Moving on from simple zero-sum games, there are a bunch of directions in which

we can go. So far, the games we’ve looked at are very restrictive. Beyond the

zero-sum property, they’re built on a set of fundamental properties which ultimately

reduce to the idea that no player ever has an information advantage over any other

player: the complete payoff matrix is known by all players; no player gets

to see the other players strategy before selecting their own; and so on.

Moving on to more interesting games, we can relax those assumptions, and allow information to be hidden. Perhaps each player can see a different part of the

payoff matrix. Perhaps they take turns, so that one player gets to see the others

strategy before selecting his own. Perhaps the game isn’t zero-sum.

Non-zero sum games turn out to be disappointing from a game theory point of

view. Given a suitable set of restrictions, you can convert a non-zero-sum game to a

zero-sum game with an additional player. In the cases where you can’t do that,

you’re pretty much stuck – the mathematical tools that work well for analyzing

zero-sum games often simply don’t work once you relax the zero-sum requirement.

The more interesting ways of exploring different realms of games comes when you

allow things to get more complex. This comes about when you allow a players strategy

selection to *alter the game*. This general takes place in a turn-taking

game, where each players strategy selection alters the game for the other player. A

simple example of this is the game of tic-tac-toe. The set of strategies of the game

for a given player at any point in time is the set of open squares on the board.

Each time a player makes a move, the game is altered for the other player.

This makes things much more interesting. The easiest way to think of it is

that now, instead of a simple matrix for the game, we end up with a tree. Each move

that a player can make creates a new game for the other player. By making each game position a tree node, and adding children nodes for each position that can follow it, you can build a tree describing the complete set of possible game positions, and thus the complete set of ways that the game could play out.