Key takeaways

  • Shanten sets distance; first-order ukeire compares the live tiles that advance the hand now.
  • Second-order ukeire expands the new hand after every first draw instead of adding every vaguely useful tile.
  • A pretty future shape rarely justifies surrendering a large immediate acceptance advantage.
  • When immediate acceptance is close, the second layer reveals which candidate more often reaches broad waits or flexible shapes.
  • Yaku, fan, tai, furiten, and defense sit outside pure ukeire and must be restored for table decisions.

What is first-order ukeire?

First-order ukeire answers: which next draws reduce the current hand’s shanten, and how many live copies remain?

If a one-shanten hand accepts 3m, 6m, and East, with 3, 2, and 2 unknown copies after visible tiles are removed, its immediate ukeire is seven. Count the union for the whole hand: the same tile cannot be counted independently for two blocks.

This first layer does not say whether each improvement reaches a two-sided, inside, single, or another one-shanten shape.

What is second-order ukeire?

Second-order ukeire asks: after one immediate effective draw changes the hand, which draws are effective from that new state?

The procedure is:

  1. List every immediate effective draw.
  2. Build the new hand for each draw.
  3. Recalculate shanten and effective tiles from that state.
  4. Remove visible copies and the tile just drawn.
  5. Compare next-step width and shape across paths.

This makes second-order ukeire a two-level decision tree, not a fixed block table. The same first draw can produce different futures after different discard candidates.

Why does shanten come first?

Suppose candidate A remains one-shanten while candidate B becomes two-shanten. B may have many “improvements,” but still requires an extra structural step. Pure efficiency normally preserves A unless B supplies necessary scoring eligibility, value, or safety.

A useful order is:

  1. Prefer lower shanten.
  2. At equal shanten, compare immediate live acceptance.
  3. When immediate acceptance is close, compare second-order ukeire.
  4. Then add resulting waits, rules value, and game state.

This prevents imagined future beauty from overwhelming real present efficiency.

How does the second layer break a tie?

Imagine two one-shanten candidates with twelve live immediate tiles:

  • Most improvements for A reach two-sided or multi-sided tenpai.
  • Most improvements for B reach inside, edge, or single waits.

The next draw sees a tie. One layer later, A usually has a wider completion set. This is the useful distinction captured by second-order analysis.

Do not represent a candidate by its best path. If only one rare draw creates a three-sided wait while every other path is narrow, that best case cannot stand for the whole candidate.

What does weighted second-order ukeire mean?

Each path can be weighted by the number of live copies that trigger it:

weighted second-order ukeire
= Σ(live copies of first draw × next-state ukeire)
  ÷ total immediate ukeire

This summarizes future shape width; it is not complete win probability. Different paths may need different numbers of draws, require a new discard choice, or overlap. Winning eligibility, value, and defense also remain outside the number.

To answer “what is the chance of winning within two or three draws,” expand a finite state tree directly rather than treating average second-order ukeire as the result.

Which shapes often have stronger future development?

Central floating tiles

Tiles 3 through 7 can connect on both sides into more two-sided, inside, and compound shapes. Even when they do not reduce shanten immediately, they may preserve valuable future development.

Four- and five-tile runs

Long runs can reassign meld and block roles after several different draws, preserving more paths than an isolated edge block.

Tiles beside a concealed pung

Neighboring tiles can create multiple decompositions involving sequences, the pung, and the pair. Count roles carefully so the pung’s tiles are not reused twice.

Clean independent blocks

Two blocks with separate effective tiles are often more stable than two that depend on the same draw. The second layer exposes candidates whose equal headline counts hide overlapping routes.

When should second-order ukeire not decide?

  • Candidates have different shanten without a clear rules-based reason.
  • Immediate acceptance differs substantially.
  • The hand is so late that immediate tenpai dominates.
  • Opponent pressure makes safety more important than offensive shape.
  • The model mishandles hand length, special forms, or visible tiles.

Second-order ukeire is a tie-breaker for close candidates, not a universal override.

Lexicographic efficiency versus finite-horizon wins

Traditional efficiency often compares shanten, immediate ukeire, and second-order ukeire in sequence. It is transparent and practical for human calculation.

A finite-horizon model asks for self-draw completion probability within a stated number of draws. It can combine new discard choices, wait quality, and multiple paths, sometimes preferring slightly less immediate acceptance that completes more often before the horizon.

The approaches are complementary:

  • Lexicographic analysis teaches a hand-calculable process.
  • A finite horizon compares near-term outcomes directly.
  • Neither automatically includes complete scoring, defense, or opponents.

A practical workflow

  1. Select a four-meld 13-tile or five-meld 16-tile model.
  2. Calculate shanten after every discard candidate.
  3. Build the immediate effective-tile union and remove visible copies.
  4. Eliminate candidates with a clear immediate disadvantage.
  5. Expand each effective draw for the close candidates.
  6. Compare next live tiles, resulting waits, and overlap.
  7. Add yaku, fan, tai, value, defense, and turn.

Scope and limitations

This article focuses on standard-hand efficiency. Special forms such as Seven Pairs use different completion units, and a draw may change the chosen route. Ukeire models do not reveal where unknown tiles physically are and should not be stated as certainty.

Practicing second-order analysis with Tingpai Lab

Before viewing the ranking, write each candidate’s immediate ukeire. If totals are close, select two or three major draws and map the resulting waits and next live tiles. Then use the explanation to identify whether the answer wins through immediate acceptance, stronger second-order shape, or higher completion probability within the stated horizon.

Continue reading: