Stolen Art• 50,000 paintings stolen from museums and private collections around the world (including 287 by Picasso, 43 by Van Gogh, 26 by Renoir, more than 100 by Rembrandt).

• "Stolen art works don't end up on the walls of criminal connoisseurs. They usually end up in storage.

• Mr Hill (former member of Metropolitan Police) : "I never pay a ransom. What I do is settle expenses and provide a finder's fee.“

• Tate Gallery paid 3 million pounds to someone who engineered the return of 2 works by Turner.

• If thieves could somehow be persuaded that no finder's fees would ever be paid, they might stop stealing works of art. "But do you know a way to persuade them that no collector, and no gallery, never mind an insurance company, will ever hand over a cent to get its treasured masterpieces returned?" he asks. "Because I don't."

•

Modeling with Game Theory

• What is the difference from previous “games” we have studied?– Decisions are made sequentially. (The thief decides

first.)– Decisions of one player is seen by another player

before his decision is made. (If the thief steals, the museum sees the art missing.)

• We can still use Game Theory to model the previous problem.

• But first, let us play a game.

Ultimatum game

• One of you is Player A and the other is Player B.

• You have £10 to divide between you.• Player A makes an offer how to divide it to

Player B.• Player B can accept or reject.• If Player B accepts, the payoff is as offered.

If Player B rejects, they both get zero.

Extensive Form Games (with perfect information)

• In both these games, decisions are made sequentially with all players knowing fully what the decisions were made prior.

• We can represent this problem by drawing a game tree. – Each node represents a player.

– Each branch represents a player’s possible decisions.

– At the end of the tree are the payoffs.

Graphing Extension Form games

A

B

a1

a2

b1

b1

b2

b2

(ua(a1,b2),ub(a1,b2))

(ua(a2,b2),ub(a2,b2))

(ua(a1,b1),ub(a1,b1))

(ua(a2,b1),ub(a2,b1))

B

B

A

Graphing Extension Form games

A

B

a1

a2

b1

b1

b2

b2

(ua(a1,b2),ub(a1,b2))

(ua(a2,b2),ub(a2,b2))

(ua(a1,b1),ub(a1,b1))

(ua(a2,b1),ub(a2,b1))

B

B

A

Players

Graphing Extension Form games

A

B

a1

a2

b1

b1

b2

b2

(ua(a1,b2),ub(a1,b2))

(ua(a2,b2),ub(a2,b2))

(ua(a1,b1),ub(a1,b1))

(ua(a2,b1),ub(a2,b1))

B

B

A

A’s decisions

Graphing Extension Form games

A

B

a1

a2

b1

b1

b2

b2

(ua(a1,b2),ub(a1,b2))

(ua(a2,b2),ub(a2,b2))

(ua(a1,b1),ub(a1,b1))

(ua(a2,b1),ub(a2,b1))

B

B

A

B’s decisions

Graphing Extension Form games

A

B

a1

a2

b1

b1

b2

b2

(ua(a1,b2),ub(a1,b2))

(ua(a2,b2),ub(a2,b2))

(ua(a1,b1),ub(a1,b1))

(ua(a2,b1),ub(a2,b1))

B

B

A

Payoffs

Stolen Art: Extension Form

A

B

Steal

Not Steal

Pay Fee

Don’t Pay

EnjoyThe art

(-20,-5)

(0,0)

(-10,5)

Museum

Thief

Museum

Stolen Art: Extension Form

A

B

Steal

Not Steal

Pay Fee

Don’t Pay

EnjoyThe art

(-20,-5)

(0,0)

(-10,5)

Museum

Thief

Museum

Players

Stolen Art: Extension Form

A

B

Steal

Not Steal

Pay Fee

Don’t Pay

EnjoyThe art

(-20,-5)

(0,0)

(-10,5)

Museum

Thief

Museum

Thief’s decisions

Stolen Art: Extension Form

A

B

Steal

Not Steal

Pay Fee

Don’t Pay

EnjoyThe art

(-20,-5)

(0,0)

(-10,5)

Museum

Thief

Museum

Museum’s decisions

A

B

Steal

Not Steal

Pay Fee

Don’t Pay

EnjoyThe art

(-20,-5)

(0,0)

(-10,5)

Museum

Thief

Museum

It costs the thief 5 to steal. (effort)

The fee=10.

The art is worth 20.

Ultimatum Game in Extensive Form

A

B

Offer(8,2)

Offer(5,5)

Accept

Accept

Reject

Reject

(0,0)

(0,0)

(8,2)

(5,5)

B

B

A

Subgame perfection• These games are called extensive form games with

perfect information.• A set of strategies is a subgame perfect equilibrium

if at every node (including those never reached), a player chooses his optimal strategy knowing that every node in the future the same will happen.

• A subgame perfect equilibrium is a Nash equilibrium at every subgame.

Backward Induction• To solve for the subgame perfect equilibria, one can start at the

end nodes.• Determine what are the decisions at the end.• Replace other earlier branches with the payoffs.• Repeat.

• What are the subgame perfect equilibria in the ultimatum game?

• If players are irrational at nodes not reached, can a player rationally choose a strategy that isn’t the subgame perfect strategy? – Coud this be a Nash equilibrium?

Gender in Ultimatum games(Solnick 2001)

• Male offers to males $4.73> to females $4.43

• Female offers to males $5.13> to females $4.31.

• Males accept $2.45 from other males<$2.82 from females.

• Females accept $3.39 from males<$4.15 from females.

Bargaining w/ shrinking pie

• Take the ultimatum game. Assume when there is a rejection the responder can make a counter-proposal.

• However, the pie shrinks after a rejection.

• What is the subgame perfect equilibrium when the pie shrinks from £10 to £6.

Bargaining w/ shrinking pie.

A

B

Offer(8,2)

Offer(5,5)

Accept

Accept

Reject

Reject

(2,4)

(5,5)

B

B

A

Size of £10 Size of £6

B

B

Offer(2,4)

Offer(3,3)

B

Accept

Accept

Reject

A

A

Reject (0,0)

(3,3)

(0,0)

(8,2)

Bargaining Discussion

• Do pies really shrink?

• From our analysis why do strikes happen?

Simple Model of Entry Deterrence

• A incumbent monopolist controls a market.

• A potential entrant is thinking of entering.

• The incumbent can expand capacity (or invest in a new technology) that is costly and not needed unless the entrant enters.

• The entrant is deterred by this and stays out.

Simple Model of Entry Deterrence

Incumbent

Entrant

Entrant

Enter

Exit

Enter

ExitExpandCapacity

Do nothing

(-10,5)

(0,20)

(0,15)

(10,10)

Patent Shelving• Other deterrents to entry: patent shelving – throw

the innovation in the closet.• Incumbant can invest in a patent. While the

technology may be better than the current that it uses, it may be too expensive to adapt existing product line. Why?

• Case studies– Hollywood: Top screen writers may rarely see a script

made into a movie.– Microsoft: Does it really take hundreds of programmers

to write word?

Patent Shelving(Incumbant, Entrant)

Incumbent

Entrant

Incumbent

Invest in patent

Do nothing

Invest in patent

Do nothing

(70,0)

(100,0)

(80,0)

(10,50)

Use

Shelve

Kidnapping Game• Criminal Kidnaps Teen.• Requests ransom and threatens to kill if not paid.• Parent decides whether or not to pay.• If parent does not pay, criminal decides whether or not

to kill hostage.• Start at end. Does the criminal kill if no ransom is paid?• What happens if there is no way to exchange ransom?• How can the hostage save himself if no ransom is paid?• What should a country do if its citizens are held for

ransom?

Kidnapping Game (parent, criminal, child)

Criminal

Don’t Kidnap

Kidnap Parent

Criminal

Exchange for Ransom

Don’t pay

Criminal

Kill

Release

(0,0,0)

(-3,10,-2)

(-10,-2,-20)

(-1,-5,-1)

Child

Identify

Refrain

(-1,-1,-3)

How reasonable is backward induction?

• May work in some simple games.

• Tic Tac Toe, yes, but how about Chess?– Too large of a tree.– Need to assign intermediate nodes.

• May not work well if players care about fairness.

Homework.

• Hold-up problem: – You get in a taxi. Should you bargain over the price at

the beginning or end of the trip? Why?

• Solve a three stage ultimatum game where in the first stage player A offers player B an offer for a $10 pie. If this offer is rejected, then the pie shrinks to $8 and player B makes the offer. If this offer is rejected, then the pie shrinks to $6 and player A makes the offer. If this final offer is rejected, then the payoffs are 0 to both players.