Lecture 10 Subgame-perfect Equilibrium 14.12 Game Theory Muhamet - PDF document

Lecture 10 Subgame-perfect Equilibrium 14.12 Game Theory Muhamet Yildiz 1

Road Map 1. Subgame-perfect Equilibrium 1. Motivation 2. What is a subgame? 3. Definition 4. Example 2. Applications 1. BankRun 2. Infinite-horizon Bargaining 2

l ~ A game 1 (2,6) T B L R R L (0,1) (3,2) (-1,3 ) (1,5) 3

Backward induction • Can be applied only in perfect information games of finite horizon. How can we extend this notion to infinite horizon games, or to games with imperfect information? 4

A subgame A subgame is part of a game that can be considered as a game itself. • It must have a unique starting point; • It must contain all the nodes that follow the starting node; • If a node is in a subgame, the entire information set that contains the node must be in the subgame. 5

~ ,-,-,- A game 1 A 2 a 1 a (1,-5) D d (4,4) (5 ,2) (3 ,3) 6

~ ~- And its subgames 1 a a 1 a 2 (1,-5) (1 ,-5) d d (3 ,3) (5 ,2) (3 ,3) 7

l ~ A game 1 (2,6) T B L R R L (0,1) (3,2) (-1,3 ) (1,5) 8

Definitions A substrategy is the restriction of a strategy to a subgame. A subgame-perfect Nash equilibrium is a Nash equilibrium whose sub strategy profile is a Nash equilibrium at each subgame. 9

l ~ Example 1 (2,6) T B L R L R (0,1) (3,2) (-1,3) (1,5) 10

A "Backward -Induction-like" method Take any subgame with no proper subgame Compute a Nash equilibrium for this subgame Assign the payoff of the Nash equilibrium to the starting node of the subgame Eliminate the subgame Yes The moves computed as a part of any (subgame) Nash equilibrium 11

In a finite, perfect-information game, ... ... the set of subgame-perfect equilibria is the set of strategy profiles that are computed via backward induction. 12

1~ A subgame-perfect equilibrium? x ___ (2,6) T B L R L R (0,1) (3,2) (-1 ,3) (1 ,5) 13

Bank Run • Alice and Bob each deposit D = $lM in a bank • Bank invests the money in a project, which pays 2r if liquidated at t= 1, 2R if waited to t=2, where R > D > r > D/2 • Either player has the option of withdrawing at either date, getting D if bank has the money • Ifthey do not withdraw, bank pays R to each 14

Bank Run A R > D > r > D!2 DW W W (r,r) DW DW W DW W (D,D) (D ,2R-D) (2R-D,D) (R,R) 15

Infinite-horizon Bargaining T = {l,2, ... , n-l,n, ... } If t is odd, 1ft is even Player 1 offers some - Player 2 offers some (xt ,Y t ), (xt,Yt), Player 2 Accept or - Player 1 Accept or Rejects Rejects the offer the offer If the offer is Accepted, - Tfthe offer is Accepted, the game ends yielding the game ends yielding t 8 (xt ,Yt), payoff (xt,Y t ), Otherwise, we proceed - Otherwise, we proceed to to date t+ 1. date t+ I. 16

n 00 t = 2n - 2k-l 1- 8 2k +! 1- 8 2n t 1 - n - W ) ) X - ---- 1+8 - 1+8 1+8 t - A SPE: At each t, • proposer offers 8/ (1 +8) to the other • and keeps 1/(1 +8) for himself; • responder accepts an offer iff • she gets at least 8/ (1 +8) . 17

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