High-Dimensioned Games

Think Patriot

The complexity of a game can be calculated using the number of ‘dimensions’ and the alternatives in each.

Tic-Tac-Toe has 3 dimensions : the grid, ‘X’ or ‘O’, and where to move next. That game is simple enough that even beginners soon learn that the game is won, lost or tied from the first move.

Checkers has a bigger board, adds another dimension in types of moves, another in King vs ordinary checker, and another in having many more pieces on the board. Still simple enough to teach young children, but adult intelligent human minds nevertheless find it very hard to grasp the higher levels of play.  

Chess adds more types of pieces and more moves to those dimensions. A lifetime of concentration is required to play at the highest levels, and the level of play continues to increase because of computer-human joint learning of the game.

3-player chess uses a more complex board, another 16 pieces, and any 2 of the 3 players can form an alliance, or pretend to.  3D chess is still two players, but now the board is 3 levels deep with 46 pieces per side.

Nobody plays 3-way or 3D chess because you can’t play enough games in a lifetime to know whether you are getting better at it or not.

Considered as games, real world contests make 3D chess as trivial as tic-tac-toe.

The huge complexities common to real-world contests are :

  • more players
  • players can make simultaneous moves, as many as they would like
  • player’s ‘pieces’ have very many possible ‘moves’, opponents may not know those in advance
  • the players do not have complete control of the moves that their own pieces make
  • moves can have multiple effects
  • pieces and players may have different capabilities
  • the board and its environment may be complex and can make ‘moves’ of their own.

Consider the game of multi-lateral international diplomacy with war in the Middle East. Maybe 20 significant players, some non-state, each dominant in their own piece of the board’s complex terrain, each having a mix of pieces of different capabilities in social, economic and force ‘spheres’ (bundles of dimensions), no piece a complete replacement for any other, each having some influence in some dimensions based on ‘natural alliance’ or by way of shared interest(s) with one or more of the others. Pieces may have volition, clan loyalties, and vengeance on their mind.  Players may have conflicting alliances, alliance partners and pieces may not respond as quickly as expected, …

Add in the many possible moves by each player’s pieces e.g. what Afghan Army member decides to shoot as many of his NATO colleagues as possible on an arbitrary day or American soldier who decides to murder a lot of civilians in the night. Involuntary pieces, civilians are victims everywhere, blood and money change the environment and loyalties, propaganda has side-effects like prolonging the war, …  The environment can add a storm to a battle or starvation to population dynamics.  Anyone who played there can add another hundred or so complexities.

Even worse, if you can imagine worse, ‘win’ is not a single dimension for any ‘side’. The US thinks it’s winning the War on Terror contest because it kills a lot of leaders and keeps some allies happy. ‘They’, whoever they are this week among the many tribes and ethnic groups and beliefs in the ME world who are causing the US or one of its ‘allies’ problems, believe they are winning because radical Islam is spreading throughout the Muslim world, its unstable allies have burgeoning populations and failing economies, and the US will not be able to continue the war forever as it is going broke. Despite the outcome being completely obvious, politicians on the losing side cannot bring themselves to end it.

Great complexity to the exponent of great complexity, non-linear indeterminacy depending on non-linear indeterminacy in a fractal environment. In this game, every move has an associated risk of beginning or adding to a cascade of failure, hard to foresee in early stages, hard-to-impossible to correct in later stages, e.g. an empire’s economy failing at the same point it’s policies have made it the maximum number of enemies.

Clauswitz was right : the deepest strategy is to play defense, let your opponent make the mistakes.

The complexity of the real-world games you allow yourself to play has a lot to do with winning.


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