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.

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Before we move beyond zero-sum games, it’s worth taking a deeper look
at the idea of utilities. As I mentioned before, in a game, the scores in
the matrix are given by something called a utility function.

Utility is an idea for how to mathematically describe preferences in terms
of a game or lottery. For a game to be valid (that is, for a game to have a meaningful analysis and solution), there must be a valid utility function that
describes the players’ preferences.

But what do we have to do to make a valid utility function? It’s
simple, but as usual, we’ll make it all formal and explicit.

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When I last wrote about game theory, we were working up to how to find
the general solution to an iterated two-player zero-sum game. Since it’s been
a while, I’ll take a moment and refresh your memory a bit.

A zero-sum game is described as a matrix. One player picks a row, and one player picks a column. Those selections are called strategies. The intersection
of the strategies in the game matrix describes what one player pays to the other. The
matrix is generally written from the point of view of one player. So if we call
our two players A and B, and the matrix is written from the viewpoint of A, then
an entry of $50 means that B has to pay $50 to A; an entry of $-50 means that A has to pay $50 to B.

In an iterated game, you’re not going to just play once – you’re going to play it repeatedly. To maximize your winnings, you may want to change strategies. It turns out that the optimal strategy is probabilistic – you’ll assign probabilities to your
strategy choices, and use that probability assignment to randomly select your
strategy each iteration. That probability assignment that dictates how you’ll
choose a strategy each iteration is called a grand strategy.

Jon Von Neumann proved that for any two-player zero-sum game, there is an optimal grand strategy based on probability. In a game where both players know exactly what the payoffs/penalties are, the best strategy is based on constrained randomness – because any deliberate system for choosing strategies can be cracked by your opponent, resulting in his countering you. The best outcome comes from assessing potential wins and losses, and developing a probabilistic scheme for optimizing the way that you play.

Once you make that fundamental leap, realizing that it’s a matter
of probability, it’s not particularly difficult to find the grand strategy: it’s just a simple optimization task, solveable via linear programming. The solution is very elegant: once you see it, once you see how to
formulate the game as a linear optimization process, it just seems completely obvious.

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