SMIRF의 기물 가치 계산

이 글은 SMIRF의 저자이자, 카파블랑카 랜덤 체스의 발명자 Reinhard Scharnagl(1953년 ~ 2015년 8월)이 발표한 체스 기물 계산법이다.

## Overview (Position Evaluation in Chess)

Many chess programs use **evaluation functions**, which work with strange **penalty** and **reward mechanisms**. Over that these often depend even on whether the system believes to be in the opening, in the middle, or in the end of a game.

**SMIRF** pursues an approach somehow different. Only such elements, which appear in the **current board situation**, have influence on an evaluation. Thus factors like move counters or not conclusively justified arguments from the experience of human play practice no more have relevance, except of being an **instrument of verification**.

In **SMIRF** the different aspects of such an evaluation function exclusivly will be derived from **theoretical considerations**.

This attempt also is permitting to formulate the problem of a position evaluation into the direction of **machine intelligence**, by breaking anything relevant down to its real basis and afterwards rejoining all in an appropriate way into a coherent system - even when the newly built up construct may appear really strange.

## Motivation for an Evaluation

In chess there originally exists the **final evaluation** only: any side either can have **lost** or **won** the game, the game ended **remis** or however still is **running**. This corresponds with the natural evaluations of a zero-sum game: the final values **(-1), (+1), (0)** and - before an ending - a **question mark (?)**. Therefore our special attention applies for that mystery, how a **current game of chess** has to be estimated.

By observating games of chess one has made the experience that their situation flow in a certain sense is **continuous**. If two involved players will act constantly in accordance with their strength, then for experts the **final result** of the game is foreseeable already quite a while before that game ends. A **game evaluation** is like that a **standardized prognosis** based on the actual existing **distribution of power**.

It obviously seems to be a promising **strategy**, whenever no worthwhile final situation can be attained, to select that under the existing possibilities, which offers a particularly good prognosis. Thereby the match of two players becomes the object of the quality of their **prognosis systems**.

During the time it turned out as expedient to make an evaluation under several criteria: it takes place on the one hand **statically**, then **dynamically** and finally **positionally**. Now these three aspects will be described in the following more detailled.

We will see, how **vague** the used terms are, grown from the practice of the game, and how necessary an appropriate **objectification** will be for any playing program.

## Aspects of an Evaluation

**static rough Evaluation**:

It already was observed very early that not rarely that one is winning a game, who finally possesses the most or the stronger chess pieces. From this experience **average exchange values** of the individual figure types crystallized, which correspond with the average potential of their individual type and for practical reasons are measured in **pawn units**. It seems to be a suitable auxiliary goal to achieve a **balance of material** as pleasant as possible e.g. by avoiding disadvantageous exchanges. Who succeeds to achieve the so-called **quality** (an advantage of approx. two pawn units) exceeds the **draw width** and often thereby already has the later victory for sure.

**dynamic Analysis**:

It does not make a lot of sense to evaluate a position in which the player actually to move can conquer a figure or realize a positional advantage. One rather would look forward, how the tangling probably would end. Therefore such completed, **calm positions** are those, which are to be evaluated in detail. From those evaluations can be derived also for the troubled situations by **combinatorial** considerations. Moreover **sacrificing combinations** could be recognized and assessed, as well as **horizon effects** could be avoided.

**positional Detail Evaluation**:

After having completed some games of chess it will become clear, that it matters, where upon the board the pieces have been placed. It depends on whether they will be **positioned** as **effective** as early and possible, and therefore being **developed**. Here the idea is, to become able to choose among as much as possible moves, for to get a bigger chance making a really good move, which serves to other auxiliary targets and also can **obstruct** the opponent to spread out. In an extreme case one side gets into a **zugzwang** situation, in which still only disadvantageous moves are possible. Another elementary goal is **to dominate** as much squares on the board as possible. At times it even will make sense to advance some certain **material sacrifices**, for to positionally optimize the situation.

## Average Values of Chess Pieces

When just learning how to play the game of chess most poeple will ask, **why** just this special piece should be worth more than that other. Additionally someone, who is willing to write a chess playing program, should ask the same question. But unfortunately the interest in getting a reasonable answer for that often will expire soon. Finally the pratice will teach us, that a game of chess hardly could be won by ignoring the valencies. One simply is **believing** in the experience based figures and finally will take over the value system of teaching masters. In the end it remains hidden away what makes the **very basis** for this differences of scales of values.

So let us have a look on which values have been stated for the separated types of chess pieces by well known **theoretician** and **practical men**:

The specified **experience based numbers** here mostly are expressed using **pawn values**. Moreover the figures differ the more, the seldom pieces from affected type usually would be exchanged into different.

There are very different views to the piece of the king. Because it is not allowed to capture him and of technical reasons an extremly high value often will be taken. But nevertheless one more positionally derived establishing of his value will make more sense (as with L. Orbán), instead of completely giving up such an idea.

## An Objectifying of Chess Piece Values (Trial)

Up to now it was not known to the author that there has been attempts made to put down the asserted values each to its only and original possible **cause** of their differences, i.e. **a)** to the prevailing different **gaits** of the individual chess pieces **and** **b)** to their possible **interaction** among themselves - independently from any current position on the board, **and also** **c)** the **relocatability** of the piece's effects.

(As however recently was communicated by **Prof. Ch. Posthoff** a comparable approach has been reported by him within the GDR magazine "Wissenschaft und Fortschritt", Berlin. This obviously underlines the relevance of that methodology.

However there are diverging starting points which **exclusivly** are based on the different gaits. This could be understood, because in principle the chess pieces functionally differ **in that point only**.)

squares
 affected
pieces

However in this context some details are not crucial, e.g.:

* *the **amount** also of simple **non-capturing moves**, like e.g. pawn moves, which depend on a lot like individual pieces not to be blocked (and which before have no further effect on their target squares)*
* *the **positions** of individual **chess pieces** on the board (because of especially the average value has to be of relevance)*
* *particularly the slow **increasing value** of a passed pawn on its way to a possible **promotion***

It would be the task of a positional detail evaluation then to weight the latter points, with the special case of a passed pawn, whose unsteady exchange value has to be determined before additionally in each case.

## Values of the stepping pieces Pawn and Knight

Let us first regard those chess pieces, which only have close effects: **pawn**, **knight** and **king**, and note down with them for each possible square of the chessboard all the corresponding effects and affected directions, then it will be noticed, that with all three piece types both aspects in each case will produce **identical** countings (which is what characterizes them as non sliding figures).

sum of affected directions = 84
sum of affected squares = 84 at 1/1
average effect value **P**:
VP = (84 + 84 \* (1 + (1/1)/4)) / 48
= 3.9375
**Pawn's piece value** (P-adjusted):
VP/VP = 3.9375/3.9375 = **1.0000**
sum of affected directions = 336
sum of affected squares = 336 at 1/1
average effect value **N**:
VN = (336 + 336 \* (1 + (1/1)/4)) / 64
= 11.8125
**Knight's piece value** (P-adjusted):
VN/VP = 11.8125/3.9375 = **3.0000**

## Values of the sliding pieces Bishop and Rook

Let us then regard those chess pieces, which also have distant effects: **bishop**, **rook** and **queen**, and note down with them for each possible square of the chessboard all the corresponding effects and affected directions, then it will be noticed, that with all three piece types both aspects in each case will produce **different** countings (which is what characterizes them as sliding figures).

sum of affected directions = 98
sum of affected squares = 280 at 1/2
average effect value **B**:
VB = (98 + 280 \* (1 + (1/2)/4)) / 32
= 12.9063
**Bishop's piece value** (P-adjusted):
VB/VP = 12.9063/3.9375 = **3.2778**
(692; 1224; 1/1)
VB = 14.1750
VA = 27.7750
 affected directions Rook
sum of affected directions = 224
sum of affected squares = 896 at 1/1
average effect value **R**:
VR = (224 + 896 \* (1 + (1/1)/4) / 64
= 21.0000
**Rook's piece value** (P-adjusted):
VR/VP = 21.0000/3.9375 = **5.3333**
(724; 1720; 1/1)
VR = 23.5500
VC = 35.9250
'*Value(**R**) = **5.8148**Value(**C**) = **8.8704'***

## Values of the royal pieces Queen and King

Let us finally regard **queen** and **king**: so we will notice, that the king in principle is a non sliding queen and that the valuating of the queen can be reduced nearly to a sum of the accounts of bishop and rook.

sum of affected directions = 420
sum of affected squares = 1456 at 1/1
average effect value **Q**:
VQ = (420 + 1456 \* (1 + (1/1)/4)) / 64
= 35.0000
**Queen's piece value** (P-adjusted):
VQ/VP = 35.0000/3.9375 = **8.8889**
sum of affected directions = 420
sum of affected squares = 420 at 1/1
average effect value **K**:
VK = (420 + 420 \* ( 1 + (1/1)/4) / 64
= 14.7656
**King's piece value** (P-adjusted):
VK/VP = 14.7656/3.9375 = **3.7500**

## Exchange Values of passed Pawns 8x8

It makes no sense to evaluate a pawn on the way to its transformation square always constantly with 1. Rather a **continuous increasing** of its possibilities to intervene into the game can be assumed until it finally can be converted (maximally) into a queen.

This increase in value primarily depends only on its individual moving on and is quite **independent** from the remaining events on the board. However it is like that, for it possibly to find **individual opponents** that way, who particularly disturb any progress. Thus for the value establishing here a special situation is present, which has to be settled somewhere after the rough evaluation but still clearly before a positional detail evaluation.

Thus we specify first the **characterisation of a passed pawn** precisely: before him no further pawn (even no own) also is on his file, and no hostile disturbing pawn on a neighbour file.

Since a passed pawn goes through exact seven distance levels (transformation square included), and because the increase in value takes place not linear but **exponentially**, the **seventh root of the queen's value**, thus about **97/71**, seems to be the obvious growing factor per step (an alternative underpromotion should be regarded as tactical maneuver, thus is no object of an evaluation).

There are however in front of it on the open file still further **individually disturbing influences**, i.e. hostile pieces or their effects. Each square disturbed in such a way forces an upgrading of the distance up to the value 7 (maximum upper limit).

From this the **exchange value** of a passed pawn thus is to be computed by following formula:

**Level = Min(7, Distance + disturbed Squares)** and finally: **Value = (97/71) ^ (7 - Level)**.

More extensive positional considerations still are of less importance here, but are reserved to the later more general **positional detail evaluation**.

## Verification and Result

Reviewing the results calculated that way and comparing them with estimates of specialists, then fortunately a clear accordance could be determined: the found values in pawn units integrate **harmoniously** into the range of traditional statements on piece values. Thus an **acceptable model** for an objective valuation seems to be discovered.

A neutral positional detail evaluation of chess positions also could be based upon this attempt. Here it is typically irrelevant whether it is related to a situation from the opening, the middle or the final game, different to many other approaches.

## Prospect and Summary

It has to be noticed particularly that this model for the determination of average piece's values will be suitable especially for the evaluation of **hypothetical** figures (e.g. from **fairy chess**) with deviating gaits. A typical example is shown within the analysis of the **Nightrider**, a knight with effects on distant squares:

sum of affected directions = 336
sum of affected squares = 616 at 1/1average effect value **D**:
VD = (336 + 616 \* (1 + (1/1)/4)) / 64
= 15.28125
**Nightrider's piece value** (P-adjusted):
VD/VP = 15.28125/3.9375 = **4.3889**

Beyond that this method as well is suitable for the adjustment of the piece's values at the transition into **Superchess**, which will be described in the following.