The one outcome that adds

Outcomes do not add is one of the first negative results on this site. Knowing who wins each part of a position tells almost nothing about who wins the whole: two first-player wins can sum to a second-player win, to another first-player win, or to a win for either side, and no rule distinguishes the cases from the outcomes alone.

That is for finite games, where there are four outcome classes. Loopy games have seven, because a position can also be drawn, or won by one player moving first and drawn otherwise. Seven classes is more room for things to go wrong, and mostly they do.

The family #

A two-node loopy game is small enough to enumerate completely. Two positions, and for each player any set of moves from each position to each position — sixteen choices for Left and sixteen for Right, so 256 games, all starting at the first node.

That family is small and it is not degenerate. Every game in it is a graph rather than a tree, which is the whole difference: a loopy game is a position graph with cycles and the recursion that evaluates finite games has nowhere to start. All seven outcome classes appear in it, and the way they are distributed is worth knowing before any cell of the table is read.

Four games is a thin class, and it is worth carrying into the table below rather than discovering there. A cell of the composition table whose row is one of those three is built from very few examples, and a cell reading one answer over sixteen pairs is a different kind of statement from one reading it over six thousand.

Where a draw comes from #

The word draw here is not the chess one. Nothing is agreed and no rule declares the game over; the players simply go round for ever, and neither is ever stuck. Whether that counts as a draw, a win, or something else is a rule from outside the game and the table below is computed under one of the three.

The analysis that finds them is retrograde: start from the positions where somebody cannot move, work backwards labelling every position from which a loss can be forced, and stop. What is left unlabelled at the end is the residue — the positions from which neither player can force the other into a corner — and calling that residue a draw is a convention rather than a computation.

The cells that hold one answer #

Seventeen of the 28 cells are determined, and they fall into three groups.

A second-player win is an identity. Add a game the second player wins to anything at all and the outcome is unchanged — all seven cells, no exceptions. That is expected and is worth confirming: a second-player win is a position the opponent can always answer in, so the mirror strategy is available and contributes nothing.

Two wins for the same player stay wins for that player. $L + L$ is $L$, $R + R$ is $R$, and the variants with a first-move qualification join in: $L + L?$ is $L$, $L + N$ is $L$. Any collection of positions Left already wins is a position Left wins.

Two draws are a draw. All 6,328 pairs of drawn games from the family, and every sum is drawn. Not one of them comes back a win for anybody, which is a stronger statement than it sounds: a sum of two games is not played by playing one and then the other, and a player who can keep going for ever in each part separately has to keep going for ever while the opponent chooses which part to move in.

That last one is the finding, and it is worth separating from the other two. The first two groups say nothing surprising: they are the statements that adding something neutral changes nothing and adding two advantages keeps the advantage. The third says that being unable to lose is preserved under sums, and that is a genuine additivity statement in a subject that has almost none.

The cells that do not #

Eleven cells hold several answers, and one of them corrects a natural over-generalisation.

$L + R$ can be a draw, a win for Left, a second-player win or a win for Right — everything except a first-player win. $N + N$ has exactly the same set. Those are the finite result carried over intact: knowing that Left wins one part and Right the other tells nothing.

And then: a draw plus a first-player win is not always a draw. The cell holds three answers — a draw, or a win for Left moving first and drawn otherwise, or the same for Right. So a drawn component does not absorb what stands beside it.

That is a claim about one row of the table, and the row is worth drawing on its own, because the twenty-eight-cell grid prints every cell at the same size and these seven are not the same size at all.

The finite version of the same failure is a whole essay: pairs of positions in the same outcome class whose sums land in different classes. Nothing about the loopy case is worse in kind. There are simply more classes for it to be undetermined among, and no values underneath to repair it with.

Why that matters for on #

The rung below this one is about a particular game. on has one position and one move, back to itself, for both players — so neither player is ever stuck, and adding anything to it leaves the whole board drawn. Whatever stands beside it, a player can always go round the loop instead, for ever.

It would be natural to generalise that to draws. It is false, and the table says so: a drawn game beside a first-player win can come out as a win for one side moving first.

The difference is what kind of draw it is. on is drawn because both players always have a move there — the loop is available every turn, unconditionally, so it functions as an infinite supply of tempo. Most drawn games are drawn for a weaker reason: neither player can force the other into a corner in that game, which does not mean either of them always has a move to spare.

So absorption is a property of the supply of moves, not of the outcome class. Reading on’s behaviour as a fact about draws is the mistake this cell exists to prevent, and the count that prevents it is 32,896 sums with a cell holding three answers in it.

on is one drawn game among 112 in this family alone, and it is the only one whose loop is available unconditionally from every position it has. That is the distinction, and it is not visible in the letter D.

The cell that is determined and unexpected #

Among the seventeen determined cells there is one that is not an identity, not two advantages combining, and not two draws — and it is worth pointing at.

A position that Left wins moving first and is otherwise drawn, added to a position Right wins outright, is always a draw. So is the same combination with the roles of the two qualifications swapped. Three cells, all determined, all giving the same answer.

The reading is that a conditional advantage and an unconditional one on opposite sides cancel exactly. Right’s outright win means Right can force a finish in that component whoever moves; Left’s conditional win means Left can force a finish only by moving first. Put them together and each player’s threat is answerable by the other’s, and the play goes round.

That is the closest thing in the table to a computation — a cell whose single answer is neither of the two inputs and had to be worked out. The other determined cells could have been guessed.

The same graph read under three conventions for what infinite play means gives three different answers, and the labelling itself does not change — only the residue does. Which is why every claim on this page is about a table computed under one convention, and why the convention is named rather than assumed.

The proof that would settle it, and why it does not #

Draw plus draw coming out drawn on every one of 6,328 pairs looks like a result with a one-line proof waiting for it, and the one-line proof is worth writing down, because the place it fails is the most instructive thing in the table.

Here it is. A drawn game is one in which neither player can be forced to run out of moves. So in a drawn $G$ each player has a survival strategy — a way of playing that guarantees they always have a move. Give a player their survival strategy in $G$ and their survival strategy in $H$, tell them to answer a move in one component with a move in the same component, and they always have somewhere to go. Neither player can be made stuck, so the sum is drawn.

It is wrong, and it is wrong at the word answer.

A disjunctive sum does not preserve alternation inside a component. A survival strategy for $G$ is a strategy for the game $G$, in which the two players move alternately. Put $G$ beside $H$ and that guarantee is gone: the opponent may move in $H$ three times running, and the player following the $G$ strategy is then required to move in $G$ three times running with no reply in between. The strategy has no instructions for that. It was never asked to survive against itself.

So the componentwise argument does not go through, and what the table reports is a measurement over the two-node family rather than a theorem with a proof behind it. That is the honest status of the one composable cell on the page, and it is worth saying, because a cell that looked as though it had an easy proof would invite a reader to assume the same for larger games.

Which is exactly why on is different #

That diagnosis also finishes the argument about absorption, and finishes it more sharply than the section above manages.

The section says absorption is a property of the supply of moves rather than of the outcome class, which is right. The reason it is right is the alternation. on offers both players a move from its only position, unconditionally, at every turn — so there is no strategy to follow, nothing that depends on what the opponent did, and nothing that can be broken by being asked to move twice in a row. A component that hands over a move whenever asked is immune to the problem the survival argument has, because it never needed a strategy in the first place.

An ordinary drawn game is not like that. Its survival depends on answering: the player keeps going because they have a reply to each of the opponent’s moves, and the reply is chosen with the opponent’s move in view. Take the opponent’s move away — let them play elsewhere — and the player is choosing in the dark.

So the distinction the essay draws between on and a general draw is not a distinction between two flavours of the same thing. It is the difference between a component that supplies moves and a component that supplies answers, and only the first survives being embedded in a sum where the questions may never come.

That is worth carrying past loopy games. Every argument in this subject that reasons about a component by giving a player a strategy there has the same hole in it, and the reason the finite theory never trips over it is that the finite theory reasons about values instead — which are defined by quantifying over every company at once, alternation included, precisely so that no such step is needed.

What survives, and what it is good for #

The draw-plus-draw cell is a small theorem with a practical reading. A player who has established that no component of a board can be lost — that in each part they can keep going for ever — has established that the whole board cannot be lost either, without any further analysis.

That is a rarer kind of statement than it sounds. Almost nothing about a sum can be established from its parts in this subject; the value can, and the value is expensive. Here is one outcome-level property that composes, available from a per-component check, and the check is cheap: run the retrograde analysis on each part and see whether the starting position was ever labelled — the same cheap check the drawn residue is defined by.

Where that appears in a real game is a ko fight in Go. A ko is a loop: the position repeats, and the rule against repetition is what stops it repeating for ever. Without the rule the component is drawn in the sense used here, and a board with two such components would be drawn as well — which is the reason the rule exists rather than an observation about it.

Seven classes, and why there are seven #

The extra three classes are worth accounting for, since a reader meeting them for the first time may suspect the taxonomy of being invented for the occasion.

A finite game has four outcome classes because each player either wins moving first or does not, giving four combinations, and every game falls into one. A loopy game has the same two questions and each of them now has three answers rather than two: moving first, the player wins, loses, or neither — the game goes on for ever.

Nine combinations, then, and seven classes — and the obvious account of the gap is that two of the nine cannot happen. It is wrong, and it is worth drawing the grid to see why.

So the collapse is in the naming rather than in the games. An outcome class records who can force a win moving first; a player who loses moving first and a player who can force nothing are different situations, and the class is deliberately blind to the difference because nobody wins either way. Eight games of the 256 sit in each of the two arrangements that difference distinguishes, and every one of them is reported as a draw.

That is worth being exact about, because a taxonomy that discards information is a different object from one that is forced. This one is forced given the question it asks — who can win moving first — and the question is the one the four finite classes ask too. Every one of those four appears in the loopy table, since a finite game is a loopy game with no loops in it. What the loopy setting adds is a third answer to each half of the question, and the additivity that survives is thinner because of it. The table has 28 cells because seven classes have 28 unordered pairs; the same table for finite games has ten.

What the solver computed, and how #

The 256 games are enumerated directly: for each of the two nodes and each of the two players, a subset of the two nodes as the move set. The outcome of each is computed by retrograde analysis from the first node.

The sum of two graph games is the product graph — a node for each pair, and a move in either coordinate leaving the other alone — which is the disjunctive sum written as a graph, and the same retrograde analysis is run on it. Every unordered pair is computed: 32,896 sums, in a fraction of a second.

The key of each cell is the unordered pair of the parts’ outcome classes, sorted. An earlier version kept the two orders apart, which split each cell in two and produced a table with asymmetries in it — and an asymmetry in a commutative operation is a sampling artefact a reader has no way to tell from a finding. The sweep also asserts on its own headline: a drawn pair coming out as anything other than a draw would be the claim the table exists to make, failing.

What a value theory would add #

Everything above is at the level of outcomes, and it is worth being explicit about what a value would buy.

For finite games the outcome table has ten cells and most of them are undetermined too — and nobody minds, because the value determines the sum completely and the outcome is read off the value at the end. The outcome table is a curiosity there; here it is the only table available.

That is what makes the drawn cell worth having. In the absence of a value theory, a composable outcome property is the whole of what can be said about a loopy sum from its parts, and there is exactly one of them beyond the trivialities.

A loopy value theory does exist in the literature, for the games with no infinite run in which the two players alternate — and it gives them canonical forms and an addition that behaves. Nothing on this site computes it, because the recursion every value here is built from requires the game to end, and that requirement is not negotiable.

Where the model stops #

Two nodes. A family with three or four positions per game has room for structures this one cannot express — a loop reachable from one side only, a loop that can be entered but not left — and the draw-plus-draw cell is a claim about the 112 drawn games in this family and not a theorem.

It is also a claim about outcomes and not about values. Loopy games have a value theory, with objects like over and under and the stoppers that behave well under addition, and nothing here touches it. The outcome-level result is what a graph and a retrograde analysis can reach; the value-level result would need the machinery that this site does not build.

And the residue is still a convention. Calling the unlabelled positions draws is a rule from outside the game, and a different rule — relabelling infinite play as a win for one side — would produce a different table with different cells determined. The one drawn here is the table for the convention this site uses, stated.

Where the ladder goes next #

The five rungs below take loopy play from the definition to the conventions. This rung asks what loopy outcomes do in a sum and finds one thing that composes. The rung above is the value theory: whether the games that behave well under addition — the ones with no infinite alternating run in them — can be given canonical forms the way finite games can.

Two neighbours are worth the trip. One part that never ends is the absorption result this page qualifies. And an outcome with no value behind it is why the whole discussion is at the level of outcomes: there is no number to add.