Game Probability Modeling

[m84uily]

I wanted to model a particular game and determine the probability for each team to win. I have no idea how to do the determination of probability part, but here's the game broken down:

There are 3 types of players, T's, D's and I's.

The amount of each type of player is as follows:
1/8 D
2/8 T
5/8 I
(game is only played in multiples of 8)

All of the T's, D's and I's are placed in a list, every turn 2 distinct players from the list are chosen randomly and interact according to the following:

T--fights I, fights D, peace T
D--kills I, peace D, fights T
I--fights I, dies D, fights T

"fights" - a coin flip determines which player goes back into the list for the next round
"peace" - both players go back into the list for the next round
"kills" - the player who is killing has a 100% chance to remove the other player from the list and return for the next round
"dies" - the player who is dying has a 100% chance to be removed from the list
The game ends when either:
-only T's remain (T win)
-all T's are removed from the list (T lose)

Where should I start in terms of getting the probability breakdown for whether T's win or lose?

 

[mfb]

There are two methods to do this:

- make a large tree diagram, keeping track of all options (e.g. after one round: [0 D 2 T 5 I or 1 D 1 T 5 I or 1 D 2 T 4 I]). This gives exact values, but takes a while both manually and with computer assistance.
- simulate 10000 (or more) games and just see how often T wins. This does not give an exact result, but if you have some programming knowledge it could be faster.

 

[m84uily]

There are two methods to do this:

- make a large tree diagram, keeping track of all options (e.g. after one round: [0 D 2 T 5 I or 1 D 1 T 5 I or 1 D 2 T 4 I]). This gives exact values, but takes a while both manually and with computer assistance.
- simulate 10000 (or more) games and just see how often T wins. This does not give an exact result, but if you have some programming knowledge it could be faster.

I did the second, I'm a bit disappointed there isn't a more clever mathy way to go about things.

 

[mfb]

There is a possible simplification: as there is just one D, "D peace D" never happens. Every selection of {D,I} leads to a death of one of them. T does not distinguish between the groups, so you can reduce the analysis to two groups: T and non-T.
That should have a reasonable tree diagram and it is much easier to evaluate, as you just have to consider four cases each time (T T, T non-T and T wins, T non-T and non-T wins, non-T non-T). Should be possible with pen and paper.

 

[CellCoree]

I would start by breaking down the game into its individual components and analyzing the probabilities for each outcome. First, we need to consider the initial distribution of players in the game. Since the game is only played in multiples of 8, we can assume that there are a total of 8 players in the game. Out of these 8 players, 1 is a D, 2 are T's, and 5 are I's. This gives us a starting probability of 12.5% for a D to be chosen, 25% for a T to be chosen, and 62.5% for an I to be chosen.

Next, we need to consider the possible interactions between the players and their corresponding probabilities. For example, when a T and an I are chosen, there is a 50% chance that the T will fight the I and a 50% chance that they will peacefully coexist. We can calculate the overall probability of each possible interaction based on the individual probabilities of the coin flips involved.

Once we have determined the probabilities for each possible interaction, we can simulate the game multiple times and track the outcomes to determine the overall probability for each team to win. This can be done using computer programs or statistical methods such as Monte Carlo simulations.

Additionally, we can also use mathematical models such as Markov chains to calculate the long-term probabilities for each team to win. This approach takes into account the potential changes in the distribution of players as the game progresses.

In summary, to determine the probability breakdown for whether T's win or lose in this game, we need to start by analyzing the initial distribution of players and the probabilities for each possible interaction. Then, we can use simulations or mathematical models to calculate the overall probabilities for each team to win.