Is God Rational? Investigating the Application of Game Theory to The Old Testament Marlene M. Merchain Senior Capstone 2005 Advisor : Dr. Karrolyne Fogel

??? The Big Question ??? Is God rational, that is, when He is a participant in a Biblical game, does he act so that He may benefit most from His decisions?

How Do We Determine If God Is Rational? God’s rationality will be determined by performing a game theory analysis on selected Biblical stories and comparing the resulting outcomes to the actual outcomes in the Bible.

Background/Basic Definitions Player- A participant in a situational event in which decisions or “moves” are made. Player- A participant in a situational event in which decisions or “moves” are made. Game- an interdependent decision situation whose outcome depends on the choices of all players. Game- an interdependent decision situation whose outcome depends on the choices of all players. Move- when a player makes a choice among a number of specified alternatives. Move- when a player makes a choice among a number of specified alternatives. Payoffs (Utilities) - numerical values that are assigned to ranked outcomes according to their level of appeal to the player, (the higher the payoff, the better the outcome). Payoffs (Utilities) - numerical values that are assigned to ranked outcomes according to their level of appeal to the player, (the higher the payoff, the better the outcome).

Background/Basic Definitions (cont’d) Games of Perfect Information- games in which all the choices of all the players are known to everyone as soon as they are made. Games of Perfect Information- games in which all the choices of all the players are known to everyone as soon as they are made. Rational- given a player’s preferences and their knowledge of the other players’ preferences, the player makes strategical choices that would better rather than worsen outcomes (outcomes that benefit the player most.) Rational- given a player’s preferences and their knowledge of the other players’ preferences, the player makes strategical choices that would better rather than worsen outcomes (outcomes that benefit the player most.)

Games To Be Examined Games Predetermined By Brams: Games Predetermined By Brams: Adam & Eve Game Adam & Eve Game Pharaoh Game Pharaoh Game Game Created by Marlene: Game Created by Marlene: Noah Game Noah Game *To minimize confusion, games are referred to by God’s opponent. *To minimize confusion, games are referred to by God’s opponent.

Game #1: Adam & Eve Adam and Eve obedient, God approving (3,2) Adam and Eve disobedient, God disapproving (2,3) Adam and Eve voluntarily restrained, God very pleased (4,1) Adam and Eve unrestrained, God very displeased (1,4) Outcome Matrix Adhere to Constraints (B) Don’t Adhere to Constraints (B’) Impose Constraints (A) Don’t Impose Constraints (A’) (A’) Adam and Eve God Key: (x,y)=(God, Adam and Eve) 4 = best ; 3 = next best ; 2 = next worst ; 1 = worst

Game #1: Adam & Eve (3,2) (2,3)(2,3)(2,3)(2,3) (4,1) (1,4) Outcome Matrix Analysis Adhere to Constraints (B) Don’t Adhere to Constraints (B’) Impose Constraints (A) Don’t Impose Constraints (A’) (A’) Adam and Eve God Key: (x,y)=(God, Adam and Eve) 4 = best ; 3 = next best ; 2 = next worst ; 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 Determining Rationality Dominant Strategy- strategy C dominates a strategy D if and only if every outcome in C is greater than its corresponding outcome in D. C dominates D. Dominant Strategy- strategy C dominates a strategy D if and only if every outcome in C is greater than its corresponding outcome in D. C dominates D. Dominance Principle- a rational player should never play a dominated strategy. Dominance Principle- a rational player should never play a dominated strategy. Saddle Point Principle- if a matrix game has a saddle point (equilibrium point), both players should play a strategy that contains it! Saddle Point Principle- if a matrix game has a saddle point (equilibrium point), both players should play a strategy that contains it! Security Level- the lowest guaranteed payoff a player can attain. Security Level- the lowest guaranteed payoff a player can attain.

Conclusions from Adam and Eve Game Game Theory Outcome: Game Theory Outcome: God imposes constraints, Adam and Eve don’t adhere Biblical Outcome: Biblical Outcome: God imposes constraints, God imposes constraints, Adam and Eve don’t adhere Determination: God acts Rationally in Adam & Eve Game Result: Expulsion from Eden

Game #2: Pharaoh New Confrontation (2,4) Help given unnecessarily (3,2) Help unforthcoming (4,1) Unaided flight successful (1,3) Outcome Matrix Pursue (B) Don’t Pursue (B’) Help Israelites (A) Don’t Help Israelites (A’) (A’) Pharaoh God Key: (x,y)=(God, Pharaoh) *This game has no pure strategy equilibrium! The AB is Pareto superior A’B’, that is AB is a better outcome for both players than A’B’, but not necessarily the best for either player Security levels can help us out!

Conclusions from Pharaoh Game Game Theory Outcome: Game Theory Outcome: No clear outcome…take your chances! However, if both players play their security levels, the outcome is God helps Israelites, Pharaoh pursues Biblical Outcome: Biblical Outcome: God helps Israelites, Pharaoh pursues Determination: God acts as Rational as possible in the Pharaoh Game God acts as Rational as possible in the Pharaoh Game Result: Pharaoh Gets A Taste of God’s Power!

Game #3: Noah God spares Noah, Noah complies (4,4) God spares Noah, Noah disobeys (1,2) God destroys Noah, Noah complies (3,3) God destroys Noah, Noah disobeys (2,1) Marlene’s Outcome Matrix Complies (Builds Ark) (B) Disobeys (Doesn’t Build Ark) (B’) Spares Noah (A) Destroys Noah (A’) (A’) Noah God Key: (x,y)=(God, Noah) *Dominant strategy for Noah ** Equilibrium point is “Pareto optimal”

Conclusions from Noah Game Game Theory Outcome: Game Theory Outcome: God spares Noah, Noah complies Biblical Outcome: Biblical Outcome: God spares Noah, Noah complies Determination: God acts Rationally in Noah Game God acts Rationally in Noah Game Result: Noah and his loved ones remained safe during the flood

Final Overview : Game #1 vs. Adam & Eve- Plays Rationally Game #1 vs. Adam & Eve- Plays Rationally Game #2 vs. Pharaoh- As Rational as possible Game #2 vs. Pharaoh- As Rational as possible Game #3 vs. Noah- Plays Rationally Game #3 vs. Noah- Plays Rationally So the final verdict is…

God is Rational!

Bibliography 1. Brams, Steven J. Biblical Games: Game Theory and the Hebrew Bible. Massachusetts: Massachusetts Institute of Technology, McCain, Roger A. Game Theory: An Introductory Sketch. (Online) Available 30 November November November Meeks, Wayne A, ed. The Harper Collins Study Bible. London: Harper Collins Publishers, Rapoport, Anatol. Two-Person Game Theory. Mineola, New York: Dover Publications, Inc., Straffin, Philip D. Game Theory and Strategy. Washington, D.C.: The Mathematical Association of America, 1993.

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Game #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’) Abraham God Key: (x,y)=(God, Abraham) *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 Game (4,4) (3,3) (2,1) (1,2) Brams’ 4a Outcome Matrix Offer Isaac (A) Don’t Offer Isaac (A’) Renege (B) Don’t Renege (B’) Abraham God Key: (x,y)=(God, Abraham) a) Abraham faithful regardless: prefers “offer” over “don’t offer”

Game #4: Abraham Game (4,4) (2,3) (3,1) (1,2) Brams’ 4b Outcome Matrix Offer Isaac (A) Don’t Offer Isaac (A’) Renege (B) Don’t Renege (B’) Abraham God Key: (x,y)=(God, Abraham) b) Abraham wavers somewhat: prefers God “renege/relent” over “don’t renege/relent”

Game #4: Abraham Game Brams’ 4c Matrix Analysis Abraham God Key: (x,y)=(God, Abraham) Offer Isaac (A) Don’t Offer Isaac (A’) Renege/ Relent (B) Don’t Renege/ Don’t Relent (B’) (4,4) (1,3) (3,1) (2,2) c) Abraham wavers seriously: Isaac’s life paramount—same as (b) except if God adamant, would prefer “don’t offer”

Conclusions from Abraham Game Game Theory Outcomes: Game Theory Outcomes:a. b. c. Biblical Outcome: Biblical Outcome: Abraham sacrifices Isaac, God spares Isaac Determination: God acts Rationally in Abraham Game Result: Abraham rewarded by God for his faithfulness

Mixed Strategy Of Pharaoh Game