Retrograde analysis — where it appears
Named by 18 essays across 4 fields — each of them below, with the objects they name alongside it.
The first theorem, and the winner it declines to name
Zermelo proved in 1913 that a finite game with no chance and no hidden information is decided before anybody sits down — every position is a win for one side or a draw, and which one is settled already. The proof is a labelling procedure, and watching it run shows exactly how little it says.
The rule that makes Go a finite game
A ko is a point in Go where a capture can be recaptured for ever, and every set of rules forbids it. That prohibition is not etiquette or tidiness — it is the hypothesis that puts Go inside the class of finite games that the theory of values is about, and removing it removes the values.
An outcome with no value behind it
Retrograde analysis labels positions in rounds, outward from the ones already lost. Whatever is still blank when nothing more can be deduced is a draw — and there is no separate test for a draw, because a draw is exactly the residue the method never reaches.
The condition the recursion rests on
Not that the moves run out, and not that the options are few. Poker Nim's heaps can grow without bound and it ends; the game called `on` has one option and never does. What every game value needs is that no infinite run of moves exists — and there are three separate ways to fail it.
One part that never ends
The game called `on` has one move and it is back to itself. Add anything to it — a star, a point, its own mirror image — and the whole board is drawn. So `off` is exactly the negative of `on` and their sum is not zero, which is the group law failing for a reason that has nothing to do with who is winning.
The class is named after memory, and that is not an accident
A 4×4 Domineering board has 6,257,129 routes through it, 5,700 distinct positions, and a deepest line eight moves long. Those three numbers are three different resources, and the smallest of them is the one that gives games their complexity class.
Three different claims are all called solved
Hex is solved in the sense that the first player provably wins, by an argument that names no move whatever. Nim is solved in the sense that a formula gives the right move from any position at any size. Between them sit strategies for one opening, and databases of a few billion positions. The word covers all four.
When never ending is a win
Retrograde analysis labels a position a win when somebody can force the opponent to be stuck, and leaves everything else blank. Calling the blanks draws is a rule from outside the game — and two other rules are available. The labelling does not change under any of them; only the residue does, and on a three-cycle that residue is every position on the board.
The one outcome that adds
Finite outcomes do not add: two first-player wins can sum to anything. Loopy play has seven outcome classes instead of four and adds even less — of the 28 cells in the table, eleven hold several answers. Two do not, and they are the two worth having: a second-player win added to anything leaves the outcome alone, and a draw added to a draw is a draw. A draw added to anything else is not.
A position with no value, and the rule that gives it one
A fortress is a cycle in the position graph, so the recursion defining a value has nowhere to bottom out and the propagation never reaches it. Chess has a rule for that — count fifty moves and call it drawn — and the rule does not merely tidy the theory up. On eleven cells of the sweep it takes away a win.
A ko is won somewhere else
The rung below shows the ko rule buying finiteness by deleting one edge. What it buys with the same edge is a fight nobody can settle by looking at it — the prohibition forces a player to spend a threat, threats are counted on the rest of the board, and every decided cell of the sweep goes to whoever is ahead on a quantity that is not in the picture.
What the play keeps coming back to
A draw is what the backward labelling never reaches, and handing every never-ending play to one player turns the draws into wins wholesale. Judge an infinite play instead by what it keeps returning to, and every draw gets a winner of its own: over the 262,144 three-node games, 15,432 send some of their draws to one player and some to the other, which no wholesale rule can do. Finding those winners takes a fixed point inside a fixed point.
Four positions, sampled
Ten names write both sides of every loopy region of two positions, and forty-eight every region of three. Four positions are over four billion graphs and cannot be counted, but they can be drawn. Three thousand regions at each of three densities: the forty-eight names cover between 95.9 and 99.5 per cent, the thirteen names invented for three positions come back at four almost all of them, and the sparsest sample meets thirty-five sides nothing earlier reproduces — a floor of eighty-three names, and a curve that grows by accretion rather than collapse.
Each size is built from the last
At three positions the sides that needed new names were sums of nothing older: no pair of stoppers, and no two-position stopper with a small game beside it, wrote any of them. At four positions the thirteen names three positions forced are available as parts, and they change the answer. Of the fifty sides the samples met that nothing earlier names, twenty-three are sums of two earlier games — fifteen of them using a three-position name — and twenty-seven are not.
Composition at every size
Twenty-seven of the fifty new four-position sides were written by no sum of two earlier games. Sums of three write sixteen of them — seven plain finite games born after day two, nine built on the names three positions forced — and leave eleven, met seventeen times in nine thousand regions. Run the same test one size down and the story the counts told falls apart: eight of the thirteen names three positions were thought to invent are sums of up to three older parts, two of them just two finite values. Every size composes. What each invents is a residue — five, then eleven.
The same rule on a star
Kōnane turned out to hold the theory's whole vocabulary, and one old game is an anecdote. Mū tōrere, a Māori game on an eight-pointed star, has the same rule for losing — a player who cannot move loses — and is small enough to solve whole. Of its 630 arrangements, not one is a win for whoever moves first or a loss for whoever must move, sixteen go to each side whoever starts, and most are drawn. A random player still loses it every time. The vocabulary was never in the rule for losing; it was in the captures.
What counting positions buys
A rule that counts arrivals at a position ends every game, which the rule of 1929 could not do, and it needs no proof beyond arithmetic: with L arrivals barred on n positions, no game is longer than (L − 1)n − 1 moves. What it does not do is change a single result. Over every game on three and four positions and a sample on five, drawing on the second arrival or the third gives exactly the labels retrograde analysis had before any rule was written. The rule pays for the game's ending and for nothing else.
The number in the rule
Draw on a repeated position and no result moves. Refuse the move instead, as Go does, and nothing is left drawn: over every game on four positions, 10,324 of 16,384 starts are drawn with no rule and every one of them acquires a winner. Which winner depends on the number written into the rule. Bar the third arrival rather than the second and 1,392 of those starts change hands. A question with a linear answer has become one that needs the history, and the extra difficulty lies entirely in the positions the labelling called drawn.
Named alongside it
The objects these essays reach for when they reach for this one.
LoopyDrawExhaustive searchOutcome classTerminationPosition graphComplexityDeterminacyDisjunctive sumKo (Go)NimNormal play