Does good chess strategy reduce to the rules of the game?

Question

I've been trying to understand what is meant by words like reduction and reductionism in different contexts. Being somewhat scientifically minded, I enthusiastically embrace reduction as a strategy of explanation when it is done one level at a time. But I get lost when people talk about collapsing all levels down to one, as if that were an end in itself. So here is a loose analogy where things are much simpler. You have the rules of chess (and the objective of winning by checkmate). Most people would agree that there are strategies and principles of play that are objectively good in the sense that they don't depend on anyone's opinion of them. They just tend to work. While nothing is guaranteed, they tend to result in wins and avoid losses.

It's not like a gymnastics competition, where you have to impress judges to score points. If you happen upon a decent chess strategy, you'll tend to beat your opponents even if you don't completely understand why it works. That's assuming they haven't already discovered even better strategies. (I'm using strategy in an expanded sense that also includes the recognition of tactical opportunities, and general rules of thumb like "castle early". What I do not mean by strategy is what game theorists mean by it--i.e. an extensive form/tree that lists the reply to every possible move your opponent could make. EDIT: In other words, I'm referring to heuristics.)

Given the rules and objective of the game, all possible games of chess can, in principle, be enumerated. The rules determine the set all possible future positions, including checkmate positions. In other words, the rules determine all legal move sequences leading to a termination of the game.

Does that mean that good chess strategy can be reduced to the rules?

Answer 1

I wouldn't say that advantageous heuristics reduce to the underlying rules of the game, although you could do this if you can define "advantageous" strictly in terms of those rules. I think we can at least say that this is a case of supervenience where a given set of effective heuristics supervene over the rules or possibly a combination of the rules and player psychology. Reducibility of one set onto another entails a supervenient relation, but whether the inverse is true is a topic of debate.

Answer 2

The short answer is a resounding no, because rules do not encompass the values of the agents that use the rules.

A theory, as often conceived, can be abstracted to a set of set-theoretic, logical, and arithmetic principles represented by a syntax. For instance, in math, foundationalists claim they can reduce the axioms of arithmetic to set-theory expressed logically. Hence, the claim is that ideas expressed in the axioms of ZFC somehow can express anything that can be expressed in arithmetic or consequentially algebra. There are connections between topoi, categories, and logic whereby partial or full reduction can occur. But games are a fundamentally different creature because they require an analysis of agency.

Take a game of chess. The rules of chess are well-understood facts, but the psychological motivation for playing is a question of value. Is a game played by two narcissistic chess masters going to proceed fundamentally different than a friendly game between a mother and daughter? Of course. The more fully the theory accounts for the nature of the game, the more the theory matches the phenomenon. Nowhere is this more apparent in contemporary psychological circles than in behavioral economics. The work of Kahneman and Tversky and the death of homo economicus show the impact of game theory to understand how scientific theory is produced and reduced.

The key to understanding reductionism is that there are two factors at play. First, there is the logical connection of the propositional expressions of the theory, but secondly, there are value-laden decisions made in developing, adopting, defending, and reducing (or resisting reduction) of theory. Even the most stringent physical theories have normative aspects as they are the product of defeasible reasoning and subject to the normativity of games of abstraction.

Hence, a set of rules operates in conjunction with the values of the agent and the choices the agent makes, which in the case of humans can often be irrational and non-deterministic. A man who let s his nephew win constantly at chess because he is dying of leukemia will play a game that cannot be predicted without taking into account his love for his kin, and THAT is not in the rules.

Answer 3

When we talk about the rules of chess, we mean the rules that govern the movements of individual pieces. In other words, we can enumerate a short list:

• All pieces capture opponents by occupying their space, except as noted for pawns
• Pawns move one space forward, except:
  • When capturing, they move on the diagonal
  • Their first move, they may move two spaces, capturing pieces they pass (en passante) without occupying their space
• Rooks move orthogonally any number of spaces, unless blocked by other pieces
• Bishops move diagonally any number of spaces, unless blocked by other pieces

...and so on. The issue of reductionism revolves around the question of whether these singular (atomistic, independent) rules are both necessary and sufficient to describe the game of chess. The rules are clearly necessary: one cannot play chess without knowing the rules of the game. But it is far less clear whether the rules are sufficient to the task. Can a given strategy — even a simple one, like using a pawn to support a bishop — be explained solely in terms of the movements of the individual pieces?

Part of the reason that reductionism is so appealing is that it actually works well in many contexts. For instance, in ballistics people regularly ignore factors like the shape of the projectile, the distribution of mass within it, its interaction with the surrounding air through friction or rotation, etc. The projectile is reduced to a point mass subject to the simple independent factors of the force of gravity and the initial velocity vector, and for the most part it works well. The omitted factors produce small variations that are easily accounted for by statistical error handling, and no one generally cares whether their cannonball lands six inches off target in one direction or another, as long as it gets where it's going.

However, in other contexts this reduction to simple independent variables produces larger errors or incoherent results. We often see this in the social sciences, where something like (say) gender might only account for 35% of the variance, and we won't get decent results unless we consider gender and ethnicity and age all together at the same time. So then we have to ask: is there some other singular 'rule' or 'variable' that this complex situation can be reduced to, or do we need to take a non-reductionistic (holistic) approach to analyzing these contexts? That question remains to be solved.

Going back to the chess analogy, we can see the three cases:

• Is strategy reducible to the simple (isolated) rules of individual pieces?
• Is strategy reducible to some other simple (isolated) rule that we have not yet discovered?
• Is strategy a holistic principle that emerges from (and is distinct from) the simple (isolated) rules of individual pieces?

We know the rules are necessary; we don't know whether they are sufficient. So pick your side...

Answer 4

Reduction is, in its relevant sense, an invertible and explanatorily adequate simplification. So, if X is reduced to Y:

• Y is simpler to understand, to handle, etc. in regard to the reductive purpose. It should not give the impression of obscurum per obscurius.

• X and Y is so related that X can be recovered or derived from Y. Y shouldn't be an arbitrary or random splitting of X, or an outcome of an irreversible processing of X.

• Y accounts for all whatever we know and would like to know about X. To wit, we should be able to understand X to our satisfaction by referring to Y.

For example, suppose that a biological phenomenon X is explained by a chemical phenomenon Y (notice that we talk about a translation from biological vocabulary to chemical vocabulary; although very interesting, it is far fetching to go into ramifications of this change). No doubt, the knowledge of Y enhances our understanding of X. But is this a reduction of a biological matter to a chemical one? We have to check further to affirm: Does Y enable us to make sense of X in its biological environment? Does Y allow us to predict sensibly about X, especially, for that environment?

This is not a question of computational resources - whether Y can actually replace X, or only potentially. It is a question of concepts, laws and representational systems. Even though we would accomplish to simulate X by Y in the most detailed manner, our understanding of X might still remain quite poor.

Many reducibility issues arise around how we ascribe simplicity and explanatory value to the reducing base, Y in our case. There may be borderline or degenerate cases in which differences between X and Y collapse as regards our assessment.

As for the chess example, does the brute force search reveal something significant to our understanding of the game? I assume that what is described is virtually a combinatorial exercise, so not a game anymore. Hence, it could only simulate game strategy, not represent a reduction of it. However, the brute force method should not be viewed as inferiority; it depends on the nature of the subject. It might well show us that a simplistic sequence of moves would suffice and there was no need to wrack our brains with chess strategy at all.