Draw — where it appears
Named by 34 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.
Loopy games
The whole theory assumes play stops. Allow a position to recur and the induction that every value rests on has nothing to stand on — and a fifth outcome appears that normal-play theory has no name for.
Hard, proved
A game is as hard as a logical formula when the formula can be drawn as the game. Here is the drawing — a quantified formula turned into a graph with a token on it — with every formula over three variables played both ways and required to agree.
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.
Start at the end and work backwards
When play can return to where it started there is no bottom for the recursion to stand on. What replaces it begins at the positions where somebody has already lost and propagates outwards — and the positions it never reaches are exactly the draws. There is no test for a draw, and there does not need to be.
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.
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 stopper and how to find one
The class a value theory for loopy games would need is the ones with no infinite alternating run, and the qualifier does the work: seventy-nine of the two hundred and fifty-six two-node loopy games qualify and seventy-two of them have a cycle. Every one has a decided outcome, and under eight tests the seventy-nine collapse to six.
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.
The gap between two answers
A draw is usually described as what the backward labelling never reached, which makes it sound like a shortfall of the algorithm. Written as one predicate the winning condition is an equation, the equation is monotone, and it has a least solution and a greatest one — and the set the two disagree about is exactly the drawn set, on every game checked.
Every play ends and no round settles
Take the finiteness hypothesis away carefully — not by adding a cycle, which has already been priced twice, but by adding infinitely many positions to a game every play of which still ends. Nothing is drawn, every line finishes, and the round the opening settles in grows with every cut: two, four, six, eight, twelve, sixteen, and no number in the column is the answer.
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.
One bit of memory
Judge an infinite play by whether one position keeps recurring and every winner can play from a table of one move per position, with nothing remembered. Ask for two positions to keep recurring and that stops being true. At a hub with two spokes a player has to alternate, and a table cannot alternate: over every three-node game, 49,487 position-and-mover pairs are won with one bit of memory and lost without it.
The rule decides who has to remember
Whether a winner needs memory is a property of the winning condition and not of the board, and the property everybody reaches for is the wrong one. Of the 128 conditions on which of three positions a play keeps returning to, 32 demand memory and 19 of those are closed under union. What separates them is measured two independent ways and the two agree on all 128: a condition needs no memory exactly when it can be rewritten as a number on each position.
Two things to hold at once, or three
Whether a condition makes a winner remember has been settled over every condition on three positions; how much it makes them remember has not. A tree built out of the condition alone, with no board in it anywhere, prices all 128: sixty-one cost nothing, fifty-eight cost two states and nine cost three. It also names the property that was nearly right — closure under union of the sets a condition rejects decides it exactly, where being writable as numbers is sufficient and reaches twenty-six.
A loop is written with two names
A region with a cycle in it has no brace expression, and every one of the 256 regions of two positions can be written anyway — as two names, the game it is when a play that never ends goes to Left and the game it is when it goes to Right. Checked against all 1,474 values born by day three, ten names cover every side, 25 regions need only finite ones, and the pair predicts every sum with a finite game, draws included: a draw arrives exactly where the two names disagree.
Four thousand nine hundred regions with no name
Two positions give 256 regions and ten names cover every side of all of them. Three positions give 262,144 graphs, 110,934 genuine loopy regions — and 4,931 of those have a side that no name in the two-position vocabulary reproduces, with 3,990 of them named on one side and blank on the other. The count the earlier essay left open comes back in the affirmative.
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.
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.
The solution that was a losing set
In 1944 von Neumann and Morgenstern defined the solution of a game by two conditions on a set of outcomes: none overrules another, and each outcome outside is overruled by one inside. They were writing about coalitions. The same two conditions, read with a move in place of an overruling, describe exactly the lost positions of a game that must end. Where play can loop, the same definition gives several answers or none, and three positions are enough to show both.
What a stable set knows about a draw
On a graph with cycles, outcome labelling calls a position lost, won or drawn, and stops. The stable sets go further in exactly one place. On every game of up to five positions they hold every lost position, no won one, and among the drawn positions are exactly the stable sets of the drawn part taken alone. So a stable set is a certificate that the player who keeps returning to it never gets stuck. It can distinguish draws that labelling cannot, and two parts with no stable set can have one between them.
A stable marriage is a stable set
Draw a graph whose positions are pairs of people, with a move from a pair to any pair one of its members prefers. Its stable sets are exactly the stable marriages, on all 46,656 profiles of three men and three women. So Gale and Shapley's theorem of 1962 is an existence theorem of the kind Richardson's was, on graphs where 49 per cent have odd cycles and his hypothesis is useless. The reason it holds is that every clique in the graph is one person's list, and four roommates show what happens when it is not.
A formula drawn as a game
Backward labelling answers who wins on any finite graph in time proportional to the arcs. Whether the same graph has a stable set is another matter: Chvátal showed in 1973 that it is NP-complete, by drawing a formula as a game whose variables are two-way pairs and whose clauses are triangles. On every one of the 255 formulas of this form on three variables the graph has a stable set exactly when the formula is satisfiable, and every position is drawn. The hard question lies entirely where the easy one says nothing.
Blocking is not enough
Everybody knows noughts and crosses is a draw, and the usual explanation is that the second player can always block. That explanation is false. Against an opponent who can only block, the first player wins, from the centre or any corner. The game is a draw only because the second player's blocks build threats of their own. On fifteen boards solved outright, it is the only one where those two things come apart: everywhere else a draw is a draw because blocking is enough.
A rule that let the game run for ever
In 1929 German chess drew a game when the same sequence of moves came three times running, and the rule was meant to make chess finite. Max Euwe, six years from the world title and trained as a mathematician, showed that it does not, using a word that had already been found twice for other reasons. Two round trips are enough. The board returns to one position thousands of times, no block of play ever comes three times running, and the rule has nothing to say.
The opponent makes it repeat
Euwe's unending game needs both players to keep to it. Give the letters to two players in turn, with one of them wanting the draw that a repetition brings, and the rule of 1929 fires within seven letters on a plan of one sentence: always play A. The word that defeats the rule exists and nobody can be held to it. Add a third round trip and the same player can no longer force anything in twenty-six letters. Existence and force turn out to be separate questions, with separate boundaries.
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.
Taking turns removes the hard part
Whether a graph has a stable set is NP-complete, and every hard instance depends on odd cycles. A real game's position graph has none. A position records whose turn it is and every move changes that, so all cycles are even, Richardson's theorem applies, and the set of all drawn positions with one player to move is a stable set of the drawn part, always. Mū tōrere and Three Men's Morris both confirm it. The hard question never reaches a game with turns, and the sets it hands over carry no information at all.
A rule that cannot see whose turn it is
The stable sets a game hands over work by telling the players apart. Forbid that, and read every board as mine and theirs, and the odd cycles come straight back. On Mū tōrere, with rotations and reflections identified too, the 892 drawn positions fall into 33 kinds, and among those there is exactly one stable set. It names sixteen shapes. A player who always hands the opponent one of them holds the draw against labelled play without knowing a colour, an orientation or an outcome, and the opening itself is one of the sixteen.
Named alongside it
The objects these essays reach for when they reach for this one.
LoopyExhaustive searchPosition graphOutcome classRetrograde analysisTerminationDeterminacyFixed pointStrategyOn, the game that never stopsEnumerationNormal play