Computing optimal strategy for finite two-player games
Simon Taylor
Bachelor of Science in Computer Science with Honours
The University of Bath
April 2009
Computing optimal strategy for finite two-player games
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I
Computing optimal strategy for finite two-player games
Computing optimal strategy for finite two-player games
Submitted by: Simon Taylor
COPYRIGHT
Attention is drawn to the fact that copyright of this dissertation rests with its author. The
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University of Bath (see http://www.bath.ac.uk/ordinances/#intelprop).
This copy of the dissertation has been supplied on condition that anyone who consults it is
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dissertation and no information derived from it may be published without the prior written
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Declaration
This dissertation is submitted to the University of Bath in accordance with the requirements
of the degree of Batchelor of Science in the Department of Computer Science. No portion of
the work in this dissertation has been submitted in support of an application for any other
degree or qualification of this or any other university or institution of learning. Except where
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Computing optimal strategy for finite two-player games
Abstract
Game theory is the study of strategic situations in which one or more players
make decisions to grant them success whilst also depending on the decision of
others in the game.
Game theory is used in many areas of research. Very commonly used in
Economics it is also used in other areas such as Biology and Political science. It
is also popular in Computer science with its use in artificial intelligence.
Although game theory had been used since the 18th century is wasn't until 1928
when John von Neumann published a series of papers that it became a unique
field of research.
The aim of this project is to create a system that solves two-player strategic
games.
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Computing optimal strategy for finite two-player games
Contents
CONTENTS...........................................................................................................................................I
LIST OF FIGURES ............................................................................................................................ II
LIST OF TABLES .............................................................................................................................III
ACKNOWLEDGEMENTS...............................................................................................................IV
INTRODUCTION................................................................................................................................ 1
LITERATURE SURVEY .................................................................................................................... 3
3.11
PRISONERS DILEMMA............................................................................................................. 3
3.12
NASH EQUILIBRIUM ............................................................................................................... 5
3.13
MIXED STRATEGIES ............................................................................................................... 7
3.14
COMPUTING MIXED STRATEGIES............................................................................................ 8
3.14.1
Computing mixed strategy for 2x2 games.................................................................... 8
3.15
GAME VALUE....................................................................................................................... 10
3.16
DOMINATING STRATEGIES ................................................................................................... 11
3.17
OTHER METHODS ................................................................................................................ 12
3.18
SIMPLEX METHOD ................................................................................................................ 12
3.19
OTHER IMPLEMENTATIONS .................................................................................................. 16
REQUIREMENTS ............................................................................................................................. 17
3.1
3.2
FUNCTIONAL REQUIREMENTS .............................................................................................. 17
NON-FUNCTIONAL REQUIREMENTS ...................................................................................... 18
CONCLUSIONS ................................................................................................................................ 19
REFERENCES................................................................................................................................... 20
BIBLIOGRAPHY .............................................................................................................................. 20
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Computing optimal strategy for finite two-player games
List of Figures
FIGURE 1.1 NORMAL AND EXTENSIVE FORM OF A STRATEGIC GAME ....................................................... 2
FIGURE 2.1 PRISONERS DILEMMA IN NORMAL FORM .............................................................................. 4
FIGURE 2.2 CAMPERS .............................................................................................................................. 5
FIGURE 2.3 STRATEGIC GAME WITH NO NASH EQUILIBRIUM ................................................................... 6
FIGURE 2.4 MATCHING PENNIES ............................................................................................................. 7
FIGURE 2.5 A 2X2 GAME.......................................................................................................................... 8
FIGURE 2.6 SELECTIONS FOR EACH PLAYER ............................................................................................ 8
FIGURE 2.7 GAME WITH DOMINATING STRATEGIES ............................................................................... 11
FIGURE 2.8 EXAMPLE GAME ................................................................................................................. 13
FIGURE 2.9 GAME AFTER ADDITION OF CONSTANT ............................................................................... 14
FIGURE 2.10 FIRST SCHEMA .................................................................................................................. 14
FIGURE 2.11 PIVOT CRITERION.............................................................................................................. 14
FIGURE 2.12 SECOND SCHEMA .............................................................................................................. 15
FIGURE 2.13 FINAL SCHEMA ................................................................................................................. 15
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Computing optimal strategy for finite two-player games
List of Tables
TABLE 1 AN EXAMPLE TABLE ......................................................... ERROR! BOOKMARK NOT DEFINED.
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Computing optimal strategy for finite two-player games
Acknowledgements
Add any acknowledgements here.
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Computing optimal strategy for finite two-player games
Chapter 1
Introduction
Games theory is the study of strategic situations in which one or more players make
decisions to grant them success whilst also depending on the decisions of the other players. It
is widely used in Social sciences, notably in Economics but it is also used in areas such as
Biology, Political science and Computer science. Players in a game can represent a variety of
different things including people, animals, plants or governments.
A strategic game consists of players and for each player a set of actions and a preference
relation over this set of actions.
The number of players in a game affects how difficult is can be to calculate optimal
strategies for each player. In one-player games it is just a case of selecting the action that
will give greatest reward. For games with two players finding the optimal strategy for each
player means looking at the highest pay off that can be achieved given the other players
strategy. With games involving more than two players it becomes more difficult to find
optimal strategies as there is the possibility of cooperation between players among other
things.
Games can be represented in one of two forms, these being extensive and normal form. In
extensive form the game is shown as a tree with nodes giving the state of the game and each
branch giving actions available from the current state. Playoffs are given at the bottom of the
tree. This representation is easier to understand and view when learning game theory,
however as games get more complex with more players and actions the tree grows to sizes
that can be difficult to draw. In normal form the game is represented as a matrix containing
each player’s payoffs. One player’s actions and payoffs are given as the rows in the matrix
and the other player as the columns. Figure 1.1 shows a strategic game in both extensive and
normal form.
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Computing optimal strategy for finite two-player games
Figure 1.1 normal and extensive form of a strategic game
As is show, although the extensive form can be read more easily even with this small game it
is large and looks a little cluttered, where as the normal form game is compact and as games
become more complex with additional actions this will become a lot easier to use than
extensive form.
The payoffs given in the game relate to the type of game being player. In zero-sum games
the payoffs for each player sum to zero, so one player gains what the other losses and so the
values sum to zero. Games where this is not the case is called non zero-sum games. Non
zero-sum games can be transformed into zero-sum games but adding an additional dummy
player. The losses of this player compensate for the total gains of the other players. Some of
the additional factors of non zero-sum make it more difficult to work with and so will be
worked with only is success is achieved with zero-sum games first.
One of the most famous strategic games is the Prisoner’s dilemma. For most people being
introduced to game theory this will be one of the first examples they may come across. It
will be used as an example to cover the basics of game theory
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Computing optimal strategy for finite two-player games
Chapter 2
Literature Survey
3.11 Prisoners dilemma
Two suspects are arrested by the police. There is insufficient evidence for a conviction of
either suspect. Having separated the prisoners, they are each visited and offered the same
deal. If one testifies (betray) against the other and the other stays silent then the betrayer will
be set free and the silent suspect will receive a 10-year sentence. If both remain silent, both
will be sentenced but will only receive six months for a minor charge. If both testify against
each other, they each receive a five-year sentence. Each prisoner must chose to testify or stay
silent. Neither prisoner will know what the other chose.
With this information we can construct a strategic game for the situation. One assumption
made during this is that both prisoners want to receive the shortest sentence possible.
In this game the players are each of the suspects.
In any game all players have a set of actions available to them. In this game both players
have the set of actions {silent, betray}. An action profile is created when the players act, an
example of this would be where both players stay silent, (silent, silent), where the active
players choice is first and the other players second.
A preference relation is a ranking of all possible action sets available to the player in order of
their highest payoff first and lowest payoff last. This however does not show how much each
action set is favoured over another. In the case of the prisoners dilemma each player will
have the preference ordering:
(betray, silent) > (silent, silent) > (betray, betray) > (silent, betray)
It is normally more convenient to represent a preference relation wait a payoff function. This
is a function with values that correspond to the rankings in the preference relation. For this
example we can take the values of jail time as the payoff for each players action, however as
the preference relation is ordinal we could simple assign values to each action profile in
order.
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Computing optimal strategy for finite two-player games
This can be show in normal form in figure 2.1
Suspect 2
Suspect 1
Silent
Betray
Silent
2, 2
0, 3
Betray
3, 0
1, 1
Figure 2.1 Prisoners Dilemma in normal form
The rows give the actions of the first suspect and the columns for the second suspect. The
numbers in this matrix are each of the player’s payoffs, with the first players being the first
value. A zero-sum game can be represented differently due to the link between the playoffs.
For a zero-sum game it is only necessary to show the first player’s payoffs as the second
player’s payoffs are found through negation of the first.
By looking at the payoff matrix it would appear that the obvious choice for both players
would be to stay silent as they would each receive the shortest sentence. However this turns
out not to be the case. Both players have a greater incentive to betray the other because if the
other remains silent they will receive no sentence. This leads to the decision that if one
player betrays then to receive the highest payoff then the second player will also have to
betray. From this it shows that although there is a high payoff for both players if they remain
silent, the incentive to betray in the hopes that the other stays silent out weight the
cooperation and so (betray, betray) becomes the action set chosen. This outcome is known as
Nash equilibrium.
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Computing optimal strategy for finite two-player games
3.12 Nash equilibrium
Named after John Nash who proposed it, is a solution concept of a game involving two or
more players. This involves the assumptions that each player knows the equilibrium
strategies of the other players and no one player has anything to gain by changing their
strategy. If each player has chosen a strategy and no player can benefit by changing their
strategy while the other keep theirs unchanged, then this current set of strategy choices and
the corresponding payoffs constitute a Nash equilibrium (Wikipedia, Nash equilibrium).
For the example of the Prisoners dilemma the action profile (betray, betray) is a Nash
equilibrium. This is because there is no action that either player could change to, to give a
greater payoff while the other remained the same.
However not every strategic game has a Nash equilibrium and this will be discussed later on
with mixed strategy. This idea of an equilibrium point was not new when Nash proposed it
but due to his work in the field it has become recognized as this. Before this time it was
known as a saddle point.
The method used for finding Nash Equilibrium is known as the minimax method. The idea
of the minimax method is that player 1 will choose the action that’s minimum payoff, given
all of player 2’s possible actions, is highest. Player 2 will act in the same way.
Player 2
1 2 3 4
1 7 2 5 1
Player 1 2 2 2 3 4
3 5 3 4 4
4 3 2 1 6
Figure 2.2 Campers
As is shown in figure 2.2 (J.D.Williams, 1966), player 1 will select action 3 as the minimum
payoff is 3 which is better than the minimum payoff for any other action. Player 2 will select
action 2 as this will give them the lowest maximum loss for any action of -3. This gives the
action profile (3, 2) as this games Nash equilibrium.
It should be noted that in this game player 2 can never win unless there are some additional
side rules added that cannot be represented in the table. In this project side rules will not be
implemented.
It isn’t uncommon to find games that have no Nash equilibrium, figure 2.3.
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Computing optimal strategy for finite two-player games
Player2
1 2
Player 1
1 2 5
2 4 3
Figure 2.3 Strategic game with no Nash equilibrium
In this game using the minimax method will return 3 and 4 for each player respectively.
Therefore no Nash equilibrium exists. This leads us on to mixed strategies.
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Computing optimal strategy for finite two-player games
3.13 Mixed strategies
So far all games discussed have seen each player picking the same action, the Nash
equilibrium. This however as just seen doesn’t always happen and this leads to a mixed
strategy equilibrium where each player doesn’t know what action the other will choose but
instead the probability with which they will choose certain actions. This can be shown with
the matching pennies game in figure 2.4.
Player 2
1
2
Player 1 1 -1
1
2
1
-1
Figure 2.4 Matching pennies
In this game, player 1 must match the face of the pennies while player 2 must make them
different. The winner receives an extra coin.
Using the minimax method it can be seen that there is no Nash equilibrium. However there
are mixed strategies for the game. It needs to be assumed that both players would prefer to
win more money than lose out. If player 1 was to choose heads with a probability greater
than a half then player 2 will choose tails with a probability 1, this would cause player 1 to
switch his strategy from heads to tails. This would continue until both players realize that
they can do no better than choosing an action with probability greater than a half, this leads
to the mixed strategy equilibrium for the game.
When the game matrix is mirrored and negated through the diagonal from the first to last
element, like in figure 2.4 both players will have the same strategy. This doesn’t help with
calculating the strategy but if the value in the columns and rows sum to zero then the value
of the game is zero and the optimal strategies involve all actions equally.
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Computing optimal strategy for finite two-player games
3.14 Computing mixed strategies
One method for computing mixed strategies uses the players’ expected utility to find an
optimal strategy. The idea is to select a strategy that will win no matter what the other player
chooses.
3.14.1 Computing mixed strategy for 2x2 games
One of the best strategies for winning with mixed strategies is to make the other player
indifferent as to which action you choose. For 2x2 zero-sum games there is an equation that
can be derived from this, that can be used to solve the mixed strategies without any need to
use simultaneous equations. The expected utility for player 1 for action 1 will be ap + d(1-p)
and for action 2 will be bp + c(1-p). Making these to equations equal and then rearranging
them will give us, p = c – d / a – b + c – d, where p is the probability of choosing the first
action and 1 – p is the probability of the second action. Putting in the values for the game in
figure 2.5 will give us p = 3 – 4 / 2 – 5 + 3 – 4 = ¼.
Player 2
1 2
Player 1 1 2 5
2 4 3
Figure 2.5 a 2x2 game
This is the probability in which player one should choose action 1. leaving us with the
probability of ¾ for action 2. The matrix is read differently for player 2 as is shown in figure
2.6.
Player 2
1
2
Player 1 1 A B
2 D C
1
2
1 A D
2 B
C
Figure 2.6 selections for each player
This will give p = 3 – 5 / 2 – 4 + 3 – 5 = ½
There are also other methods for calculating the mixed strategies of a 2x2 game. One of
them is done by finding the odds of the game. Firstly Nash equilibrium need to be looked
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Computing optimal strategy for finite two-player games
for, if one exists then the odds will not need to be looked of as they will be 1 for the row and
column of the equilibrium point. One problem with this method is that it will normally give
incorrect strategies if there is a Nash equilibrium. If no Nash equilibrium exists then the
mixed strategy will be found.
The method for finding the odds for a player is to subtract the 2nd value for each action from
the 1st. The signs of these odds are not important. The odds of each action are the result of
the subtraction from the opposite action.
In the example from figure 2.5 the odds for player 1’s first action will be 4 – 3 = 1 and for
the second action 2 – 5 = -3. This gives the odds for this game for player 1 as 1:3, which is
equivalent to action 1 being a probability of ¼. The same can be done for player 2 where
action 1 will be 5 – 3 = 2 and for action 2 will be 2 – 4 = -2. This giving the odds or 2:2
which can be reduced to 1:1, which is the equivalent of action 1 being probability ½.
So for this game the mixed strategies would be for player 1:
¼ A1 + ¾ A2
And for player 2:
½ A1 + ½ A2
Working the values out will give player 1’s average payoff as 4.5 and player 2’s average
payoff as -4.5. This shows that the strategies are optimal as both values are equivalent with
signs being ignored; this gives the value of the game.
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Computing optimal strategy for finite two-player games
3.15 Game value
The value of any game is the payoff that can be won when both players are playing their
optimal strategies. If this value is positive then on average player 1 will win, if negative
player 2 will win and if it is zero then the game is considered fair and both players have the
same chances of winning.
If a game has a Nash equilibrium then this value will be the games value, however not all
games have this and so the mixed strategies will give average payoffs for the players and this
will be the value.
Sometimes a game can be changed by either adding a constant to the payoffs or by
multiplying them. This will not affect game play but will affect game value. The change in
game value will be the change made to the payoffs. For example if 3 is added then the new
game value would be 3 higher than its original value. Adding constants can change the
fairness of the game while multiplication can represent something like a change in currency.
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Computing optimal strategy for finite two-player games
3.16 Dominating strategies
In 2xm games where m is larger than 2 finding dominating strategies is important. If no
Nash equilibrium exists then reducing the size of the game makes calculating its mixed
strategies easier. A strategy can be dominated by another if, for every action chosen by the
other player the payoff is greater in the other.
Player 2
1 2
1 2 5
Player 1 2 4 3
3 3 6
4 5 4
5 4 4
Figure 2.7 Game with dominating strategies
In this game there exist a number of dominant strategies for player 1. The 1st and 3rd action
being compared shows that action 3 dominates action 1 for both player 2’s actions. Also
action 4 dominates action 2 and 5. This allows for the game to be reduced to a 2x2 game.
This can now be solved as a 2x2 game with the probability of the removed actions being 0.
This however is not the case for all 2xm games, some will not reduce to 2x2 and others will
not reduce at all.
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Computing optimal strategy for finite two-player games
3.17 Other Methods
There are other methods for solving or helping to solve matrix games. Some of these are
Graphical which can help reduce the game. Trial and error, which again is useful in reducing
a game. Gaussian elimination is a more complex method for solving games and makes use of
3 matrices. However for the purpose of this project the Simplex method will be used
3.18 Simplex method
The following algorithm for solving finite games is called the simplex method and is used
for solving zero-sum games, mxn games (J.D.Williams, 1966)
1. Add a constant to all elements of the game matrix, if necessary, to insure that the value is
positive. After computation this constant must be subtracted from the value of the new
matrix game to get the value of the original matrix game. The strategies are not affected.
2. Create the first schema by augmenting the game matrix with a new column of -1s on the
lower edge, a new row containing 1s on the right edge and zero in the lower right corner.
Label Player 1’s strategies on the left from x1 to xm and Player 2’s strategies on the top
from y1 to yn.
3. Select any entry from the interior of the schema to be the pivot, say row p column q,
subject to the following properties:
3.1. The number in the final row of the potential pivot column, r, must be negative.
3.2. The potential pivot, p, must be positive.
3.3. Compute for each potential pivot the quantity – r x c / p, where c is the final value
in the column containing the potential pivot. This value is known as the pivot
criterion.
3.4. Find the smallest criterion in each valid column (Those containing a negative value
in the final row).
3.5. Find the largest criterion from the set of smallest column criterion. The element
corresponding to this criterion is the pivot.
4. The numbers for the next schema are found as follows:
4.1. Number corresponding to the pivot is the value, D, of the preceding schema.
4.2. Numbers that correspond to those of the preceding pivot row stay the same.
4.3. The numbers that correspond to those of the preceding pivot column are the same
value negated, except the pivot.
4.4. All other values, except the pivot, are computed from N x p – pR x pC / D, where
N is the original number and pR and pC are the numbers that have rows and
columns in common with p and N.
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Computing optimal strategy for finite two-player games
4.5. The next value of D is the pivot value, p, of the preceding schema. N.B. D is 1 for
the first schema.
5. Player 1's strategies are represented by the rows from the original game matrix and
Player 2's strategies are, similarly, represented by the columns from the original game
matrix.
The two strategies represented by the row and column containing the pivot are part of
the optimal strategy for each player respectively. However, when pivoting the strategies
swap the action they represent. Example, p = a[1][4], This means that Player 1's
action, represented by row 1 is part of his optimal strategy but is now represented by the
value in the final element of column 4. Similarly Player 2's action that is represented by
column 4 is now represented by the value in the final element of row 1.
6. Check the final row of the new schemata for negative numbers.
6.1. If one exists, return to step 3.
6.2. If not, the computation is complete and the strategies can be found
The numbers in the final column that are part of the optimal strategy for Player 1 are the
odds for the action they represent.
The numbers in the final row that are part of the optimal strategy for Player 2 are the odds
for the action they represent.
The value of the probabilities is easily calculated by dividing each individual odd by the sum
of the odds. Example, if Player 1's optimal strategy is represented by 0:1:2, the probability
for action 1 is 0, for action 2 is 1/3 and for action 3 is 2/3. The game value calculated from D
divided by the final element in the matrix.
The game in figure 2.8 (J.D.Williams, 1966) will be used as a worked example of the
simplex method
Player 2
1
2
3
Player 1 1 6
0
3
2 8 -2 3
3 4
6
5
Figure 2.8 Example game
To begin with the matrix must be cleared of any negative numbers by adding a constant
equal to the lowest negative value. This must be added to every element in the matrix. In this
example the lowest value is -2 and so 2 will be added to every element. This gives:
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Computing optimal strategy for finite two-player games
Player 2
Player 1 1
1
2 3
8
2 5
2 10 0 5
3
6
8 7
Figure 2.9 Game after addition of constant
This matrix is now augmented with a column of 1s, a row of –1s and a 0 where the two meet.
Player 2
1
2
3
8
2
5
1
Player 1 2 10
0
5
1
6
8
7
1
-1
-1
-1 0
1
3
Figure 2.10 First schema
The next step is to find the pivot point. This involves finding the largest of the smallest
column pivot criterion. As is shown in figure 2.11 the largest pivot criterion from the set of
smallest pivot criterion is 1/7 in the final row and column, this will be the initial pivot point.
1/8
1/2 1/5
1/10
1/6
0
1/5
1/8 1/7
Figure 2.11 Pivot criterion
The pivot column values are then negated and the pivot row is left as they are. The pivot
point itself takes a new value D, which for the first pivot is 1. After this D takes the value of
the previous pivot.
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Computing optimal strategy for finite two-player games
Player 2
1
2
3
1 26 -26
-5 2
Player 1 2 40 -40
-5 2
3
6
8
0
1
-1
1
5
1
Figure 2.12 Second schema
Figure 2.12 shows the completed second schema. All values that are not part of the pivots
row or column are found using the formula, N x P – R x C / D, where N is the value of the
pervious schema, p is the pivot value and C and D are the values that share the same row and
column as N and P.
For the first element this would be 8 x 7 – 6 x 5 / 1.
This method is repeated until there are no negative numbers left in the final row. As there is
still a negative in the final row in the example, another pivot point needs to be found from
any column that still has a negative in the final row. In this example that is only the first
column.
Player 2
1
1 -26
2
3
0
-10
4
Player 1 2
7
-40
-5
2
3
-6
80
1
4
1
0
5
6
Figure 2.13 Final Schema
Once all negatives in the final row have been eliminated the final schema will have been
found. Now strategies are ready to be taken from the matrix. Player 1’s mixed strategy is
action 2 and action 3 in the ratio 1:5 or 0:1:5. Player 2’s mixed strategy is action1 and action
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Computing optimal strategy for finite two-player games
3. in the ratio 1:2 or 1:0:2. The game value if found by dividing D, which is 40 by the final
element in the matrix, which is 6 giving us 6 2/3.
3.19 Other implementations
Software that finds equilibrium is not widely available even though Game theory has its
usefulness in many areas of academic research. Some programming languages had a simplex
function for them. There is also a set of tools known as Gambit which can be found online.
There are a few applications that can be placed on certain calculators.
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Computing optimal strategy for finite two-player games
Chapter 3
Requirements
In this section I will outline the requirements for the system to be created from. These
requirements will define the constraints on the implementation and operation of the system.
It will be important to keep an up to date set of requirements to make sure that the system
has a structure to follow. In small projects it can be easy to create a set of requirements and
then not make use of it. This can allow for alterations to be made to the requirements at the
end of the project so that it fits the final system, however this is bad practice and will be
avoided.
This system has one function and that is to compute the optimal strategies for each player in
a finite zero-sum game given a set of payoffs from the user. This systems main use is going
to be as an aid in the teaching of game theory and as research as it will be too simple to be of
use in industry.
After research it was decided that the simplex method was to be implemented in this system
and so the requirements incorporate that.
3.1 Functional requirements
1. User input
1.1.
Game data is separated by whitespace
1.2.
Any nxm matrix can be input
1.3.
Payoff values can be signed
1.4.
Payoffs can be fractions
1.5.
Selected output file
2. System operation
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Computing optimal strategy for finite two-player games
2.1.
Look for Nash equilibrium
2.1.1.
2.2.
Get size of the game matrix
2.2.1.
2.3.
If Nash equilibrium exists then return and end program
If size is 2x2 then use method for computing 2x2 games
If any other size use simplex method
2.3.1.
Make matrix elements positive if needed
2.3.2.
Create first schema
2.3.3.
Find pivot and perform calculations
2.3.4.
search for negatives in final row
2.3.5.
repeat from 2.3.3 if a negative is found otherwise return solutions
3. System output
3.1.
results will be written to a user specified output file and to the command line
3.2 Non-functional requirements
1. The system will be written in java
2. The system should run efficiently
3. The interface should be easy to learn and use
4. The system must be reliable
4.1.
If an error occurs the system should exit and an error message displayed
4.2.
Error messages must be understandable
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Computing optimal strategy for finite two-player games
Chapter 4
Conclusions
During this project I did not manage my time properly and as such I did not give enough
time to complete the code. I am not a very competent coder and so when problems arose I
had difficulty finding and fixing the error. This was not helped by not going to get help with
some of it and leaving it so late that it couldn’t be finished. Had I been able to complete the
code then I may have been able to complete this document to a much better level.
Although I have increased my capabilities as a coder it is still not sufficient at the moment to
be of use to myself and so I will need to work harder to achieve this.
Had I managed my time properly and began the project when I should have, I could have
finished the code and been able to create a high standard project.
Working on this project has improved my knowledge on game theory and helped with my
research abilities. I am interested in the field of game theory and with the outcome of this
project I am disappointed with myself for not working my hardest to get a working system
and to have it work successfully.
In the future as well as making the system function fully and work, there would be some
other functionality that would have been nice to add.
A graphical interface would allow for easier understanding of the system and could allow for
the games to be represented in both normal and extensive form.
Allowing for more than 2 players. There are examples in game theory where a game may
include more than just 2 players.
Ability to solve non zero-sum games.
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Computing optimal strategy for finite two-player games
References
WIKIPEDIA, The Free Online Encyclopaedia,
http://en.wikipedia.org/wiki/Game_theory
http://en.wikipedia.org/wiki/Nash_equilibrium
GARG, R – Online Game Theory Text, available at
http://www.cse.iitd.ac.in/~rahul/cs905
OSBORNE, MARTIN J (1997), – An Introduction to Game Theory
http://www.socsci.mcmaster.ca/~econ/faculty/osborne
WILLIAMS, J.D (1966), McGraw Hill Book Company – The Compleat Strategyst
Bibliography
MCKELVEY, RICHARD, MCLENNAN, ANDREW, TUROCY, THEODORE – Gambit:
Software tools for Game Theory, available at http://gambit.sourceforge.net/
PORTER, NUDELMAN, SHOHAM (2004) – Simple Search Methods for Finding a Nash
Equilibrium
EPPERSON, JAMES (2002) – An Introduction to Numerical Methods and Analysis
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