1 perfect correlated equilibria in stopping games yuval heller tel-aviv university (part of my ph.d....

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Perfect Correlated Equilibria in Stopping Games

Yuval HellerTel-Aviv University

(Part of my Ph.D. thesis supervised by Eilon Solan)

http://www.tau.ac.il/~helleryu/

3rd Israeli Game Theory Conference

December 2008

Introduction:

Stopping games perfect correlated ()-equilibrium Main Result

Summary2

Proof OutlineReductions

Finite trees & absorbing games

Stochastic variation of Ramsey theorem

Equilibrium construction

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Stopping Games(Undiscounted, Multi-player, Discrete time)

Finite set of players: I

Unknown state variable: (state space)

Filtration: F=(Fn)

At each stage n the players receive a symmetric partial

information about the state : Fn()

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Stopping Games(undiscounted, multi-player, discrete time)

Stage 1 - everyone is active

Stage n: All active players simultaneously declare whether they

stop or continue

A player that stops become passive for the rest of the game

Player’s payoff depends on the history of players’ actions

while he has been active and on the state variable

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Literature: 2-player zero-sum Stopping Games

Dynkin (1969) – introduction, value where

simultaneous stops are not allowed

Neveu (1975) – value when each player prefers the

other to stop

Rosenberg, Solan & Vieille (2001) – use of

randomized strategies, value with payoffs’

integrability

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Literature: 2-player non-zerosum Stopping Games

Existence of approximate Nash equilibrium when the

payoffs have a special structure: Morimoto (86), Mamer

(87), Ohtsubo (87, 91), Nowak & Szajowski (99), Neumann,

Ramsey & Szajowski (02)

Recently, Shmaya & Solan (04) proved existence

assuming only integrability

Multi-player stopping games: no existence results

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Stopping Games - Applications

Most applications in the literature: Payoffs: Specific assumptions, such as monotony Discount factor 2 players

Multi-player variations are natural

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Struggle of survival in a declining market

At each turn, each firm loses money

A firm can stay or exit the market for good Partial production is inefficient Market is more profitable with less firms

Which firms survive? What is the exit order?

Ghemawat & Nalebuff (1985)...

Steel market in 70’s and 80’s

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Research & Development

Race for developing a patent

At each turn, continue spending money on research or leave the race

The first firm to complete the patent earns a lot Stochastic function of spent money

Fudenberg & Tirole (1985)…

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War of attrition

Attrition wars among animals: Becoming the leader (alpha-male) Territory Maynard-Smith (1982), Nalebuff & Riley (1985)…

2nd price auctions where all bidders pay Krishna & Morgan (1997)….

Political Sciences – lobbying Bulow & Klemperer (2001)

Introduction:


Summary11





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Perfect Equilibrium

Nash equilibrium may be sustained by non-credible threats of punishment Punisher receives a low payoff

The stronger concept of perfect equilibrium (Selten, 1965, 1975) has been studied. Examples: Fine & Li (1989): uniqueness in discounted 2-player games

with monotone payoffs Mashiah-Yaakovi (2008) – existence of ()-perfect

equilibrium when simultaneous stops aren’t allowed

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Correlated Equilibrium

Aumann (1974): An equilibrium in an extended game with a correlation device Device D sends each player i a private signal mi M i

(M=i M i) before the game starts according to (M)

The extended game G(D)

Consistent with Bayesian decision making (Aumann, 87)

Other appealing properties: computability, linear equations, closed and convex set

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Correlated Equilibrium in Sequential Games

Two main versions: Normal-form: signals are sent only before the game starts

Extensive-form: signals are sent at each stage

Equilibrium: normal-form extensive-form

Correlation among players is natural in many setups: Countries negotiate actions

Firms choose strategies based on market’s history

A manager coordinates the actions of his workers

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Normal-Form Correlation (1)

Sometimes players may coordinate before play starts but coordination along the play is costly / impossible:

Example (1) - war of attrition in nature: Commonly modeled as stopping games Coordination before play starts is implemented by

evolution of phenotype roles E.g.: Shmida & Peleg, 1997

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Normal-Form Correlation (2)

Example (2) - News playing among day traders: Monthly employment report will be published at noon Several minutes elapse before market adjusts

New information gradually arrives during that time Quick trading can be profitable See e.g., Christie-David, Chaudhry & Khan (2002)

Traders of a firm can coordinate their actions in advance Coordination along the play is costly (time limit) Traders may have different payoffs

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()-Perfect Correlated Equilibrium

– A bound for the probability of: An event E

Correlation device sends a signal in M’M

>0 – A bound for the maximal profit a player can

earn by deviating at any stage and after any history,

conditioned on that E and m M’

Extending the definitions for finite games: Myerson (1986), Dhillon & Mertens (1996)

Introduction:


Summary18





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Main Result

For every >0, a multi-player stopping game admits

a normal-form uniform perfect correlated

()-equilibrium with a universal correlation device Uniform: An approximate equilibrium in any long enough

finite game and in any discounted game with high enough

discount factor

Universal device – doesn’t depend on game payoffs

Corollary: Uniform perfect correlated equilibrium payoff

Introduction:






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Summary

1. Terminating games: game terminates at the first stop

2. Tree-like games (Shmaya & Solan, 03):

for every n, Fn is finite A finite collection of matrix payoffs

3. Deep enough in the tree: with high probability any matrix payoff either:

Repeats infinitely often Never occurs

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Reductions

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Reductions

Reductions require 2 properties from the equilibrium

(,-unrevealing - expected payoff of each player

“almost” doesn’t changeWith probability of at-least 1-, changes by less than

Universal - The correlation device D(G,,)

depends only on |I| and D(G,,)=D(|I|,)

Introduction:


Proof Outline




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Summary

Reductions

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Games on Finite Trees

Equivalent to an absorbing game: A stochastic game with a single non-absorbing state. 2 special properties: Recursive game – Payoff in non-absorbing states is 0 Single non-absorbing action profile

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An adaptation of a result of Solan & Vohra (2002):

A game on a finite tree has one of the following:

1. Non-absorbing equilibrium (game never stops)

2. Stationary absorbing equilibrium. Adaptations:

Perfection

Limit minimal per-round terminating probability

3. A special distribution: allows to construct a correlated

-equilibrium. Adaptations: unrevealing, universal device

Introduction:


Proof Outline




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Summary

Reductions

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Ramsey Theorem (1930)

A finite set of colors

Each 2 integers (k,n) are colored by c(k,n)

There is an infinite sequence of integers

k1<k2<k3<… such that: c(k1,k2) =c(ki,kj) for all i<j

0 1 2 3 4 5 6 7 8 9 10 11 12k1

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Stochastic Variation of Ramsey Theorem (Shmaya & Solan, 04)

Coloring each finite sub-tree.

There is an increasing sequence of stopping times: 1<2<3<…, such that: Pr(c(1,2) =c(,) =….)>1-

Low probability

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Introduction:


Proof Outline




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Summary

Reductions

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Equilibrium Construction

Each finite tree is colored according to: Whether it has a non-absorbing perfect equilibrium, an

absorbing perfect equilibrium, or a special distribution

The equilibrium payoff

The maximal payoffs when a player stops alone

If c implies that each game on finite tree has a perfect

equilibrium, concatenate the equilibria to obtain an

approximate perfect equilibrium of G

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Equilibrium Construction

Last case: c implies that a special distribution exists

This allow to construct an approximate unrevealing

perfect correlated equilibrium with a universal

correlation device An adaptation of the protocol of Solan and Vohra (01)

Introduction:


Proof Outline




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Summary

Reductions

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Summary and Future Research

Summary: every multi-player stopping game admits an

approximate normal-form uniform perfect correlated

equilibrium with a universal correlation device

Future research: Using this notion of equilibrium in the study of other dynamic

games

Structure of uniform perfect correlated equilibrium payoffs in

specific applications

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Questions & Comments?

Y. Heller (2008), Perfect correlated equilibria

in stopping games, mimeo.

http://www.tau.ac.il/~helleryu/

Introduction:


Proof Outline




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Summary

Reductions

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Reduction to Terminating Games

Proposition: Every game that stops immediately admits a (,)-

unrevealing perfect correlated ()-equilibrium with a universal correlation device

Every stopping game admits the same kind of

()-equilibrium

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Proof Outline

Induction on the number of players

Given a stopping game G, we define an auxiliary terminating

game G’: The payoff to I \ S when a coalition S stops is the

equilibrium payoff in the induced stopping game

G’ admits an unrevealing perfect correlated

()-equilibrium with a universal correlation device

Concatenation gives such an equilibrium in G

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Tree-like Games

Shmaya & Solan (2002) showed that any stopping game can be approximated by a tree-like stopping game, with the same set of approximate equilibria Small perturbations of the payoffs don’t change the set of

approximate equilibria we can assume that the payoff process has a finite range

Each set FnFn can be identified with a node in a tree

Tree-like Games – Shmaya & Solan’s Proof Outline

kth partition: Discretization of the game:

Depth: k

Precision:

Refinement of all previous partitions Defines the kth approximating game on a tree

The game on finite tree that begins on m and ends on l will be played on the m+l approximating game

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Tree-like Games

F4F4

F1

F2

F4

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Deep Enough in the Tree

Fn

v1

v1v1

v1

v1, v3, v5 occur infinitely often, all other vV do not occur at all

v1

v1

v1

G (Fn): The induced game

that begins at the node Fn

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Lemma - Induced Games

Let: G - a terminating game, - a stopping time

Every induced game G (Fn), where Fn is in the range of

, admits an unrevealing perfect correlated ()-equilibrium with a universal correlation device

G admits the same kind of (C·)-equilibrium

Corollary: We can assume to be “deep enough”

Proof Outline

Until the players follow an equilibrium in a finite stopping game with absorbing states {F} with payoffs {xF

equilibria payoffs of G (F)

After players follow ()-equilibrium of G (F)

Relying on that the equilibrium is unrevealing and with a universal correlation device

Illustration….

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Proof Outline

F

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x1

x2 x3

V(|I|,) - universal correlation device

x4

x5 x6

x7

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gi: maximal payoff player i can get by stopping alone

The special distribution over (nodes · players): A stopping player i and a node with maximal payoff

(Rii,n=gi)

The distribution gives each player i at-least gi

Each stopping player has a punisher j that stops when

Rji,n<gi

Allows to construct a correlated -equilibrium

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Stationary Absorbing Equilibrium: Adaptations

Perfection - using a perturbed tree with probability to

ignore players’ requests to stop

Limiting the minimal per-round terminating probability

(adapting the methods of Shmaya & Solan, 2004) If there is a player i with a payoff below gi, then can’t be too

small or player i stops when his payoff is gi

Otherwise either case 3 applies, or there is a node where at-least 2

players stop with a non-negligible probability Recursive trimming of such nodes gives the needed limit

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Last case: c implies that a special distribution exists

Let ik be the k-th time that player i’s maximal payoff

occur with the requirement ik> j

k-1 for all i, j

Using the fact that we are “deep enough” in the tree

An approximate unrevealing perfect correlated

equilibrium with a universal correlation device is

constructed as follows…

Equilibrium Construction: Protocol Description

`

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Equilibrium Construction: Protocol Description

A quitter i’ is secretly chosen according to the special distribution

A number l’ is chosen uniformly in {1,T’}

i’ receives the signal l’

A number l is chosen uniformly in {l’+1,l’+T} 1<<T<<T‘

The punisher of i’ receives the signal l

Each other player receives a signal l+1

Approximate unrevealing perfect correlated equilibrium: each player stops at l (when l is his signal) modulo 1+T+T’

Introduction:


Proof Outline




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Summary

Reductions

1 perfect correlated equilibria in stopping games yuval heller tel-aviv university (part of my ph.d....

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