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### Extensive Game with Imperfect Information

Part I: Strategy and Nash equilibrium

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 (nature’s choice is non-observable to other players)

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Nature’s choice is unknown to third partyExtensive game with imperfect information and chances

- Definition: An extensive game
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.

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 i’s pure strategies.

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.

Mixed strategy and Behavioral strategy: an example

β1(φ)(L)=1; β1(φ)(R)=0;

β1({(L,A),(L,B)})(l)=1/2; β1({(L,A),(L,B)})(r)=1/2

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.

non-equivalence between behavioral and mixed strategy amid imperfect recallEquivalence 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.

Nash equilibrium

- DEFINITION: The Nash equilibriumin mixed strategies is a profile σ* of mixed strategies so that for each player i,

ui(O(σ*-i, σ*i))≥ ui(O(σ*-i, σi))

for every σi of player i.

- A Nash equilibrium in behavioral strategies is defined analogously.

(L,R) is a Nash equilibrium

According to the profile, 2’s information set being reached is a zero probability event. Hence, no restriction to 2’s belief about which history he is in.

2’s 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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The importance of off-equilibrium path beliefsbelief

- 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 (β,μ) consisting of a profile of behavioral strategies and a belief system.

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.

Perfect Bayesian equilibrium

- Definition: An assessment (β,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

Sequential equilibrium

- Definition. An assessment (β,μ) is consistent if there is a sequence ((βn,μn))n=1,… of assessments that converge to (β,μ) and has the properties that each βn 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 ((βn,μn))n=1,… )

The assessment (β,μ) in which β1=L, β2=R and μ({M,R})(M)= for any (0,1) is consistent

Assessment (βε,με)with the following properties

βε1 = (1-ε, ε,(1-)ε)

βε2 = (ε,1- ε)

με({M,R})(M)= for all ε

As ε→0, (βε,με)→ (β,μ)

For ≥1/2, this assessment is also sequentially rational.

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Back to the motivating example1

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Two similar gamesGame 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

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. (β,μ) being consistent is neither sufficient nor necessary for μ to be structurally consistent.

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 player’s payoff depends upon the value of , the message m, and the action a taken by the receiver.

Signaling games

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

Spence’s education game

- Players: worker (1) and firm (2)
- 1 has two types: high ability H with probability p Hand 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 worker’s payoff is w – e/

Pooling equilibrium

- e H = e L = e* L pH (H - L)
- w* = pHH + pLL
- 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

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.

The signal is costly

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

When does signaling work?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 equilibrium—not every sequential equilibrium satisfies this criterion

Two sequential equilibria with outcomes: (R,R) and (L,L), respectively

(L,L) is supported by belief that, in case 2’s information set is reached, with high probability 1 chose M.

If 2’s 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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An example where a sequential equilibrium is unreasonableIf 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 2’s belief is unreasonable and the equilibrium is not appealing under scrutiny.

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Why the second equilibrium is not reasonable?Spence’s education game

- All the pooling equilibria are eliminated by the Cho-Kreps intuitive criterion.
- Let e satisfy w* – e*/ L > H – e/ L and w* – e*/ H > 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 firm’s belief upon such a deviation should construe that the deviator is a high type rather than a low type.
- The pooling equilibrium break down!

Spence’s 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 >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.

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