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
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.