What Is One Ply of Search Actually Worth?
Our board-game opponents — Connect 4, Checkers, Othello and Chess — all think the same way: negamax with alpha-beta pruning, looking a fixed number of moves ahead. The one knob that makes them stronger is search depth. So how much does one more ply buy? We ran 2,400 headless self-play games to find out — and the honest answer is it depends enormously on which ply.
· 8 min read · all numbers from a re-runnable, seeded benchmark · Updated September 2026
The Numbers
What the first extra ply is worth (depth 1 → 2)
…and every ply after the first
Note the different vertical scales: the first ply above tops out near 1,000 Elo, the later plies here rarely clear 230. Connect 4's zig-zag is a real parity effect — in a game this shallow, whether the search horizon ends on your move or your opponent's changes what it sees, so odd and even depths behave differently.
Per engine, at a glance
Front-Loaded, Not a Curve
It's tempting to expect a tidy law — "each ply is worth X% less than the last." The data refuses. What actually happens is that the first ply does almost all the work, because a depth-1 search only avoids one-move blunders and otherwise plays nearly at random; the jump to depth 2 turns a coin-flip opponent into a real one. After that, each ply is chasing rarer and rarer mistakes while costing exponentially more, and how much it finds depends entirely on the game.
Chess is the exception that proves the point. Its first two plies are almost equal (+321 then +315) rather than front-loaded, because its branching factor is so large that even a depth-2 search is still missing a great deal — there's plenty left for the next ply to find. Connect 4 is the opposite extreme: a small, near-solved game where, once past the blunder-avoiding first ply, extra depth barely changes the result until it can see the whole forced line.
That's the real contrast with the companion Go study. There, Monte-Carlo playouts scaled up on one smooth curve — each doubling bought a predictable, even growing, amount of strength. Depth-limited negamax has no such curve: it is front-loaded and game-specific. Same goal, two search paradigms, two completely different shapes of return on compute.
The Harness
Each engine's shipped search() is deterministic, so to get a distribution of games we start every one from a random opening — a handful of random legal moves — then let the two depth-limited engines play it out. Crucially, each opening is played twice with the sides swapped, so any imbalance in the random start cancels exactly. Draws score half a point. Depth, not milliseconds, is the knob — so the numbers are hardware-independent and reproduce from a seed. (Draw rates run high in Checkers and Chess, so read those Elo gaps as approximate.)
- Connect 4: depths 1–7, 120 colour-balanced games per adjacent pair; the same engine that runs at lkforge.com/games/connect4/ (commit c045890).
- Checkers: depths 1–7, 120 colour-balanced games per adjacent pair; the same engine that runs at lkforge.com/games/checkers/ (commit 1fb8a04).
- Othello: depths 1–6, 120 colour-balanced games per adjacent pair; the same engine that runs at lkforge.com/games/othello/ (commit 78c7144).
- Chess: depths 1–4, 120 colour-balanced games per adjacent pair; the same engine that runs at lkforge.com/games/chess/ (commit 93d8643).
Seeded (mulberry32, master seed 20260819); counts are deterministic per seed. Elo per step = 400·log₁₀(p/(1−p)) from the colour-balanced score p (draws = ½).
The companion piece — where more search bought more, not less — is Does Thinking Twice as Long Make a Go AI Twice as Good?. The algorithms themselves are broken down in Six Games, Three Classic Algorithms, and why they're search, not neural nets, in Your Game Opponent Isn't an LLM.
Play the Engines
Every engine in this study is live in your browser, three strength levels, no sign-up.
Common questions
How much stronger is one more ply of search?
It depends enormously on which ply. Across LK Forge's four negamax board-game engines (2,400 self-play games), the first extra ply — going from depth 1 to depth 2 — is worth the most, up to +953 Elo in Connect 4, because a depth-1 engine barely beats a random mover. Every later ply adds far less.
Why is the first ply worth so much more than the later ones?
A depth-1 search only avoids one-move blunders and otherwise plays almost at random, so jumping to depth 2 turns a coin-flip opponent into a real one. After that, each additional ply adds a small, game-specific, non-monotonic amount rather than following a clean curve.
Does every game front-load the gains the same way?
No. Three of the four do, but Chess does not: its first two plies are almost equal (+321 then +315) because its branching factor is so large that even a depth-2 search still misses a great deal, leaving plenty for the next ply to find.
Which engines and depths were tested?
Connect 4 (depths 1 to 7), Checkers (1 to 7), Othello (1 to 6) and Chess (1 to 4) — the same engines that run on LK Forge — with 120 colour-balanced games per adjacent depth pair.
How is the Elo-per-ply figure calculated?
From the colour-balanced score p between two depths, Elo = 400 x log10(p / (1 minus p)), counting draws as half a point. The runs are deterministic per seed (mulberry32, master seed 20260819).
Does adding more depth ever stop helping?
Effectively yes for shallow games. In near-solved Connect 4, once past the blunder-avoiding first ply, extra depth barely changes the result until the search is deep enough to see the whole forced line to the end.