac{T_g^{(k)}-\mathsf E(T_g^{(k)})}{\mathsf V(T_g^{(k)})}\le x\right)-\Phi(x)\right\rvert\le\frac{0.6\mathsf E(\lvert B_1^{(k)}\rvert^3)}{\mathsf V(B_1^{(k)})^{\frac 32}},$$
for all $g\in\mathbb N$.
Coming back to our case: Since $B_1^{(k)}$ takes finitely many values, it is clearly $L^3$. And it is also clear that $\mathsf V(B_1^{(k)})>0$.
Therefore, for $g$ large enough, Berry-Esseen gives that the CDF of $T_g^{(k)}$ is, "close" to the CDF of a normal distribution $\mathcal N(\mathsf E(T_g^{(k)}), \mathsf V(T_g^{(k)}))$.
Second step
You have to determine $\mathsf E(T_g^{(k)})$ and $\mathsf V(T_g^{(k)})$. Note that these two quantities only depend on whether player $k$ is of type H or type L. To determine $\mathsf E(T_g^{(k)})$ and $\mathsf V(T_g^{(k)})$, I recommend to first answer the following question ("the big question"): "What is the probability that a player of type L/H gets first/second/third/fourth place in a game of $64$ (or more, generally, $n$) players such that $k$ players other than the player himself are of H type?"
Denote by $h$ the total number of H players (in this case $h=29900000$) and let $l$ be the total number of L players (in this case $l=100000$).
Let $N^{(k)}$ denote the number of H-players in the first game of player $k$, where player $k$ is not counted should he be a H-player. Then, by the modeling assumption, $N^{(k)}$ is a random variable with hypergeometric distribution with population size $h+l$ and $h$ "success states" as well as $l$ "non-success states" and $63$ draws.
From the law of total expectation, we then get
$$\mathsf E(B_1^{(k)})=\sum_{j=0}^{63}\mathsf E(B_1^{(k)}\vert N^{(k)}=j)\frac{\binom{h}{j}\binom{l}{63-j}}{\binom{h+l}{63}}.$$
Once you answer the big question above (this can be done approximately using simulations and maybe even exactly using combinatorial arguments), you immediately get $\mathsf E(B_1^{(k)}\vert N^{(k)}=j)$, from which you can deduce the value of $\mathsf E(B_1^{(k)})$. Analogously, you can determine $\mathsf V(B_1^{(k)})$.
Third step
Without loss of generality we may assume that players $1,\dots, l$ are of L-type and players $l+1,\dots, l+h$ are of H-type.
We therefore have $E_L\overset{\text{Def.}}=\mathsf E(T_g^{(1)}=\dots=\mathsf E(T_g^{(l)})$, as well as $E_H\overset{\text{Def.}}=\mathsf E(T_g^{(l+1)}=\dots=\mathsf E(T_g^{(h)})$ and $V_L\overset{\text{Def.}}=\mathsf V(T_g^{(1)})=\dots=\mathsf V(T_g^{(l)})$, as well as $V_H\overset{\text{Def.}}=\mathsf V(T_g^{(l+1)})=\dots=\mathsf V(T_g^{(h)})$.
From the argument in the first step, the "top 1" (or "top 500") performance of L-type players is just the maximum ("500"-th biggest) of $l$ random variables that are $\mathcal N(E_L, V_L)$-distributed. These random variables are not independent (if one person looses, the other wins), however it seems like a useful approximation to at first assume that these are independent and to do the same for the H-type players.
The maximum of iid normal distributions hcan be tackled analytically, see for example this question and this question, or approximately using Monte-Carlo-type simulations.
References
[1] Achim Klenke, Wahrscheinlichkeitstheorie. 3. Auflage. Springer-Verlag Berlin/Heidelberg (2013).