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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