Empire games

Strategy in discovering then taking over cities.  Addictive, mainly as I tire of software which the alternative geek stuff.

The technique is the same, all players are working on a decision tree, in the computer it is graph.  Basically it is a problem  of finding a homomorphic graph of the opposition, under uncertainty.  Noded weightings are equivalent to price, Alpha Zero is a price maker, it plays these games.

The homomorphic graph is a simpler version of the real thing.  If you took the larger graph and 'compressed' it a bit, you get the smaller, and it should be homomorphic.  Important concept, it means we can price balance two pricing graphs;  In the computer, they are like a specially designed digit system, in nested block you get optimum quant with an ordering, it has some linear properties.  So a compact representation of a stable queue looks like a digit sequence'

For example, consider two of these nested block sequences.  Together, two of these graphs can be shaped to be homomorphic as long as parties can carry the round off error, which can be be optimum.  How do we reshape them? Add our dummy bets in each, compress part of one, expand part of another until they are nearly equal.  Yhis is how the market maker works.

So, you see, there is a perfectly natural method in the computer to match any two sequences as long as they both minimize transaction counts as as goal.  The issue in the computer is the work need to encode the nested block, lot of sorting.  But there is carry over, the entire sequence may nor be collected by holding the previous graph as a starting point there will be fewer sort moves.  We can mostly be adiabatic, no one will notice the sorting pauses.

It is pricing that makes it work, the AI folks call it weighting value on a node.