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n -Player Stochastic Games with Additive Transitions

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Frank ThuijsmanJános Flesch & Koos VriezeMaastricht University

European Journal of Operational Research 179 (2007) 483–497

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- Model
- Brief History of Stochastic Games
- Additive Transitions
- Examples

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as = (a1s , a2s , , ans) joint action

rs(as) = (r1s(as), r2s(as), , rns(as)) rewards

ps(as) = (ps(1|as), ps(2|as), , ps(z|as)) transitions

1

s

z

- Infinite horizon
- Complete Information
- Perfect Recall
- Independent and Simultaneous Choices

rs(as)

ps(as)

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F

3

0, 0, 0

0, 0, 0

N

L

2

R

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

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

0, 0, 0

0, 0, 0

0, 0, 0

0, 0, 0

T

(1, 0, 0, 0)

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

1

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

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

0, 0, 0

0, 0, 0

1

B

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

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

1, 3, 0

0, 3, 1

3, 0, 1

(0, 1, 0, 0)

(0, 0, 1, 0)

(0, 0, 0, 1)

2

3

4

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general strategy i : N×S ×H→Xi

(k, s, h) →Xis

Markov strategy fi : N×S→Xi

(k, s) →Xis

stationary strategy xi : S→Xi

(s) →Xis

opponents’ strategy -i , f-i and x-i

mixed actions

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-Discounted rewards (with 0 << 1)

is() = Es((1-) k k-1Rik)

Limiting average rewards

is() = Es(limK→K-1Kk=1Rik)

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-Discounted minmax

vis = inf -isup i is()

Limiting average minmax

vis = inf -isup i is()

Highest rewards player i can defend

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= (i)iN is an -equilibrium if

is(i, -i) ≤is() +

for all i, for all i and for all s.

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Any -equilibrium?

0, 0

(0, 0, 0, 1)

0, 0

3, -1

0, 0

0, 0

2,1

(0, 0.5, 0.5, 0)

(1, 0, 0, 0)

(0, 1, 0, 0)

(0, 0, 1, 0)

(0, 0, 0, 1)

2

1

3

4

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Shapley, 1953

0-sum, “discounted”

Everett, 1957

0-sum, recursive, undiscounted

Gillette, 1957

0-sum, big match problem

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Fink, 1964 & Takahashi, 1964

n-player, discounted

Blackwell & Ferguson, 1968

0-sum, big match solution

Liggett & Lippmann, 1969

0-sum, perfect inf., undiscounted

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Kohlberg, 1974

0-sum, absorbing, undiscounted

Mertens & Neyman, 1981

0-sum, undiscounted

Sorin, 1986

2-player, Paris Match, undiscounted

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Vrieze & Thuijsman, 1989

2-player, absorbing, undiscounted

Thuijsman & Raghavan, 1997

n-player, perfect inf., undiscounted

Flesch, Thuijsman, Vrieze, 1997

3-player, absorbing example, undiscounted

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Solan, 1999

3-player, absorbing, undiscounted

Vieille, 2000

2-player, undiscounted

Solan & Vieille, 2001

n-player, quitting, undiscounted

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ps(as) = ni=1is pis(ais)

pis(ais) transition probabilities controlled by player i in state s

istransition power of player i in state s

0 ≤is ≤1 and iis = 1 for each s

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ps(as) = ni=1is pis(ais)

21 = 0.7

p21(1)=(1, 0, 0)

p21(2)=(0, 1, 0)

11 = 0.3

(1, 0, 0)

(0.3, 0.7, 0)

p11(1) = (1, 0, 0)

p11(2) = (0, 0, 1)

(0.7, 0, 0.3)

(0, 0.7, 0.3)

1

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- 0-equilibria for n-player AT games (threats!)
- 0-opt. stationary strat. for 0-sum AT games
- Stat. -equilibria for 2-player abs. AT games
- Result 3 can not be strengthened,
neither to 3-player abs. AT games,

nor to 2-player non-abs. AT games,

nor to give stat. 0-equilibria

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Additive Transitions

induce

a Complete Ordering of the Actions

If ais is “better” than bis against some strategy,

Then ais is “better” than bis against any strategy.

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Consider strategies ais and bis for player i

If, for some strategya-iswe have

t Sps(t | ais , a-is ) vit ≥ t Sps(t | bis , a-is ) vit

Then for all strategies b-iswe have

t Sps(t | ais , b-is ) vit ≥ t Sps(t | bis , b-is ) vit

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If t Sps(t | ais , a-is)vit ≥ t Sps(t | bis , a-is)vit

Then

ist Sps(t | ais) vit + j ijst Sps(t | a-js) vit ≥

ist Sps(t | bis) vit + j ijst Sps(t | a-js) vit

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ist Sps(t | ais) vit + j ijst Sps(t | b-js) vit ≥

ist Sps(t | bis) vit + j ijst Sps(t | b-js) vit

And therefore

t Sps(t | ais , b-is)vit ≥ t Sps(t | bis , b-is)vit

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The “best” actions for player i in state s

are those that maximize the expression

t Sps(t | ais) vit

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Let G be the original AT game and

let G* be the restricted AT game,

where each player is restricted

to his “best” actions.

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Now v*i≥ vi for each player i.

In G* : t Sps(t | a*s) v*it = v*isi, s, a*s

In G : t Sps(t | bis , a*-is ) vit < visi, s,

a*-is , bis

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If x*iyields at least v*i in G*,

then x*iyields at least v*i in G as well.

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0.5

(1, 0, 0)

(0, 1, 0)

0, 0

0, 0

(1, 0, 0)

(1, 0, 0)

(0.5, 0.5, 0)

-3, 1

-1, 3

0, 0

0, 0

0.5

(0.5, 0, 0.5)

(0, 0, 1)

(0, 0.5, 0.5)

(0, 1, 0)

(0, 0, 1)

2

1

3

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NO stationary 0-equilibrium

0, 0

0, 0

(1, 0, 0)

(0.5, 0.5, 0)

-3, 1

-1, 3

0, 0

0, 0

(0.5, 0, 0.5)

(0, 0.5, 0.5)

(0, 1, 0)

(0, 0, 1)

2

1

3

B, L

T, L

T, L

T, R

B, R

-1, 3

0, 0

0, 0

-3, 1

-2, 2

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1-/2

/2

stationary -equilibrium

with >0

0, 0

0, 0

0

(1, 0, 0)

(0.5, 0.5, 0)

-3, 1

-1, 3

0, 0

0, 0

1

(0.5, 0, 0.5)

(0, 0.5, 0.5)

(0, 1, 0)

(0, 0, 1)

2

1

3

equilibrium rewards ≈ ((-1-, 3-), (-3,1), (-1,3))

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non-stationary 0-equilibrium

Player 1: B, T, T, T, ….

Player 2: R, R, R, R, ….

0, 0

0, 0

(1, 0, 0)

(0.5, 0.5, 0)

-3, 1

-1, 3

0, 0

0, 0

(0.5, 0, 0.5)

(0, 0.5, 0.5)

(0, 1, 0)

(0, 0, 1)

2

1

3

equilibrium rewards ((-2, 2), (-3, 1), (-1, 3))

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NO stationary -equilibrium

with >0

0, 0

p

q

1- q

(0, 0, 0, 1)

0, 0

3, -1

0, 0

0, 0

2,1

1- p

(0, 0.5, 0.5, 0)

(1, 0, 0, 0)

(0, 1, 0, 0)

(0, 0, 1, 0)

(0, 0, 0, 1)

2

1

3

4

p > 1 -

q > 0

p <

q = 0

q > 0

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non-stationary 0-equilibrium

Player 1: T, B, B, B, ….

Player 2: R, R, R, R, ….

0, 0

(0, 0, 0, 1)

0, 0

3, -1

0, 0

0, 0

2,1

(0, 0.5, 0.5, 0)

(1, 0, 0, 0)

(0, 1, 0, 0)

(0, 0, 1, 0)

(0, 0, 0, 1)

2

1

3

4

equilibrium rewards ((2, 1), (0, 0), (3, -1), 2, 1))

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alternative 0-equilibrium

Player 1: T, T, T, T, ….

Player 2: L, R, R, R, ….

0, 0

(0, 0, 0, 1)

0, 0

3, -1

0, 0

0, 0

2,1

(0, 0.5, 0.5, 0)

(1, 0, 0, 0)

(0, 1, 0, 0)

(0, 0, 1, 0)

(0, 0, 0, 1)

2

1

3

4

equilibrium rewards ((2, 1), (2, 1), (3, -1), 2, 1))

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F

3

0, 0, 0

0, 0, 0

N

L

2

R

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

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

0, 0, 0

0, 0, 0

0, 0, 0

0, 0, 0

T

(1, 0, 0, 0)

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

1

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

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

0, 0, 0

0, 0, 0

1

B

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

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

1, 3, 0

0, 1, 3

3, 0, 1

How to share 4 among three people

if only few solutions are allowed?

(0, 1, 0, 0)

(0, 0, 1, 0)

(0, 0, 0, 1)

2

3

4

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F

3/2, 1/2, 2

3, 0, 1

N

L

R

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

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

0, 0, 0

0, 1, 3

4/3, 4/3, 4/3

2, 3/2, 1/2

T

(1, 0, 0, 0)

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

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

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

1/2, 2, 3/2

1, 3, 0

1

B

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

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

1, 3, 0

0, 1, 3

3, 0, 1

(0, 1, 0, 0)

(0, 0, 1, 0)

(0, 0, 0, 1)

2

3

4

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F

3/2, 1/2, 2

3, 0, 1

3, 0, 1

N

L

R

1/3 *

2/3 *

0, 0, 0

0, 1, 3

0, 1, 3

4/3, 4/3, 4/3

2, 3/2, 1/2

T

1/3 *

1 *

2/3 *

1/2, 2, 3/2

1, 3, 0

1, 3, 0

B

NO stationary -equilibrium

1/3 *

2/3 *

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F

3/2, 1/2, 2

3, 0, 1

N

L

R

1/3 *

2/3 *

0, 0, 0

0, 1, 3

4/3, 4/3, 4/3

2, 3/2, 1/2

T

1/3 *

1 *

2/3 *

1/2, 2, 3/2

1, 3, 0

B

non-stationary 0-equilibrium

1/3 *

2/3 *

Player 1 on B: 1, ¾, 0, 0, 0, 0, 1, ¾, 0, 0, 0, 0, ….

Player 2 on R:0, 0, 1, ¾, 0, 0, 0, 0, 1, ¾, 0, 0, ….

Player 3 on F:0, 0, 0, 0, 1, ¾, 0, 0, 0, 0, 1, ¾,….

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F

3/2, 1/2, 2

3, 0, 1

N

L

R

1/3 *

2/3 *

0, 0, 0

0, 1, 3

4/3, 4/3, 4/3

2, 3/2, 1/2

T

1/3 *

1 *

2/3 *

1/2, 2, 3/2

1, 3, 0

B

equilibrium rewards (1, 2, 1)

1/3 *

2/3 *

Player 1 on B: 1, ¾, 0, 0, 0, 0, 1, ¾, 0, 0, 0, 0, ….

Player 2 on R:0, 0, 1, ¾, 0, 0, 0, 0, 1, ¾, 0, 0, ….

Player 3 on F:0, 0, 0, 0, 1, ¾, 0, 0, 0, 0, 1, ¾,….

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- 0-equilibria for n-player AT games (threats!)
- 0-opt. stationary strat. for 0-sum AT games
- Stat. -equilibria for 2-player abs. AT games
- Result 3 can not be strengthened,
neither to 3-player abs. AT games,

nor to 2-player non-abs. AT games,

nor to give stat. 0-equilibria

Center for the Study of Rationality

Hebrew University of Jerusalem

?

frank@math.unimaas.nl

Center for the Study of Rationality

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GAME VER

Center for the Study of Rationality

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GAME VER

Center for the Study of Rationality

Hebrew University of Jerusalem

GAME VER

Center for the Study of Rationality

Hebrew University of Jerusalem

GAME VER

Center for the Study of Rationality

Hebrew University of Jerusalem