Developing an Artificial Intelligence for Whist - Fongboy.com

Developing an Artificial Intelligence for Whist
Abstract
This paper details the development of a simple artificial
intelligence for playing the card game Whist.
A brief introduction to the game is given, and details
are given for the design decisions that were
made in the design of the AI. The results have some
possible applications toward other similar games.
1 Introduction
Whist is an old card game that has been falling into obscurity
due to the popularity of the game Bridge. However,
Whist is still an interesting game to study because of its
differences from Bridge. Whist is a true multiplayer game
with more than 2 players. Also, bid values are to be met
exactly so there are extra complications for avoiding going
over a bid.
There are many variations to the game Whist and currently
there is no widely accepted official version of the
game. The version studied in this paper is similar to the
variation known as "Oh Hell!" [1].
Prior works in this area include the Ginsberg Intelligent
Bridge Player (GIB) [2]. Although bridge is significantly
different from Whist, the trick taking structure of Bridge
makes some of the techniques used in the GIB to be also
useful for Whist.
2 Rules of Whist
Since Whist is a game with many variations, there needs to
be some explanation of the rules of the variation that is studied
in this paper.
2.1 General Game Structure
This variation of Whist is a game played with five players
with no partnerships. A set number of rounds are played in
each game. Each player takes turns dealing, with the dealer
position passing in clockwise order. The number of cards
dealt to each player varies with each round. In the first
round, each player is dealt ten cards. In each subsequent
round, one less card is dealt to each player. This continues
down to one card and then repeats from one card back to ten
cards. After all players' cards are dealt, the next card in the
deck is turned face up. The suit of this card determines the
Jason Fong
University of California, Los Angeles
Computer Science
jfong@cs.ucla.edu
trump suit for the round. Each player receives a score at the
end of each round. The winner is the player with the highest
cumulative score at the end of the 20 rounds.
2.2 Turn Structure
The rules of play in each turn are similar to other trick taking
games such as Bridge and Spades. In a turn, each player
plays one card, going in clockwise order. The first player in
a turn may play any card in his hand. This card determines
the lead suit and the subsequent players must play a card of
the same suit if possible. If a player does not have a card of
the same suit, then any card may be played.
The player that played the highest card of the same suit as
the lead suit wins the trick. However, if a card of the trump
suit was played, then the highest card of the trump suit will
take precedence and win the trick. The winner of a trick
plays the first card of the next trick.
2.3 Bidding
After the cards for a round are dealt and before play begins,
each player looks at his cards and makes a bid declaring
how many tricks they will take in the round. The goal of
each round is to take the exact number of tricks that were
bid. Taking too few or too many tricks are both considered a
failure to meet the bid.
The sum of all bids is not permitted to equal the number
of cards dealt to each player in the current round. The effect
of this is to ensure that at least one player will not be able to
meet their bid.
Bidding starts with the person following the dealer and
proceeds in clockwise order. Since the dealer is the last to
bid, his bid may be restricted because his bid cannot make
the sum of the bids equal the cards dealt. After the dealer
bids, he plays the first card of the first trick.
2.4 Scoring
At the end of a round, each player's score is calculated and
added to their cumulative score. A player receives one point
for each trick taken. If the number of tricks taken is exactly
equal to the number bid for, then the player receives an additional
ten points. There is no extra bonus on top of the ten
points for making a bid for zero tricks.

From this scoring structure we see that making or missing
a bid is the most important component of the final score.
This implies that stopping other players from making their
bids will factor into an effective strategy. Also, the ability of
an opponent to stop a player from making his bid needs to
be considered.
3 Designing a Search for Whist
We can design a Whist AI as a search on a tree. Each node
represents a particular play order. Internal nodes of the tree
are order of plays for partial games. Leaf nodes are order of
plays for complete games. A search would attempt to find
the child that is most likely to lead to a leaf with a favorable
result for the player to move. This child represents the next
move to make for the current player.
3.1 A Complete Search
At a particular point in a round there are a set of cards that
are known and a set of cards that are unknown. The known
set consists of the cards whose location is known. These are
the cards that either in your hand, the trump-determining
card, or cards that have already been played. The unknown
set consists of the cards whose location is unknown. These
are the cards that are in the hands of the opponent players.
A complete search would consider an opponent to be able
to play any unknown card. This method will completely
search all of the possible outcomes of the game since it covers
all of the possible configurations of the opponents'
hands. However, the cost for this search is very expensive.
The worse case for this search occurs in the rounds with
10 cards dealt to each player. At the level where the first
opponent is to move there will be a branching factor of 41.
This is the number of unknown cards since there are 10
known cards in the AI player's hand and one known card in
the trump-determining card. In each subsequent opponent
turn, the branching factor will reduce by one since the number
of unknown cards will decrease by one whenever an
opponent makes a move.
An approximate number of leaves in this search tree can
be calculated as
56
41! × 10!
≈ 10
(1)
where 41! is the number of permutations for the order of
play of the opponents' cards, and 10! is number of permutations
for the order of play of the AI player's cards. This results
in a very large search tree, so it is desirable to reduce
this if possible.
3.2 A Simplified Search
The size of the search tree can be reduced if the branching
factor at the opponents' moves could be reduced. The
branching factor is high because at each opponent's turn, it
is assumed that any unseen card can be played. The number
of possible moves can be significantly reduced if the cards
in the opponents' hands can be seen.
In order reduce the size of the search, we will make a
simplifying assumption and assume that the AI player is
omniscient and can see all of the cards in all the hands. This
will significantly reduce the branching factor of the search
and allow the search to reach a greater depth within a reasonable
amount of time. In section 5.3 we will consider how
to remove the omniscience assumption.
With this assumption in place, the number of leaves in the
search tree can be calculated as:
5
32
( 10 ! ) ≈ 6×
10
(2)
where 10! is the number of permutations for the order of
play of each player's cards, raised to the 5 th power since
there are five players. This analysis is not entirely correct
since at each turn players must follow the lead suit and cannot
truly play any card in their hand. However, the equation
in (2) simplifies this analysis and can be considered a loose
upper bound on the number of leaves.
The resulting search tree is significantly smaller than the
tree in the complete search. However, this is still a very
large tree and cannot be searched completely. In order to
make a decision for the next move in a reasonable amount
of time, we can cut off the search at a particular depth and
use a heuristic evaluation function.
4 Designing a Heuristic Evaluation
In order to design a heuristic evaluation function, we need to
determine the factors that contribute to the value of a particular
game position. Since the goal of Whist is to end with
the highest score, we need to consider factors that contribute
to earning points.
The biggest contributor to a player's score is the 10 points
that are earned when a bid is successfully met. Thus, the
primary component of the value of the heuristic evaluation
should be an estimate of the likelihood of a player making
his bid. An estimate of the number of tricks that will be taken
compared to the player's bid value will give an estimate
of the likelihood of making a bid.
4.1 Card Classifications
In order to estimate how many tricks will be taken, we can
consider the cards held in a hand and classify them according
to their ability to win or lose tricks. The ability to lose
tricks needs to be considered since there will be situations
where a player needs to avoid going over his bid.
Conveniently, the omniscience assumption also makes it
easier to classify the cards in a hand. Since we can see all
cards, we can easily determine if a card will be guaranteed
to win or to lose a trick if it is played as the lead card. The
card is analyzed for when it is the lead card since considering
the card in the general case for any play position will
result in card classifications that are too narrowly defined.
For instance, a guaranteed winner in the general case will
usually only be the highest trump card. This is because even
if a player has the highest card of a non-trump suit, it cannot
win if that card's suit was not led. The classification being
based on lead cards is not too restrictive since the classification
also applies to the case where the card is played later in
a trick, but the lead suit matches the card's suit.
The cards are classified into five categories. The first two
categories are the aforementioned guaranteed winners and

guaranteed losers. The next two categories are likely winners
and likely losers. These are similar to the guaranteed
winners and losers, except that the opponents' intentions are
taken into consideration. If an opponent is able to win a
trick but is not forced to win it, he may not actually decide
to take the trick if he is in danger of going over his bid.
Similarly, if an opponent is able to avoid winning a trick but
is not forced to do so, he may still decide to take the trick if
he needs more tricks to make his bid.
The likely winner category is a looser form of the guaranteed
winner category where a card is considered a winner if
no player that may intend to take a trick can beat the card.
The likely loser category is similarly related to the guaranteed
loser category. An opponent's intentions are determined
by considering how many tricks are needed to made his bid
and the number of "guaranteed" category of cards. The
"likely" category of cards is not considered since this would
lead to an infinite recursion.
The fifth category contains the neutral cards. These include
all the cards that are not part of the first four categories.
These cards are not neutral in the sense that they do not
contribute at all to the ability to successfully make a bid.
They are neutral in the sense that they are flexible and can
be used for either winning trick or losing a trick. They are
flexible because they are not dominating winner cards or
weak loser cards. Thus, depending on when the player decides
to use these cards, they can be put to either use.
4.2 Calculating a Heuristic Value
A heuristic value can be obtained by estimating the resulting
score for the round. The first component of this is an estimate
of the likelihood of making a bid. This can be obtained
by using the card classifications to estimate the resulting
number of tricks over or under the bid.
We begin with the tricks left to make the bid. The number
of guaranteed winner cards is subtracted from this. Next, the
number of likely winner cards multiplied by 0.75 is subtracted.
Not all of the likely winner cards will win, so only
some fraction of those cards will be considered as tricks
won.
If the resulting value is positive, then there is probably a
need to take additional tricks to make the bid. In this case,
the number of neutral cards multiplied by 0.25 is subtracted.
If the previous result was negative, then there is probably
not a need to take additional tricks since there is a danger of
going over the bid. Some small portion of the neutral cards
still needs to be considered since opponents players are
likely to try to force the taking of additional tricks when one
is in danger of going over a bid. In this case, the number of
neutral cards multiplied by 0.1 is subtracted.
This results in the following equations:
t′ r = b − t − cgw
− 0.
75clw
(3)
⎧t′
′
r − 0.
25cn
if tr
> 0
t r = ⎨
(4)
⎩ t′
− 0.
1 ′
r cn
if tr
< 0
where b is the bid, t is the number of tricks already taken,
cgw is the number of guaranteed winners, clw is the number
of likely winners, cn is the number of neutral cards, t'r is the
number of tricks remaining to take to make a bid after guaranteed
winners and some portion of the likely winners are
taken, and tr is the number of tricks remaining to be taken
after consider all card categories.
Equations (3) and (4) did not use the number of guaranteed
loser and likely loser cards in the calculation. This is
because the tricks lost will not change the number of tricks
needed to make a bid. The purpose of the guaranteed loser
and likely loser categories is to take away cards from the
other categories that do factor into the calculation.
The expected value of points earned for making a bid is
estimated as:
ch
− tr
pb
= 10 ⋅
(5)
ch
where ch is the number of cards remaining the hand. This
formulation results in 10 points for meeting a bid exactly,
and reduces the value with each trick over or under the bid.
The number of tricks over or under is weighed against the
cards remaining to play since there is more flexibility in
play when there are more cards to choose from. This flexibility
makes it more likely that an unfavorable outcome can
be avoided. Thus, the penalty to the estimated value of the
bid points is reduced when there are more cards remaining
to play.
In addition to the estimated points from making a bid,
there are some additional minor components to consider.
One point is added for each trick taken. One point is also
added for each opponent who is likely to miss a bid. An
opponent is considered to be likely to miss a bid if the opponent's
tr value is less than -2. The choice of this value is a
bit arbitrary. It was chosen since it is usually hard to recover
from a position where one is likely to take two tricks over
the bid.
Taking these extra considerations results in the final heuristic
value can be obtain as follows:
pt = b − tr
(6)
h = pb
+ pt
+ m
(7)
where pt is the estimated number of points from tricks taken,
m is the estimated number of opponents that will miss their
bids, and h is the final heuristic value.
4.3 Choosing Card Category Weights
The weights of the card categories were obtained by testing
various weights in a few sample games. The values chosen
for consideration were based on what I thought was reasonable
for the outcomes of each card type. However, these
choices are still fairly arbitrary.
A more rigorous approach would create multiple instances
of the Whist AI with different category weights and
allow the instances to compete against each other. This
would provide more empirical evidence as to which weights
are more effective. However, this may not give results that
actually work well against human players since the play
against AI opponents may be biased toward idiosyncrasies

of the AI's style of play. This approach toward obtaining the
card category weights was not taken due to time constraints.
5 Determining the Next Move
The next move is determined by performing a search on the
game tree and obtaining values for each of the children of
the root. The children of the root node represent each of the
possible cards that can be played for the next move.
5.1 Search Methodology
The search is performed by using the Max n algorithm [3].
This is similar to a minimax search, but adapted for handling
more than two players. In a game with n players, the
evaluation of each node gives an n-tuple. The n-tuple is a
five element array of values where each element corresponds
to the evaluation of the node from the viewpoint of
each player. Each player's goal is to maximize the value of
his element of the tuple.
When a node is evaluated, each of the children's values
are considered and the entire tuple of the child with the
highest evaluation for the current player is backed up to the
current node. The child values are obtained by the previously
described heuristic evaluation function if the search
did not reach the leaves. If the leaves are reached, then the
child value is obtained by calculating the actual score for the
current player. One point is given for each trick taken and
ten points are given for making the player's bid. In order to
adjust for the desire to make opponents miss their bids, an
additional point is given for each opponent player that fails
to make his bid.
After the Max n search is performed to a fixed depth, the
root node will have backed up values for its children. These
values represent the estimated value of each of the possible
moves that can be made. The card that should be played
next is the card that is played in the child node with the
highest backed up value.
5.2 Search Optimization Considerations
As a round of Whist progresses, the branching factor of the
search tree decreases. This is due to the fact that as cards are
played, the players have fewer cards in their hands to chose
from. This reduction in possible moves reduces the branching
factor of the tree. Thus, for a given amount of time, we
can usually search deeper when there are fewer cards in the
players' hands. This is taken advantage of by adapting the
search depth to the number of cards in the players' hands.
During the last trick each player has only one card in their
hand. This results in forced moves, so there is no need to
search at the bottom five levels of the search tree. When the
search reaches a node of height six, the last five moves are
played and the final game state is evaluated at the leaf node.
This value is then backed up to the height six node.
In some situations, a player may have only one legal
move. This can be the case when a player has only one card
of the lead suit. When this occurs, the search need not be
performed since there is no choice of move.
5.3 Removing the Omniscience Assumption
The preceding analysis on heuristics and search methodology
has been made under the assumption of an AI player
with an omniscient view of the game. In a real game of
Whist we cannot assume that we can cheat and see all cards.
In order to remove the omniscience assumption we can
take a probabilistic approach and consider which card is
most likely to lead to a favorable result. While a round is in
progress we can remember all of the cards that have been
played. We also know the cards in our hand and the card
that was used to determine the trump suit. This leads to a set
of cards whose locations are currently unknown.
A Monte-Carlo method can then be used to arrive at an
educated guess as to which card will lead to a favorable
result. The set of unknown cards are used to randomly create
the opponents' hands. The previously described search is
used to determine the best card to play in this game configuration.
This is repeated many times with different random
dealings of the opponent cards. A count is kept of how
many times each of the AI player's cards were chosen as the
best next move. When this process is complete, the move to
make is the card that was chosen the most times as the next
best move.
5.4 Search Depth vs. Deal Variations
The quality of the move chosen is affected by the depth of
each search and the number of iterations in the Monte-Carlo
simulation. Thinking time can be allocated to either of these
processes. In order to improve the quality of the move chosen,
we need to consider how to allocate the thinking time.
When there are many cards remaining in each hand, the
number of iterations is favored. In this case there are many
more possible configurations of the opponent hands. This
leads to a need for more samples to arrive at a reasonable
result. Searching deeper is likely to be very expensive due to
the high branching factor when there are many cards in each
player's hand. Also, the heuristic evaluations are likely to be
fairly inaccurate since are many turns left in the game. Thus,
searching more iterations is likely to be more valuable than
searching deeper.
When there are few cards remaining in each hand, the
search depth is favored. In this case there are fewer possible
configurations of the opponent hands. This leads to fewer
samples giving a similar coverage as the case with more
cards and more samples. Searching deeper will result in the
search reaching the leaves more often. These positions are
more valuable since the information is based on a final
game state and not a heuristic estimate. Also, even when a
leaf is not reached, the heuristic evaluations are more accurate
since there are few turns left in the game.
With these considerations in mind, the search depth and
iterations of the Monte-Carlo simulation are adjusted based
on the number of cards remaining in each hand.
6 Evaluation Methodology
Since the variation of Whist studied in this paper is not the
official standard of Whist, there is no known existing pro-

gram that plays this game. Thus, a somewhat subjective
method is needed to evaluate the effectiveness of the Whist
AI.
The evaluation was performed by having the AI play
games against a focus group of four human players. This
group consisted of players that are proficient at playing the
studied variation of Whist. Comments were taken from the
human players concerning blunders or brilliant moves made
by the AI. The frequency of the AI winning rounds was also
observed. The reasoning behind the AI's moves was observed
by analyzing a report generated by the AI at each
move. This report contained evaluations of each possible
next move and evaluations of each opponent's position. This
gives some insight into the reasoning behind the AI's moves
and whether it is making some decisions based on faulty
inferences.
7 Results
Play against the focus group suggests that the AI plays at
the level of an average player. The AI plays acceptably well,
but does not dominate the game and win consistently. However,
it also does not fall hopelessly behind in score.
The test trials were possibly effected by opponent bias
against the AI player. It is possible that the human players
were focused on defeating the AI player and stopping its
bids. In this case the AI player may have been hindered by
not being paranoid enough with respect to the opponent
players' actions.
8 Possible Improvements
There are a number of different aspects that can be improved
to increase the AI player's strength.
8.1 Heuristic Improvements
The heuristic evaluation can be possibly improved in a number
of ways. The weights of the various card categories can
be adjusted as previously described. The neutral card category
could be divided into cards that are neutral winners and
neutral losers. This can be determined by considering such
things as a card's rank relative to other remaining cards of
the same suit, or how many cards of the same suit have already
been played.
Protected high cards can also be considered. If a player
has low cards of the same suit as a high card, the high card
is not as dangerous when trying to avoid going over a bid.
This is because the player can use the low card when forced
to play the suit and try to get rid of the high card at a later
point when the suit is not led.
Improvements in the heuristic can lead to more intelligent
choices of moves.
8.2 Intelligent Dealing of Opponent Hands
The opponent hands can possibly be dealt with more intelligence
than just a random deal. The cards can be dealt to
match the bids that were made by the opponents. However,
there needs to be care taken to consider that an opponent's
tricks remaining can be incorrect if one of his winner or
loser cards gave the opposite result. This will result in the
number of tricks remaining to be incorrect for the opponent
player.
It is possible to adjust for this by observing the cards
played and attempting to predict when an opponent gets the
wrong result for a card played. However, this is quite a
complicated endeavor that time restrictions made impractical.
8.3 Automated Bidding
The issue of automated bidding was considered. A possible
algorithm is to consider each bid possibility as a separate
search and get the backed up value at the root for each of
these searches. The best bid is then the bid made in the
search with the best backed up value.
This approach was attempted but the results were not
very effective. The AI tended to make bids that were not
very reasonable for the cards in its hand. This is the result of
not considering other important factors when making a bid.
These other factors could include low cards protecting high
cards, long suits, short suits, ability to give or take the lead.
These are considerations that affect an intelligent bid but are
not considered in the heuristic evaluation function.
In order to focus on the AI's card play ability, I chose to
manually enter the AI player's bid. This helps to remove the
concern of bid quality affecting the card play results. This is
not too unreasonable for an evaluation method based on
play with humans since there is not a need to quickly make
many bids.
9 Conclusions
A somewhat respectable AI Whist player was produced. It is
not a dominating player, but it plays reasonably well and
leaves much room for future improvements. The card classification
system appeared to be effective for Whist. This
suggests possible applications in other trick taking games.
The method of peeking at opponent cards combined with the
Monte-Carlo simulation appeared to be effective for Whist.
This suggests possible applications in other imperfect information
card games.
References
[1] Wikipedia, "Oh Hell,"
http://en.wikipedia.org/wiki/Oh_Hell
[2] Ginsberg, M. L., "GIB research page,"
http://www.cirl.uoregon.edu/ginsberg/gibresearch.html
[3] Luckhardt, C. A., and Irani, K. B., An algorithmic solution
of N-person games, Proceeding of the National
Conference on Aritificial Intelligence (AAAI-86), Philadelphia,
PA, August, 1986, pp. 158-162