Find optimal strategy and AI for the game 'Proximity'?
- by smci
'Proximity' is a strategy game of territorial domination similar to Othello, Go and Risk.
Two players, uses a 10x12 hex grid. Game invented by Brian Cable in 2007.
Seems to be a worthy game for discussing a) optimal algorithm then b) how to build an AI.
Strategies are going to be probabilistic or heuristic-based, due to the randomness factor, and the insane branching factor (20^120).
So it will be kind of hard to compare objectively. A compute time limit of 5s per turn seems reasonable.
Game: Flash version here and many copies elsewhere on the web
Rules: here
Object: to have control of the most armies after all tiles have been placed. Each turn you received a randomly numbered tile (value between 1 and 20 armies) to place on any vacant board space. If this tile is adjacent to any ally tiles, it will strengthen each tile's defenses +1 (up to a max value of 20). If it is adjacent to any enemy tiles, it will take control over them if its number is higher than the number on the enemy tile.
Thoughts on strategy: Here are some initial thoughts; setting the computer AI to Expert will probably teach a lot:
minimizing your perimeter seems to be a good strategy, to prevent flips and minimize worst-case damage
like in Go, leaving holes inside your formation is lethal, only more so with the hex grid because you can lose armies on up to 6 squares in one move
low-numbered tiles are a liability, so place them away from your main territory, near the board edges and scattered. You can also use low-numbered tiles to plug holes in your formation, or make small gains along the perimeter which the opponent will not tend to bother attacking.
a triangle formation of three pieces is strong since they mutually reinforce, and also reduce the perimeter
Each tile can be flipped at most 6 times, i.e. when its neighbor tiles are occupied. Control of a formation can flow back and forth. Sometimes you lose part of a formation and plug any holes to render that part of the board 'dead' and lock in your territory/ prevent further losses.
Low-numbered tiles are obvious-but-low-valued liabilities, but high-numbered tiles can be bigger liabilities if they get flipped (which is harder). One lucky play with a 20-army tile can cause a swing of 200 (from +100 to -100 armies). So tile placement will have both offensive and defensive considerations.
Comment 1,2,4 seem to resemble a minimax strategy where we minimize the maximum expected possible loss (modified by some probabilistic consideration of the value ß the opponent can get from 1..20 i.e. a structure which can only be flipped by a ß=20 tile is 'nearly impregnable'.)
I'm not clear what the implications of comments 3,5,6 are for optimal strategy.
Interested in comments from Go, Chess or Othello players.
(The sequel ProximityHD for XBox Live, allows 4-player -cooperative or -competitive local multiplayer increases the branching factor since you now have 5 tiles in your hand at any given time, of which you can only play one. Reinforcement of ally tiles is increased to +2 per ally.)
