# ECO326H1 Lecture Notes - Subgame, Ultimatum Game, Nash Equilibrium

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26 Jan 2013
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Topic 7 Extensive Forms Games: Bargaining
Nov 5th, 2012 (Chapter 7.1-7.2, 7.6-7.7)
Last Lecture:
Definition:
An Extensive Form Nash Equilibrium is a profile of extensive form strategies that are mutual best responses in the
original game.
Example 1:
Apple
Accommodate Fight


Subgame after history
- The same set of players
- The same set of actions
- Histories  such that  is a continuation of
Definition:
A Subgame Perfect Equilibrium (SPE) is a profile of strategies that induce a Nash Equilibrium / i.e. they are mutual
best responses in every Subgame.
- Initial game = Subgame after ;
- Every SPE is NE;
- NOT every NE is SPE
Refer to example 1:
- (In, Accommodate if In) is a SPE, also an NE in the subgame , also an NE in the subgame {In};
- However, (Out, Fight if Out) is not a NE in the subgame {In}, therefore it is a NE not a SPE.
Example 2 Ultimatum Game
- Two Players
- Player 1 makes an offer
- Player 2 either:
Accepts: Payoff 
Rejects: Payoff 
Subgame after history , player 2 best responses:
- Accepts if
- Rejects if
Two Nash Equilibriums:
(In, Accommodate if In)
(Out, Fight if Out)
𝑥𝑥
Player 1
𝑥
Player 2
Accepts Rejects
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There is a unique SPE in the ultimatum game:
- Player 1 offers ;
- Player 2 accepts after ANY offer
Example 3 Two Period Bargaining (A. Rubinstein)
Period 1: Player 1 makes an offer (Pie size = 1)
Player 2 Accepts or Rejects. If Player 2 rejects:
Period 2: Player 2 makes a counter offer (Pie size = )
Player 1 Accepts or Rejects. If Player 1 rejects: game ends.
Player 1
Player 2
Accepts Rejects

Player 2
Player 1
Accepts Rejects

1. Subgame 
- SPE is player 2 offers and player 1 accepts always.
2. Erase the solved subgame determined the unique payoff in that subgame, substitute that payoff into the game.
Player 1
Player 2
Accepts Rejects

3. In this game, SPE is:
- Player 1 offers ,
- Player 2 accepts if
rejects if
- Unique payoff: 
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