Entry game ■ Two Nash equilibria (In, Accommodate )is subgame-perfect (Out, Fight )is not subgame-perfect because it does not induce a Nash equilibrium in the subgame beginning at Incumbent challenger Incumbent n Out A Incumbent A 1,2 0,0 Accommodate is the nash 0,0 equilibrium in this subgame
Entry game ◼ Two Nash equilibria ➢ ( In, Accommodate ) is subgame-perfect. ➢ ( Out, Fight ) is not subgame-perfect because it does not induce a Nash equilibrium in the subgame beginning at Incumbent. 26 Challenger In Out Incumbent A F 1, 2 2, 1 0, 0 Incumbent A F 2, 1 0, 0 Accommodate is the Nash equilibrium in this subgame
Find subgame perfect Nash equilibria backward induction Starting with those smallest subgames Then move backward until the root is reached Challenger n Out Incumbent A F 1,2 The first number is the payoff of the challenger 2,1 0,0 The second number is the payoff of the incumbent
Find subgame perfect Nash equilibria: backward induction ◼ Starting with those smallest subgames ◼ Then move backward until the root is reached 27 Challenger In Out Incumbent A F 1, 2 2, 1 0, 0 The first number is the payoff of the challenger. The second number is the payoff of the incumbent
Find subgame perfect Nash equilibria backward induction a Subgame perfect Nash equilibrium(DG, b) Player 1 plays D, and plays g if player 2 plays E Player 2 plays E if player 1 plays C Player 1 C Player 2 E Player 1 3,1
Find subgame perfect Nash equilibria: backward induction ◼ Subgame perfect Nash equilibrium (DG, E) ➢ Player 1 plays D, and plays G if player 2 plays E ➢ Player 2 plays E if player 1 plays C 28 Player 2 E F Player 1 G H 3, 1 1, 2 0, 0 Player 1 C D 2, 0
Existence of subgame-perfect nash equilibrium Every finite dynamic game of complete and perfect information has a subgame-perfect Nash equilibrium that can be found by backward induction
Existence of subgame-perfect Nash equilibrium ◼ Every finite dynamic game of complete and perfect information has a subgame-perfect Nash equilibrium that can be found by backward induction. 29
Sequential bargaining(2. 1. of Gibbons) Player 1 and 2 are bargaining over one dollar. The timing is as follows At the beginning of the first period player 1 proposes to take a share s, of the dollar, leaving 1-S, to player 2 Player 2 either accepts the offer or rejects the offer(in which case play continues to the second period) At the beginning of the second period, player 2 proposes that player 1 take a share s, of the dollar, leaving 1-s2 to player 2 Player 1 either accepts the offer or rejects the offer(in which case play continues to the third period) At the beginning of third period, player 1 receives a share s of the dollar, leaving 1-s for player 2, where 0<s<1 The players are impatient. They discount the payoff by a fact 5, Where 0<8<1
Sequential bargaining (2.1.D of Gibbons) ◼ Player 1 and 2 are bargaining over one dollar. The timing is as follows: ◼ At the beginning of the first period, player 1 proposes to take a share s1 of the dollar, leaving 1-s1 to player 2. ◼ Player 2 either accepts the offer or rejects the offer (in which case play continues to the second period) ◼ At the beginning of the second period, player 2 proposes that player 1 take a share s2 of the dollar, leaving 1-s2 to player 2. ◼ Player 1 either accepts the offer or rejects the offer (in which case play continues to the third period) ◼ At the beginning of third period, player 1 receives a share s of the dollar, leaving 1-s for player 2, where 0<s <1. ◼ The players are impatient. They discount the payoff by a fact , where 0< <1 30