PPT Cours Games

8/9/2019 PPT Cours Games

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Elements of Game Theory

S. Pinchinat

Master2 RI 2011-2012

S. Pinchinat (IRISA) Elements of Game Theory Master2 RI 2011-2012 1 / 64
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Introduction

EconomyBiologySynthesis and Control of reactive SystemsChecking and Realizability of Formal SpecicationsCompatibility of InterfacesSimulation Relations between SystemsTest Cases Generation...

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In this Course

Strategic Games (2h)

Extensive Games (2h)



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Bibliography

R.D. Luce and H. Raiffa: Games and Decisions (1957) [LR57]K. Binmore: Fun and Games. (1991) [Bin91]

R. Myerson: Game Theory: Analysis of Conict. (1997) [Mye97]M.J. Osborne and A. Rubinstein: A Course in Game Theory.(1994) [OR94]Also the very good lecture notes from Prof Bernhard von Stengel(search the web).

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Strategic Games

Representions

r T w 1, w 2 x 1, x 2B y 1, y 2 z 1, z 2

Player 1 has the rows (Top or Bottom) and Player 2 has the columns(left or right):

S 1 = {T , B } and S 2 = {, r }

For example, when Player 1 chooses T and Player 2 chooses , the

payoff for Player 1 (resp. Player 2) is w 1 (resp. w 2), that isu 1(T , ) = w 1 and u 2(T , ) = w 2


S i G
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Strategic Games

Example

The Battle of Sexes

Bach StravinskyBach 2, 1 0, 0

Stravinsky 0, 0 1, 2

Strategic Interaction = players wish to coordinate their behaviors buthave conicting interests.A Coordination Game

Mozart Mahler

Mozart 1, 1 0, 0Mahler 0, 0 2, 2

Strategic Interaction = players wish to coordinate their behaviorsand have mutual interests.


St t i G
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Strategic Games

The Prisoners DilemmaThe story behind the name prisoners dilemma is that of two prisonersheld suspect of a serious crime. There is no judicial evidence for this crimeexcept if one of the prisoners confesses against the other. If one of themconfesses, he will be rewarded with immunity from prosecution (payoff 0),whereas the other will serve a long prison sentence (payoff 3). If bothconfess, their punishment will be less severe (payoff 2 for each).

However, if they both cooperate with each other by not confessing atall, they will only be imprisoned briey for some minor charge that can beheld against them (payoff 1 for each). The defection from thatmutually benecial outcome is to confess, which gives a higher payoff nomatter what the other prisoner does, which makes confess a dominating

strategy (see later). However, the resulting payoff is lower to both. Thisconstitutes their dilemma.

confess dont ConfessConfess 2, 2 0, 3

Dont Confess 3, 0 1, 1


Strategic Games
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Strategic Games

The Prisoners Dilemma

confess dont ConfessConfess 2, 2 0, 3

Dont Confess 3, 0 1, 1

The best outcome for both players is that neither confess.Each player is inclined to be a free rider and to confess.


Strategic Games
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Strategic Games

A 3-player Games

L R T 8 0B 0 0

L R 4 00 4

L R 0 00 8

L R 3 33 3

M 1 M 2 M 3 M 4

Player 1 chooses one of the two rows;Player 2 chooses one of the two columns;Player 3 chooses one of the four tables.


Strategic Games Denitions and Examples
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(Finite) Strategic GamesA nite strategic game with n-players is = (N , {S i }i N , {u i }i N ) where

N = {1, . . . , n} is the set of players.S i = {1, . . . , mi } is a set of pure strategies (or actions) of player i .u i : S I R is the payoff (utility) function.

S := S 1 S 2 . . . S n is the set of proles.s = ( s 1, s 2, . . . , s n )

Instead of u i , use preference relations:s i s for Player i prefers prole s than prole s

= ( N , {S i }i N , {u i }i N ) or = (N , {S i }i N , { i }i N )


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Comments on Interpretation

A strategic game describes a situation wherewe have a one-shot eveneach player knows

the details of the game. the fact that all players are rational (see futher)

the players choose their actions simultaneously and independently.


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g p

Comments on Interpretation

A strategic game describes a situation wherewe have a one-shot eveneach player knows

the details of the game. the fact that all players are rational (see futher)

the players choose their actions simultaneously and independently.

Rationality: Every player wants to maximize its own payoff.


Strategic Games Dominance and Elimination of Dominated Strategies
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g g

Notations

Use s i S i , or simply j S i where 1 j mi .Given a prole s = ( s 1, s 2, . . . , s n ) S , we let a counter prole be anelement like

s i := ( s 1, s 2, . . . , s i 1, empty, s i +1 , . . . , s n )

which denotes everybodys strategy except that of Player i , and writeS i for the set of such elements.

For r i S i , let (s i , r i ) := ( s 1, s 2, . . . , s i 1, r i , s i +1 , . . . , s n ) denote thenew prole where Player i has switched from strategy s i to strategy r i .


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g g

Dominance

Let s i , s i S i .

s i strongly dominates s i if

u i (s i , s i ) > u i (s i , s i ) for all s i S i ,

s i (weakly) dominates s i if

u i (s i , s i ) u i (s i , s i ), for all s i S i ,

u i (s i , s i ) > u i (s i , s i ), for some s i S i .


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Example of Dominance

The Prisoners Dilemmac d

C 2, 2 0, 3D 3, 0 1, 1

Strategy C of Player 1 strongly dominates strategy D.Because the game is symmetric, strategy c of Player 2 stronglydominates strategy d.

Note also that u 1(D , d ) > u 1(C , c ) (and also u 2(D , d ) > u 2(C , c )),so thatC dominates D does not mean C is always better than D.


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Example of Weak Dominance

r T 1, 3 1, 3B 1, 1 1, 0

(weakly) dominates r .




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Elimination of Dominated Strategies

If a strategy is dominated, the player can always improve his payoff bychoosing a better one (this player considers the strategies of the otherplayers as xed).Turn to the game where dominated strategies are eliminated.

c dC 2, 2 0, 3D 3, 0 1, 1

The game becomes simpler.Eliminating D and d, shows (C,c) as the solution of the game, i.e. arecommandation of a strategy for each player.


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Iterated Elimination of Dominated Strategies

We consider iterated elimination of dominated strategies.The result does not depend on the order of elimination:If s i (strongly) dominates s i , it still does in a game where some

strategies (other than s

i ) are eliminated.In contrast, for iterated elimination of weakly dominated strategiesthe order of elimination may matterEXERCISE: nd examples, in books

A game is dominance solvable if the Iterated Elimination of DominatedStrategies ends in a single strategy prole.


Strategic Games Nash Equilibrium
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Motivations

Not every game is dominance solvable, e.g. Battle of Sexes.

Bach StravinskyBach 2, 1 0, 0

Stravinsky 0, 0 1, 2

The central concept is that of Nash Equilibrium [Nas50].


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Best Response and Nash Equilibrium

Informally, a Nash equilibrium is a strategy prole where each playersstrategy is a best response to the counter prole.Formally:Given a strategy prole s = ( s 1, . . . , s n ) in a strategic game

= ( N , {S i }i N , { i }i N ), a strategy s i is a best response (to s i )if (s i , s i ) i (s i , s i ), for all s

i S i

A Nash Equilibrium in a prole s = ( s 1 , . . . , s

n ) S such thats

i is best response to s

i , for all i = 1 , . . . , n.

Player cannot gain by changing unilateraly her strategy, i.e. with theremained strategies kept xed.


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Illustration

r

T 22

10

B 031

1


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Illustration

r

T 22

10

B 031

1

draw best reponse in boxes

r

T 221

0

B 031

1


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ExamplesBattle of Sexes: two Nash Equilibria.

Bach StravinskyBach 2,1 0, 0

Stravinsky 0, 0 1,2

Mozart-Mahler: two Nash Equilibria.

Mozart MahlerMozart 1,1 0, 0Mahler 0, 0 2,2

Prisoners Dilemma: EXERCISE

c dC 2, 2 0, 3D 3, 0 1, 1


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Dominance and Nash Equilibrium (NE)

A dominated strategy is never a best response it cannot be part of a NE.

We can eliminate dominated strategy without loosing any NE.

Elimination does not create new NE (best response remains whenadding a dominated strategy).

Proposition

If a game is dominance solvable, its solution is the only NE of the game.

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Mixed Strategy Nash Equilibrium


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Extended Strategic Games

We have seen that NE need not exist when players deterministicallychoose one of their strategies (e.g. Matching Pennies)If we allow players to non-deterministically choose, then

NE always exist (Nash Theorem)By non-deterministically we mean that the player randomises hisown choice.We consider mixed strategies instead of only pure strategies

considered so far.


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Mixed StrategiesA mixed (randomized) strategy x i for Player i is a probability distributionover S i . Formally, it is a vector x i = ( x i (1), . . . , x i (mi )) with

x i ( j ) 0, for all j S i , andx i (1) + . . . + x i (mi ) = 1

Let X i be the set of mixed strategies for Player i .

X := X 1 X 2 . . . X n is the set of (mixed) proles.

A mixed strategy x i is pure if x i ( j ) = 1 for some j S i and x i ( j ) = 0for all j = j . We use i , j to denote such a pure strategy.The support of the mixed strategy x i is

support( x i ) := { i , j | x i ( j ) > 0}

We use e.g. x i (counter (mixed) prole), X i (counter proles), etc.S. Pinchinat (IRISA) Elements of Game Theory Master2 RI 2011-2012 24 / 64

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Interpretation of Mixed Strategies

Assume given a mixed strategy (probability distribution) of a player.This player uses a lottery device with the given probabilities to pickeach pure strategy according to its probability.The other players are not supposed to know the outcome of thelottery.A player bases his own decision on the resulting distribution of payoffs, which represents the players preference.


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Expected Payoffs

Given a prole x = ( x 1, . . . , x n ) of mixed strategies, and acombination of pure strategies s = ( s 1, . . . , s n ), the probability of combinaison s under prole x is

x (s ) := x 1(s 1) x 2(s 2) . . . x n (s n )

The expected payoff of Player i is the mapping U i : X I R dened by

U i (x ) :=s S

x (s ) u i (s )

It coincides with the notion of payoff when only pure strategies areconsidered.


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Mixed Extension of Strategic Games and Mixed StrategyNash Equilibrium

The mixed extension of = (N , {S i }i N , {u i }i N ) is the strategicgame

(N , {X i }i N , {U i }i N )

(CONVENTION: we still use u i instead U i .)

A mixed strategy Nash Equilibrium of a strategic game = ( N , {S i }i N , {u i }i N ) is a Nash equilibrium of its mixedextension.


Mixed Strategy Nash Equilibrium 2-player Strategic Games
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Focus on 2-player Games

Sets of strategies {1, . . . , n} and {1, . . . , m} respectively;

Payoffs functions u 1 and u 2 are described by n m matrices:

U 1 :=

u 1(1, 1) u 1(1, 2) . . . u 1(1, m)u 1(2, 1) u 1(2, 2) . . . u 1(2, m)

. . . . . . . . . . . .

u 1(n, 1) u 1(n, 2) . . . u 1(n, m)

U 2 . . .

x 1 = ( p 1, . . . , p n ) X 1 and x 2 = ( q 1, . . . , q m ) X 2 for mixedstrategies; that is p j = x 1( j ) and q k = x 2(k ).Write (U 1x 2) j the j -th component of vector U 1x 2.

(U 1x 2) j = mk =1 u 1( j , k ) q k is the expected payoff of Player 1when playing row j .

we shall write x 1(U 1x 2) instead of x T 1 (U 1x 2).


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Given a mixed prole x = ( x 1, x 2),

The expected payoff of Player 1 is

x 1(U 1x 2) =n

j =1

m

k =1p j u 1( j , k ) q k

The expected payoff of Player 2 is

(x 1U 2)x 2 =n

j =1

m

k =1

p j u 2( j , k ) q k



h h h
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Existence: the Nash Theorem

Theorem (Nash 1950)Every nite strategic game has a mixed strategy NE.

We omit the proof in this course and refer to the literature.

Remarks

The assumption that each S i is nite is essential for the proof.The proof uses Brouwers Fixed Point Theorem.

TheoremLet Z be a subset of some space I R N that is convex and compact, andlet f be a continuous function from Z to Z . Then f has at least onexed point, that is, a point z Z so that f (z ) = z .



Th B R P
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The Best Response Property

Theorem

Let x 1 = ( p 1, . . . , p n ) and x 2 = ( q 1, . . . , q m ) be mixed strategies of Player 1 and Player 2 respectively. Then x 1 is a best response to x 2 iff for all pure strategies j of Player 1,

p j > 0 (U 1x 2) j = max {(U 1x 2)k | 1 k n}

Proof Recall that (U 1x 2) j is the the expected payoff of Player 1 whenplaying row j . Let u := max {(U 1x 2) j | 1 j n}. Then

x 1U 1x 2 = n j =1 p j (U 1x 2) j =n j =1 p j [u [u (U 1x 2) j ]]

=n j =1 p j u

n j =1 p j [u (U 1x 2) j ] = u

n j =1 p j [u (U 1x 2) j ]

Since p j 0 and u (U 1x 2) j 0, we have x 1U 1x 2 u . The expectedpayoff x 1U 1x 2 achieves the maximum u iff n j =1 p j [u (U 1x 2) j = 0, thatis p j > 0 implies (U 1x 2) j = u .



C f h B R P
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Consequences of the Best Response Property

Only pure strategies that get maximum, and hence equal, expected payoff can be played with postive probability in NE.

Proposition

A prople (x

1 , x

2 ) is a NE if and only if there exists w 1, w 2 I R such that for every j support (x 1 ), (U 1x

2 ) j = w 1, and for every j / support (x 1 ), (U 1x

2 ) j w 1.for every k support (x 2 ), (x

1 U 2)k = w 2. and

for every k / support (x

2 ), (x

1 U 2)k w 2.



Fi di Mi d N h E ilib i (1)
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Finding Mixed Nash Equilibria (1)We characterize NE: suppose we know the supports support (x 1 ) S 1 andsupport (x 2 ) S 2 of some NE (x

1 , x

2 ). We consider the following systemof constraints over the variables p 1, . . . , p n , q 1, . . . , q m , w 1, w 2:

Write x 1 = ( p 1, . . . , p n ) and x

2 = ( q 1, . . . , q m ).

(U 1x

2 ) j = w 1 for all 1 j n with p j = 0(x 1 U 2)k = w 2 for all 1 k m with q k = 0

n j =1 p j = 1mk =1 q k = 1

EXERCISE : what is missing ?

Notice that this system is a Linear Programming system however thesystem is no longer linear if n > 2 . Use e.g. Simplex algorithm



Fi di g Mi d N h Eq ilib i (1)
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Finding Mixed Nash Equilibria (1)We characterize NE: suppose we know the supports support (x 1 ) S 1 andsupport (x 2 ) S 2 of some NE (x

1 , x

2 ). We consider the following systemof constraints over the variables p 1, . . . , p n , q 1, . . . , q m , w 1, w 2:

Write x 1 = ( p 1, . . . , p n ) and x

2 = ( q 1, . . . , q m ).

(U 1x

2 ) j = w 1 for all 1 j n with p j = 0(x 1 U 2)k = w 2 for all 1 k m with q k = 0

n j =1 p j = 1mk =1 q k = 1

EXERCISE : what is missing ?

Notice that this system is a Linear Programming system however thesystem is no longer linear if n > 2 . Use e.g. Simplex algorithmFor each possible support (x 1 ) S 1 and support (x

2 ) S 2 solve the system Worst-case exponential time (2 m + n possibilities f or the supports).



Finding Mixed Nash Equilibria (2)


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Finding Mixed Nash Equilibria (2)Battle of Sexes

b s

B 21

00

S 001

2

As seen before, we already have twoNE with pure strategies (B , b ) and(S , s ).

We now determine the mixed strategy probabilities of a player so as tomake the other player indifferent between his or her pure strategies,because only then that player will mix between these strategies. This is aconsequence of the Best Response Theorem: only pure strategies that getmaximum, and hence equal, expected payoff can be played with positiveprobability in equilibrium.



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b s

B 21

00

S 001

2

Suppose Player I plays B with prob.1 p and S with prob. p . The bestresponse for Player II is b when p 0,

whereas it is s when p 1. There is some probability so thatPlayer II is indifferent.



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b s

B 21

00

S 001

2

Fix p (probability) the mixed strategy of Player I. The expected payoff of Player II when she plays b is 2(1 p ), and it is p when she plays s .

She is indifferent whenever 2(1 p ) = p , that is p = 2 / 3.

If Player I plays B with prob 1/ 3 and S with prob 2/ 3, Player II has anexpected payoff of 2/ 3 for both strategies.



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b s

B 21

00

S 001

2

If Player I plays the mixed strategy(1/ 3, 2/ 3), Player II has an expectedpayoff of 2/ 3 for both strategies.



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b s

B 21

00

S 00 12


Then Player II can mix between b and s . A similar calculation shows thatPlayer I is indifferent between C and S if Player II uses the mixed strategy(2/ 3, 1/ 3), and Player I has an expected payoff of 2/ 3 for both strategies.



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b s

B 21

00

S 00

12


Then Player II can mix between b and s . A similar calculation shows thatPlayer I is indifferent between C and S if Player II uses the mixed strategy(2/ 3, 1/ 3), and Player I has an expected payoff of 2/ 3 for both strategies.

The prole of mixed strategies ((1/ 3, 2/ 3), (2/ 3, 1/ 3)) is the mixed NE.



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The difference trick method

The upper envelope method



Matching Pennies
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Matching Pennies

Head TailHead 1, 1 1, 1Tail 1, 1 1, 1

x

1 (Head ) = x

1 (Tail ) = x

2 (Head ) = x

2 (Tail ) = 12

is the unique mixed strategy NE.

Here, support( x 1

) = {1

Head , 1

Tail } and support( x 2

) = {2

Head , 2

Tail }.

EXERCISE: apply previous techniques to nd it yourself.



Complements
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Complements

Particular classes of games, e.g. symmetric games, degeneratedgames, ...Bayesian games for Games with imperfect information.Solutions for N 3: there are examples where the NE has irrationalvalues. See Nash 1951 for a Poker game....


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Strategic Games (2h)

Extensive Games (2h)


Extensive Games (with Perfect Information)
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By Extensive Games,

we implicitly mean

Extensive Games with Perfect Information.


Extensive Games (with Perfect Information) Denitions and Examples

Example
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p

ny ny ny

(2, 0)(1, 1)

(0, 2)

1

2 2

(2, 0)

2

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

H = {,(2, 0), (1, 1), (0, 2), ((2 , 0), y ), . . .}.Each history denotes a unique node in the game tree, hence we often usethe terminology decision nodes instead.



1
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13

23

a b c

X Y Z

d e

P Q

1

2 2

3, 4

0, 15 3, 0

2, 3

1, 3

0, 5 4, 2

4, 5

EXERCISE: Tune the denition of an extensive game to take chance nodesinto account.



Extensive Games (with Perfect Information)
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( )An extensive game (with perfect information) is a tupleG = ( N , A, H , P , { i }i =1 ,..., n ) where

N = {1, . . . , n} is a set of players.Write A = i Ai , where Ai is the set of action of Player i .H A is a set of (nite) sequences of actions s.t.

The empty sequence H . H is prex-closed.

The elements of H are histories; we identify histories with the decisionnodes they lead to. A decision node is terminal whenever it is(reached by a history) of the form h = a1a2 . . . aK and there is noaK +1 A such that a1a2 . . . aK aK +1 H . We denote by Z the set of terminal histories.P : H \ Z N indicates whose turn it is to play in a givennon-terminal decision node.Each i Z Z is a preference relation.

We write A(h) AP (h) for the set of actions available to Player P (h) atdecision node h.



Strategies and Strategy Proles
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A strategy of a player in an extensive game is a plan.Formally, let G = ( N , H , P , { i }i =1 ,..., n ) be an extensive game(from now on, we omit the set A of actions).

A strategy of Player i is (a partial mapping) s i : H \ Z A whose domainis {h H \ Z | P (h) = i } and such that s i (h) A(h).Write S i for the set of strategies of Player i .

Note that the denition of a strategy only depends on the game tree(N , H , P ), and not on the preferences of the players.



Example
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ny ny ny

(2, 0)(1, 1)

(0, 2)

1

2 2

(2, 0)

2

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

Player 1 plays at decision node (she starts the game), and this is theonly one; she has 3 strategies s 1 = (2 , 0), s 1 = (1 , 1), s 1 = (0 , 2).Player 2 takes an action after each of the three histories, and in each

case it has 2 possible actions (y or n); we write this as abc (a, b , c {y , n}), meaning that after history (2 , 0) Player 2 choosesaction a, after history (1, 1) Player 2 chooses action b , and afterhistory (0, 2) Player 2 chooses action c .



Example
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ny ny ny

(2, 0)(1, 1)

(0, 2)

1

2 2

(2, 0)

2

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

Player 1 plays at decision node (she starts the game), and this is theonly one; she has 3 strategies s 1 = (2 , 0), s 1 = (1 , 1), s 1 = (0 , 2).Player 2 takes an action after each of the three histories, and in eachcase it has 2 possible actions (y or n); we write this as abc (a, b , c {y , n}), meaning that after history (2 , 0) Player 2 choosesaction a, after history (1, 1) Player 2 chooses action b , and afterhistory (0, 2) Player 2 chooses action c . Possible strategies for Player2 are yyy , yyn, yny , ynn, ... There are 23 possibilities.



Example
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ny ny ny

(2, 0) (1, 1)

(0, 2)

1

2 2

(2, 0)

2

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

Player 1 plays at decision node (she starts the game), and this is theonly one; she has 3 strategies s 1 = (2 , 0), s 1 = (1 , 1), s 1 = (0 , 2).Possible strategies for Player 2 are yyy , yyn, yny , ynn, ... There are23 possibilities.In general the number of strategies of a given Player i is in O (|Ai |m )where m is the number of decision nodes where Player i plays.



Remarks on the Denition of Strategies
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Strategies of players are dened even for histories that are notreachable if the strategy is followed.

1

AB

D C

F E

1

1

2

In this game, Player 1 has four strategies AE , AF , BE , BF : By BE ,we specify a strategy after history e.g. AC even if it is specied thatshe chooses action B at the beginning of the game.



Reduced Strategies
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A reduced strategy of a player species a move for the decision nodesof that player, but unreachable nodes due to an earlier own choice.

It is important to discard a decision node only on the basis of theearlier own choices of the player only.A reduced prole is a tuple of reduced strategies.


Extensive Games (with Perfect Information) Relation to Strategic Games

Outcomes
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The outcome of a strategy prole s = ( s 1, . . . , s n ) in an extensive gameG = ( N , H , P , { i }i N ), written O(s), is the terminal decision node thatresults when each Player i follows the precepts s i :

O (s ) is the history a1a2 Z s.t. for all (relevant) k > 1, we have

s P (a 1 ... a k 1) (a1 . . . ak 1) = ak



Example of Outcomes
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O (AE , C ) has payoff v !...

O (BE , C ) has payoff v 4...O (BF , D ) has payoff v 4...

1

AB

D C

F

E

1

1

2

v 1 v 2

v 3

v 4



Strategic Form of an Extensive Game
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The strategic form of an extensive game G = ( N , H , P , { i }i N ) is thestrategic game ( N , {S i }i N , { i }i N ) where

i S i S i is dened by

s i s whenever O (s ) i O (s )

As the number of strategies grows exponentially with the number m of decision nodes in the game tree recall it is in O (|Ai |m ) , strategic formsof extensive games are in general big objects.



Extensive Games as Strategic Games
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C D AE v 1 v 3AF v 2 v 3BE v 4 v 4BF v 4 v 4

Reduced Strategies:C D

AE v 1 v 3

AF v 2 v 3B v 4 v 4

1

AB

D C

F

E

1

1

2

v 1 v 2

v 3

v 4


Extensive Games (with Perfect Information) Nash Equilibrium

Nash Equilibrium in Extensive Games
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Simply use the denition of NE for the strategic game form of the

extensive game.



Examples of Nash Equilibria
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1

AB

R L

1

2

0, 0 2, 1

1, 2

Two NE (A, R ) and (B , L) with payoffs are (2, 1) and (1, 2) resp.(B , L) is a NE because:

given that Player 2 chooses L after history A, it is always optimal forPlayer 1 to choose B at the beginning of the game if she does not,then given Player 2s choice, she obtains 0 rather than 1.



Examples of Nash Equilibria1
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1

AB

R L

1

2

0, 0 2, 1

1, 2

Two NE (A, R ) and (B , L) with payoffs are (2, 1) and (1, 2) resp.(B , L) is a NE because:

given that Player 2 chooses L after history A, it is always optimal forPlayer 1 to choose B at the beginning of the game if she does not,then given Player 2s choice, she obtains 0 rather than 1.

given Player 1s choice of B , it is always optimal for Player 2 to play L.

The equilibrium (B , L) lacks plausibility.The good notion is the Subgame Perf ect Equilibrium



Subgame Perfect Equilibrium
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We dene the subgame of an extensive game G = ( N , H , P , { i }i N ) thatfollows a history h as the extensive game

G (h) = ( N , h 1H , P |h , { i |h }i N )

whereh 1H := {h | hh H }P |h (h ) := P (hh ) ...h i |h h whenever hh i hh

G (h) is the extensive game which starts at decision node h.



G and G (h ), for h = A
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1

AB

D C

F E

1

1

2

v 1 v 2

v 3

v 4 D

C

F E

1

2

v 1 v 2

v 3



Subgame Perfect (Nash) Equilibrium
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A subgame perfect equilibrium represents a Nash equilibrium of everysubgame of the original game. More formally,

A subgame perfect equilibrium (SPE) of G = ( N , H , P , { i }i N ) is a

strategy prole s

s.t. for every Player i N and every non-terminalhistory h H \ Z for which P (h) = i , we have

O |h (s i |h , s

i |h ) i |h O |h (s

i |h , s i )

for every strategy s i of Player i in the subgame

G (h).



Example
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1

AB

R L

1

2

0, 0 2, 1

1, 2

NE were (A, R ) and (B , L), but the only SPE is (A, R ).



Another Example
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ny ny ny

(2, 0)(1, 1)

(0, 2)

1

2 2

(2, 0)

2

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

Nash Equilibrium are:((2, 0), yyy ), ((2 , 0), yyn), ((2 , 0), yny ), and ((2 , 0), ynn) for thedivision (2, 0).((1, 1), nyy ), ((1 , 1), nyn), and ((1 , 1), nyn) for the division (1, 1)....

EXERCISE: What are the SPE?



Computing SPE of Finite Extensive Games (with PerfectInformation): Backward Induction
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Information): Backward Induction

Backward Induction is a procedure to construct a strategy prole which isa SPE.

Start with the decision nodes that are closest to the leaves, consider a

history h in a subgame with the assumption that a strategy prole hasalready been selected in all the subgames G (h, a), with a A(h). Amongthe actions of A(h), select an action a that maximises the (expected)payoff of Player P (h).

This way, an action is specied for each history of the game G , whichdetermines an entire strategy prole.



Backward Induction Example1
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1

AB

R L

1

2

0, 0 2, 1

1, 2



Backward Induction Example1
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1

AB

R L

1

2

0, 0 2, 1

1, 22, 1

2, 1



Theorem: Backward Induction denes an SPE.
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We prove inductively (and also consider chance nodes): consider anon-terminal history h. Suppose that A(h) = {a1, a2, . . . , am } each a j leading to the subgame G j , and assume, as inductive hypothesis, that thestrategy proles that have been selected by the procedure so far (in thesubgames G j s) dene a SPE.



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We prove inductively (and also consider chance nodes): consider anon-terminal history h. Suppose that A(h) = {a1, a2, . . . , am } each a j leading to the subgame G j , and assume, as inductive hypothesis, that thestrategy proles that have been selected by the procedure so far (in thesubgames G j s) dene a SPE.

If h is a chance node, then the BI procedure does not select a particularaction from h. For every player, the expected payoff in the subgame G (h)is the expectation of the payoffs in each subgame G j (weighted withprobability to move to G j ). If a player could improve on that payoff, she

would have to do so by changing her strategy in one subgame G j whichcontradicts the induction hypothesis.



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We prove inductively (and also consider chance nodes): consider anon-terminal history h. Suppose that A(h) = {a1, a2, . . . , am } each a j leading to the subgame G j , and assume, as inductive hypothesis, that thestrategy proles that have been selected by the procedure so far (in thesubgames G

j s) dene a SPE.

Assume now that h is a decision node (P (h) N ).

Every Player i = P (h) can improve his payoff only by changing his strategy

in a subgame, which contradicts the induction hypothesis.



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We prove inductively (and also consider chance nodes): consider a

non-terminal history h. Suppose that A(h) = {a1, a2, . . . , am } each a j leading to the subgame G j , and assume, as inductive hypothesis, that thestrategy proles that have been selected by the procedure so far (in thesubgames G j s) dene a SPE.

Assume now that h is a decision node (P (h) N ).

Player P (h) can improve her payoff, she has to do it by changing herstrategy: the only way is to change her local choice to a j together withchanges in the subgame G j . But the resulting improved payoff would onlybe the improved payoff in G j , itself, which contradicts the inductionhypothesis.



Consequences of the Theorem
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Backward Induction denes an SPE.(In extensive games with perfect information)



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CorollaryBy the BI procedure, each players action can be chosen deterministically,

so that pure strategies suffice. It is not necessary, as in strategic games, toconsider mixed strategies.

S. Pinchinat (IRISA) Elements of Game Theory Master2 RI 2011-2012 62 / 64 Extensive Games (with Perfect Information) Nash Equilibrium

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CorollaryBy the BI procedure, each players action can be chosen deterministically,

so that pure strategies suffice. It is not necessary, as in strategic games, toconsider mixed strategies.

CorollarySubgame Perfect Equilibrium always exist.For game trees, we can use SPE synonymously with strategy proleobtained by backward induction.

S. Pinchinat (IRISA) Elements of Game Theory Master2 RI 2011-2012 62 / 64 Extensive Games (with Perfect Information) Nash Equilibrium

An example of an exam (2009)II
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3,5,0

II III

TM

B

l c r L R

2,1,3 0,2,4 2,3,1

1,2,3

1,4,2

1

How many strategy proles does this game have?2 Identify all pairs of strategies where one strategy strictly, or weakly,dominates the other.

3 Find all Nash equilibria in pure strategies. Which of these aresubgame perfect?S Pinchinat (IRISA) Elements of Game Theory Master2 RI 2011-2012 63 / 64


What we have not seen
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Combinatorial game theory

A mathematical theory that studies two-player games which have aposition in which the players take turns changing in dened ways ormoves to achieve a dened winning condition.

S Pinchinat (IRISA) Elements of Game Theory Master2 RI 2011-2012 64 / 64 Extensive Games (with Perfect Information) Nash Equilibrium

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A mathematical theory that studies two-player games which have aposition in which the players take turns changing in dened ways ormoves to achieve a dened winning condition.Game trees with imperfect informationA player does not know exactly what actions other players took up tothat point. Technically, there exists at least one information set withmore than one node.


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A mathematical theory that studies two-player games which have aposition in which the players take turns changing in dened ways ormoves to achieve a dened winning condition.Game trees with imperfect informationA player does not know exactly what actions other players took up tothat point. Technically, there exists at least one information set withmore than one node.Games and logicImportant examples are semantic games used to dene truth,

back-and-forth games used to compare structures, and dialoguegames to express (and perhaps explain) formal proofs.


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A mathematical theory that studies two-player games which have aposition in which the players take turns changing in dened ways ormoves to achieve a dened winning condition.Game trees with imperfect informationA player does not know exactly what actions other players took up tothat point. Technically, there exists at least one information set withmore than one node.Games and logicImportant examples are semantic games used to dene truth,

back-and-forth games used to compare structures, and dialoguegames to express (and perhaps explain) formal proofs.others ...


M.J. Osborne and A. Rubinstein.

A Course in Game Theory .
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MIT Press, 1994.

Ken Binmore.

Fun and Games - A Text on Game Theory .

D. C. Heath & Co., 1991.

Roger B. Myerson.

Game Theory: Analysis of Conict .

Harvard University Press, September 1997.

R.D. Luce and H. Raiffa.

Games and Decisions .

J. Wiley, New York, 1957.S Pinchinat (IRISA) Elements of Game Theory Master2 RI 2011-2012 64 / 64


J.F. Nash.Equilibrium points in n-person games.Proceedings of the National Academy of Sciences of the United States of
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America, 36:4849, 1950.

Andres Perea.Rationality in extensive form games .Boston : Kluwer Academic Publishers, 2001.

E. Gradel, W. Thomas, and T. Wilke, editors.Automata, Logics, and Innite Games: A Guide to Current Research[outcome of a Dagstuhl seminar, February 2001] , volume 2500 of Lecture Notes in Computer Science .Springer, 2002.

S Pinchinat (IRISA) Elements of Game Theory Master2 RI 2011 2012 64 / 64
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