# Game Probability Modeling

1. Jun 25, 2014

### 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?

2. Jun 25, 2014

### Staff: Mentor

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.

3. Jun 25, 2014

### m84uily

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

4. Jun 25, 2014

### Staff: Mentor

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.