Combinatorial Game Theory
who wins, and by how much
Generator

The Richman value of 1

The Richman value of 1
The Richman value of 1. One position under the bidding rule: both players hold a share of a fixed sum of money, the higher bid wins the right to move and pays the bid to the other player. Every position then has a critical fraction rather than an outcome class, and the fraction is the average of the two answers the two possible auction winners would give.

One position under the bidding rule: both players hold a share of a fixed sum of money, the higher bid wins the right to move and pays the bid to the other player. Every position then has a critical fraction rather than an outcome class, and the fraction is the average of the two answers the two possible auction winners would give.

7 essays call bidding-tree. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

The positions it draws

22 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
↑ — Richman value 1/2 L Nobody has to move
↓ — Richman value 1/2 R Nobody has to move
−1 — Richman value 1/4 R Nobody has to move
−2 — Richman value 1/8 R Nobody has to move
∗ — Richman value 1/2 N Nobody has to move
0 — Richman value 1/2 P Nobody has to move
1 — Richman value 3/4 L Nobody has to move
1 | −1 — Richman value 1/2 N Nobody has to move
1/2 — Richman value 5/8 L Nobody has to move
2 — Richman value 7/8 L Nobody has to move
2 | 0 — Richman value 11/16 N Nobody has to move
3 — Richman value 15/16 L Nobody has to move
{0, {1 | 0} | {∗ | −1}} — Left's chance 1/2 N The best chance is the wrong move
{0, {1 | 0} | ↓} — Left's chance 9/16 N The best chance is the wrong move
↑ — Left's chance 1/2 L Left always wins, and loses more often than not · The coldest position has the biggest swing
∗ — Left's chance 1/2 N A coin needs no tie-break
0 — Left's chance 1/2 P A coin needs no tie-break · The best chance is the wrong move · The coldest position has the biggest swing
1 — Left's chance 3/4 L A coin needs no tie-break · The coldest position has the biggest swing
1 | −1 — Left's chance 1/2 N A coin needs no tie-break · The coldest position has the biggest swing
1 | 0 — Left's chance 5/8 N The best chance is the wrong move
2 — Left's chance 7/8 L Left always wins, and loses more often than not
3 — Left's chance 15/16 L Left always wins, and loses more often than not

Where it is called

Changing this generator changes every one of these figures.

What the auction can and cannot see. Values under both conventions. The Richman value is the share of the money the second player needs; a half means the position itself decides nothing and whoever has more money wins. Every infinitesimal on the list, and zero with them, comes out at a half. Where it stops

Nobody has to move

Every convention here rests on one sentence nobody examines — the players move alternately. Replace it with an auction and a position stops having an outcome class and starts having a number: the share of the money the second player needs. The 22 values born by day two collapse to seven of those numbers, eight of them landing on exactly a half; the new number respects the game order on all 179 comparable pairs, and is not determined by the parts under addition on 14 of 49.

 The money played out, and it never mattered. The bidding rule played move by move with a countable pool of chips, at every way of splitting it. The verdict is constant across the splits and opposite under the two ways of resolving equal bids, so what settles these positions is the tie-break rather than the money. Where it stops

The auction never gets to the money

The critical fraction is computed and never played. Played out with a countable pool of chips — twelve positions, four pool sizes, every split of the chips, every bid answered — the verdict does not move with the money on a single one of the forty-eight sweeps, and the rule for equal bids settles all forty-eight. The reason is one line long: declining every auction wins, and bidding nothing declines.

 The same number, from a rule that needs no tie-break. The number computed twice: once as the critical share of a pot under the auction, and once as the probability that Left wins when a fair coin decides who moves at each turn. They agree on every position, and only the second derivation survives being played out. Where it stops

A coin needs no tie-break

The same recursion has a second derivation: a fair coin decides who moves at each turn, a player whose turn it is with no move has lost, and both play to win. Written from those rules it comes out identical on every position — and it needs no rule for equal bids, because there are no bids. The number is a probability, it belongs to Left rather than Right, and the empty position is the one where the coin decides everything.

 A position Left always wins, and not always. Values grouped by the outcome class alternating play assigns them, with the range of probabilities the coin gives Left inside each class. A class that alternating play calls a win for Left every time holds no position the coin makes certain. Where it stops

Left always wins, and loses more often than not

Alternating play answers with one of four classes and the coin answers with a chance, and the two do not have to agree. Over the twenty-two values born by day two they never disagree and the margin is exactly nothing — the lowest chance on a position Left wins whoever moves is a half. Over the 1,474 born by day three, seven of them sit at seven sixteenths, and seven mirror them on the other side.

 Every chance the coin gives, by day 2. The probabilities the coin produces over all the values born by a given day, drawn on the unit interval. They fall on a grid of dyadic fractions, every interior point of it is reached, and the two ends never are — so no position is ever a certainty under random turns. Where it stops

Every chance but a certainty

The coin's number lands on a grid of dyadic fractions, and which points of that grid arrive is a count rather than a guess. Over the 1,474 values born by day three it reaches every one of the fifteen interior sixteenths and neither end — no position is ever certain. The groups sharing a chance run 1, 2, 4, 8 on the small pool, which looks like doubling, and 1, 2, 4, 20 on the large one, which is not.

 What the flip is worth, and what is at stake. Left's chances if Left moves and if Right moves, with the gap between them beside the position's temperature. The two are answers to the same question computed by different routes, and they do not order the positions the same way. Where it stops

The coldest position has the biggest swing

How much a flip is worth is the gap between the coin's two branches, and it is a rival to the temperature — both answer how much is at stake. They disagree at once: the empty position has the lowest temperature there is and a swing of one, twice the hottest thing born on day two. On the small pool the two quantities look like a perfect three-way correspondence, and 255 of day three's values break it.

 The coin's move is often a blunder. Positions where Left has a choice and at least one option wins under alternating play, with how often the option maximising Left's chance under random turns is an option that loses the alternating game outright. Where it stops

The best chance is the wrong move

Maximising a probability and denying an opponent a reply are different objectives, and on 189 of the 904 day-three positions where Left has a choice and a winning move, the option the coin prefers is one that loses the alternating game outright. The smallest case is two options and one line of arithmetic: five eighths beats a half, and a half is the move that wins.

The whole library · The position index · The figures that play back