Today we’re going to take our first baby-step into the land of surreal games.

A surreal *number* is a pair of sets {L|R} where every value in L is less than every value in R. If we follow the rules of surreal construction, so that the members of L sets are always strictly less than members of R sets, we end up with a totally ordered field (almost) – it gives us something essentially equivalent to a superset of the real numbers. (The reason for the almost is that technically, the surreals form a *class* not a set, and a field must be based on a set. But for our purposes, we can treat them as a field without much trouble.)

But what happens if we take away the restriction about the < relationship between the L and R sets? What we get is a set of things called *games*. A game is a pair of sets L and R, where each member of L and R is also a game. It should be obvious that every surreal number is also a game – but there are many more games than there are surreal numbers, and most games are *not* surreal numbers.

Games lose some of the nice properties of the surreal numbers. They are *not* a field. They are *not* totally ordered. In fact, they’re not even all positive or negative. They’re very strange things.

So why would we want to break the restriction on the surreals that gives us games? Naturally, because games have useful applications in modeling many things – in particular, games (in the non-mathematical sense – games like Go, Chess, Checkers, Poker, etc).

Let’s take a bit more of a detailed look at games, and how they interact.

Game arithmetic is *exactly* the same as surreal arithmetic: addition, subtraction, multiplication, negation – even division (which we haven’t looked at yet) are all defined in the same way of surreal numbers and games.

But: while surreal numbers are always either positive, negative, or zero, games can also be *fuzzy*. Remember, games are *not* fully ordered. That means that there are pairs of games (a,b) where ¬a≤b and ¬b≤a – that is, where the two games *cannot* meaningfully be compared. Fuzzy games are games that can’t be compared to zero.

What does a fuzzy game look like? The simplest example is: {1|-1}. Try to use the definition of “≤” on that game with zero – it doesn’t work.

Games also have some strange behaviors with respect to multiplication. If a, b, and c are games, then (as you would expect for numbers), if x×z=y×z then x=y. *But*, with games, x=y doesn’t mean that x×z=y×z. Nasty, that, eh?

So what are these beasts useful for? Part of Conway’s motivation was trying to analyze the game of Go (aka Wei-Chi). Go is one of the oldest strategic games in the world; it’s been played for thousands of years in China, Japan, and Korea. Go is the Japanese name, which is generally used here in the US; Wei-Chi is the chinese name for it. It’s a thoroughly fascinating game.

Go is a two-player game where the players have a 17×17 grid. Each move, a player puts a piece of their own color on one of the intersections on the grid. The goal of the game is to surround territory using your pieces. Whoever has the most territory at the end wins. Mechanically, it’s about as simple as a game can get. Strategically, it’s unbelievably deep and complex. It’s frequently compared to Chess in terms of depth and strategy. It’s a wonderful game.

One particularly odd thing about Go – particularly when you compare it to Chess – is how hard it is to analyze mathematically. Computer Go players are still absolutely lousy in comparison to humans, because it’s so hard to get a handle on the strategy in a way that gives a computer player a reasonable chance.

Like Chess, but even moreso, Go has three strategic stages, and the way you play in those stages is very different. There’s the opening; the mid-game; and the end-game. Conway’s games are useful for modeling go. They’re not particularly helpful in terms of strategy in the opening – but the opening doesn’t need that much help. It’s typically quite stylized: there are particular board positions that are extremely valuable, and in their first moves, each player typically stakes a claim to one of them. And then they start to sketch out territories very loosely according to a mostly predictable pattern. Then, at some point, someone makes a hostile move, and bingo, you’re into the mid-game, and that’s when strategy gets incredibly hard.

You can describe a Go game using Conway’s games. Each board position is a game {L|R} where L is the set of possible moves for the first player, and R is the set of possible moves for the right player. Each turn, a player takes the number corresponding to the current board position, and picks one of the numbers from the appropriate set. That number becomes the new board position, and is passed to the other player.

One of the things that makes this work so well for Go is that in the end-game, a go-board is mostly “filled in” – that is, most the territory is controlled by one player or the other, and there are just a few pockets of free space left. Using Conway’s games, you can take each of those free territories, and model *them* as games of their own. Then the game representing the full board position is the *sum* of the games representing the remaining free territories.

The observation that the Go endgame could be analyzed a set of subgames was a prime motivator for Conway in the creation of the surreals. A representation of the games that allowed you to analyze the subgames was very desirable – but that representation also needed to be able to view the subgames are parts of the combined game. Conway was brilliant enough to see that the {L|R} pseudo-numeric structure with addition gave him a representation where the subgames could be combined using addition! From there, he just followed a fairly typical mathematicians analysis of the structure, and pushed it to see what it could do – and discovered the restriction that gave him an alternate construction of the real numbers. But the games actually came first!

Anyway – once you’ve got the construction of surreal numbers and games, you can use

them to do things like analyze the kinds of games, like sprouts, that Conway always plays

with using surreal numbers and games. In fact, you can prove most of the well-known results

about those (recreational) games using properties of the (numeric) games.

Suppose you want to analyze a simple game. For the surreal numbers, we’ll assume it must have the following properties: (note that this is a simple example, and that by changing these restrictions, we can analyze more complicated games, and the price of a more complicated analysis – but also notice that this framework is almost enough to allow Chess to be analyzed this way!)

- There are two players. To make things easy, we’ll called them L and R.
- The game contains no randomizer, like a die.
- There is no hidden information: both players are in possession of
*all*

of the information about the current state of the game, and the possible future

moves. - Players alternate turns.
- The game ends when there are no legal moves left for one player; when this happens, the other player wins

Let g_{x} be the game representing the game after *x* moves. (That is, each player has moved x/2 times.). If g_{x}>0, then it means that L will win: all possible games in which R wins have been eliminated. If g_{x}<0, then R will win, all possible games in which *L* wins have been eliminated.