1. Extensive Games with Imperfect Information
Chapter 10
Extensive Games with
Imperfect Information

2. Extensive Games with Imperfect Info
In this section, we model situations in which players move sequentially but may or may not be informed about previous player’s actions.
With perfect information, we needed to specify players, terminal histories, when players had the move, preferences and payoffs.
With imperfect information, we need to add to that list: a specification about each player’s information about the history at every point at which she moves.

3. Extensive Games with Imperfect Info
Definition: an information set for a player is a collection of decision nodes satisfying:
The player has the move at every node in the information set, and
When 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 has (or has not) been reached

4. Extensive Games with Imperfect Info
Example 314.2: Battle of the Sexes I U2 CP II II U2 CP U2 CP
2,1
0,0
1,2
0,0
Nash equilibria?
Subgame Perfect Nash Equilibria ?

5. Extensive Games with Imperfect Info
Other Examples
315.1: Card Game

6. Extensive Games with Imperfect Info
Definition: Strategy in an Extensive game of Imperfect information: 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 ) (ie, the set of actions available to player i at information set Ii ).
Definition: Mixed Strategy in an Extensive game of Imperfect information: A mixed strategy of a player in an extensive game is a probability distribution over the player’s pure strategies.

7. Extensive Games with Imperfect Info
Definition: Nash Equilibrium in an Extensive game of Imperfect information: The mixed strategy profile, a*, in an extensive game is a (mixed strategy) Nash equilibrium if, for each player i and every mixed strategy ai of player i, player i’s expected payoff to a* is at least as large as her expected payoff to (ai,a*-i) according to a payoff function whose expected value represent player i’s preferences over lotteries.

8. Extensive Games with Imperfect Info
Example (Commitment and Observability) II X Y X
(3,2)
(1,1) I
(4,3)
(2,4) Y
NE of Simultaneous game: (Y,Y)
SPNE outcome of extensive game (I moves first, II second) of perfect information: (X,X)

9. Extensive Games with Imperfect Info
Example (Commitment and Observability)
-- Imperfect Information Game:
3,2
1,1
4,3
2,4 Y X X Y II
1-e X e X X I Y
Chance
Chance Y
1-e Y e II X Y X Y
3,2
1,1
4,3
2,4
Nash Equilibrium for e < 1/4 ?

10. Extensive Games with Imperfect Info
317.1 / / 323.1: Entry Game
Motivates a need for “beliefs” of players.

11. Extensive Games with Imperfect Info
In simultaneous (strategic) games of perfect information, our equilibrium concept was the Nash equilibrium (NE). When we moved to sequential games, we introduced a slightly stronger equilibrium called subgame perfect Nash equilibium (SPNE), primarily to rule out non-credible threats. In static games of imperfect information, we needed a further “refinement” which we called Bayesian Nash Equilibrium (BNE). Finally, in extensive games of imperfect information, we make a further refinement and introduce Perfect Bayesian Nash Equilibrium, or (PBE) for short.

12. Extensive Games with Imperfect Info
A note on terminology:
Perfect Bayesian Equilibrium or PBE
Weak Sequential Equilibrium or WSE (Osborne’s terminology)
PBE = WSE

13. Perfect Bayesian Equilibrium*
Gibbons Game, Page 176 I R
(1,3) L M II II L’ L’ R’ R’
(2,1)
(0,2)
(0,0)
(0,1)

14. Perfect Bayesian Equilibrium*
Gibbons Game, Page 176 – Strategic Form
So 2 NE in pure strategies: (L,L’) and (R,R’)
Subgame Perfect NE? Since the game has
no subgames besides the whole game,
NE=SPNE.
But player II playing R’ is based on a non-credible threat since L’ is always optimal if player II gets to move.
So we rule out (R,R’) with the following 2 refinements. II L’ R’ L
(2,1)
(0,0) I M
(0,2)
(0,1) R
(1,3)
(1,3)

15. Perfect Bayesian Equilibrium*
Requirement 1: At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game. For a nonsingleton information set, a belief is a probability distribution over the nodes in the information set; for a singleton information set, the player’s belief puts probability one on the single decision node.

16. Perfect Bayesian Equilibrium*
Requirement 2: Given their beliefs, the players’ strategies must be sequentially rational. That is, at each information set the action taken by the player with the move (and the player’s subsequent strategy) must be optimal given the player’s belief at that information set and the other players’ subsequent strategies (where a “subsequent strategy” is a complete plan of action covering every contingency that might arise after the given information set has been reached)

17. Perfect Bayesian Equilibrium*
Gibbons Game, Page 176 I R
(1,3) L M II II
[1-p]
[p] L’ L’ R’ R’
(2,1)
(0,2)
(0,0)
(0,1)
(p,1-p) are the beliefs of player II. p is the probability that player II puts on the history that player I has played L and (1-p) is the probability that player II puts on the history that player I has played M.

18. Perfect Bayesian Equilibrium*
Gibbons Game, Page 176 I R
(1,3) L M II II
[1-p]
[p] L’ L’ R’ R’
(2,1)
(0,2)
(0,0)
(0,1)
So, given player II’s beliefs, the expected payoff to player II from playing R’ is 0*p + 1*(1-p) = 1-p. The expected payoff to player II from playing L’ is 1*p + 2*(1-p) = 2-p.
Since 2-p > 1-p, for ALL p, then R’ can never be played in equilibrium.
So (R,R’) does not satisfy the requirement #2.

19. Perfect Bayesian Equilibrium*
Requirements 1 and 2 insist that the players have beliefs and act optimally given these beliefs, but not that these beliefs be reasonable. We need further requirements.
Definition: for a given equilibrium in a given extensive game, an information set is on the equilibrium path if it will be reached with positive probability if the game is played according to the equilibrium strategies, and is off the equilibrium path if it is certain not to be reached if the game is played according to the equilibrium strategies.

20. Perfect Bayesian Equilibrium*
Requirement 3: At information sets on the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies.
Requirement 4 :At information sets off the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies where possible.

21. Perfect Bayesian Equilibrium*
Gibbons Game, Page 176 I R
(1,3) L M II II
[1-p]
[p] L’ L’ R’ R’
(2,1)
(0,2)
(0,0)
(0,1)
So in the SPNE where players play (L,L’), player 2’s belief must be p = 1. “Given player 1’s equilibrium strategy (L), player 2 knows which node in the information set has been reached.”
As a second example, suppose in a mixed strategy equilibrium, player I played L with probability q1, M with probability q2, and R with probability 1-q1-q2. Then, this would force player II to set p = q1/(q1+q2), by Bayes rule.

22. Perfect Bayesian Equilibrium*
Definition: A Perfect Bayesian Equilibrium consists of strategies and beliefs satisfying 1 through 4.
Crucial new features of a PBE: beliefs are elevated to the level of importance of strategies in our definition of an equilibrium. An equilibrium consists of strategies and beliefs for each player at each information set at which the player has the move.
Players must have reasonable beliefs both on and off the equilibrium path.
* taken from Robert Gibbons, “Game Theory for Applied Economists”, Princeton University Press

23. Signaling Games
Signaling Games: a dynamic game of imperfect information involving a sender and a receiver, S and R.
Nature draws a type for the S from a set of possible types. S observe his type and chooses a message to send to R. R observes the message (not the type) and then choose an action.
Payoffs are then realized.

24. Signaling Games
Sender may be a worker who signals to a potential employer (Reciever) with an education choice based on his type, possibly his productivity level. See Spence (1973): Job Market Signaling.

25. Signaling Games Consider the following signaling game: Sender t1z
Receiver
Receiver
Nature
1-z a1 a1 m1 m2
Sender t2 a2 a2
If both types of senders send the same message in an equilibrium, we call this equilibrium “Pooling”
If a sender sends a different message depending on his type, we called this equilibrium “Separating.”

26. Signaling Games
Denote (p,1-p) and (q,1-q) denote R’s beliefs at each of his (two) information sets. a1 a1
Sender t1 m1 m2
[p]
[q] a2 a2 z
Receiver
Receiver
Nature
1-z a1 a1
[1-p]
[1-q] m1 m2
Sender t2 a2 a2
This game has four possible (pure strategy) equilibria: 1) Pooling on m1, 2) pooling on m2, 3) Separating with t1 playing m1 and t2 playing m2, 4) Separating with t1 playing m2 and t2 playing m1.

27. Signaling Games Lets add payoffs and solve an example. Sender t1
2,1
1,3 u u
Sender t1 L R
[p]
[q] d d
0,0
4,0
0.5
Receiver
Receiver
Nature
1,0
2,4
0.5 u u
[1-p]
[1-q] L R
Sender t2 d d
1,2
0,1
Solve for all PBE of this game ...

28. Signaling Games Need to consider the following candidate equilibria:
1) Pooling on (L,L); 2) Pooling on (R,R);
3) Separating on (L,R); and 4) Separating on (R,L).
2,1
1,3 u u
Sender t1 L R
[p]
[q] d d
0,0
4,0
0.5
Receiver
Receiver
Nature
1,0
2,4
0.5 u u
[1-p]
[1-q]
Sender t2 L R d d
1,2
0,1
Pooling on (L,L)  p=1/2
Pooling on (R,R)  q=1/2
Separating on (L,R)  p=1, q=0
Separating on (R,L)  q=1, p=0

29. The Market for Lemons Akerlof models used car market.
Buyers value good cars at bG and lemons at bL.
Sellers value good cars at sG and lemons at sL.
Sellers know the true quality of the car and offer a price, P.
Buyers only know a certain proportion of cars, q, are good and a proportion, 1-q, are lemons.

30. The Market for Lemons Safe assumption: bG>bL and sG > sL.
Assume that under perfect information, trade would take place for all cars:
bL > sL
bG > sG
Under imperfect information, buyers buy if E[V] = q*bG+(1-q)*bL  P
In a separating equilibrium, owners of good cars do not sell their cars and owners of lemons sell their cars. This happens if:
bL = P < sG
this is easily satisfied (see homework 5)
How about a pooling equilibrium where all cars are sold at the price P.
Buyers Require: P  q*bG+(1-q)*bL
Sellers Require: P  sG
So: sG  q*bG+(1-q)*bL
this is hard to satisfy (see homework 5) especially if q is small.
For reasonable valuations and probabilities,
a pooling equilibrium will not exist. bG sG bL
If q is small,
q*bG+(1-q)*bL
might be here sL