Sunday, 28 August 2016

Why Chess? Part 1: Smaller Boards

Chess fascinates me; it's a very pure board game, and it has existed in some recognizable form for centuries. There's a couple of general questions I have about chess that I return to occasionally, but haven't formally approached until now.

1) Why is the game that we consider chess the orthodox game and not some variant or alternative?

2) How robust is existing artificial intelligence to changes in chess?

Chess, at least in the western world, fills a niche in the game market. It's a competitive game between two players with no random chance and no physical requirement. The rules of the game take an hour to learn, a day to understand, a lifetime to master. In game design terms, chess has a great deal of depth, but limited complexity.

It's not the only such game. The game of Go, for example, has fewer rules, comparable depth, and an effective and convenient method to handicap.

Also, what about variants of chess? Would the game be less interesting or deep if the knight moved in a 3-and-1 jump instead of a 2-and-1 jump? What if the board dimensions were different? Chess has changed before, however slowly. Is it just an accident of history that this is the particular game we got, or is there something optimal about it?

I've been playing games of different variations of chess starting with those available on the Android app Chess Variants. Several of the variants avialable in this app are games that use the same kinds of pieces as orthodox chess but fewer of them, and on a smaller, simpler board. One clear pattern has emerged after 50 plays across 3 variants: I'm horrible. I am embarrassingly bad at chess compared to a computer opponent.

Mini chess (original).

The first variant I tried was Martin Gardner's 5x5 variant, made in 1969. I played as white (first) against an AI opponent (to be discussed in part 2) about 20 times. The result was 0 wins, 2 draws, and a heap of losses. The app allows you to play as black, or even as both for local play, but I'm stubborn.

The layout, shown in Figure 1, suggests a major advantage to black to a naive player like myself. There are 7 possible moves white can make at the beginning of the game. All of these moves let a piece be captured right away. What else could explain my tremendous defeat?

Figure 1 – Martin Garner's original Minichess

I looked up the game to see if my suspicions on white's disadvantage were correct. They were not. It turns out the Gardner's 1969 version of 5x5 chess is a weakly solved problem. Each player can play perfectly and always draw or win. If both are playing perfectly, the game ends in a draw. Furthermore, the perfect play algorithm (also called the 'oracle'), is a mere ~10 pages of if-then instructions.

The "weakly" in weakly solved refers to the fast that a perfect strategy exists and is known from the starting position of the game, but not necessarily every position. So if you play sub-optimally to start a game, the 'oracle' doesn't necessarily cover your situation.

A paper by Mhalla, M., & Prost, F. (2013) outlines this perfect strategy and talks about the winning rates of each side in a sample of historical correspondence games. The white player won slightly more of these games than black did.

Mini chess (updated).

This is a 1989 update to Gardner's 5x5 chess. It is identical except that the knight and bishop have been swapped on the black side.

I played 5 games on this version after playing the 1969 version. Games lasted longer and were closer, but with a small sample and an experience confound, I'm not confident in explaining why.

Micro Chess

This seems to be as minimal as chess gets without being a chess puzzle instead of a game. The layout, shown in Figure 2, only has one pawn per side. The pawn can be moved two squares in its first move. However, the pawn cannot be promoted to a queen. This seems reasonable as no player starts with a queen.

Figure 2 – Micro Chess

There is a one move checkmate for black:
White bishop to C2
Black knight to C3
White is in checkmate.

I was unable to win any match at this (again as white) either, but I did reach a draw quite often. My suspicion is that the knight is the most powerful piece, and that this change in the relative strength of pieces could cause an AI opponent to make poor trades if it was using piece valuations from classic chess. No such luck.

The Alpha-Beta Algorithm

The AI used in this app employs the alpha-beta tree algorithm, which is good for quick computation, like on a phone, because it eliminates large sets of possible actions quickly. It's also non-deterministic; it will choose different actions in identical situations if there is no single obviously best solution.

The alpha-beta algorithm is not specific to chess. It could be used for any discrete choice system, like Shogi or Go. However, it relies on a heuristic scoring system. To use the algorithm, an AI needs a way to evaluate how good or bad a consequence of that action is. Last time, I talked about throwing off the AI by assuming it would under-value a knight. That is, I was assuming that the value assigned to the consequence "lose your knight" would be copied over from classic chess to micro chess, and that this value could be exploited.

The Alpha-Beta algorithm also has some tuning parameters that determine the quality of the decisions made by it. By changing the number or depth (turns removed from the current situation) of the possibilities that the algorithm will evaluate. These parameters can be manipulated indirectly in the Chess Variants app by a difficulty setting with four options: easy, medium, hard, and fast.

The last setting, fast, was the one I was using. In this case, the algorithm will evaluate solutions until 1 second of processor time has been used. This means the algorithm becomes harder to beat when it is provided more processing power, so I can blame my equipment for my losses. The AI may have been playing a close-to-perfect game on the simplified boards, because the space of possible states was much smaller. I also noticed that in orthodox chess matches, the AI got better as the game progressed and pieces were removed from the board. I could frequently take the opponent's queen without a sacrifice only to be crushed in the end-game.

For existing pieces, the value of retaining each piece, and having them in various positions relative to an opposing king is well established by humans. That, I suspect, is why Jocli, the platform used to make all these chess variants online, only has variants that use pieces that exist in orthodox chess and which alter the board - because the heuristics for these (combinations of) pieces are well-established.

Next, I intend to look into Chess 960, which is a variant played on an 8x8 board in which the starting positions of non-pawn pieces are randomized. There are 960 possible starting arrangements that fit the restrictions (King is between rooks, bishops are on opposite colours, black mirrors white, etc. ), hence the name.

An AI may not be able to adapt a few opening strategies from orthodox chess to chess 960, but it fares well by treating any starting as a game already in progress (even if that progress was impossible). If you introduced a common chess puzzle piece, such as the grasshopper or princess, would an entirely new set of heuristics be needed?

Relevant links:

Mhalla, M., & Prost, F. (2013). Gardner’s minichess variant is solved. ICGA Journal, 36(4), 215-221. The platform used for the Chess Variants app. Also playable in a web browser. including Gardner's Minichess, and Microchess. list of unorthodox chess pieces used in puzzles and variants.

Still to explore:
The Duke
Nightmare Chess
Alice Chess
The Encyclopedia of Chess Variants, by David Pritchard
The Classified Encyclopedia of Chess Variants, by John Derek Beasley