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Eilon Solan. , . Tel Aviv University. Omri N. Solan. Tel Aviv University. with. Multiplayer Absorbing Games. . I. = a finite set of players.. . A. i. = a finite set of actions of player . i. . . A := A. ID: 458635

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## Presentations text content in The Modified Stochastic Game

The Modified Stochastic Game

Eilon Solan, Tel Aviv University

Omri N. SolanTel Aviv University

with

Slide2Multiplayer Absorbing Games

I = a finite set of players. Ai = a finite set of actions of player i. A := A1 × … × AI. r : A → RI = nonabsorbing payoff function. p : A → [0,1] = probability of absorption. r* : A → RI = absorbing payoff function.

0

1

1

0

*

*

Example: the Big Match

Slide3Multiplayer Absorbing Games

I = a finite set of players. Ai = a finite set of actions of player i. A := A1 × … × AI. r : A → RI = nonabsorbing payoff function. p : A → [0,1] = probability of absorption. r* : A → RI = absorbing payoff function.

Discounted Payoff

γλi(x) = λri(x) + (1- λ)(p(x)r*i(x) + (1-p(x)) γλi(x)

Expected absorbing payoff

Slide4Multiplayer Absorbing Games

r : A → RI = nonabsorbing payoff. p : A → [0,1] = prob. of absorption. r* : A → RI = absorbing payoff.

Discounted Payoff

γλi(x) = λri(x) + (1- λ)(p(x)r*i (x) + (1-p(x)) γλi(x)

λri(x) + (1- λ)p(x)r*i (x)

1-(1- λ)(1-p(x))

γλi(x) =

λ

ri(x) + (1- λ)p(x)r*i (x)

λ + (1- λ)p(x)

=

Slide5Multiplayer Absorbing Games

Discounted Equilibrium

λri(xλ) + (1- λ)p(xλ)r*i (xλ)

γλi (xλ) =

If

p(xλ) = o(λ) then lim γλ(xλ) = r(x0), and x0 is a uniform ε-equilibrium.

r : A → RI = nonabsorbing payoff. p : A → [0,1] = prob. of absorption. r* : A → RI = absorbing payoff.

If p(xλ) = ω(λ) then lim γλ(xλ) = lim r*(xλ), and xλ is a uniform ε-equilibrium provided λ is sufficiently small.

≥ vλi (=min-max value)

λ

+ (1-

λ

)p(x

λ

)

Slide6Multiplayer Absorbing Games

Discounted Equilibrium

λri(xλ) + (1- λ)p(xλ)r*i (xλ)

γλi (xλ) =

r : A → RI = nonabsorbing payoff. p : A → [0,1] = prob. of absorption. r* : A → RI = absorbing payoff.

If p(xλ) = Θ(λ) then lim γλ(xλ) is a convex combination of r*(xλ) and r(xλ). When |I|=2, we have (a) r(x0) ≥ lim γλ(xλ) or (b) lim r*(xλ1,x02) ≥ lim γλ(xλ) or (c) lim r*(x01,xλ2) ≥ lim γλ(xλ). There is a uniform ε-equilibrium (Vrieze and Thuijsman 89).

≥ vλi (=min-max value)

λ

+ (1-

λ

)p(x

λ

)

Slide7Modified Absorbing Games

Modified Discounted Payoff

λRi (x) + (1- λ)p(x)r*i (x)

Γλi (x) :=

r : A → RI = nonabs. payoff. Ri(x) := min{ ri(x), v0i }. p : A → [0,1] = prob. of absorption. r* : A → RI = absorbing payoff.

λ + (1- λ)p(x)

Theorem: Vλi := min max Γλi (xi,x-i) satisfies V0i=v0i.

x-i

xi

Theorem

: The modified game admits a discounted stationary equilibrium.

Slide8Modified Stochastic Games 0

0

2

Attempt 1

: The modified payoff is the minimum between the stage payoff and the stage max-min value.

Original game:

0

1

Modified game:

The max-min value changed!

Slide9Modified Stochastic Games 1

Let v1i,…,vLi be the different limit max-min value of player i.tλ(s1,σ;l) := E [ Σn=1 λ(1-λ)n-11 ]uλi(s1,σ;l) := E [ Σn=1 λ(1-λ)n-1ri(sn,an) 1 ]Γλi (s1,σ) := Σl=1L min{uλi(s1,σ;l) , vli tλ(s1,σ;l) }

∞

{v0i(s(n)) =vli}

The modified game is the normal-form game ( I, Σi, (Γλi (s1,σ)){i in I} ) .

{v0i(s(n)) =vli}

s1,σ

The max-min value

s

1,σ

Note

: The modified game depends on the initial state.

Slide10Modified Stochastic Games 2

Let τk be the k-th time in which the limit max-min value of player i changes:τ0 := 1 τk+1 := min{ n > τk : v0i(s(n)) ≠ v0i(s(τk)) }tλ(s1,σ;k) := E [ Σn=1 λ(1-λ)n-11 | H(τk) ]uλi(s1,σ;k) := E[ Σn=1 λ(1-λ)n-1ri(sn,an) 1 | H(τk) ]Γλi (s1,σ) := E[ Σk=0 min{ uλi(s1,σ;k) , v0i(s(τk)) tλ(s1,σ;k) } ]

∞

s1,σ

{τk ≤ n < τk+1}

{τk ≤ n < τk+1}

∞

∞

Slide11Results

Γλi (s1,σ) := Σl=1L min{uλi(s1,σ;l) , vli tλ(s1,σ;l) } Γλi (s1,σ) := E[ Σk=0 min{ uλi(s1,σ;k) , v0i(s(τk)) tλ(s1,σ;k) } ]

∞

Theorem: In both modified games, V0i (s1) =v0i (s1).

Theorem: The first modified game admits a discounted stationary equilibrium (that depends on the initial state). The second modified game admits a more complex equilibrium.

Question: Does there exist a stationary strategy that is almost optimal for all initial states?

Theorem

: Analog results hold for min-max modification.

Slide12Monovex Sets

Definition: A set X in Rd is monovex if for every x,y in X there is a continuous monotone path from x to y in X.

Theorem: In the first modified game, if the other players play stationary strategies, then player i has an optimal stationary best response. Moreover, the set of his stationary best responses is monovex.

Question

: Is every monovex set contractible?

Slide13Monovex Sets

Definition: A set X in Rd is monovex if for every x,y in X there is a continuous monotone path from x to y in X.

Theorem: Every upper semi-continuous set-valued function from a compact convex subset of Rd to itself with monovex nonempty values has a fixed point.

Question

: Is every monovex set contractible?

Slide14Merci

תודה רבה

Thank you

شكرا

To be continued

IHP, February 15, 2016, 10AM