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Extensive Game with Imperfect Information

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### Extensive Game with Imperfect Information

### Part II: Belief and Sequential Equilibrium perfect recall

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 <N,H,P,fc,(Ti),(ui)> 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 perfect recall

- 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 perfect recall

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 perfect recall

- 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 perfect recall

- 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 perfect recall

- 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 perfect recall

- 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 β perfect recall1=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 perfect recall

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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 perfect recall

- 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 perfect recall

- 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 perfect recall

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

Spence’s education game perfect recall

- 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 perfect recall

- 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 perfect recall

- 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 perfect recall

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 perfect recall

- 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 unreasonable1,1 respectively

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Beer or QuicheIf 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 is worse for her than following the equilibrium, whatever the response of player 2.

- 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 is worse for her than following the equilibrium, whatever the response of player 2.

- 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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