Game-Theoretic Rough Sets

Game-Theoretic Rough Sets

Joseph P. Herbert and JingTao Yao

Department of Computer Science

University or Regina

CANADA S4S 0A2

[email protected]

http://www.cs.uregina.ca/~jtyao


J T Yao Game-Theoretic Rough Sets 2

Introduction

• Calculating the regions in the rough set

approximation space is a key issue in rough set

data analysis.

• Large boundary region is one of the barriers

for rough applications.

• The probabilistic rough set model was created

to provide a means to better define the regions.

• Finding correct values for region thresholds

remains a major challenge.


Challenges of Probabilistic Models

• Two major challenges for providing a

complete probabilistic model:

– Model the effects of the thresholds on

classification ability.

– Purely data-driven thresholds.

• The first challenge is the focus of this

research and perhaps will lead to new

insights when facing the second challenge.


The Approach

• Game theory is a mathematical-based infrastructure to govern competition in games between two or more parties.

• It can be used to analyze the effects that different threshold values have on classification ability.

• It may also gives us a fresh perspective on rough set analysis by:

– Giving us the ability to observe the trade-off between accuracy and precision.

– Giving us a means of finding the relationships between this trade-off and threshold values.


Game Theory

• Each player has a set of actions (strategies) with

expected payoffs (benefits) as a result of taking that

action. A payoff function determines the utility of

a chosen action and returns a payoff value.

• All players are assumed to be rational.

– Rational players choose strategies that improve their

position in the game.

– Rational players choose strategies that maximize their

ability to win while minimizing the other players’ ability

to do the same.


Payoff Tables

• Games are formulated into payoff tables, showing players

involved, possible strategies, and expected payoffs.

Player 2

S2,1 … S2,n

Player 1

S1,1 p1,1, p2,1 … p1,1, p2,n

… … … …

S1,n p1,n, p2,1 … p1,n, p2,n


Prisoners’ Dilemma Example

• Two prisoners are held by police under suspicion of burglary.

• They are questioned separately by the police.

• The prisoners have two actions: confess and implicate the

other, or don’t confess (remain silent).

• Depending on each prisoners chosen action, each will serve a

certain period of time in jail.

• A set of players:

• A set of actions:

prisoner2prisoner1,O

confesstdon'confess,S


Prisoners’ Dilemma

• Payoff Table:

prisoner 2

Confess Don’t Confess

prisoner 1

Confessp1 serves 10 years,

p2 serves 10 years

p1 serves 0 years,

p2 serves 20 years

Don’t

Confess

p1 serves 20 years,

p2 serves 0 years

p1 serves 1 years,

p2 serves 1 years


Rough Sets

• Sets derived from imperfect, imprecise, and

incomplete data may not be able to be

precisely defined.

• Sets have to be approximated

– Using describable concepts to approximate

known concept

– 1.76 cm => 1.7, 1.8


Visualizing Rough Sets

Let T = (U, A), , UX AB

AxxAaprB)(

• Upper Approximation: +

AxxAaprB

)(

• Boundary Region:

)()( AaprAaprABNDB

• Negative Region:

• Lower Approximation:

)( AaprUANEGB


Probabilistic Model

• Inclusion into regions are given by

probability thresholds:

– If , decide

– If , decide

– If and ,

decide

| [ ]P A x | [ ]P A x

][| xAP

][| xAP

)( APOS

)( ANEG

)( ABND

)( APOS

)( ANEG

)( ABND


Decision-Theoretic Rough Sets (DTRS)

• Rough set theory may not be suitable when the boundary

region is too large.

• DTRS, proposed by Y.Y Yao in 1990, is a robust

approach to the probabilistic model for two reasons:

– Uses easily understandable notions of risk or cost to determine

threshold values.

– Many application domains could make use of these concrete

notions in order to satisfy decision needs.

• Calculated cost (risk) using the Bayesian decision

procedure provides minimum cost α, β threshold values

for region division.


Classification with DTRS

• Let be the conditional probability of an object

being in state give the description .

• Classification is performed through a set of actions:

where , , and represent the three actions to

classify objects into the positive, negative, and boundary

regions respectively.

• Objects are classified into a set of states. In this case,

either the set or its complement .

)|( xj

wP x

jw x

BNP

aaa ,,

pa

Na B

a

AC

A


Cost of Classification

• A loss is incurred when taking a classification action.

• Let denote the loss incurred for taking action

when an object is in , and let denote the loss

incurred by taking the same action when the object belongs

to .

• They are provided by the user based on their beliefs

regarding the consequences of a classification action.

Aai

|

C

iAa |

ia

A

CA


Loss Functions

• These losses from classification actions are given

by loss functions.

• Functions and can be considered as the

costs for false-positives and false-negatives.

Function Description Function Description Function Description

Classify into positive

region, object

belongs to .

Classify into negative

region, object

belongs to .

Classify into boundary

region, object belongs

to .

Classify into positive

region, object

belongs to .

Classify into negative

region, object

belongs to .

Classify into boundary

region, object belongs

to .

PP

PN

NP

NN

BP

BN

A A A

CA

CA

CA

PN

NP


Expected Loss• We have expected loss values for each action:

– : Classifying an object into the positive region.

– : Classifying an object into the negative region.

– : Classifying an object into the boundary region.

• They can be expressed as:

–

–

–

• By finding the lowest expected cost, we can determine which action to perform.

NR

BR

PR

xAPxAPxaRRC

PNPPPP|||

xAPxAPxaRRC

NNNPNN|||

xAPxAPxaRRC

BNBPBB|||


Threshold Calculation

• If the following hold:

we can calculate the , , and thresholds:

PNBNNNNPBPPP ,

,

PNPPBNBP

BNPN

,

BNBPNNNP

NNBN

.

PNPPNNNP

NNPN


Decision Rules

• The inequalities help us derive decision rules:

– (PP) : If and , decide

– (NP) : If and , decide

– (BP) : If and , decide

• The Bayesian decision procedure leads to minimum risk

decision rules:

– (PN) : If and , decide

– (NN) : If and , decide

– (BN) : If and , decide

][| xAP ][| xAP

][| xAP ][| xAP

][| xAP ][| xAP

)( APOS

)( ANEG

)( ABND

)( APOS

)( ANEG

)( ABND

NPRR

PNRR

BPRR

BNRR

PBRR NB

RR


A Simple DTRS Program

• DTRS Demonstration

DTRS_Software/DTRS.jar


RSA from a Game Theory Perspective

• Use game theory to analyze the effects that different

threshold values have on classification ability.

– Connect the user’s notions of cost to the classification

ability of rough sets.

– This connection determines the consequences of a

fluctuation in expected cost.

• Game theory provides a means for the user to change

their beliefs regarding the types of decisions they can

make.

– Rather than changing probabilities to improve

classification, they are provided suggested strategies for

changing the cost of a classification action.


The Formulation Process

1. Game formulation.

• Defines the game.

2. Strategy formulation.

• Defines possible actions.

3. Payoff measurement.

• How effective are these actions?

4. Competition Implementation.

• Create payoff tables.

5. Result Acquisition.

• Result interpretation.


1. Game Formulation

• A single game is defined as:

• We define the connections between the following:

– A set of players, O : a set of classification measures.

– A set of strategies, S : a set of risk modifications.

– Action payoffs, F : new classification measure values.

• Improved classification is the goal, therefore, each player

will represent a measure trying achieve a maximum value.

FSOG ,,

,O

: Approximation accuracy

: Approximation precision


Measures• Approximation accuracy

| ( ) |

| ( ) |

apr A

apr A

| ( ) |

| |

apr A

A

| |

| ( ) |

A

apr A

• Approximation precision

Deterministic: Non-deterministic:


Strategies• Each measure is competing with the other to win the “game”

(tradeoff). The goal of the game is to improve classification ability.

• Each player (measure) has a set of strategies it can use to achieve better payoff.

• Individual strategies for a player i, when performed, are called actions:

• In essence, the strategy for a measure should only consist of actions that improve the measure’s value, i.e., acquire a maximal value for approximation accuracy as possible.

mi

aaS ,,1


Payoffs

• Each action performed results in a payoff.

• Over successive turns, each player wishes to increase

payoff, i.e., for accuracy: where is the payoff

(accuracy value) after the first action is performed and

is the payoff after the second action is performed

• To achieve payoff increases, we need a formulation of the

strategies that directly effect the classification ability.

1a

2a

21 aa


2. Strategy Formulation

• Actions need to have direct influence on accuracy and

precision.

• To increase accuracy:

– Make as large as possible.

– Maintain the size of .

• To increase precision:

– Make as large as possible.

Aapr

Aapr

Aapr


Strategy Formulation

• Recalling rules (PN, NN, BN), in order to increase

the size of the lower approximation (positive

region), we need to decrease the expected loss

.

• To do this, we need to decrease one or both of the

loss functions and

PR

xAPxAPxaRRC

PNPPPP|||

PP

PN


Determining Valid Strategies

• We begin to see that by modifying loss functions,

we change the expected cost of an action.

• This, in turn, has an effect on the lower and upper

approximations. These approximations determine

the values of accuracy and precision.

• In total, three actions can be used for improving

approximation accuracy in the game theory

formulation.


Determining Valid Strategies

Action (Strategy) Goal Method Result

DecreaseDecrease

orLarger POS region

IncreaseIncrease

orSmaller NEG region

IncreaseIncrease

orSmaller BND region

1a

3a

2a

NR

BR

PR PP

PN

NP

NN

BP

BN

N

R

B

R

P

R


POS

BND

NEG

POS

BND

NEG

POS

BND

NEG

POS

BND

NEG

Action -RP

Action +RN

Action +RB …

…

…


3. Payoff Measurement• A payoff function measures the effectiveness of an action j

performed by a player i:

• Since each player has three actions they can perform, the

payoffs for and are:

• For example, player (accuracy) choosing action would

get the following:

jji

a,

3,12,11,11

,, P

3,32,21,22

,, P

ja

jj

a,1


Payoff Measurement• The payoff functions imply that there relationships

between the measures selected as players, the action they perform, and the probabilities used for region classification.

• These properties can be used to formulate guidelines regarding the amount of flexibility the user’s loss functions may have in order to maintain a certain level of consistency in the data analysis.

• Both the payoffs and the actions performing them are conveyed in a payoff table.


4. Competition Implementation• A payoff table expresses players, strategies, and payoffs into

intuitive tabular form.

• Each cell of the payoff table consists of a payoff pair:

.

• In order to find an optimal solution to a game, we determine whether

there is equilibrium within the table.

• is an equilibrium if for any action where ,

and .

ji ,2,1,

*

,2

*

,1,

ji

ka jik ,

ki ,1

*

,1 kj ,2

*

,2


Payoff Table

• Payoff table for our formulation:

– Two players: and

– Three actions / player: , , and

– Nine payoff pairs:

NR B

RP

R

NR

BR

PR

NR

BR

PR

1,11,1,

1,33,1,

3,11,3,

3,22,3,

3,33,3,

2,11,2,

2,22,2,

2,33,2,

1,22,1,


5. Results Acquisition• The final step is to interpret the results from the observations made on the

execution of the game.

• We can calculate how much the loss function needs to be modified in order acquire the level of accuracy or precision indicated with the payoff table.

• The new values for the loss functions must still satisfy the inequalities discussed earlier:

• Action can be reduced any amount and rule (PN) will be satisfied, however, rules (NN) and (BN) must also be satisfied, so there is some tolerance in the amount of change that can take place:

– (PN) : If and , decide

– (NN) : If and , decide

– (BN) : If and , decide

PR

PNBNNNNPBPPP ,

)( APOS

)( ANEG

)( ABND

NPRR

PNRR

BPRR

BNRR

PBRR

NBRR


Loss Function Tolerance

• The new value for expected cost, denoted must

satisfy:

• This results in an allowable change of to and

to .

• The following tolerance values indicate how much a user

may change their loss functions:

*

PR

PBPPNP

RRRRRR **

,

PP

PN

PPt

PNt

PN

BNPN

PN

PP

PPBP

PPtt

minmax,


Example• Let the following be a series of loss functions for correct

classifications, boundary classifications, and incorrect

classifications respectively:

• The inequalities hold. We acquire the following:

• We can increase by 50% and increase by 25% and

maintain the same classification ability.

• These tolerance values can be used to determine the new values of

expected cost.

8,6,4 NPPNBNBPNNPP

4

1

8

68,

2

1

4

46 minmax

PNPPtt

PP

PN


Concluding Remarks• We provide a preliminary study on using game theory for

determining the relationships between loss functions, expected loss, threshold values, and classification ability.

• Loss function modification is now guided by the competitive process between classification measures.

• Through the formulation of expected loss, strategies we choose have a direct effect on rough set analysis.

• Tolerance values are given to the user where they are used to determine how much a chosen expected cost action should change.

• Game theory can be a powerful method for governing rough set analysis when adjustable criteria, such as loss functions, are used to influence classification ability.

• A learning method governed by game theory is in development to calculate threshold values from the actual data.


Game-theoretic Rough Sets

• Combining game theory with rough set theory.

• Game theory for threshold completion in

probabilistic rough sets.

• Risk determination for DTRS with game theory

Game-Theoretic Rough Sets

Joseph P. Herbert and JingTao Yao

Department of Computer Science

University or Regina

CANADA S4S 0A2

[email protected]




Where is Regina?

Game-Theoretic Rough Sets

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