Extensive Game with Imperfect Information

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Extensive Game with Imperfect Information

• Part I Strategy and Nash equilibrium

2
Adding new features to extensive games

• A player does not know actions taken earlier
• non-observable actions taken by other players
• The player has imperfect recall--e.g. absent
  minded driver
• The type of a player is unknown to others
  (natures choice is non-observable to other
  players)

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Player 1s actions are non-observable to Player 2
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Imperfect recall Absent minded driver
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Natures choice is unknown to third party
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Extensive game with imperfect information and
chances

• Definition An extensive game ltN,H,P,fc,(Ti),(ui)
  gt consists of
• a set of players N
• a set of sequences H
• a function (the player function P) that assigns
  either a player or "chance" to every non-terminal
  history
• A function fc that associates with every history
  h for which P(h)c a probability distribution
  fc(.h) on A(h), where each such probability
  distribution is independent of every other such
  distribution.
• For each player i, Ti is an information partition
  and Ii (an element of Ti) is an information set
  of player i.
• For each i, a utility function ui.

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Strategies

• DEFINITION A (pure) strategy of player i in an
  extensive game is a function that assigns to each
  of i's information sets Ii an action in A(Ii)
  (the set of actions available to player i at the
  information set Ii).
• DEFINITION A mixed strategy of player i in an
  extensive game is a probability distribution over
  the set of player is pure strategies.

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Behavioral strategy

• DEFINITION. A behavioral strategy of player i in
  an extensive game is a function that assigns to
  each of i's information sets Ii a probability
  distribution over the actions in A(Ii), with the
  property that each probability distribution is
  independent of every other distribution.

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Mixed strategy and Behavioral strategy an example
(L,l) ½
(L,r) ½
(R,l) 0
(R,r) 0
ß1(f)(L)1 ß1(f)(R)0 ß1((L,A),(L,B))(l)1/2
ß1((L,A),(L,B))(r)1/2
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non-equivalence between behavioral and mixed
strategy amid imperfect recall

• Mixed strategy choosing LL with probability ½ and
  RR with ½.
• The outcome is the probability distribution
  (1/2,0,0,1/2) over the terminal histories. This
  outcome cannot be achieved by any behavioral
  strategy.

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Equivalence between behavioral and mixed strategy
amid perfect recall

• Proposition. For any mixed strategy of a player
  in a finite extensive game with perfect recall
  there is an outcome-equivalent behavioral
  strategy.

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Nash equilibrium

• DEFINITION The Nash equilibrium in mixed
  strategies is a profile s of mixed strategies so
  that for each player i,
• ui(O(s-i, si)) ui(O(s-i, si))
• for every si of player i.
• A Nash equilibrium in behavioral strategies is
  defined analogously.

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Part II Belief and Sequential Equilibrium
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A motivating example
Strategic game
L R
L 2,2 2,2
M 3,1 0,2
R 0,2 1,1
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The importance of off-equilibrium path beliefs

• (L,R) is a Nash equilibrium
• According to the profile, 2s information set
  being reached is a zero probability event. Hence,
  no restriction to 2s belief about which history
  he is in.
• 2s choosing R is optimal if he assigns
  probability of at least ½ to M L is optimal if
  he assigns probability of at least ½ to L.
• Bayes rule does not help to determine the belief

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belief

• From now on, we will restrict our attention to
  games with perfect recall.
• Thus a sensible equilibrium concept should
  consist of two components strategy profile and
  belief system.
• For extensive games with imperfect information,
  when a player has the turn to move in a
  non-singleton information set, his optimal action
  depends on the belief he has about which history
  he is actually in.
• DEFINITION. A belief system µ in an extensive
  game is a function that assigns to each
  information set a probability distribution over
  the histories in that information set.
  
• DEFINITION. An assessment in an extensive game is
  a pair (ß,µ) consisting of a profile of
  behavioral strategies and a belief system.

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Sequential rationality and consistency

• It is reasonable to require that
• Sequential rationality. Each player's strategy is
  optimal whenever she has to move, given her
  belief and the other players' strategies.
• Consistency of beliefs with strategies (CBS).
  Each player's belief is consistent with the
  strategy profile, i.e., Bayes rule should be
  used as long as it is applicable.

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Perfect Bayesian equilibrium

• Definition An assessment (ß,µ) is a perfect
  Bayesian equilibrium (PBE) (a.k.a. weak
  sequential equilibrium) if it satisfies both
  sequential rationality and CBS.
• Hence, no restrictions at all on beliefs at
  zero-probability information set
• In EGPI, the strategy profile in any PBE is a SPE
• The strategy profile in any PBE is a Nash
  equilibrium

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Sequential equilibrium

• Definition. An assessment (ß,µ) is consistent if
  there is a sequence ((ßn,µn))n1, of assessments
  that converge to (ß,µ) and has the properties
  that each ßn is completely mixed and each µn is
  derived from using Bayes rule.
• Remark Consistency implies CBS studied earlier
• Definition. An assessment is a sequential
  equilibrium of an extensive game if it is
  sequentially rational and consistent.
• Sequential equilibrium implies PBE
• Less easier to use than PBE (need to consider the
  sequence ((ßn,µn))n1, )

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Back to the motivating example

• The assessment (ß,µ) in which ß1L, ß2R and
  µ(M,R)(M)? for any ??(0,1) is consistent
• Assessment (ße,µe) with the following properties
• ße1 (1-e, ?e,(1-?)e)
• ße2 (e,1- e)
• µe (M,R)(M) ? for all e
• As e?0, (ße,µe)? (ß,µ)
• For ?1/2, this assessment is also sequentially
  rational.

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Two similar games
Game 1 has a sequential equilibrium in which both
1 and 2 play L
Game 2 does not support such an equilibrium
Game 1
Game 2
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Structural consistency

• Definition. The belief system in an extensive
  game is structurally consistent if for each
  information set I there is a strategic profile
  with the properties that I is reached with
  positive probability under and is derived from
  using Bayes rule.
• Remark Note that different strategy profiles may
  be needed to justify the beliefs at different
  information sets.
• Remark There is no straightforward relationship
  between consistency and structural consistency.
  (ß,µ) being consistent is neither sufficient nor
  necessary for µ to be structurally consistent.

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Signaling games

• A signaling game is an extensive game with the
  following simple form.
• Two players, a sender and a receiver.
• The sender knows the value of an uncertain
  parameter ? and then chooses an action m
  (message)
• The receiver observes the message (but not the
  value of ?) and takes an action a.
• Each players payoff depends upon the value of ?,
  the message m, and the action a taken by the
  receiver.

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Signaling games

• Two types
• Signals are (directly) costly
• Signals are directly not costly cheap talk game

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Spences education game

• Players worker (1) and firm (2)
• 1 has two types high ability ? H with
  probability p H and low ability ? L with
  probability p L .
• The two types of worker choose education level e
  H and e L (messages).
• The firm also choose a wage w equal to the
  expectation of the ability
• The workers payoff is w e/?

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Pooling equilibrium

• e H e L e ? ?L pH (?H - ?L)
• w pH?H pL?L
• Belief he who chooses a different e is thought
  with probability one as a low type
• Then no type will find it beneficial to deviate.
• Hence, a continuum of perfect Bayesian equilibria

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Separating equilibrium

• e L 0
• ?H (?H - ?L) e H ?L (?H - ?L)
• w H ?H and w L ?L
• Belief he who chooses a different e is thought
  with probability one as a low type
• Again, a continuum of perfect Bayesian equilibria
• Remark all these (pooling and separating)
  perfect Bayesian equilibria are sequential
  equilibria as well.

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When does signaling work?

• The signal is costly
• Single crossing condition holds (i.e., signal is
  more costly for the low-type than for the
  high-type)

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Refinement of sequential equilibrium

• There are too many sequential equilibria in the
  education game. Are some more appealing than
  others?
• Cho-Kreps intuitive criterion
• A refinement of sequential equilibriumnot every
  sequential equilibrium satisfies this criterion

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An example where a sequential equilibrium is
unreasonable

• Two sequential equilibria with outcomes (R,R)
  and (L,L), respectively
• (L,L) is supported by belief that, in case 2s
  information set is reached, with high probability
  1 chose M.
• If 2s information set is reached, 2 may think
  since M is strictly dominated by L, it is not
  rational for 1 to choose M and hence 1 must have
  chosen R.

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Beer or Quiche
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Why the second equilibrium is not reasonable?

• If player 1 is weak she should realize that the
  choice for B is worse for her than following the
  equilibrium, whatever the response of player 2.
• If player 1 is strong and if player 2 correctly
  concludes from player 1 choosing B that she is
  strong and hence chooses N, then player 1 is
  indeed better than she is in the equilibrium.
• Hence player 2s belief is unreasonable and the
  equilibrium is not appealing under scrutiny.

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Spences education game

• All the pooling equilibria are eliminated by the
  Cho-Kreps intuitive criterion.
• Let e satisfy w e/ ?L gt ?H e/ ?L and w
  e/ ?H gt ?H e/ ?L (such a value of e clearly
  exists.)
• If a high type work deviates and chooses e and is
  correctly viewed as a good type, then she is
  better off than under the pooling equilibrium
• If a low type work deviates and successfully
  convinces the firm that she is a high type, still
  she is worse off than under the pooling
  equilibrium.
• Hence, according to the intuitive criterion, the
  firms belief upon such a deviation should
  construe that the deviator is a high type rather
  than a low type.
• The pooling equilibrium break down!

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Spences education game

• Only one separating equilibrium survives the
  Cho-Kreps Intuitive criterion, namely e L 0
  and e H ?L (?H - ?L)
• Why a separating equilibrium is killed where e L
  0 and e H gt ?L (?H - ?L)?
• A high type worker after choosing an e slightly
  smaller will benefit from it if she is correctly
  construed as a high type.
• A low type worker cannot benefit from it however.
• Hence, this separating equilibrium does not
  survive Cho-Kreps intuitive criterion.