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 b1 (ua(a1,b1),ub(a1,b1)) a1 B b2 A (ua(a1,b2),ub(a1,b2)) A b1 (ua(a2,b1),ub(a2,b1)) a2 B B b2 (ua(a2,b2),ub(a2,b2)) Graphing Extension Form games Players b1 (ua(a1,b1),ub(a1,b1)) a1 B b2 A (ua(a1,b2),ub(a1,b2)) A b1 (ua(a2,b1),ub(a2,b1)) a2 B B b2 (ua(a2,b2),ub(a2,b2)) Graphing Extension Form games A’s decisions b1 (ua(a1,b1),ub(a1,b1)) a1 B b2 A (ua(a1,b2),ub(a1,b2)) A b1 (ua(a2,b1),ub(a2,b1)) a2 B B b2 (ua(a2,b2),ub(a2,b2)) Graphing Extension Form games B’s decisions b1 (ua(a1,b1),ub(a1,b1)) a1 B b2 A (ua(a1,b2),ub(a1,b2)) A b1 (ua(a2,b1),ub(a2,b1)) a2 B B b2 (ua(a2,b2),ub(a2,b2)) Graphing Extension Form games Payoffs b1 (ua(a1,b1),ub(a1,b1)) a1 B b2 A (ua(a1,b2),ub(a1,b2)) A b1 (ua(a2,b1),ub(a2,b1)) a2 B B b2 (ua(a2,b2),ub(a2,b2)) Stolen Art: Extension Form Pay Fee (-10,5) Steal Museum Don’t Pay A (-20,-5) Thief Not Steal B Museum Enjoy (0,0) The art Stolen Art: Extension Form Players Pay Fee (-10,5) Steal Museum Don’t Pay A (-20,-5) Thief Not Steal B Museum Enjoy (0,0) The art Stolen Art: Extension Form Thief’s decisions Pay Fee (-10,5) Steal Museum Don’t Pay A (-20,-5) Thief Not Steal B Museum Enjoy (0,0) The art Stolen Art: Extension Form Museum’s decisions Pay Fee (-10,5) Steal Museum Don’t Pay A (-20,-5) Thief Not Steal B Museum Enjoy (0,0) The art 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 A (-20,-5) Thief Not Steal B Museum Enjoy (0,0) The art Ultimatum Game in Extensive Form Accept (8,2) Offer (8,2) B Reject A (0,0) A Accept (5,5) Offer (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. 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? 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. Size of £10 Size of £6 Accept (8,2) Offer Accept (2,4) Offer (2,4) (8,2) B A Reject A B Reject (0,0) A Accept (5,5) Accept (3,3) Offer (5,5) B B B Offer A (3,3) Reject B Reject (0,0) Bargaining Discussion • Do pies really shrink? • The main government labour union in Israel went on strike in September shutting down most of the country. • From our analysis why do strikes happen? Hold-up problem • A Contractor is hired to construct a building. • Unexpected need emerges (new colour). • Contractor can charge cost of change or high price. • Client can agree or try to find outside help. • Client is held up. • Can one “solve” this with more explicit contracts? • Reputation effects. Hold-up problem: (contractor, client) Give In (1300,0) High price Client Search (0,-100) Contractor Outside Normal price (0,1300) Note: High price is 1300 more than normal (competitive). Searching costs 1400. Supplier hold-up problem • If one company is supplying another company a good used in production (such as a supplier of coal to an electric company), then the supplier can hold-up the buyer company. • This works if the buyer company decides to make an investment to adjust its products to make better use of the supplier’s product. • Once the investment is made, the supply can raise its prices. Supplier hold-up problem • The investment by the buyer costs him 500. • The gross gain to the buyer is 1500. • The net gain is 1500-500=1000. • The supplier can raise the price by 750 • This would reduce the net gain of the buyer by 750 to 250. • If the buyer switches to a new supplier, the buyer’s investment (of 500) is lost to him and the supplier loses 1000 worth of previous business with him. Holdup payoffs:(Buyer, Supplier) Keep Price (1000,0) Keep Make investment Supplier Supplier (250,750) Raise price Buyer Buyer New (-500,-1000) Supplier Don’t invest (keep Supplier) (0,0) Buyer’s investment costs 500 – only useful for that supplier. Saves buyer 1500 (net 1000). Supplier can raise price by 750. Supplier losing the Buyer’s business costs him 1000. Supplier hold-up problem • Now the investment is 1000 (instead of 500). • The gross gain to the buyer remains 1500. • The net gain changes to 1500-1000=500. • The supplier can still raise the price by 750 • This would reduce the net gain of the buyer by 750 to -250. (rather than +250) • If the buyer switches to a new supplier, the buyer’s investment (of 1000) is lost to him and the supplier loses 1000 worth of previous business with him. Holdup payoffs:(Buyer, Supplier) Keep Price (1000,0) (500,0) Keep Make investment Supplier Supplier (250,750) (250,750) Raise price Buyer Buyer New (-500,-1000) Supplier (-1000,-1000) Don’t invest (keep Supplier) (0,0) What if investment now costs 1000? Potential savings 500. What happens? Another reason for a government to allow Vertical Integration. 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 (-10,5) Scorpion Carry 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 (-10,5) Expand Entrant Exit Capacity (0,15) Incumbent Enter (10,10) Entrant Do nothing Exit (0,20) 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 – Lucent buys Chromatis for $4.8 billion never uses product. Lucent wants to prevent Nortel from buying it. – 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 (70,0) Incumbent Invest in patent Shelve (80,0) Incumbent Invest in patent (10,50) Entrant Do nothing Do (100,0) nothing 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 • Israel and Iran. – Israel is a nuclear power and Iran is close to becoming one. Will Israel attack Iran like they did Iraq? – Iran warns Israel that an attack will mean a harsh response. Is this credible? – Why would Israel not want a MAD situation? – Could it make sense for missile defence rather than offensive attack. – The Israeli spy satellite Ofek 6 malfunctioned and was destroyed on launch. This may make a window where Israel will be blind. How may this increase the chance of an attack? 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) (-10,-2,-20) Kidnap Parent Don’t pay Kill Criminal 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.