Game Theory: The Mathematics of Competition

6th Edition = Chapter 165th Edition = Chapter 15

Game Theory - definitions

• Strategies – courses of action a player might choose– Pure Strategy – a course of action which does

not involve randomized choices – pick a strategy and stay with it

– Mixed Strategy – randomizes the strategies to get the best outcome

• Outcomes – the consequences of the course of action

Game Theory

• Game Theory – using mathematical tools to study situations involving conflict and co-operation

• Game Theory - analyzes the rational choice of strategies How players select strategies to obtain preferred outcomes

• Game Theory – analyzing situations in which there are at least 2 players in conflict because of different goals

Applications of Game Theory

• Labor – Management Disputes• Resource Allocation Decisions• Military Choices in international conflict• Threats by animals

Definitions - Continued

• Saddlepoint – When MaxMin and MiniMax Values are the same (=), Same Result(outcome)– Complex Games have saddlepoints e.g. Chess just don’t

know where it is.– No Saddlepoint Games - Poker

• Value of the Game Where the two strategies intersect

• Zero Sum Game – payoff to one player is the negative payoff to the other player

Definitions - continued

• Conflict between players– Total – One player WINS while the other loses.– Partial – Players can benefit from some kind or

form of co-operation

Henry and Lisa- Strategy

• MaxMin Strategy – the Maximum value of the minimum choices

• MiniMax Strategy– the minimum value of the maximum choices

• Both – Worst Case analysis• Each player is guaranteed at Least the value

of their MaxMin and MiniMax strategies

Two Person – Total Conflict –Mixed Strategy

• Baseball !!!

PitcherFast Curve

Batter Fast Curve

Two Person – Total Conflict –Mixed Strategy

EF=(.300)(1-P)+.200P= .300-.300P + .200P= .300-.100P

EC = .100(1-P) + .500P= .100 -.100P + .500P= .100 + .400P

EF = EC.300 - .100P = .100 + .400P.300 - .100 = .400P + 100P.200 = .500PP = .200/.500 P=2/51-P = 3/5

PitcherFast Curve

Batter Fast 0.300 0.200 1-QCurve 0.100 0.500 Q

1-P P

Two Person – Total Conflict –Mixed Strategy

EF=(.300)(1-Q)+.100Q= .300-.300Q + .100Q= .300-.200Q

EC = .200(1-Q) + .500Q= .200 -.200Q + .500Q= .200 + .300Q

EF = EC.300 - .200Q = .200 + .300Q.300 - .200 = .300Q + 200Q.100 = .500QQ = .100/.500 Q=1/51-Q = 4/5

PitcherFast Curve

Batter Fast 0.300 0.200 1-QCurve 0.100 0.500 Q

1-P P

Partial Conflict Games

• Partial Conflict – Variable Sum Games. Different payoffs as the outcome changes

• Non-Cooperative – No binding agreement is possible or can be enforced

• Ordinal Games – Players rank the outcomes from best to worst

Prisoner’s Dilemma

• 2 people accused of a crime – both held incommunicado (Harry and Joe)

• Each have two choices:– Stay quiet– Tell on your partner

Prisoner’s Dilemma – cont.

• Harry needs to rank the possible outcomes from low to high

4. Harry tells on Joe and Joe stays quiet – Harry might get to go home!! (Joe’s going to Jail)

3. Harry remains quiet and so does Joe – possible both get off

2. Harry tells on Joe and Joe tells on Harry – both going to Jail

1. Harry is quiet and Joe tells on him – Joe gets off and Harry goes to jail for a long time

Prisoner’s Dilemma

JOEConfess Silent

Harry Confess (2,2) (4,1) Silent (1,4) (3,3)

John Nash

• Nash Equilibrium –When no player can benefit by departing unilaterally from the strategy associated with an outcome

John Nash

• Nash Equilibrium –When no player can benefit by departing unilaterally from the strategy associated with an outcome

Chicken – Partial Conflict

• Each Player has 2 choices:1. Keep going2. Swerve out of the way

Chicken - continued

• Frank vs Mustang Sally• Frank’s ordinal Choices

4. Frank keeps going – Sally swerves – Sally is the chicken – Frank “wins”

3. Frank swerves - Sally Swerves – both chicken – both alive

2. Frank swerves – Sally keeps going – Frank is the chicken and Sally “wins”

1. Frank keeps going – Sally keeps going (disaster – both dead)

Chicken - continued

SallySwerve Don't

Frank Swerve (3,3) (2,4)Don't (4,2) (1,1)

Chicken - continued

• Nash Equilibrium at (4,2) and (2,4) • There is no dominate strategy in Chicken

making it a very dangerous game – can’t tell what you opponent will do

• “Best” outcome at (3,3) but no way to get there – until T.O.M.

Partial Conflict – important Points

• Dominant Strategy – the strategy that will give the highest average result

• (x,y) = x+y = value of the game• (1,1) = disaster• (3,3) = compromise• (4,x) = best for row player – won’t change• (1,x) = worst for row player – nash

equilibrium not possible

TOM – Theory of Moves

• John Neumann• Based on Game

Theory• Postulate – players

will think AHEAD• Elucidates on different

kinds of Power

Tom - Continued

• Oskar Morgenstern• Games in extended

form – sequential choice for players.

• Many games only depend on the final state reached

• Payoffs only if you stay

TOM

• Backward induction – reasoning process in which players working backward from the last possible move in a game, anticipate each other’s rational choices

• Survivor – payoff selected at each state as a result of backward induction

• Block(age) – when it is not rational to move beyond this point in a game

TOM - Outcomes

• Non-myopic Equilibria (NME) regardless of who moves first the same outcome is reached. The consequence of both players looking ahead and anticipating where the move – countermove process will end up

• Indeterminate – the result of the game depends on who moves first – the outcome is different depending on who goes first

Samson

• Great Warrior4. Samson Don’t tell – Delilah Don’t nag

(party all the time)3. Samson Tell – Delilah Nag2. Samson Tell – Delilah Don’t Nag1. Samson Don’t Tell – Delilah Nag

Delilah

• Paid for Info4. Delilah Don’t Nag - Samson Tells (no

work involved)3. Delilah Nag – Samson Tells (have to work

but get results)2. Delilah Don’t Nag - Samson Don’t Tell1. Delilah Nag – Sampson Don’t Tell

(disaster)

Samson vs Delilah

SamsonDon't Tell Tell

Delilah Don't Nag (2,4) (4,2)Nag (1,1) (3,3)

Samson vs Delilah

SamsonDon't Tell Tell

Delilah Don't Nag (2,4) (4,2)Nag (1,1) (3,3)

Delilah Starts (4,2)->(3,3)->(1,1)->(2,4)->(4,2)

Samson Starts (4,2)->(2,4)->(1,1)->(3,3)->(4,2)

Larger Games

• Truel – Duel with Three People• Each Player has a gun with One bullet –

everyone is a perfect shot – no communications between players

• Goals1. Survive2. Survive with as few opponents left as possible

Larger Games with TOM

• Modify Rules1. Take Turns firing – One Player at a time

“moves”• Now must “think ahead”• Two choices

1. Shot2. Don’t shot

Order Power

• A player has order power – if that player can force the other player to move first

• Only beneficial when the outcome is indeterminate

Samson vs Delilah

SamsonDon't Tell Tell

Delilah Don't Nag (2,4) (4,2)Nag (1,1) (3,3)

Delilah Starts (1,1)->(2,4)->(4,2)->(3,3)->(1,1)

Samson Starts (1,1)->(3,3)->(4,2)->(2,4)->(1,1)

Cycling

• TOM Rule changes5’ If at any state a player whoes turn it is to

move has received his best payoff (4) that player will not move!

– Moving Power – one player has the ability to force the other player to STOP! Then

6’ at some point in cycling the player must stop

Row vs Column

ColumnS1 S2

Row S1 (2,4) (4,1)S2 (1,2) (3,3)

Rows turn (2,4)->(1,2)->(3,3)->(4,1)->(2,4)

Column has moving power!! Tell Row has to stop!!