Lecture 4n

8/13/2019 Lecture 4n

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Lecture 4TOPICS IN GAME THEORY

Jorgen Weibull

September 18, 2012


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An extensive-form game: captures the dynamics of the interaction in

question

- who moves when, who observes what, and who cares about what

(Bernoulli functionsover plays)

A normal-form game: represents the game structure in a mathemati-

cally convenient way

- everybodys strategy sets and payofffunctions


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1 What is an extensive-form game?

Kuhn (1950, 1953), Selten (1965, 1975), Kreps and Wilson (1982)

=hN,A,, P,I, C,p,r,vi where:

1. N={1,...,n} the (finite) set ofpersonal players

2. A directed (and connected) tree:

(a) A (fi

nite) set A ofnodes, with a root, a0 A

(b) A predecessor function:, :A\ {a0} A

(c) From each nodea6=a0, the root can be reached by a finite number

of iterations of


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(d) a0

precedes a ifa0

= ( (... (a))). Write a0

< a

(e) A A the terminal nodes, nodes that precedes no node (a /

(A))

3. T the set ofplays , ways to go from a0 to A

4. P={P0, P1,...,Pn} the player partitioningof non-terminal nodes

5. I= iNIi, where each Ii is the information partitioning ofPi A

into information sets I Ii. Regularity conditions:

(a) Each play intersects every information set at most once

(b) All nodes in an info set have the same number of outgoing branches


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6. C ={CI :I I}, each CIbeing the choice partitioning at I

(a) notationc < a

7. p:P0 [0, 1] the probabilities of natures random moves

8. r : T D the result (or outcome) mapping, assigning material out-

comes (money, food,...) to plays

9. v :T Rn the combined Bernoulli function (6= payofffunction)

=hN,A,, P,I, C, pi the game form

=hN,A,, P,I, C, p , ri the game protocol


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Example: A Seller-Buyer Game

Player 1: A used-car seller. Player 2: a representative buyer

Quality of the car is either H (high) or L (low), observed only by the

seller

The seller chooses either a low price, pL, or a high price, pH(and can

condition on the quality)

Seeing the price, the buyer chooses either B (buy) or N.(not buy)

What is =hN,A,, P,I, C,p,r,vi?


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What is = (N,A,, P,I, C,p,r,v)?


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Example: Is the following an extensive form?

AA

L R

1

A BAB

B B

3

3

2

2


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1.1 Subgames

The follower set

F(a) =n

a0 A:a a0o

Subrootsare nodes a for which:

F(a) I6= I F(a) .

A subgame of is the tree starting at a subroot a, endowed with thesame partitionings etc. and denoted a (in particular, a0 = is a

subgame)


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Payofffunctions i:S R

i (s) =X

T

( , s) vi ()

Recall that a pure strategy thus is more than what people usuallythink...

2.2 Mixed strategies

Mixed strategies xi Xi = (Si). As if each player randomizes

before starting to play.

Mixed-strategy profiles

x= (x1,...,xn) X= (S) =i (Si)


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Realization probabilities:

( , x) =XsS

hjNxj

sji

( , s)

Payofffunctions

i (x) =X

T

( , x) vi ()


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Example:

C F

(1,3) (0,0)(2,2)

A E

1

2

Game 4

( 1 (x) =x11+ 2x12x212 (x) = 3x11+ 2x12x21


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2.3 Behavior strategies

Local strategies: statistically independent randomizations over choice

sets,

yiI YiI= (CI)

Behavior strategies: functions yi that assign local strategies to infor-

mation sets I Ii,

yi Yi=IIiYiI

- as if players randomize as the play proceeds

Behavior strategy profiles:

y Y =iNYi


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Realization probabilities: ( , y) = the product of all choice probabil-

ities along

Payofffunctions

i (y) =X

T

( , y) vi ()

In the preceding example: Y = X because each player has only one

info set

Definition 2.1 Outcomeof strategy profile = induced probability distribu-

tion over plays (or end-nodes)

Definition 2.2 Pathof strategy profile = the set of plays assigned positive

probabilities


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3 Perfect recall and Kuhns theorem

Mixed strategies: global randomizations performed at the beginning

of the play of the game

Behavior strategies: local randomizations performed during the course

of play of the game

Equivalence in terms of realization probabilities?


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Definition 3.1 (Kuhn 1950,1953) An extensive form hasperfect recall

if

c < a c < a0

for each playeri N, pair of information setsI, J Ii, choicec CIand

nodesa, a0 J.


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Note:

- All perfect-information games have perfect recall

- If each player has only one information set, then perfect recall

- Bernoulli values and payoffs are irrelevant for the definition of perfect

recall

Informally:

Theorem 3.1 (Kuhns Theorem) If has perfect recall, then, for each

mixed strategy, a realization-equivalent behavior strategy.


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To state this more exactly, and prove it: Consider a player i in a finite

extensive form .

Definition 3.2 A (behavior-strategy) mixture, wi, is a finite-support ran-

domization over the players behavior strategies: wi

Wi, whereWi is theset of probability vectorswi =

wi

y1i

,...,wi

yki

for somek N and

y1i ,...,yki Yi.

Every behavior strategy yi Yi can be viewed as a (degenerate)

behavior-strategy mixture, the mixture wi that assigns unit probability

to yi.

Every mixed strategy xi Xi can be viewed as the mixture wi that

assigns probabilityxih [0, 1] to the (degenerate) behavior strategyyhi

that assigns unit probability to the choices made under pure strategy

h Si.


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Definition 3.3 Mixtureswi, w0i Wi arerealization equivalentif the real-

ization probabilities under the profilesw0i, w

00iandwi, w

00iare identical

for all mixture profilesw00 W =nj=1Wj.

Theorem 3.2 (Kuhn 1950, Selten 1975) Consider a playeriin a finite ex-

tensive form with perfect recall. For each behavior-strategy mixturewi Wi there exists a realization-equivalent mixturew

0i Wi that assigns

unit probability to a behavior strategyyi Yi.


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Proof sketch:

1. Consider those of is information sets I that are possible under wi in

the sense that I is on the path ofwi, w00i for some w00 W

2. Note that conditional probabilities over the nodes in any of i0s infor-

mation sets Ido not depend on is own strategy

The behavior induced at information sets by a mixed-strategy profile

Equivalence in terms of realization probabilities

We henceforth will assume perfect recall.


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4 The five NF-games associated with an EF-form

game

Five normal forms associated with any given extensive form:

1. The pure-strategy normal form

2. The mixed-strategy normal form

3. The behavior-strategy normal form

4. The quasi-reduced normal form

5. The reduced normal form


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The pure-strategy normal-form: G=hN,S,i where

The strategy set of player i is Si =IIiCI

S=iN

Si is the set ofpure-strategy strategy profiles s= (s

i)

iN

:S Rn is the combined payoff function, i (s) R the payoffto

player i under s

The mixed-strategy normal-form: G=hN,X, i where

X=iNXi is the set ofmixed-strategy profiles x= (xi)iN where

Xi=4 (Si)

:X Rn is the combined payofffunction, i (x) R the payoffto

player i under x


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Thereduced normal-form gameis the pure-strategy normal-form gameG=hN,S,iwhere all equivalent and redundant pure strategies have

been removed.


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Example:

(2,1)

(1,2)

(3,2) (0,0)

A B

C D

E F

1

1

2

Pure-strategy normal form:

C DAE 2, 1 2, 1AF 2, 1 2, 1BE 1, 2 3, 2

BF 1, 2 0, 0


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Quasi-reduced (and reduced) normal-form representation:

C D

A 2, 1 2, 1BE 1, 2 3, 2BF 1, 2 0, 0


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5 Solution concepts

Now we are in a position to define and analyze different solution con-

cepts for extensive-form games.

Focus on behavior strategies in finite EF games with perfect recall

(recall Kuhns Theorem)

Solution concepts: Nash equilibrium, subgame perfect equilibrium, se-

quential equilibrium, perfect Bayesian equilibrium, and extensive-form

perfect equilibrium


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5.1 Nash equilibrium

Let G= (N,Y, ) be the behavior-strategy NF representation of

Q1: Do NE in G always exist?

Mathematical difficulty: the best-reply correspondence in G need not

be convex-valued


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Proposition 5.1 NE ofG= (N,Y, ).

Proof: Consider mixed-strategy NF G = (N,X, ) and use Kuhns Theo-

rem!


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Q2: Howfi

nd NE directly in the extensive form?

1. For any node a, information set I, and behavior-strategy profile y, let

(a, y) be the probability that nodeais reached wheny is played, and

let

(I, y) =XaI

(a, y)

2. Definition: I I is on the path ofy Y if (I, y)>0

3. ConsiderIon the path ofy. By Bayes rule, the conditional probabil-itiesover the node in I are

Pr (a|y) = (a, y)

(I, y) a I


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Definition 5.1 Suppose thatI Ii is on the path ofy Y. A behavior

strategyyi Yi is a best reply to y at I if:XaI

Pr (a|y) ia (yi , yi)

XaI

Pr (a|y) ia

y0i, yi y0i Yi

whereia (, yi) is the conditional payoff-function for playeri when play

starts at nodea I.

Definition 5.2 A behavior-strategy profile y Y is sequentially rational

on its own path in if, for all players i N, yi Y

i is a best reply toy

at all information sets I Ii on its path.


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Proposition 5.2 (van Damme, 1984) A behavior-strategy profiley is a NE

ofG iff it is sequentially rational on its own path in .

Proof sketch:

1. Suppose that y is not a NE ofG. Then it prescribes a suboptimalmove somewhere on its own path

2. Suppose that y does notprescribe a best reply to itself at some infoset on its path


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By a slight generalization of Kuhns algorithm:

Proposition 5.4 Every finite EF game with perfect recall has at least one

SPE.

Proof sketch in class: use a generalized version of Kuhns algorithm.


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Example 5.1 Consider a battle-of-the-sexes game in which player 1 hasan outside option (go to a cafe with a friend):

1

L R

a a bb

(3,1) (0,0) (1,3)(0,0)

(2,v)

A B

2

1

Find all subgame perfect equilibria! (Does the valuev R matter?)


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However, SPE is sensitive to details of the EF form.

Example 5.2 Add a payoff-equivalent move to the entry-deterrence game.

In this extensive-form game, s = (A, F) becomes subgame perfect (al-

though still implausible):

1

A

F F YY

0

0

2

2

2

2

0

0

1

3

E1 E2

2


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Subgame perfection ; sequential rationality at all singleton-information

sets

Example 5.3 Seltens horse0

0

0

3

2

2

0

0

1

4

4

0

1

1

1

L

LL

L

R R

R

R

1

2

3

y = (L,R,R) is a NE and hence a SPE.


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But 2s move is not a best reply to y

In all solution concepts: a strategy is not only a contingent plan for

the player in question, but also others expectation about the players

moves.


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5.3 Sequential equilibrium

Refine SPE by generalizing the stochastic dynamic programming ap-

proach from 1 decision-maker to n decision-makers

Kreps and Wilson (1982)

Definition 5.4 A belief system is a function :AA [0, 1] such thatXaI

(a) = 1 I I

Definition 5.5 An belief system is consistent with a behavior-strategy

profile y if sequence of interior behavior-strategy profiles yt y such

thatPr |yt ().


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Reconsider:

(1) The above battle-of-sexes with an outside option

(2) The above entry-deterrence game with added move

(3) Seltens horse (see next slide)


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Example 5.4 Reconsider Seltens horse

0

0

0

3

2

2

0

0

1

4

4

0

1

1

1

L

LL

L

R R

R

R

1

2

3

Wefirst note that the arguably unsatisfactory subgame-perfect equilibrium

s= (L,R,R) is not a sequential equilibrium. The reason is that player 2s

(pure) behavior strategy, s2=R, is not sequentially rational, sinces02=L


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is a better reply than R, at 2s node, against s (yielding a payoff of 4

instead of 1)

Next, we note that all sequential equilibria are of the form y = (R,R,y3),where y3 assigns probability 3/4 to L. Under such a strategy profile,

3s information set is not on the path, and any conditional probabilities

(a) and (b) (both non-negative and summing to 1) that player 3 may

attach to her 2 nodes are part of a consistent belief system. Her strategy

Lis sequentially rational iff (a) 1/3, and she is indifferent between her

strategiesL andR iff (a) = 1/3, and in that case, any strategy for her

is sequentially rational. However, if she would play R with a probability

>1/4, then player 2s strategy in y, R, would not be sequentially rational.

Hence, sequential equilibrium requires thaty3L 3/4.


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5.4 Perfect Bayesian equilibrium

Relaxing the consistency requirement in sequential equilibrium:

Definition 5.8 A belief system is weakly consistent with a behavior-

strategy profile y if agrees with the conditional probability distributionPr ( |y), induced byy, on each information set on ys path.

Definition 5.9 A behavior-strategy profiley is a (weak) perfect Bayesian

equilibrium (PBE)if there exists a weakly consistent belief systemunder

which y is sequentially rational.

Clearly SE PBE NE

There are more restrictive definitions of PBE, but no consensus among

game theorists


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6 Properness implies sequentiality

The normal-form refinementpropernesshas a beautiful implication for

extensive-form analysis [van Damme (1983), Kohlberg and Mertens

(1986)]:

Proposition 6.1 LetG be a finite game with mixed strategy extension G.

For every proper equilibrium x

in G and every EF game with G as itsNF, there exists a realization-equivalent SEy in .

Proof sketch: Let G, G,

and x

be as stated

1. a sequence of t-proper profiles xt int [ (S)] with t 0 and

xt x


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2. For each xt a realization-equivalent behavior-strategy yt in

3. Since xt int [ (S)], each info set in is on the path ofyt

4. Since xt x

, yt y

Y. Sufficient to verify that y

is a SE!

5. For each t N, let t =

|yt

. Then = limtt is a belief

system consistent with y

6. Suppose that y is not sequentially rational under : player i,

information set I Ii and y0i Yi such that y

0iI6=y

iI andX

aI

(a) ia

y0i, yi

>

XaI

(a) ia (y)

7. But this leads to a contradiction.


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7 Forward induction

Discussion in class of the outside-option game in the light of forward

induction reasoning (see Kohlberg and Mertens, 1986):

1

L R

a a bb

(3,1) (0,0) (1,3)(0,0)

(2, v)

A B

2

1


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THE END

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Lecture 4n