RSG is a board game played on 37 hexagonal cells arranged in a star shape.
We used data to ask how deep it really is, compared with Reversi.
This is the plain-language edition of the full academic report. It takes about 10 minutes to read.
Forty years ago, I had an idea for a new board game. Back then you had to build the board and the stones yourself, and I never managed to turn the idea into something real.
Time passed, and we reached an era in which anyone can build an app with the help of AI, so at last the idea could take shape. Rethinking it as "how would I build this today?", I shrank the original design (a larger board of 73 cells) down to a smaller star-shaped board of 37 cells, better suited to the way people play now.
But that raised a worry.
"On a board this small, won't a computer searching a few moves ahead simply settle on the best move every time, leaving the game shallow in the end?"
So I had AI players compete against each other in three Reversi-family games, and used the data to find out whether RSG is genuinely deep, or shallow. This page is the plain-language edition of that analysis, told in the simplest words I could find.
RSG (ReverStarGo) is a board game that combines elements of Reversi and Go with rules of its own. Its main features are:
In short: Reversi + the "surround and capture" idea from Go + strategic elements of its own.
RSG is published as a free browser app (links at the bottom of this page). The rules are explained in full, with diagrams, in Chapter 1 of the academic report.
We all say things like "this game has real depth" or "that one is shallow". But how would you put a number on it?
This study starts from the following question.
"If you play the same game 10,000 times, how different are those games from one another?"
If every game unfolds differently, the range of viable strategies is wide — which sounds like depth.
If the same shapes come round again and again, the range is narrow.
To measure this we use Heaps' law, a long-established relationship. It was originally used in linguistics to describe how quickly new words appear as you read further into a text.
Don't worry about the details of the formula. What matters is β (beta).
So β is an index of how slowly a game runs out of fresh strategic ground.
Here is where it gets interesting.
Set two AI players that move completely at random against each other, and every game record comes out different, so β lands close to 1.0. That is not because there is strategy — it is because there is no strategy, so anything goes. (In this study, the weakest AI level — the one closest to random play — produced β ≈ 1.0 in all three games.)
The absolute value of β on its own tells you nothing about whether a game is deep. A high value may simply mean "no strategy at all", and a low value may simply mean "the openings settled early".
So how can we get at the real depth of a game?
This study takes the following approach.
Within the same game, vary how far ahead the AI looks, from shallow (D1) to deep (D5), and watch how β moves.
In other words, what this study calls "the depth of a game" is not the absolute value of β, but the pattern of change in β as the search depth increases. That pattern is exactly what leads to the "J-shaped behavior" described in Chapter 4.
Studying RSG on its own would never answer "but is it really deep?". So we ran three games from the Reversi family under identical conditions and compared them.
Reversi on a board almost the same size as RSG in cell count. But both the shape of the board (square) and the rules differ. It is the control group that is "close enough in size to compare with RSG".
The subject of this study. Its cell count is almost identical to 6×6 Reversi, but the board is a star shape of hexagonal cells and the rules are its own.
The standard 8×8 Reversi that almost everyone knows. About 1.7 times larger than RSG, with roughly twice as many moves.
This is the crucial part. We implemented the same AI architecture for all three games. There are 10 levels (Lv.1–Lv.7 and D1–D5; because Lv.6 = D2 and Lv.7 = D5 overlap, that makes 10 distinct levels), and at each level we ran 10,000 games of self-play between opponents of the same level. That is 300,000 games across the three titles.
If the two AI players differ in strength, the stronger one simply wins and the results skew. By looking only at matches between equals, we can observe the intrinsic range of strategy the game offers at that level.
Measuring β while varying the search depth (D1–D5) revealed a curious pattern shared by all three games.
In all three games, β falls and then rises again as the search deepens. This study calls that movement "J-shaped behavior" (a term borrowed from economics and other fields).
Figure 1 below shows the whole picture.
In Figure 1, the left half of the horizontal axis is the evaluation-function variants (Lv.1–Lv.5, AI players that do not look ahead), and the right half is the Negamax search depth (D1–D5, AI players that do). In all three games you can see β drop sharply from Lv.5 to D1, bottom out at D2, and then climb again from D3 onwards.
That "down, then up" movement is what J-shaped behavior actually is. All three games show the same shape.
The interesting part is that all three games show a J-shape, but the magnitude differs. Figure 1 already shows it: blue (8×8) recovers the most, green (6×6) the least. Here are the β values for the whole D series.
| Search depth | RSG | 8×8 Reversi | 6×6 Reversi |
|---|---|---|---|
| D1 | 0.369 | 0.445 | 0.000 |
| D2 | 0.370 | 0.380 | 0.010 |
| D3 | 0.462 | 0.664 | 0.058 |
| D4 | 0.424 | 0.663 | 0.021 |
| D5 | 0.539 | 0.683 | 0.097 |
β values from a 7-point regression over n ≥ 500 (checkpoints at 500 / 1,000 / 2,000 / 4,000 / 6,000 / 8,000 / 10,000 games).
Now compare the size of the rise from D2 to D3, the point where the J turns upward.
| Game | β at D2 | β at D3 | Rise | Class |
|---|---|---|---|---|
| 8×8 Reversi | 0.380 | 0.664 | +0.284 | large J-shape |
| RSG | 0.370 | 0.462 | +0.092 | moderate J-shape |
| 6×6 Reversi | 0.010 | 0.058 | +0.048 | small J-shape |
In other words, how much new strategy opens up when you look further ahead differs sharply between the three games. In 8×8 Reversi, searching deeper opens up a great deal (large J-shape). RSG opens up a fair amount (moderate). 6×6 Reversi opens up very little, however deep you search (small J-shape).
The standard game. A large board, and a correspondingly wide range of strategy.
The board is about 58% the size of 8×8 Reversi and games are about 55% as long, yet the qualitative feature — J-shaped behavior — is the same as in 8×8, and the range of β values is at a level comparable to it. And compared with 6×6 Reversi (36 cells, about 32 moves), which is almost exactly the same size, RSG (37 cells, about 33 moves) sits at a clearly deeper level despite near-identical size and game length.
Almost the same 36 cells as RSG, and a square-board relative of the same family, yet its J-shape is far smaller. Searching deeper does not open up much strategy space.
If the story were as simple as "the bigger the board, the deeper the game", then RSG (37 cells) and 6×6 Reversi (36 cells) ought to come out at roughly the same depth.
In practice the two are clearly separated — by more than 0.36 in β — and they fall into different J-shape groups.
Game length tells the same story. Here is the average number of moves per game:
| Game | Average moves per game |
|---|---|
| RSG | 36.04 (including passes; 31.61 actual placements) |
| 6×6 Reversi | 34.62 |
| 8×8 Reversi | 62.69 |
Measured over 10,000 D5 self-play games per title. The round figures given in the game cards in Chapter 3 ("about 33 moves", and so on) are nominal board capacities; the numbers here are what the games actually ran to.
RSG and 6×6 Reversi run for almost the same number of moves, yet RSG's β is more than five times that of 6×6 Reversi at D5 (0.539 vs 0.097) — and the gap is wider still at shallower depths. That is the clearest answer to the natural objection, "surely a short game must be a shallow one".
The depth of a game is not decided by the size of the board alone.
The shape of the board (a star of hexagonal cells) and the rules (Capture, Call, Ko) widen the strategy space considerably. On a far smaller board, RSG landed in the same "J-shape" group as 8×8 Reversi (moderate J-shape).
This study has its limits.
This study was carried out by an independent researcher on personal PCs (Intel Core i5, 16GB and 8GB of memory). Looking ahead is done by Negamax search, and each additional ply increases computation time exponentially.
Collecting 10,000 games of 8×8 Reversi at D5 (five plies of lookahead) took roughly 56 hours — about 2.3 days. D6 and beyond would mean leaving a PC running for days or weeks, which was not realistic, so D5 is where this round of work stops.
Q1. What is good about β being close to 1?
It means "games tend to unfold differently every time" — that is, the game is slow to grow stale and offers many strategic options. But a high β does not automatically make a game fun; it is strictly an index of the range of strategy.
Q2. Does a low β mean 6×6 Reversi is a "weak" game?
No. What this study observed is that with a smaller board the strategy space is narrower, so once the AI looks even a little way ahead the routes it judges promising are limited, and games converge on similar shapes.
It is also worth noting that 6×6 Reversi has been solved (Feinstein and others) and is known to be a win for White. The AI in this study searches only to D5, so it is nowhere near that conclusion, but the deeper it searches, the closer it presumably moves in that direction.
Q3. "A level comparable to 8×8 Reversi" — does that mean they are equal?
No. In absolute terms 8×8 Reversi is higher, and especially from three plies of lookahead onwards (D3–D5) its β values go considerably higher. "A comparable level" means that RSG belongs to the "moderate J-shape" group, close to 8×8 Reversi (large J-shape), rather than to the small J-shape of 6×6 Reversi. The point of this study's claim is that RSG falls in that range at all, on a board about 58% the size and with about half the number of moves.
Q4. Measurements come from AI play — isn't human strategy different?
Yes, it is. What this study measures is the strategy space that emerges when the same AI framework plays the game. How a game feels to a human, and how much fun it is, are different questions. That said, collecting large volumes of human game records is difficult, so using AI is also a practical compromise.
Q5. Why does the study go no deeper than D5?
On the author's personal PC, collecting 10,000 games at D5 took about 56 hours (roughly 2.3 days). Beyond D6 the computation time grows exponentially further, which is simply not realistic. If a more powerful computing environment becomes available, this will be revisited in a revised edition. 6×6 Reversi in particular may show a J-shape that rises much further with deeper search, and that remains an interesting open question.
Q6. Why call it a "J" shape?
Looking at the graph plainly, the curve drops and then climbs. "J-curve" and "J-shaped" are used worldwide in economics, sociology and medicine for exactly this pattern of an initial decline followed by a later rise, and this study follows that convention. (Japanese readers have pointed out that the hiragana character "し" is arguably an even better visual match — and they are right 😊)
Q7. What is good about a large J-shape?
A large J-shape means "when you look further ahead, a lot of new strategy appears". From a player's point of view, it is a game where the harder you think, the more there is to discover. A small J-shape means the opposite: the best move settles beyond a certain point, and thinking deeper produces few new lines. So the magnitude of the J-shape reflects how much room the game leaves a player to dig into it. (Again, a large J-shape does not automatically mean "more fun" — it is an index of the breadth of strategic options.)
All of the data behind this study (CSV records of 300,000 games) is kept in a verifiable form. Peer review, corrections and counter-arguments are all welcome.
Note that the analysis in this report covers 300,000 same-level self-play games run under uniform conditions. Including experiments between opponents of differing levels, the total number of games played comes to roughly 2 million, but this study uses same-level matches only, so that the comparison holds conditions constant.
The date of authorship remains April 2026, as that is when the data was collected.
This English edition follows the Japanese plain-language edition v5.2. It adds a little context for readers outside Japan — the table of average game lengths in section 5-1, and this revision history — but no figure and no conclusion differs from the Japanese edition.