Is God Rational?Is God Rational?Investigating the Application of Investigating the Application of

Game Theory to The Old Game Theory to The Old TestamentTestament

Marlene M. MerchainMarlene M. MerchainSenior Capstone 2005Senior Capstone 2005

Advisor : Dr. Karrolyne FogelAdvisor : Dr. Karrolyne Fogel

??? The Big Question ?????? The Big Question ???

Is God rational, that is, when He Is God rational, that is, when He

is a participant in a Biblical game, is a participant in a Biblical game,

does he act so that He may does he act so that He may

benefit most from His decisions?benefit most from His decisions?

How Do We Determine If How Do We Determine If God Is Rational?God Is Rational?

God’s rationality will be God’s rationality will be determined by performing a determined by performing a

game theory analysis on game theory analysis on selected Biblical stories and selected Biblical stories and

comparing the resulting comparing the resulting outcomes to the actual outcomes to the actual outcomes in the Bible.outcomes in the Bible.

Background/BasicBackground/Basic DefinitionsDefinitions

Player- Player- A participant in a situational event A participant in a situational event in which decisions or “moves” are made.in which decisions or “moves” are made.

Game-Game- an interdependent decision situation an interdependent decision situation whose outcome depends on the choices of all whose outcome depends on the choices of all players.players.

Move-Move- when a player makes a choice among when a player makes a choice among a number of specified alternatives.a number of specified alternatives.

Payoffs (Utilities)Payoffs (Utilities)-- numerical values that numerical values that are assigned to ranked outcomes according are assigned to ranked outcomes according to their level of appeal to the player, (the to their level of appeal to the player, (the higher the payoff, the better the outcome).higher the payoff, the better the outcome).

Background/BasicBackground/Basic DefinitionsDefinitions (cont’d)(cont’d)

Games of Perfect InformationGames of Perfect Information- games - games in which all the choices of all the players in which all the choices of all the players are known to everyone as soon as they are known to everyone as soon as they are made.are made.

RationalRational- given a player’s preferences - given a player’s preferences and their knowledge of the other players’ and their knowledge of the other players’ preferences, the player makes strategical preferences, the player makes strategical choices that would better rather than choices that would better rather than worsen outcomes (outcomes that benefit worsen outcomes (outcomes that benefit the player most.)the player most.)

Games To Be ExaminedGames To Be Examined

Games Predetermined By Brams:Games Predetermined By Brams:Adam & Eve GameAdam & Eve GamePharaoh GamePharaoh Game

Game Created by Marlene:Game Created by Marlene:Noah GameNoah Game

*To minimize confusion, games are referred to by God’s *To minimize confusion, games are referred to by God’s opponent. opponent.

Game #1: Adam & EveGame #1: Adam & Eve

Adam and Eve Adam and Eve obedient, obedient,

God approvingGod approving

(3,(3,22))

Adam and Eve Adam and Eve disobedient, disobedient,

God God disapprovingdisapproving

(2,(2,33))Adam and Eve Adam and Eve

voluntarily voluntarily restrained, restrained,

God very God very pleasedpleased

(4,(4,11))

Adam and Eve Adam and Eve unrestrained, unrestrained,

God very God very displeaseddispleased

(1,(1,44))

Outcome MatrixOutcome Matrix

Adhere to Adhere to Constraints (B)Constraints (B)

Don’t Adhere to Don’t Adhere to Constraints (B’)Constraints (B’)

Impose Impose ConstraiConstrai

nts nts

(A)(A)

Don’t Don’t Impose Impose

ConstrainConstraintsts

(A’)(A’)

Adam and Adam and EveEve

GodGod

Key:Key:

(x,(x,yy)=(God, )=(God, Adam and EveAdam and Eve))

4 = best ; 3 = next best ; 2 = next worst ; 1 = worst4 = best ; 3 = next best ; 2 = next worst ; 1 = worst

Game #1: Adam & EveGame #1: Adam & Eve

(3,(3,22)) ((22,,33))

(4,(4,11)) (1,(1,44))

Outcome Matrix AnalysisOutcome Matrix Analysis

Adhere to Adhere to Constraints (B)Constraints (B)

Don’t Adhere to Don’t Adhere to Constraints (B’)Constraints (B’)

Impose Impose ConstraiConstrai

nts nts

(A)(A)

Don’t Don’t Impose Impose

ConstrainConstraintsts

(A’)(A’)

Adam and Adam and EveEve

GodGod

Key:Key:

(x,(x,yy)=(God, )=(God, Adam and EveAdam and Eve))

4 = best ; 3 = next best ; 2 = next 4 = best ; 3 = next best ; 2 = next worst ; 1 = worstworst ; 1 = worst

Our outcome AB’ is an

equilibrium point!

AB’ is the highest ‘x’ in its column and highest ‘y’ in its row

Some Strategies & Tools for Some Strategies & Tools for Determining RationalityDetermining Rationality

Dominant Strategy-Dominant Strategy- strategy strategy CC dominatesdominates a a strategy strategy DD if and only if every outcome in if and only if every outcome in CC is greater than its corresponding outcome in is greater than its corresponding outcome in DD.. CC dominates dominates DD..

Dominance Principle- aDominance Principle- a rationalrational player player should never play a dominated strategy.should never play a dominated strategy.

Saddle Point Principle-Saddle Point Principle- if a matrix game if a matrix game has a saddle point (equilibrium point), both has a saddle point (equilibrium point), both players should play a strategy that contains players should play a strategy that contains it!it!

Security Level-Security Level- the lowest guaranteed the lowest guaranteed payoff a player can attain.payoff a player can attain.

Conclusions from Adam and Conclusions from Adam and Eve GameEve Game

Game Theory OutcomeGame Theory Outcome::

God imposes constraints, God imposes constraints,

Adam and Eve don’t adhereAdam and Eve don’t adhere Biblical OutcomeBiblical Outcome::

God imposes constraints, God imposes constraints,

Adam and Eve don’t adhereAdam and Eve don’t adhere

Determination: Determination:

God acts God acts RationallyRationally in Adam & in Adam & Eve GameEve Game

Result: Expulsion from Eden

Game #2: PharaohGame #2: Pharaoh

New New ConfrontationConfrontation

(2,(2,44))

Help given Help given unnecessarilyunnecessarily

(3,(3,22))

Help Help unforthcoming unforthcoming

(4,(4,11))

Unaided flight Unaided flight successfulsuccessful

(1,(1,33))

Outcome MatrixOutcome Matrix

Pursue Pursue (B) (B)

Don’t Pursue Don’t Pursue (B’) (B’)

Help Help IsraelitIsraelit

es es

(A)(A)

Don’t Don’t Help Help

IsraeliteIsraelitess

(A’)(A’)

PharaohPharaoh

GodGod

Key:Key:

(x,(x,yy)=(God, )=(God, PharaohPharaoh))

*This game *This game has no pure has no pure

strategy strategy equilibriumequilibrium

!!

The AB is The AB is Pareto Pareto

superior superior A’B’, that is A’B’, that is

AB is a AB is a better better

outcome for outcome for both both

players players than A’B’, than A’B’,

but not but not necessarily necessarily the best for the best for

either either playerplayer

Security levels can help Security levels can help us out!us out!

Conclusions from Pharaoh Conclusions from Pharaoh GameGame

Game Theory OutcomeGame Theory Outcome::No No clearclear outcome…take your chances! However, outcome…take your chances! However, if if both players play their security levels, the both players play their security levels, the outcome is outcome is God helps Israelites, Pharaoh God helps Israelites, Pharaoh pursuespursues

Biblical OutcomeBiblical Outcome::God helps Israelites, God helps Israelites, Pharaoh pursuesPharaoh pursues

Determination: Determination: God acts as God acts as RationalRational as possible in the as possible in the

Pharaoh GamePharaoh Game

Result: Pharaoh Gets A Taste of God’s Power!

Game #3: Noah Game #3: Noah

God spares God spares Noah,Noah,

Noah compliesNoah complies

((4,4,44))

God spares God spares Noah, Noah Noah, Noah disobeys disobeys

(1,(1,22))

God destroys God destroys Noah, Noah Noah, Noah complies complies

(3,(3,33))

God destroys God destroys Noah, Noah Noah, Noah

disobeys disobeys

(2,(2,11))

Marlene’s Outcome MatrixMarlene’s Outcome Matrix

Complies Complies (Builds Ark) (Builds Ark) (B) (B)

Disobeys Disobeys (Doesn’t Build (Doesn’t Build

Ark) Ark) (B’)(B’)Spares Spares

Noah Noah

(A)(A)

Destroys Destroys Noah Noah

(A’)(A’)

NoahNoah

GodGod

Key:Key:

(x,(x,yy)=(God, )=(God, NoahNoah))

*Dominant *Dominant strategy for strategy for

NoahNoah

** ** Equilibrium Equilibrium point is point is “Pareto “Pareto optimal”optimal”

Conclusions from Noah Conclusions from Noah GameGame

Game Theory OutcomeGame Theory Outcome::

God spares Noah,God spares Noah,

Noah compliesNoah complies Biblical OutcomeBiblical Outcome::

God spares Noah, God spares Noah,

Noah compliesNoah complies

Determination: Determination:

God acts God acts RationallyRationally in Noah in Noah GameGame

Result: Noah and his loved ones remained safe during the

flood

Final Overview :Final Overview :

Game #1 vs. Adam & Eve- Plays Game #1 vs. Adam & Eve- Plays RationallyRationally

Game #2 vs. Pharaoh- As Rational Game #2 vs. Pharaoh- As Rational as possibleas possible

Game #3 vs. Noah- Plays RationallyGame #3 vs. Noah- Plays Rationally

So the final verdict is…So the final verdict is…

God is God is Rational!Rational!

BibliographyBibliography

1.1. Brams, Steven J. Brams, Steven J. Biblical Games: Game Theory and the Biblical Games: Game Theory and the Hebrew BibleHebrew Bible. Massachusetts: Massachusetts Institute of . Massachusetts: Massachusetts Institute of Technology, 2003.Technology, 2003.

2.2. McCain, Roger A. McCain, Roger A. Game Theory: An Introductory SketchGame Theory: An Introductory Sketch. . (Online) Available(Online) Available http://william-king.www.drexel.edu/top/eco/game/game.htmlhttp://william-king.www.drexel.edu/top/eco/game/game.html 30 November 2004 30 November 2004..

3.3. Meeks, Wayne A, ed. Meeks, Wayne A, ed. The Harper Collins Study BibleThe Harper Collins Study Bible. . London: Harper Collins Publishers, 1993London: Harper Collins Publishers, 1993

4.4. Rapoport, Anatol. Rapoport, Anatol. Two-Person Game TheoryTwo-Person Game Theory. Mineola, . Mineola,

New York: Dover Publications, Inc., 1966New York: Dover Publications, Inc., 1966 5.5. Straffin, Philip D. Straffin, Philip D. Game Theory and StrategyGame Theory and Strategy. .

Washington, D.C.: The Mathematical Association of Washington, D.C.: The Mathematical Association of America, 1993.America, 1993.

GraphicsGraphicsComplements of:Complements of:

Biblical Clip Art:Biblical Clip Art:

http://www.wels.net/wmc/html/clip_art_--_volume_2http://www.wels.net/wmc/html/clip_art_--_volume_2__part_a.html__part_a.html

Sistine Chapel Frescos: Sistine Chapel Frescos: http://sun.science.wayne.edu/~mcogan/Humanitieshttp://sun.science.wayne.edu/~mcogan/Humanities/Sistine/Panels/index.html/Sistine/Panels/index.html

Biblical GamesBiblical Games Book cover: Book cover: http://www.amazon.com/exec/obidos/tg/detail/-/026http://www.amazon.com/exec/obidos/tg/detail/-/0262523329/102-7691842-8123321?v=glance2523329/102-7691842-8123321?v=glance

Game #3: Abraham GameGame #3: Abraham Game

Abraham faithful a. (4,4) God merciful b.(4,4) Isaac saved c. (4,4)

Abraham faithful a. (3,3) God adamant b.(2,3) Isaac sacrificed c. (1,3)

Abraham resistant a.(2,1) God merciful b.(3,1) Isaac saved c. (3,1)

Abraham resistant a.(1,2) God adamant b.(1,2) Isaac’s fate c. (2,2) uncertain

Bram’s Outcome Matrices

Offer Isaac (A)

Don’t Offer Isaac (A’)

Renege/

Relent

(B)

Renege/

Relent

(B’)

AbrahamAbraham

GodGod

Key:Key:(x,(x,yy)=(God, )=(God, AbrahamAbraham))

*Renege-

To fail to carry out a promise or commitment:

**Note: This matrix is a composition of 3 different matrices with 3 distinct views of Abraham’s possible mentalities toward this situation

Game #4: Abraham GameGame #4: Abraham Game

((4,4,44)) (3,(3,33))

(2,(2,11)) (1,(1,22))

Brams’ 4a Outcome Matrix

Offer Isaac (A)

Don’t Offer Isaac (A’)

Renege

(B)

Don’t Renege

(B’)

Abraham

God

Key:Key:(x,(x,yy)=(God, )=(God, AbrahamAbraham))

a) Abraham faithful

regardless:

prefers “offer” over “don’t

offer”

Game #4: Abraham GameGame #4: Abraham Game

((4,4,44)) (2,(2,33))

(3,(3,11)) (1,(1,22))

Brams’ 4b Outcome Matrix

Offer Isaac (A)

Don’t Offer Isaac (A’)

Renege

(B)

Don’t Renege

(B’)

Abraham

God

Key:Key:(x,(x,yy)=(God, )=(God, AbrahamAbraham))

b) Abraham wavers

somewhat:

prefers God “renege/relent”

over “don’t renege/relent”

Game #4: Abraham GameGame #4: Abraham GameBrams’ 4c Matrix Analysis

AbrahamAbraham

GodGod

Key:Key:(x,(x,yy)=(God, )=(God, AbrahamAbraham))

Offer Isaac (A)

Don’t Offer Isaac (A’)

Renege/ Relent

(B)

Don’t Renege/

Don’t Relent (B’)

((4,4,44)) (1,(1,33))

(3,(3,11)) ((2,2,22))

c) Abraham wavers seriously:

Isaac’s life paramount—same

as (b) except if God adamant, would prefer “don’t offer”

Conclusions from Abraham Conclusions from Abraham GameGame

Game Theory OutcomesGame Theory Outcomes::a. a. bb. .

c. c.

Biblical OutcomeBiblical Outcome::Abraham sacrifices Isaac, Abraham sacrifices Isaac,

God spares IsaacGod spares Isaac

Determination: Determination:

God acts God acts RationallyRationally in Abraham in Abraham GameGame

Result: Abraham rewarded by God for his

faithfulness

Mixed StrategyMixed Strategy

Of Pharaoh GameOf Pharaoh Game