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11/12/11 Game theory Lecture 5 Three firms •  Backward induc;on equilibrium: (OUT; OUT; AAF) •  There is however a lot more Nash equilibria, e.g. (IN; IN; AFA) – non‐credible threat by Tesco. 1 11/12/11 Standard form BILLA AAA AAF AFA FAA AFF FAF FFA FFF IN Albert IN OUT ‐1 5 6 ‐1 5 6 ‐1 5 6 ‐1 5 6 ‐1 5 6 ‐1 5 6 2 0 1 2 0 1 ‐1 5 6 ‐1 5 6 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 AAA AAF AFA FAA AFF FAF FFA FFF OUT Albert IN OUT ‐2 2 0 5 4 4 0 ‐1 7 5 4 4 ‐2 2 0 3 1 2 ‐2 2 0 5 4 4 0 ‐1 7 3 1 2 0 ‐1 7 5 4 4 ‐2 2 0 3 1 2 0 ‐1 7 3 1 2 Dr Strangelove •  A U.S. Air force commander orders thirty B‐52’s to launch a nuclear aTack on Soviet Union. •  He shuts off all communica;ons with the planes and with the base •  U.S. president invites the Russian ambassador to the war room and explains the situa;on •  They decide to call the Russian president Dimitri 2 11/12/11 Dr Strangelove •  What is the outcome if U.S. does not know about doomsday device? •  What is the outcome if they know? •  Commitment must be observable •  What if Soviet Union has an op;on to deac;vate doomsday device? •  Commitment must be irreversible Thomas Schelling: The power to constrain an adversary depends upon the power to bind oneself Credible commitments (threats) •  In non‐strategic situa;ons more op;ons never makes you worse off •  In strategic situa;ons some;mes it is not the case •  You can change your opponent’s behavior by simply elimina;ng some of your op;ons 3 11/12/11 Examples •  1066: William the Conqueror ordered his soldiers to burn the ships aaer landing to make retreat impossible •  1519: Hernan Cortes sunk his baTleships aaer landing in Mexico for the same reason •  Sun‐tzu w The art of war, 400 BC: At the cri9cal moment, the leader of an army acts like one who has climbed up a height, and then kicks away the ladder behind him Treasure •  5 pirates Barbarossa, Blackbeard, Angelica Teach, Teague and Jack Sparrow find a big box containing 200 golden coins. •  Pirates obey a strict hierarchy based on their strength and wit. The order of importance is the one given above. •  First Barbarossa proposes a split among them 5. They all vote democra;cly. •  If majority accepts the proposal passes •  If majority rejects, they kill Barbarossa throwing him in the sea and then Blackbeard proposes a split 4 11/12/11 Treasure Sequen;al game – we will find an equilibrium using backward induc;on: 4 step: there are only two pirates lea (Teague and Jack Sparrow), Teague proposal is to take all for himself – a draw, the proposal passes 3 step: Sparrow, Teague and Teach are the ones lea. Teach proposes: she cannot give more than he had to Teague but can give more to Sparrow 5 11/12/11 Treasure 2 step: Sparrow, Teague, Teach and Blackbeard are lea. Blackbeard proposes: he has to give more compared to the 3 step to at least one of the rest. So he proposes something to Teague and zero to the rest. A draw and the proposal passes. Treasure 1 step: Now that everybody is s;ll alive, Barbarossa has to give more to at least 2 of the rest. He proposes something to Teach and Sparrow and his proposal passes. 6 11/12/11 Extensive form games (tree) vs strategic form games (table) •  Strategic form game has three components: –  A set of players –  For each player: •  Set of ac;ons •  Payoff func;on for every profile of ac;ons •  Extensive form game contains more informa;on: –  The order of moves –  Ac;ons available at different stages of the game –  Informa;on available during the game A tree •  Nodes: –  Decision nodes and nature nodes –  A root and leaves • 
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Edges (branches) Player labels Ac;on labels Payoffs Informa;on sets (to be seen in a second) 7 11/12/11 Strategies in extensive form games •  Pure strategy of a player is a plan of ac;ons in every decision node of this player (Tesco example) •  In backward induc;on equilibrium every player plays op;mally in every decision node (plays a sequen;ally ra;onal strategy) •  Equilibrium is a profile of strategies that ... •  Equilibrium outcome of a game is what players get in equilibrium Another example •  Kodak is contempla;ng entering the instant photography market and Polaroid can either fight the entry or accommodate 8 11/12/11 •  Set of Nash equilibria = {(In, A), (Out,F)} •  (Out, F) is sustained by an incredible threat by Polaroid •  Backward induc;on equilibrium eliminates equilibria based upon incredible threats •  Nash equilibrium requires ra;onality •  Backward inducPon requires sequenPal raPonality –  Players must play op;mally at every point in the game Extensive form games with imperfect informa;on •  We have seen extensive form games with perfect informa;on –  Every player observes the previous moves made by all the players •  What happens if some of the previous moves are not observed? •  We cannot apply backward induc;on algorithm any more •  Consider the following game between Kodak and Polaroid –  Kodak doesn’t know whether Polaroid will fight or accommodate (the doTed line is an informa;on set – nodes that cannot be dis;nguished from one another by the player) –  We cannot determine the op;mal ac;on for Kodak at that informa;on set 9 11/12/11 Subgame Perfect Nash Equilibrium •  A subgame is a part of game tree, such that: –  It starts with a single decision node –  It contains all subsequent decision nodes (successors) –  If it contains a decision node in some informa;on set, then it must contain all decision nodes in this informa;on set subgame Not a subgame Subgame Perfect Nash Equilibrium •  Player’s pure strategy is a plan of ac;on in each informa;on set of this player •  A profile of strategies consPtutes SPNE if it induces a Nash equilibrium in each subgame 10 11/12/11 Bank run Bank run •  Two investors have each deposited D with a bank. •  The bank has invested these deposits in a long‐term project. •  If the bank is forced to liquidate its investment before the project matures, a total of 2r can be recovered, where D>r>D/2 •  If the bank allows the investment to reach maturity, the project will pay out a total of 2R, where R>D •  There are two dates at which investors can make withdrawals from the bank –  Date 1 is before the bank’s investment matures –  Date 2 is aaer •  There is no discoun;ng 11 11/12/11 A nonstandard way to represent the game • 
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If both investors make withdrawals at date 1 then each receives r If only one investor makes a withdrawal at date 1 then the investor receives D, the other receives 2r‐D, a and the game ends Finally, if neither investor makes a withdrawal at date 1 then the project matures and the investors make withdrawal decisions at date 2 withdraw withdraw r, r don't 2r‐D, D don't D, 2r‐D next stage R>D>r>D/2 • 
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If both investors make withdrawals at date 2 then each receives R and the game ends If only one investor makes a withdrawal at date 2 then that investor gets 2R‐D, the other receives D, and the game ends. Finally, if neither investor makes a withdrawal at date 2 then the bank returns R to each investor and the game ends. withdraw don't withdraw don't R, R 2R‐D, D D, 2R‐D R, R Solving backwards withdraw don't withdraw don't R, R 2R‐D, D D, 2R‐D R, R R>D>r>D/2 •  Since R>D, “withdraw” strictly dominates “don’t”, hence (withdraw, withdraw) is a Nash equilibrium in strictly dominant strategies in this game) •  Since there is no discoun;ng we just put (R,R) in the game of first stage: withdraw withdraw r, r don't 2r‐D, D don't D, 2r‐D R, R •  Two Nash equilibria of this game: –  (don’t, don’t) leading to payoff (R,R) –  (withdraw, withdraw) leading to payoff (r,r) [bank run] 12 11/12/11 Backward induc;on Eq. vs SPNE •  Backward inducPon equilibrium in extensive games of perfect informa;on •  SPNE more general – also for games with imperfect informa;on •  From now on I will always say SPNE and will not make the dis;nc;on any more Ul;matum game •  Player 1 proposes a split of $100 between him and another player (one unit is assumed to be $10) •  Player 2 accepts the proposal (then they get what they proposed/accepted) or rejects (they both get nothing) •  What are players’ strategies? How many of them there are? •  Find SPNE in this game. How many of them there are? 13 11/12/11 Ul;matum game The only SPNE: •  (0, AAAAAAAAAAA) •  (1,OAAAAAAAAAA) Ul;matum game •  How about Nash Equilibria? •  Is (5,OOOOOAAAAAA) Nash equilibrium? •  Can you give an example of a threat in this game? •  Is this an empty threat or not? •  How many Nash equilibria there are? 14 11/12/11 Cen9pede game What is a SPNE of this game? Cri;que of SPNE and backward induc;on: what kind of ra;onality it requires? Sta;c vs dynamic games •  Sta;c games –  Players move simultaneously and –  Choose only one ac;on –  Nash equilibrium – ac;on profile (one ac;on – best response – for each player) •  Dynamic games –  Players move sequen;ally –  They choose one ac;on in each of their informa;on set –  SPNE – strategy profile (one strategy – best response – for each player) 15