M9302 Mathematical Models in Economics
Instructor: Georgi Burlakov
3.1.Dynamic Games of Complete but Imperfect Information
Lecture 3
24.03.2011
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How to solve the GT problem?
oPerfect Bayesian Equilibrium (PBNE)
in static games of complete information
in dynamic games of complete information
in static games of incomplete information
in dynamic games of incomplete information
oStrategic Dominance
oNash Equilibrium (NE)
Solution Concepts:
o  Subgame-Perfect Nash Equilibrium (SPNE)
o Bayesian Nash Equilibrium (BNE)
o  Backwards Induction

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Revision
oWhat is information set?
oAn information set for a player is a collection of decision nodes satisfying:
othe player has the move at every node in the information set, and
owhen the play of the game reaches a node in the information set, the player with the move does not
know which node in the information set is reached
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Revision
oWhat does the extensive form representation of a game specifies?
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1.Who are the PLAYERS.
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o2.1. When each player has the MOVE.
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o2.2. What each player KNOWS when she is on a move.
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o2.3. What ACTIONS each player can take.
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o3. What is the PAYOFF received by each player.
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Dynamic games of complete but imperfect information
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oInformally, the games of this class could be described as follows:
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oFirst, Players 1 and 2 simultaneously choose actions a1 and a2 from feasible sets A1 and A2,
respectively
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oSecond, players 3 and 4 observe the outcome of the first stage, (a1, a2), and then simultaneously
choose actions a3 and a4 from feasible sets A3 and A4, respectively.
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oFinally, based on the resulting combination of actions chosen in total, each player receives a
given payoff ui(a1,a2,a3,a4) for i=1,2,3,4

Dynamic Games of Complete and Imperfect Information
oStandard assumptions:
oPlayers move at different, sequential moments
o– it is DYNAMIC
oThe players’ payoff functions are common knowledge
o– it is COMPLETE INFORMATION
oAt each stage of the game players move simultaneously
o– it is IMPERFECT INFORMATION
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oThe aim of the third lecture is to show:
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o1. What is the difference between perfect and imperfect information?
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o2. How to solve games of complete but imperfect information?
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Dynamic games of complete but imperfect information

Perfect vs. Imperfect Information
oWhat is perfect information?
owhen at each stage the player with the move knows the full history of the game thus far
oWhen each information set is a singleton
oThen, what is imperfect information?
oWhen there is at least one non-singleton information set
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How to solve dynamic games of imperfect information?
oIn a game of complete and perfect information BI eliminates noncredible threads. Why?
oBecause each decision node represents a contingency in which a player might be called on to act.
oThe process of working backwards thus amounts to forcing each player to consider carrying out each
threat

How to solve dynamic games of imperfect information?
oIn a game of imperfect information BI does not work so simply. Why?
oBecause working backwards would eventually lead us to a decision node in a non-singleton
information set
oThen the player does not know whether or not that decision node is reached
oThe player is forced to consider what it would eventually do if a node is really reached not in a
contingency in which she is called on to act

How to solve dynamic games of imperfect information?
oHow to deal with the problem of nonsingleton information sets in BI?
oWork backwards until a nonsingleton information set is encountered, then:
oSkip over it and proceed the tree until a singleton information set is found and solve for the
subgame emanating from it - SGPNE
oForce the player with the move at the information set to consider what she would do if that
information set was reached – Bayesian NE

Dynamic games of Complete but Imperfect Information – key terms
oSubgame – a piece of a game that remains to be played beginning at any point at which the complete
history of the game thus far is common knowledge among the players, i.e.:
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obegins at a singleton information set
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oincludes all the decision and terminal nodes following but not preceding the starting singleton
decision node
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odoes not cut any (non-singleton) information sets.

Dynamic games of Complete but Imperfect Information – key terms
oStrategy – a complete plan of action – it specifies a feasible action which the player will take
in each stage, for every possible history of play through the previous stage.

Dynamic games of Complete but Imperfect Information – SGPNE
o(Selten 1965) Subgame-perfect Nash Equilibrium (SGPNE)– a Nash equilibrium is subgame perfect if
the players’ strategies constitute a Nash equilibrium in every subgame.

oBI fails to eliminate noncredible threads in the games of imperfect information because of the
non-singleton information sets.
oTherefore a stronger solution concept called subgame-perfect N.E. is applied.
oSGPNE includes not only the best response to the unique action played in the first stage but full
plan of action (strategy) how it would be best to respond to any possible action in the unobserved
part of the game (subgame).
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Dynamic games of Complete but Imperfect Information – Summary