sequential rationality and weak perfect bayesian equilibriumhrtdmrt2/teaching/gt_2016_19/l6.pdf ·...

Sequential Rationality andWeak Perfect Bayesian Equilibrium

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

June 16th, 2016

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 Formalizing the Game

2 Systems of Beliefs and Sequential Rationality

3 Weak Perfect Bayesian Equilibrium

4 Exercises


Formalizing the Game

On the Agenda




4 Exercises




I Let me fix some Notation:- set of players: I = {1, 2, · · · ,N}- set of actions: ∀i ∈ I, ai ∈ Ai , where each player i has a set of actions Ai .- strategies for each player: ∀i ∈ I, si ∈ Si , where each player i has a set ofpure strategies Si available to him. A strategy is a complete contingent planfor playing the game, which specifies a feasible action of a player’sinformation sets in the game.

- profile of pure strategies: s = (s1, s2, · · · , sN) ∈∏N

i=1 Si = S.Note: let s−i = (s1, s2, · · · , si−1, si+1, · · · , sN) ∈ S−i , we will denotes = (si , s−i ) ∈ (Si , S−i ) = S.

- Payoff function: ui :∏N

i=1 Si → R, denoted by ui (si , s−i )- A mixed strategy for player i is a function σi : Si → [0, 1], which assigns aprobability σi (si ) ≥ 0 to each pure strategy si ∈ Si , satisfying∑

si∈Siσi (si ) = 1.

- Payoff function over a profile of mixed strategies:

ui (σi , σ−i ) =∑

si∈Si

∑s−i∈S−i

[∏j 6=i

σj (sj )

]σi (si )ui (si , s−i )

C. Hurtado (UIUC - Economics) Game Theory 1 / 17



I An extensive form game is defined by a tuple ΓE = {I, χ, p,A, α,H, h, i , ρ, u}1. A finite set of I players: I = {1, 2, · · · ,N}2. A finite set of nodes: χ3. A function p : χ→ χ ∪ {∅} specifying a unique immediate predecessor of

each node x such that p(x) is the empty-set for exactly one node, called theroot node, x0. The immediate successors of node x are defined ass(x) = y ∈ X : p(y) = x .

4. A set of actions, A, and a function α : χ\{x0} → A that specifies for eachnode x 6= x0, the action which leads to x from p(x).

5. A collection of information sets, H, that forms a partition of χ, and afunction h : χ→ H that assigns each decision node into an information set.

6. A function i : H → I assigning the player to move at all the decision nodes inany information set.

7. For each H a probability distribution function ρ(H). This dictates nature’smoves at each of its information sets.

8. u = (u1, · · · , uN), a vector of utility functions for each i .


Systems of Beliefs and Sequential Rationality

On the Agenda




4 Exercises




I A limitation of the preceding analysis is subgame perfection is powerless indynamic games where there are no proper subgames.




I A limitation of the preceding analysis is subgame perfection is powerless indynamic games where there are no proper subgames.

E\I F AOut 0,2 0,2In1 -1,-1 3,0In2 -1,-1 2,1




I We need a theory of "reasonable" choices by players at all nodes, and not just atthose nodes that are parts of proper subgames.

I One way to approach this problem in the above example is to ask: could Fight beoptimal for Firm I when it must actually act for any belief that it holds aboutwhether Firm E played In1 or In2? Clearly, no.

I Regardless of what Firm I thinks about the likelihood of In1 versus In2, it isoptimal for it to play Accommodate.

I This motivates a formal development of beliefs in extensive form games.

Definition

A system of beliefs is a mapping µ : χ→ [0, 1] such that, for all h ∈ H,∑x∈H

µ(x) = 1.

I In words, a system of beliefs, µ, specifies the relative probabilities of being at eachnode of an information set, for every information set in the game.




I Let E [ui |H, µ, σi , σ−i ] denote player i’s expected utility starting at her informationset H if her beliefs regarding the relative probabilities of being at any node, x ∈ His given by µ(x), and she follows strategy σi while the others play the profile ofstrategies σ−i .

DefinitionA strategy profile, σ, is sequentially rational at information set H, given a system ofbeliefs µ, if

E[ui(H)|H, µ, σi(H), σ−i(H)

]≥ E

[ui(H)|H, µ, σ̃i(H), σ−i(H)

]for all σ̃i(H) ∈ ∆(Si(H))

I A strategy profile is sequentially rational given a system of beliefs if it issequentially rational at all information sets given that system of beliefs.

I In words, a strategy profile is sequentially rational given a system of beliefs if thereis no information set such that once it is reached, the actor would strictly prefer todeviate from his prescribed play, given his beliefs about the relative probabilities ofnodes in the information set and opponents’ strategies.


Weak Perfect Bayesian Equilibrium

On the Agenda




4 Exercises




I With these concepts in hand, we now define a Weak Perfect Bayesian Equilibrium(WPBE). The idea is straightforward:

- strategies must be sequentially rational, and beliefs must be derived fromstrategies whenever possible via Bayes rule.

P(A|B) = P(B|A)P(A)P(B)

DefinitionA profile of strategies, σ, and a system of beliefs, µ, is a Weak Perfect BayesianEquilibrium (WPBE), (σ, µ), if:1. σ is sequentially rational given µ2. µ is derived from σ through Bayes rule whenever possible. That is, for any

information set H such that P(H|σ) > 0, and any x ∈ H,

µ(x) = P(x |σ)P(H|σ)




I Keep in mind that strictly speaking, a WPBE is a strategy profile-beliefs pair.

I However, we will sometimes be casual and refer to just a strategy profile as aWPBE.

I This implicitly means that there is at least one system of beliefs such the pairforms a WPBE.

I The "weak" in WPBE is because absolutely no restrictions are being placed onbeliefs at information sets that do not occur with positive probability in equilibrium.

I To be more precise, no consistency restriction is being placed; we do require thatthey be well-defined in the sense that beliefs are probability distributions.

I As we will see, in many games, there are natural consistency restrictions one wouldwant to impose on out of equilibrium information sets as well.



Weak Perfect Bayesian Nash EquilibriumI Consider the following two person game played by an Entrant E and an Incumbent

I (where the first of the two payoffs given belongs to the Entrant (E) and thesecond to the Incumbent (I)).

a. Derive the Subgame Perfect Nash Equilibrim (SPNE) in this game.b. Derive both pure and mixed Weak Perfect Bayesian Nash Equilibria in this game.



Weak Perfect Bayesian Nash Equilibrium

a. Derive the Subgame Perfect Nash Equilibrim (SPNE) in this game.

E\I L ROut 3/2, 2 3/2, 2In1 1, 0 0, 1In2 1, 2 2, 1

I In pure strategies the NE is E: Out and I: L

I No NE in mix strategies (why?)



Weak Perfect Bayesian Nash Equilibriumb. Derive both pure and mixed Weak Perfect Bayesian Nash Equilibria in this game.

I We first solve for the value of p for which I chooses L over R. L produces anexpected payoff of 2(1− p) and R produces an expected payoff of 1.

I Therefore, L is at least as good as R for I if

2− 2p ≥ 1 ⇐⇒ p ≤ 12




b. Derive both pure and mixed Weak Perfect Bayesian Nash Equilibria in this game.

I If I chooses L, then E chooses Out. This is our first equilibrium:

E : Out

I : L, p ≤ 12





I We next try to construct an equilibrium in which I chooses R. This results in Echoosing IN2, which requires p = 0.

I We need p ≥ 1/2 to support I’s choice of R over L. So there is no pure strategyWPBNE in which I chooses R.





I Finally, we consider the use of a mixed strategy by I. Let σ denote the probabilitythat I chooses L. E’s expected payoff from choosing IN1 is σ and his expectedpayoff from choosing IN2 is σ + 2(1− σ), which is strictly greater than σ forσ < 1. It is therefore possible to choose σ so that E is indifferent between IN1 andIN2, only by setting σ = 1, which leads us back to the equilibrium derived above.





I We next consider the possibility that E chooses Out in a mixed strategyequilibrium.

I We need32 ≥ 2− σ ⇐⇒ σ ≥ 1

2C. Hurtado (UIUC - Economics) Game Theory 15 / 17



I

I We therefore have derived one more type of WPBNE:

E : Out

I : σ ≥ 12 , p = 1

2


Exercises

On the Agenda




4 Exercises


Exercises

Exercises

I Consider the following two person game played by an Entrant E and an IncumbentI (where the first of the two payoffs given belongs to the Entrant (E) and thesecond to the Incumbent (I)).

a. Derive the Subgame Perfect Nash Equilibrim (SPNE) in this game.



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