ECO326H1 Lecture Notes - Subgame, Ultimatum Game, Nash Equilibrium

52 views6 pages
26 Jan 2013
School
Department
Course
Professor
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
Unlock document

This preview shows pages 1-2 of the document.
Unlock all 6 pages and 3 million more documents.

Already have an account? Log in
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: 
Unlock document

This preview shows pages 1-2 of the document.
Unlock all 6 pages and 3 million more documents.

Already have an account? Log in

Get OneClass Notes+

Unlimited access to class notes and textbook notes.

YearlyBest Value
75% OFF
$8 USD/m
Monthly
$30 USD/m
You will be charged $96 USD upfront and auto renewed at the end of each cycle. You may cancel anytime under Payment Settings. For more information, see our Terms and Privacy.
Payments are encrypted using 256-bit SSL. Powered by Stripe.