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• Please, be advised that all overheads provided before the lectures are incomplete
• USE OVERHEADS PROVIDED BEFORE THE LECTURE ONLY AS A GUIDELINE
• Much of this module’s content is interactive and you will learn more if you do not see some examples/applications before the class
• Please, make sure that you take notes during lectures and seminars
Disclaimer
Extensive Form Games: Part 1
L5
Plan for Today
• Last time we talked about Normal Form Games • Today we will consider Extensive Form Games• We will introduce Extensive Form Games of Complete and Perfect
Information• We will talk about equilibrium concepts which could be
applied to these game
Extensive Form Games: Definition
Differences between Normal Form and Extensive Form Games
Normal Form Games Extensive Form Games• players in the game• moves available to each player• payoff received by each player
for each combination of moves that could be chosen by the players
• players in the game• when each player has to move• what each player can do at
each of his/her opportunities to move
• payoff received by each player for each combination of moves that could be chosen by the players
Extensive Form Games: Definition
Differences between Normal Form and Extensive Form Games
Normal Form Games Extensive Form Games• players in the game• moves available to each player• payoff received by each player
for each combination of moves that could be chosen by the players
• players in the game• when each player has to move• what each player can do at
each of his/her opportunities to move
• payoff received by each player for each combination of moves that could be chosen by the players
Extensive Form Games: Perfect Information
Worker 1
Worker 2
Worker 1
10, 0
5, 5
15, 0 0, 10
• Decision tree• Players do not move
simultaneously• When moving, each
player is aware of all the previous moves (perfect information)
• A (pure) strategy for player is a mapping from player ’s nodes to actions
Shirk Work
Shirk Work
Shirk Work
Extensive Form Games: Backward Induction
Worker 1
Worker 2
Worker 1
10, 0
5, 5
15, 0 0, 10
Shirk Work
Shirk Work
Shirk Work
• When we know what will happen at each of a “node’s children”, we can decide the best action for the player who is moving at that node
• Worker 1 will end game in the first stage by shirking –equilibrium
In-Class Poll
NOTE: not all extensive form games have
perfect information
While I appreciate your positive approach, try
harder next time as LoLhardly reflects your
thinking here
Great thoughts! Thanks to everyone who took
this seriously!
Backward Induction: A Limitation
Player 1
Player 2 Player 2
15, 10 10, 15 20, 5
Let us play the following game
0, 5
RL
L’ R’ L’ R’
In-class results: Player 1
In-class results: Player 2
Backward Induction: A Limitation
Player 1
Player 2 Player 2
15, 10 10, 15 20, 5
• If there are ties, then the decision rule applied to how they are broken determines what happens higher up in the tree
• Multiple equilibria…0, 5
RL
L’ R’ L’ R’
Extensive Form Games: Strategy
A strategy for a player is a complete set of actions – it specifies a feasible action for the player in every contingency in which the player might be called on to act
Mind the difference between action and strategy!!!
Extensive vs Normal FormPlayer 1 (P1)
Player 2(P2)
Player 2(P2)
15, 10 10, 15 20, 5 0, 5
RL 15, 10 15, 10 10, 15 10, 15
20, 5 0, 5 20, 5 0, 5LR
L’L’ L’R’ R’L’ R’R’
L’L’ = P2 moves Left if P1 moves LeftP2 moves Left if P1 moves RightL’R’ = P2 moves Left if P1 moves Left and P2 moves Right if P1 moves RightR’L’ = P2 moves Right if P1 moves Left and P2 moves Left if P1 moves RightR’R’ = 2 moves Right if 1 moves Left and 2 moves Right if 1 moves Right
Player 2 has two actions but four strategies, because there are two different contingencies: (i) observing P1 play Left and (ii) observing P1 play Right
L’ L’R’ R’
Extensive vs Normal Form
Player 1
Player 2 Player 2
15, 10 10, 15 20, 5 0, 5
RL
• Nash equilibria of this normal form game include
• (R, L’L’)• (R, R’L’)• (L, R’R’) • MANY mixed-strategy
equilibria
15, 10 15, 10 10, 15 10, 15
20, 5 0, 5 20, 5 0, 5LR
L’L’ L’R’ R’L’ R’R’
R’L’ R’L’
Equilibria in Extensive Form Games
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
10, 20 10, 20 25, 15 25, 15
10, 20 10, 20 25, 15 25, 15
15, 10 5, 0 15, 10 5, 0
15, 10 0, 5 15, 10 0, 5
LL LR RL RRLL
RLRR
LR
Equilibria in Extensive Form Games
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
10, 20 10, 20 25, 15 25, 15
10, 20 10, 20 25, 15 25, 15
15, 10 5, 0 15, 10 5, 0
15, 10 0, 5 15, 10 0, 5
LL LR RL RRLL
RLRR
LR
All scenarios when Player 1 moves Left
Equilibria in Extensive Form Games
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
10, 20 10, 20 25, 15 25, 15
10, 20 10, 20 25, 15 25, 15
15, 10 5, 0 15, 10 5, 0
15, 10 0, 5 15, 10 0, 5
LL LR RL RRLL
RLRR
LR
All scenarios when Player 1 moves Right
Equilibria in Extensive Form Games
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
10, 20 10, 20 25, 15 25, 15
10, 20 10, 20 25, 15 25, 15
15, 10 5, 0 15, 10 5, 0
15, 10 0, 5 15, 10 0, 5
LL LR RL RRLL
RLRR
• Solving this game by backward induction we get (RL,LL)
LR
Equilibria in Extensive Form Games
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
10, 20 10, 20 25, 15 25, 15
10, 20 10, 20 25, 15 25, 15
15, 10 5, 0 15, 10 5, 0
15, 10 0, 5 15, 10 0, 5
LL LR RL RRLL
RLRR
• Pure-strategy Nash equilibria of this game are (LL, LR), (LR, LR), (RL, LL), (RR, LL)
• But the only backward induction solution is (RL, LL)
• WHY?
LR
Subgame Perfect Equilibrium
Formally, a subgame in an extensive-form game
(i) Begins with a decision node that is a singleton information set (but is not the game’s first decision node
(ii) Includes all the decision and terminal nodes following in the game tree (but no nodes that do not follow ), and
(iii) Does not cut any information sets (i.e., if a decision node ’ follows in the game tree, then all other nodes in he information set containing ′ must also follow , and so must be included in the subgame
Subgame Perfect Equilibrium
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
Each node in a (perfect-information) game tree, together with the remainder of the game after that node is reached, is called a subgame
A strategy profile is a subgame perfect equilibrium if it is an equilibrium for every subgame
Subgame Perfect Equilibrium
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
(RR, LL) and (LR, LR) are not subgame perfect equilibria because (*R, **) is not an equilibrium
15, 10 5, 0
15, 10 0, 5
*L *R*L*R
5, 00, 5
***L*R
Subgame Perfect Equilibrium
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
(LL, LR) is not subgame perfect because (*L, *R) is not an equilibrium
15, 10 5, 0
15, 10 0, 5
*L *R*L*R
5, 00, 5
***L*R
Subgame Perfect Equilibrium
Player 1
Player 2 Player 2
Player 110, 20 25, 15 15, 10
5, 0 0, 5
*R is not a credible threat
15, 10 5, 0
15, 10 0, 5
*L *R*L*R
5, 00, 5
***L*R
Lecture 5: Takeaways of the Day
Today we learned
• We have introduced Extensive Form Games
• We have seen how a game can be represented in a Normal Form and in an Extensive Form
• So far, we considered games of Complete and Perfect information
• We introduced the concept of subgame perfect equilibrium
Next time
• We will talk about Extensive Form Games for games of Incomplete Information
• We will talk about Bayesian Nash Equilibrium
• We will then switch to Iterated Games
• Please, complete the task at http://ncase.me/trust/ fornext week
“Must-Read” List
• Osborne, M.J. and A. Rubinstein (1994) A Course in Game Theory, MIT Press. Chapter 6.
• Osborne, M.J. (2004), An Introduction to Game Theory, Oxford University Press Chapters 5-7. [any edition of this book is suitable]
If you want to know more…
• Kreps, D. M. (1990) A Course in Microeconomic Theory, Princeton, NJ: Princeton University Press, Chapter 11
• Kreps, D. M. and R. Wilson (1982) “Sequential Equilibrium,” Econometrica, 50, pp.863-894
SEE YOU NEXT TIME!
