This paper was published in the late nineties. Right before [Deep B...
### MiniMax and Alpha-Beta pruning
#### MiniMax
MiniMax is a de...
### Chess as a search problem
You can think of chess as a decision...
#### How big is the chess game tree?
On average, the tree has a br...
In the 1970s Simon and Chase performed a number of very interesting...
- [Programming a Computer for Playing Chess by Claude E. Shannon (1...
A *quiescent position / Turing dead position* is a position where t...
The Historical Development of Computer Chess and its
Impact on Artificial Intelligence
David Heath and Derek Allum
Faculty of Science and Computing,
University of Luton,
Park Square,
Luton LU1 3JU
United Kingdom
david.heath@luton.ac.uk
derek.allum@luton.ac.uk
Abstract
In this paper we review the historical
development of computer chess and discuss its
impact on the concept of intelligence. With the
advent of electronic computers after the Second
World War, interest in computer chess was
stimulated by the seminal papers of Shannon
(1950) and Turing (1953). The influential paper
of Shannon introduced the classification of chess
playing programs into either type A (brute force)
or type B (selective). Turing’s paper (1953)
highlighted the importance of only evaluating
’dead positions’ which have no outstanding
captures. The brute force search method is the
most popular approach to solving the chess
problem today. Search enhancements and
pruning techniques developed since that era have
ensured the continuing popularity of the type A
method. Alpha-beta pruning remains a standard
technique. Other important developments are
surveyed. A popular benchmark test for
determining intelligence is the Turing test. In the
case of a computer program playing chess the
moves are generated algorithmically using rules
that have been programmed into the software by
a human mind. A key question in the artificial
intelligence debate is to what extent computer
bytes aided by an arithmetic processing unit can
be claimed to ’think’.
Introduction
With the advent of computers after the end of the
Second World War, interest in the development
of chess playing programs was stimulated by two
seminal papers in this area. The paper by
Shannon (1950) remains even to this day to be
central importance while the paper by Turing
(I 953) is equally influential.
The minimax algorithm was first applied in a
computer chess context in the landmark paper of
Shannon. He also introduced the classification of
chess playing programs into either type A or B.
Type A are those that search by ’brute force’
alone, while type B programs try and use some
considerable selectivity in deciding which
branches of the game tree require searching.
Alpha-beta pruning was first formulated by
McCarthy at the Dartmouth Summer Research
Conference on Artificial Intelligence in 1956.
However, at this stage no formal specification of
it was given, but it was implemented in game
playing programs of the late 1950s. Papers by
Knuth and Moore (1975) and Newborn (1977)
have analysed the efficiency of the method and it
has been proved that the algorithm returns
exactly the same move as that obtained by full
minimaxing or, alternatively, a move of the same
value.
The success of type A ’brute force’ programs
using exhaustive search, minimaxing and alpha-
beta pruning, transposition tables and other
search enhancements has had the unfortunate
effect of minimising interest in the development
of type B programs. The work of Simon and
Chase (1973) established that most humans only
consider a handful of plausible moves. The
power and ability of the chess grandmaster
resides in his ability to select the correct subset
of moves to examine.
In contrast the brute force programs search the
entire spectrum of initial moves diverging from
some given position referred to as the root.
These initial moves fan out, generating a game
tree which grows exponentially with depth.
Apart from the work of Botvinnik et.al in recent
years, there has been no significant progress in
developing a type B strategy program which
63
From: AAAI Technical Report WS-97-04. Compilation copyright © 1997, AAAI (www.aaai.org). All rights reserved.
would reduce the initial span of the search tree.
The successful development of algorithms that
introduced selectivity into the search engine, so
that the program followed similar lines of
thought to those of a chess grandmaster, would
considerably reduce the amount of searching
required.
Turing’s paper (1953) highlighted the
importance of only evaluating ’dead positions’
which have no outstanding captures. It is from
this paper the term ’Turing dead’ is taken. Most
chess programs search to a quiescent position
(Turing dead) and then evaluate the position as
function of the material balance and other
features. This evaluation function encapsulates
chess-specific knowledge typically relating to
pawn structures and king safety.
History
The historical evolution of computer chess
programming techniques and knowledge can be
conveniently discussed in three broad eras. Each
era is characterised by its own particular
developments, some of which can be directly
linked to increased processor power, the
availability of new hardware devices and others
to algorithmic advances.
The boundary between each era is not always
precise as it is sometimes not easy to draw a
clear dividing line across a time continuum. Any
such process is artificial. However, these broad
historical eras are (Donskoy and Schaeffer
1990):
(i) 1st era (1950 - c1975)
(ii) 2ndera(c1975-c1985)
(iii) 3rd era (c1985 onwards)
The first pioneering era as stated above runs
from 1950-c1975. Here there is a definite point
at which this history commences, marked by the
publication of Shannon’s paper and the advent of
electronic computers. These computers, although
originally regarded as revolutionary and
powerful, had but a fraction of the computing
power of the current generation of
microprocessors. Indeed hardware limitations
characterise the first era of computer chess
development, requiring highly selective
techniques in order to produce moves in an
acceptable time. The earliest programs were,
therefore, Shannon type B.
The first significant chess playing program
was by Bernstein (1957) and ran on an IBM 704
computer, capable of performing approximately
42,000 operations per second. This was not a
’brute force’ program as it only selected the best
seven moves for consideration using heuristics
based on chess lore. Compared to the
sophisticated brute force programs of today
which generate the full span of moves at the
root, this is a very limited range of moves.
The Bernstein program was built around the
strategy of working to a plan. As it was
incapable of doing a full width search due to
time considerations, it selected its quota of seven
moves for deeper analysis by seeking the answer
to eight questions. Once the moves were
selected, they were analysed to a search depth of
4 ply. The potential replies by the opponent were
also selected on the basis of seeking answers to
the same set of questions.
Bernstein’s program played at a very
elementary level and the first program to attain
any recognisable standard of play was that of
Greenblatt (1968). For a number of years this
remained the most proficient chess program and
played at an Elo strength of approximately 1500.
It had carefully chosen quiesence rules to aid
tactical strength and was also the first program to
use transposition tables to reduce the search
space. However, the Greenblatt program also
used an initial selection process to minimize the
size of the game-tree as the computing hardware
of that era was incapable of achieving the
computing speeds of today. Again this program,
because of its selectivity at the root node, falls
into the first era.
The first program to achieve full width search
and make ’brute force’ appear a viable
possibility was Chess 4.5. This program was
developed for entry into the ACM 1973
Computer Chess contest by Slate and Atkin,
using the experience they had gained in
programming earlier selective search chess
programs. The techniques used by Slate and
Atkin (1977) are still in use today although they
have been refined and improved over the years.
These standard techniques initiated what may be
classed as the second technology era giving rise
to programs typically searching to a fixed depth
as fast as possible and then resolving the horizon
problem by extending checks at the cutoff depth
and considering captures only in a quiescence
search.
64
The second era of computer chess also saw
more emphasis placed on the development of
dedicated chess hardware. Typical of such
developments was Ken Thompson’s chess
machine Belle which won the ACM North
American Computer Chess Championship in
1978. The special purpose hardware increased
the speed of Belle enabling it to analyse 30
million positions in 3 minutes and search
exhaustively to 8 or 9 ply in middlegame
positions. It also had an extensive opening book.
Belle won the 3rd World Computer Chess
Championship in 1983, achieving a rating of
over 2200 in 1980 and was the first program to
receive a Master rating. It was for Belle that
Thompson devised his first end game database,
solving the KQKR problem and hence partially
removing the perceived weakness at that time of
computers in endgame positions.
In 1975 Hyatt commenced work on Blitz
which was then entered in the ACM 1976 North
American Computer Chess Championship.
Initially Blitz was selective and relied on a local
evaluation function to discard moves. However,
the availability of the world’s fastest
supercomputer, the Cray, enabled the program
appropriately renamed Cray Blitz, to use brute
force and in 1981 it was searching approximately
3000 nodes per second and consistently
achieving six ply searches. This rate of analysis
was improved by the use of assembly language
and the availability of the Cray XMP computer
with multiprocessing facilities, allowing 20,000 -
30,000 nodes to be searched per second in 1983.
The third stage, aptly named algorithmic by
Donskoy and Schaeffer (1990), extends onwards
from the mid 1980s to the current time. This has
seen some considerable activity in the refinement
of the basic tree searching algorithms used (see
later). It has also seen the emergence of personal
computers on a scale originally unenvisaged.
The widespread practice of incorporating vast
opening books and more cd-roms for end games.
This current phase has been most fruitful and
seen a considerable increase in the playing
strength of chess programs to the point where the
top commercial software programs are generally
recognised as being superior to humans at speed
chess and in sharp tactical positions. Under strict
tournament conditions the most highly rated
player of all time Kasparov has now lost a game,
but not the match, against Deep Blue.
Move Ordering Techniques
The previous section has outlined the historical
development of computer chess. The transition
to Shannon B type programs to Shannon A is not
solely attributable to increased computing power.
It also partly arose out of increased
understanding of the alpha-beta algorithm which
was subjected to deep analysis by Knuth and
Moore (1975). Other techniques for increasing
the efficiency of the search and pruning
mechanisms also became more prominent at the
beginning of the second era.
Producing a cutoff as soon as possible with the
alpha-beta algorithm considerably reduces the
size of the search tree. Consequently, move
ordering is an important aspect of achieving
maximum speed since, if we know the best move
in a certain situation producing it early rather
than late, will have beneficial results. The worst
case is where the moves are examined in such an
order as to produce no cutoffs, generating the
maximum number of nodes to be analysed. This
is the maximal tree which grows exponentially
with depth, d, so that the number of nodes
examined assuming a uniform branching factor,
b, is given by b
d.
The best situation is where the
moves are all well-ordered and provide
immediate cutoffs, producing the minimal tree
which, although it also grows exponentially with
depth, is very much reduced in size by the
cutoffs. In between these two extremes we have
game trees generated by chess programs which,
initially are unordered, but become progressivley
more ordered as the depth of search increases.
Algorithmic developments and various
heuristics are moving the programs closer to the
minimal tree. This progressive improvement in
reaching closer and closer to the minimal tree is
borne out by Belle, which it is estimated came
within a factor of 2.2 of the minimal tree,
Phoenix within a factor of 1.4 and Zugzwang
within an impressive factor of 1.2 (Plaat 1996).
Iterative deepening is a useful device for
allowing moves to be re-ordered prior to the
depth being increased as well as providing a
method for limiting search depth in response to
time constraints. Originally introduced in 1969,
it has become standard practice in brute force
programs.
The killer heuristic is another move ordering
technique using information from the alpha-beta
pruning algorithm to facilitate it. Whenever a
certain move causes a cutoff in response to
65
another move, then it is likely that it is able to
refute other moves in a similar position. Such a
move is a ’killer move’ and can be stored in a
killer table so that it can be tried first in order to
cause another cutoff. Captures and mate threats
provide the commonest form of such ’killer
moves’. In this context the null move can
provide a list of the most powerful moves for the
opponent and assist in the move ordering
process.
The killer heuristic was described in detail by
Slate and Atkin (1977) although it had been used
in earlier chess programs. However, the benefits
of this technique are controversial with Hyatt
claiming an 80% reduction while Gillolgly
observed no such significant reduction in size
(Plaat 1996).
A generalisation of the killer heuristic is the
history heuristic introduced by Schaeffer (1989).
This extends the ideas of the killer heuristic to
include all the interior moves which are ordered
on the basis of their cutoff history.
Tree Searching Algorithms
Chess programs have used till now almost
exclusively one of a variety of related algorithms
to search for the best move. Constant
improvements in the search algorithm have been
sought to reduce the number of nodes visited,
and thus speed up the program.
We now briefly describe a number of
variations upon the alpha-beta algorithm. These
depend on having some knowledge, albeit
imperfect, of what is the most likely principal
variation, the series of moves down the tree
consisting of the best play.
The usual call to alpha-beta at the root node
has an infinite window. In aspiration search
some initial estimate of a likely value is made
together with a window, or band of values this
likely value could lie in. Clearly the success of
this algorithm depends on the initial estimates. If
a value is returned which lies outside the
aspiration search window then a re-search needs
to be done with full alpha-beta width. There is
thus a trade-off to be made between a greater
degree of pruning using the limited window and
the amount of re-search which has to be done.
Principal variation search (PVS) is a variation
of alpha-beta search where the search at a
particular level is ordered according to some
evaluation function. The node which is found to
be the most likely member of the principal
variation is searched with a full width alpha-beta
search, while others are searched with a minimal
window where [3 = (x + 1.
With perfect move ordering, all moves outside
those estimated to be part of the principal
variation will be worse than those estimated to
be on it. This is proved when the minimal
window search fails low. Should this search fail
high, a re-search has to be done with full alpha-
beta width.
A refinement on PVS is NegaScout. This is
very similar to PVS and incorporates the
observation that the last two plies of a tree in a
fail-soft search always return an exact value.
There have been a number of further
derivations of the so-called zero window alpha-
beta algorithm, which use a zero (actually unity)
window at all nodes, unlike PVS which uses zero
window only away from the principal variation.
One of these is described by Plaat et al. (1996a)
and has been given the name MTD(f). It relies
on a large memory to hold a transposition table
containing records of previous visits to nodes.
We recall that in fact games like chess really take
the form of graphs rather than trees, so that a
particular board position might be visited from
many different paths. The memory should be as
large as feasible, since it serves to hold as many
as possible of the results of visiting nodes.
Since the memory enhanced test algorithms
operate with zero window, they, of necessity,
need to do many re-searches. It is the use of
memory to help speed up the production of
results that makes these algorithms competitive
in speed. Although Plaat et al. (1996a) claim
their version is faster in practice than PVS, R.
Hyatt, as he wrote in the rec.games.
chess.computer Usenet newsgroup, in April
1997, still uses PVS for his Crafty chess
program. The following pseudo-code from Plaat
et al. (1996a) illustrates their MTD(f) version.
function MTDF(node-type, f,d)-+
g := f;
upperbound := +o0;
lowerbound :=-oo;
repeat
if g = lowerbound then
[3 :=g+ I;
else
[3 := g;
g:=ABWM(root,[3-1,[3, depth);
if g < [3 then
upperbound := g;
66
else
lowerbound := g;
until lowerbound >-- upperbound;
return g;
The algorithm works by calling ABWM a
number of times with a zero window search.
Each call returns a bound on the minimax value.
These bounds are stored in upperbound and
lowerbound, forming an interval around the true
minimax value for that search depth. When the
upper and lower bounds collide, the minimax
value is found.
ABWM (AlphaBetaWithMemory) is
conventional alpha-beta algorithm which has
extra lines to store and retrieve information into
and out of a transposition table.
Interest has also been shown in the B*
algorithm as well as those involving conspiracy
numbers. These are more complex than
algorithms involving alpha-beta searches, and
seem to get closer to the Shannon B kind of
algorithm where some expert knowledge of the
game is programmed in to guide the machine.
Impact on AI and Philosophy
In the pioneering era, both computer chess and
AI were emerging disciplines. At that early stage
computer chess was regarded as a testbed for AI
search and problem solving techniques. The
early chess programs were primitive but are now
much more sophisticated and have achieved
spectacular results. However, as recognised by
Donskoy and Schaeffer (1990), the dominantion
of AI by computer chess has now changed and it
is currently viewed as a small part of AI with its
methods, identified primarily as brute force,
regarded within the AI community as simplistic.
However, the impact of computer chess on AI
can be gauged by the number of methods
developed within the chess environment that are
domain independent and applicable elsewhere. It
is, for instance, undeniable that computer chess
has made significant contributions to search
techniques which find applications in other areas
such as theorem proving and problem solving in
general. It has also emphasised the development
of special hardware to solve the chess problem, a
viewpoint which is echoed in other areas of AI
(Horacek 1993).
The computer scientists and AI enthusiasts that
initiated the pioneering era of computer chess
were partly motivated by a desire to investigate
the processes of human reasoning. They argued
that if it was possible for computers to solve the
chess problem, it must be possible for computers
to tackle other difficult problems relating to
planning and applications to the economy. This
preoccupation with intelligence formed the core
of Turing’s work, commencing with the creation
of Turing machines and then the formulation of
the Turing test which still remains of central
importance in the AI debate today.
Turing himself worked in the period 1940-45
at Bletchley Park as an Enigma code-breaker. An
indifferent chess player himself, he was,
however, in the constant company of the best
British chess players of that era, thus creating the
environment and ideas for the final phase of his
work. The luring test states that if the responses
of a computer are indistinguishable from those of
a human it possesses intelligence. Consequently,
it can be argued that a computer playing chess
possesses intelligence as, in many cases, its
moves are identical to those of chess experts.
This is the strong AI viewpoint.
It was precisely to repudiate these claims of
intelligence that Searle (1980) put forward his
famous Chinese Room argument against strong
AI. Chess programs produce their moves
algorithmically. Suppose another human being
having no prior knowledge of chess per se,
performs the same algorithms, does he also
understand the game? The weak AI viewpoint is
that computers lack any such intelligence or
understanding. This viewpoint is supported by
Penrose (1995) who cites an example of
position in which Deep Thought blundered
because of its lack of understanding of a
particular position, despite being capable of
deceiving us into thinking that it really has some
chess intelligence.
Conclusion
The strength of the top chess playing programs is
continually increasing. In this paper we have
reviewed the main techniques that have enabled
programmers to achieve such a spectacular
increase in playing strength commencing with
the earliest exploratory programs to the
sophisticated software of today. During this
period, we have seen significant increases in
processing power which, despite the prevalent
von Neumann architecture of today, is still
increasing in power. Consequently, we foresee
67
the possibility of further increases in playing
strength.
Future work will probably refme the
traditional Shannon A type algorithms even
more. For example, new avenues of research still
exist on application-independent techniques of
exploiting the fact that computer chess trees are
really graphs (Plaat 1996b). However, there is
limit on the extent to which this can proceed.
The consideration of Shannon B type algorithms
is an area requiring further investigation and
development.
References
Botvinnik, M.; Cherevik, D.; Vladimirov, V.,
and Vygodsky, V. 1994. Solving Shannon’s
Problem: Ways and Means, Advances in
Computer Chess 7, van der Herik H. J, Hersberg
I.S, and Uiterwijk, J.W.H.M.(eds) Maastricht,
University of Limburg.
Donskoy, M.V and Schaeffer, J. 1990.
Perspectives on Falling from Grace, in
Computers, Chess and Cognition, Marsland, T
and Schaeffer, J. (eds) New York, Springer
Verlag.
Horacek, H. 1993. Computer Chess, its
Impact on Artificial Intelligence, ICCA Journal
16(1):31-36.
Knuth, D.E, and Moore, R.W. 1975. An
analysis of alpha-beta
pruning. Artificial
Intelligence 6(4):293-326.
Newborn, M.M. 1977. An analysis of alpha-
beta pruning. Artificial Intelligence 6:137- 153.
Penrose, g. 1995. in Shadows of the Mind,
London, Vintage.
Plaat, A.1996. Research Re:Search and Re-
search. Ph.D.diss., Dept. of Computer Science,
Erasmus University.
Plaat, A., Schaeffer, J., Pijls, W., and de Bruin,
A. 1996a. Best-first fixed-depth minmax
algorithms, Artificial Intelligence 87:255-293.
Plaat, A., Schaeffer, J., Pijls, W., and de
Bruin, A. 1996b. Exploiting Graph Properties of
Game Trees, Proceedings of the 13th National
Conference on Artificial Intelligence, Portland,
Oregon.
Searle, J.R. 1980. Minds, brains amd
programs, in
The behavioural and brain
sciences, Vol 3. Cambridge, Cambridge
University Press.
Schaeffer, J.1989. The history heuristic and
alpha-beta search enhancements in practice,
IEEE Transactions on Pattern Analysis and
Machine Intelligence, 11(1):1203-1212.
Shannon, C.E. 1950. Programming a
Computer for Playing Chess. Philosophical
Magazine 41(7): 256-275.
Simmon, H.A and Chase, W.G. 1973. Skill in
Chess. American Scientist, 6 l, 482-488.
Slate, D and Atkin, L.1977. Chess 4.5:The
North Western University Chess Program, in
Chess Skill in Man and Machine, P.Frey(ed.),
82-118. New York, Springer Verlag.
Turing, A. M. 1953. Digital computers
applied to games, in Faster than Thought,
Bowden, B.V.(ed). London, Pitman.
68
An end game database is a simply a database containing an array of pre analyzed chess endgame positions.
KQKR is an end game position. It means King Queen vs King Rook
A transposition table is a set of recently-visited positions in the game. This allows you to save some time by looking up positions in the transposition table instead of having to recompute them.
They are referring to the 1996 game between Kasparov and Deep Blue. In 1997, Deep Blue would end up beating Kasparov.
- [Programming a Computer for Playing Chess by Claude E. Shannon (1950)](http://vision.unipv.it/IA1/aa2009-2010/ProgrammingaComputerforPlayingChess.pdf)
- [Digital Computers applied to Games by Alan Turing (1953)](https://www.dropbox.com/s/9y75a3otxd46lti/Turing%2C%20Digital%20Computers%20Applied%20to%20Games.pdf?dl=1)
### Chess as a search problem
You can think of chess as a decision tree. At each turn you have a set of possible legal moves (on average you have 35 legal moves at your disposable).
Each branch of the tree represents a move (e.g. *move Pawn from c2 to c4*). Each node represents a state of the chess board. Once you decide to go down one branch it’s time for your opponent to play. On average a game of chess has about 80 turns.
Winning a chess game means navigating your way down this decision tree such that you get to a node where you checkmate your opponent (or he quits).
In iterative deepening the program starts with a one ply search, then increments the search depth and does another search. This process goes on until we run out of time. In case of an unfinished search, the program always has the option to fall back to the move selected in the last iteration of the search.
A *quiescent position / Turing dead position* is a position where there are no more tactical moves (e.g. moves where a piece is captured or there is a promotion) left to be made.
In this context, evaluating a position essentially means giving it a score of how good it is for you.
What this section is essentially saying is that as a general rule if you get to a position where there are still some outstanding moves that could drastically change the balance of the game you should keep searching deeper. This helps you avoid certain blunders that might result from having limits to how deep you can search (due to the size of the tree). For instance, maybe you got down to a position where you just captured a pawn, which could be interpreted as a good outcome, however if in the next move you opponent captures your queen, that might not be such a good outcome after all.
#### How big is the chess game tree?
On average, the tree has a branching factor of 35 (this is the average number of of legal moves available to a player) and a height of 80 (number of turns in a game).
A tree with such characteristics will have $35^{80}$ or roughly $10^{123}$ leaf nodes. That is a very, very big number. This makes pure brute force search intractable (it is estimated that there are about $10^{82}$atoms in the observable universe). That is the reason why you need *pruning* algorithms like Alpha-beta. These algorithms reduce the size of the tree you have to evaluate in order to make your next move.
Even today, there is no program that can perfectly predict the outcome of any given chess game given the state of the board (assuming both players play perfectly). This means that unlike checkers or tic tac toe, chess remains an “unsolved” game.
In the 1970s Simon and Chase performed a number of very interesting studies with the goal trying to understand how great chess players think.
In one of their experiments they showed that when chess masters looked at a board for 5 seconds, they could reproduce it with enormous accuracy, while beginners could only recall very few pieces. This difference could not be explained by masters' greater memory, for, in randomized positions, the effect disappeared, with masters and beginners able to reproduce only a few pieces of the board. The conclusion was that Masters seemed to have an easier time recalling plausible board positions.
The Elo rating system is a method for calculating the relative skill levels of players in games such as chess. Here are a few Elo ratings:
- Beginner chess player: < 1400 (this is debatable)
- Magnus Carlsen (World Chess Champion): 2851
- Komodo 9 chess engine (one of the most powerful modern chess engines): 3340
This paper was published in the late nineties. Right before [Deep Blue beat Kasparov](https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov) - the first time a computer defeated the reigning world chess champion. The advancements in computer chess were seen as a sign that artificial intelligence was catching up to human intelligence.
In this paper the authors give an overview of the history of computer chess, the different approaches to the problem and its significance to the question of to what extent computers ‘think’ or be seen as intelligent.
It’s also interesting to relate this paper to the recent [events in the game of Go](https://en.wikipedia.org/wiki/AlphaGo). How should we interpret the achievements of AlphaGo? How is it different than Deep Blue?
*Garry Kasparov, Russian Chess Grandmaster*
### MiniMax and Alpha-Beta pruning
#### MiniMax
MiniMax is a decision rule used in decision theory and game theory for minimizing the possible loss for the worst case scenario.
**Example**
Let’s say you are in the middle of a game, where you have 2 possible moves. Let’s also assume you have some way of evaluating how favorable the state of the game is for you at any point. Such algorithm takes in a state of the game and outputs a number from 1 to 10 (10 means you win, 1 means your opponent wins).
Here is the game tree for such game:
In such tree the **MiniMax** value is $3$. Given that your opponent plays perfectly, that is the best you can do. You would perhaps like to get to node with the $8$, but if you decide to go to the right subtree, in the next move, your opponent will choose to go left (because he wants to get that number to be as low as possible) and you will end up with $2$.
Now that you know the MiniMax value, you know which move to play.
#### Alpha-beta
[Alpha-beta](https://en.wikipedia.org/wiki/Alpha%E2%80%93beta_pruning) is an algorithm that seeks to decrease the number of nodes you have to evaluate when running the MiniMax algorithm in a search tree.
For instance, in the previous example we evaluated 4 leaf nodes. However, we didn’t really have to evaluate the leaf node all the way to the right. As soon as we saw that the value of the third node was $2$, we knew we would be better off by going left, and therefore we don’t even need to look at that last node.
