Arimaa: Chess, but Better
By Graham Everhart
Few games around the world have enjoyed such a universal and ubiquitous following as chess.
In the 1990s, chess was considered the last bastion of human dominance over computers. Until, of course, it wasn’t. In 1997, the IBM supercomputer Deep Blue defeated chess champion Garry Kasparov in a series of six games.
Since then, it’s only gotten easier for computers to beat humans. The highest ever chess rating achieved by a human was 2882 by Magnus Carlsen. Today’s top chess programs can reach 3400 ratings when run on powerful computers. Meanwhile, the human tournaments are now considered “mind sports” and include live commentary, computer analysis, and intimate profiles of the competitors that treat them more like gladiators than game players. It’s not about what’s on the board—it’s about the spectacle and the struggle around it. Even if matches produce twelve consecutive draws, like last year’s championship.
22 years ago, Indian-American engineer Omar Syed watched the Deep Blue vs. Garry Kasparov match, and had an idea. Humans didn’t need to learn how to beat computers at chess—they needed a new game that they would always be better than computers at. It should be simple enough for humans to learn, yet complex enough that computers would not be able to “brute-force” their way to victory by considering every possible combination of moves, as Deep Blue had.
Arimaa is playable with the same board and pieces as chess, although it’s more fun to play with a custom Arimaa set. Here’s what you need to know:
- All pieces have animal names. Instead of pawns, knights, bishops, rooks, queens, and kings, Arimaa has rabbits, cats, dogs, horses, camels, and elephants, in that order.
- All pieces move the same way. You get four “moves” each turn, and each move consists of moving a piece to an adjacent space. You can split those moves between pieces any way you like.
- You don’t capture enemy pieces by landing on them. Instead, you have to get them onto a “trap”—one of four special spaces clustered around the middle of the board—and then isolate them from any adjacent friendly pieces.
- Larger animal pieces outrank smaller animal pieces. For instance, horses outrank cats, dogs, and rabbits, but not elephants, camels, or other horses. Elephants outrank all other pieces but themselves, while rabbits don’t outrank any pieces. Outranking an adjacent enemy piece confers three benefits:
- The larger piece “freezes” the smaller piece. The smaller piece cannot move unless it’s adjacent to a friendly piece.
- The larger piece can “push” the smaller piece. At the cost of two moves, the larger piece moves into the smaller piece’s space and pushes it to an adjacent space.
- The larger piece can “pull” the smaller piece. At the cost of two moves, the larger piece moves to an adjacent space and pulls the smaller piece into its old space.
- There is no fixed starting position in Arimaa. Before a game begins, both players set up their pieces in any configuration on their first two rows.
- To win, a player must get one of their rabbits to the other side of the board.
There are a few more minor nuances of these rules, but that’s pretty much it.
Here’s a video that explains all that and more.
Straightforward, right? Not if you’re a computer like Deep Blue.
When your turn begins in a game of chess, you will have, on average, about 35 different legal moves to choose from. You want to choose the move that gives you the biggest advantage, and you determine that by anticipating which of the 35 or so legal moves your opponent is likely to make, which in turn will influence your next move, which will influence their next move, and so on. Each move you add to your calculations increases their complexity by a factor of 35 or so. This is referred to as chess’s “branching factor.” Branching factor is not terribly important for human players, as humans are excellent at weeding out obviously poor moves and focusing on the few potentially good ones. But it is for computers.
Because chess’s branching factor is a relatively low 35, Deep Blue was able to consider all possible board positions of moves further in advance than Kasparov, giving it more insights into which move should be played in the present. This was how it won.
Go, the East Asian stone-laying game, has a branching factor of about 250. Arimaa, however, has an average branching factor of about 17,000! If the number of calculations a computer has to do goes up by a factor of 17,000 for each additional move the computer considers, the computer can only “think” a few moves in advance—theoretically giving humans the advantage.
But with the advent of neural networks and machine learning, computers are learning to avoid wasting time considering poor moves, freeing them up to think deeper about the good ones. This has enabled them to beat humans at games with high branching factors.
In March 2016, a Google program defeated 18-time Go champion Lee Sedol four games to one. And in April 2015, the program bot_Sharp defeated three top human Arimaa players seven games to two. The game supposed to be hard for computers is now easy for computers.
But chess and go are both ancient games centuries old. Arimaa is only as old as the Class of 2020, and new strategies and tactics are being discovered all the time. If you feel like chess is dead—if you feel like there are no new things about the game to discover—you’re not alone. Join me.