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
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
b1
a1
B
(ua(a1,b1),ub(a1,b1))
b2
(ua(a1,b2),ub(a1,b2))
AA
b1
a2
(ua(a2,b1),ub(a2,b1))
B
B
b2
(ua(a2,b2),ub(a2,b2))
Graphing Extension Form games
Players
b1
a1
B
(ua(a1,b1),ub(a1,b1))
b2
(ua(a1,b2),ub(a1,b2))
AA
b1
a2
(ua(a2,b1),ub(a2,b1))
B
B
b2
(ua(a2,b2),ub(a2,b2))
Graphing Extension Form games
A’s decisions
b1
a1
B
(ua(a1,b1),ub(a1,b1))
b2
(ua(a1,b2),ub(a1,b2))
AA
b1
a2
(ua(a2,b1),ub(a2,b1))
B
B
b2
(ua(a2,b2),ub(a2,b2))
Graphing Extension Form games
B’s decisions
b1
a1
B
(ua(a1,b1),ub(a1,b1))
b2
(ua(a1,b2),ub(a1,b2))
AA
b1
a2
(ua(a2,b1),ub(a2,b1))
B
B
b2
(ua(a2,b2),ub(a2,b2))
Graphing Extension Form games
Payoffs
b1
a1
B
(ua(a1,b1),ub(a1,b1))
b2
(ua(a1,b2),ub(a1,b2))
AA
b1
a2
(ua(a2,b1),ub(a2,b1))
B
B
b2
(ua(a2,b2),ub(a2,b2))
Stolen Art: Extension Form
Pay Fee
(-10,5)
Steal Museum Don’t Pay
(-20,-5)
A
Thief
Not Steal
B
Museum
Enjoy
The art
(0,0)
Stolen Art: Extension Form
Players
Pay Fee
(-10,5)
Steal Museum Don’t Pay
(-20,-5)
A
Thief
Not Steal
B
Museum
Enjoy
The art
(0,0)
Stolen Art: Extension Form
Thief’s decisions
Pay Fee
(-10,5)
Steal Museum Don’t Pay
(-20,-5)
A
Thief
Not Steal
B
Museum
Enjoy
The art
(0,0)
Stolen Art: Extension Form
Museum’s decisions
Pay Fee
(-10,5)
Steal Museum Don’t Pay
(-20,-5)
A
Thief
Not Steal
B
Museum
Enjoy
The art
(0,0)
It costs the thief 5 to steal.
(effort)
The fee=10.
The art is worth 20.
Pay Fee
(-10,5)
Steal Museum Don’t Pay
(-20,-5)
A
Thief
Not Steal
B
Museum
Enjoy
The art
(0,0)
Ultimatum Game in Extensive
Form
Accept
Offer
(8,2)
B
(8,2)
Reject
(0,0)
AA
Offer
(5,5)
Accept
(5,5)
B
B
Reject
(0,0)
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 only accept $2.45 or more from other
males< $3.39 from females.
• Females only accept $2.82 or more 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.
Size of £10
Size of £6
Accept (8,2)
Offer
(8,2)
B
Offer
(2,4)
Reject
Accept (2,4)
A
Reject
B
AA
Offer
(5,5)
Accept
Accept (3,3)
(5,5)
B
B
(0,0)
B
Offer A
(3,3)
Reject
B
Reject
(0,0)
Bargaining Discussion
• Do pies really shrink?
• Last year the Israeli universities went on
strike. This was quite costly for all of us.
• From our analysis why do strikes happen?
Frog and the Scorpion
• Frog and Scorpion were at the edge of a river
wanting to cross.
• The Scorpion said “I will climb on you back and
you can swim across.”
• Frog said “But what if you sting me.”
• Scorpion answered, “Why would I do that? Then
we both die.”
• What happened?
• Scorpion stung. The frog who cried “Now we are
both doomed! Why did you do that?”
• “Alas,” said the Scorpion, “it is my nature.”
Frog and the Scorpion
payoffs=(Frog,Scorpion)
Sting
Carry
(-10,5)
Scorpion
Refrain
(5,3)
Frog
Refuse
(0,0)
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
Enter
Expand
Capacity
Entrant
Exit
(0,15)
Incumbent
Do nothing
(-10,5)
Enter
(10,10)
Exit
(0,20)
Entrant
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)
Use
Invest in patent
Incumbent
(70,0)
Shelve
(80,0)
Incumbent
Do nothing
Invest in patent (10,50)
Entrant
Do
nothing
(100,0)
War Games
• Cold War Strategy: MAD, mutually assured
destruction. Both the US and USSR had enough
nuclear weapons to destroy each other.
• What does the game tree look like?
• The US put troops in Germany and said that if
West German were attacked it would mean
nuclear war.
• Would this have happened?
• Why didn’t USSR invade?
New War Games
• US and North Korea.
– North Korea is manufacturing a bomb.
– US is threatening an attack.
– US has troops in North Korea. Bush is
considering reducing the numbers. Why?
Bible Games:
(Adam & Eve, God)
Adam and Eve decide
whether or not to eat the
forbidden fruit from the
tree of knowledge.
If they eat, God knows
and decides upon a
punishment.
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)
Exchange for Ransom
(-3,10,-2)
Kidnap Parent
Don’t pay
Criminal
Criminal
(-10,-2,-20)
Kill
Criminal
Identify (-1,-5,-1)
Release
Child
Don’t Kidnap (0,0,0)
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.