SELFISH AGENTS ON THE INTERNET

courageouscellistΤεχνίτη Νοημοσύνη και Ρομποτική

29 Οκτ 2013 (πριν από 3 χρόνια και 5 μήνες)

45 εμφανίσεις

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

SELFISH AGENTS ON THE INTERNET

© Peter van Emde Boas

© Peter van Emde Boas

Noam Nisan

Algorithms for Selfish Agents

Mechanism Design for Distributed Computing

Proc. STACS’99, Springer LNCS 1563, pp 1
--
15

Invited paper. See also STOC’99 and SOFSEM 2000

Amir Ronen

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

The Internet according to
Nissan & Ronen

Distributed Computation by a
Network

of
Agents

Agents

are
Unreliable

and
Uncontrolable
; not

Byzantine

but
Selfish
:

Game Theoretical Solution

for Controling

their Behavior.

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

The Internet as seen by a
Game Theoretician

Distributed Computation by a
Network

of
Agents


Agents

are
Unreliable

and
Uncontrolable
; not

Byzantine

but
Selfish
:
Quisque sibi proximus....

Game Theoretical Solution

for Controling

this Zoo.

© Scott Adams

© Scott Adams

© the Games Workshop

© the Games Workshop

© the Games Workshop

© the Games Workshop

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Sample Scenarios


Resource Allocation


Disc Space, Processing Power,
Bandwidth, .....


Routing


Trafic density, Bandwidth, Congestion


Electronic Trade


Auctions, Leveraging, Brokering

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Approach


Joint Work
Nissan

&
Ronen


Based on
Mechanism Design


Solutions based on
Known

Mechanisms


Related to Applications of Game
Theory in
Distributed AI

(
Agents
) and
Theory of
Communications Networks

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Model

n

Players
:
1, ... , n

:
Agents

Each Agent has a
private input

t
i

: its
Type
, and an

reported

input

t
i

or
action

a
i


Output Specification
:
F: t = ( , t
i

, ) => o


W


set of

allowed

outcomes
.

Valuations

v
i

(t
i

,o)

of outcome for the agents

Objective function

g: W => R

which must be

optimized

Payments

p
i

which are added yielding
Utilities
:

u
i

= v
i

(t
i

,o) + p
i


Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Mechanism

The
Mechanism

M

has as
inputs

the

reported types

t
i

or actions
a
i


It
computes

an
outcome

o =

o( , t
i
’, )

and
payments

p
i
( , t
i


, )

[
o( , a
i

, )

and
p
i
( , a
i
, )

].


The agents score their utilities:

u
i

:= v
i
(o( , a
i
, ), t
i
) + p
i
( , a
i
, )

So the Mechanism generates for every tuple of types

a Game

This mechanism should have
Dominating Strategies
:

whatever the inputs are

all players in the resulting game
possess a
Dominating Strategy
. Moreover the output
should be
allowed

and
minimize

the
Objective Function
.

Then it is called an
implementation

or
solution
.

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Domination Conditions

a

: tuple of actions
( ,a
i
, )

a
-
i

: tuple of actions
( ,a
i
, )

of the remaining agents.

(a
-
i
,a
i
)

: tuple of actions decomposed with agent
i

playing


her
dominating strategy
.

(a
-
i
,a
i
’)

: tuple of actions with an alternative for agent
i


The
dominating

strategies a
i

satisfy (for all agents i)


u
i

= v
i
(o(a
-
i
,a
i
), t
i
) + p
i
(a
-
i
,a
i
) ≥ u
i
’ = v
i
(o(a
-
i
, a
i
’), t
i
) + p
i
(a
-
i
, a
i
’)



o p
i

o’ p
i


Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Examples

In the examples presented
actions

amount to
reporting

a type
. Subsequently the
mechanism

computes an

outcome

and
payments

to the agents.

What must be shown is that the outcome is
allowed

assuming that the agents report
truthfully
. Moreover

a single agent

can not
improve its utility

by reporting

a
lie
......
It doesn’t make a difference whether the others

report the truth or a lie
.


Maximum
: serving the most valuable client

Threshold
: caching some object provided there is


enough interest in it.

Shortest Path
: how to pay the cable
-
owners

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Truthful Implementation

The action of an agent consists of
reporting its type
,

more specifically its
true type
.


Playing the truth

is the
dominating strategy


FACT:
If there exists a mechanism then there exists

also a Truthful Implementation
.


Proof:
Just simulate the hypothetical implementation

based on the actions derived from the reported types
.

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Maximum

Agents are
clients

asking to be
served
.

Each client
ascribes

value
T
i


to being served.

The mechanism must serve the client with the

highest value
. However the clients may lie by

reporting
T
i
’ ≠ T
i

The solution evidently will
serve the client who

reports the highest value
, but a which price??


Non Solutions
:

Free Service

(payment
0
)


This will invite all clients to
exagerate

Full Payment

(
p
i

= T
i

) ; this will invite clients to


underreport
, leading potentially to selection


of the wrong one.....

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Solution of Maximum

Serve the client reporting the highest value and charge

him the value reported by the second highest client
.

This is the so
-
called
Vickrey Auction
.


Claim
:
Reporting the Truth is a Dominating Strategy
.


Proof
: Supose Client
i

is not the highest client.

If Client
i

is
not served
, then his
utility

u
i

= 0

and remains

so unless he
outbids

the highest client
k

at value
T
k

.

But then he
pays

T
k
resulting in utility
T
i

-

T
k

≤ 0

.

Similarly the highest client has no influence on the price

paid, unless he
underreports

so much that he looses
both

service

and the
positive utility

collected by being served.

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Threshold

Object
X

should be
loaded in cache
, provided the

agents using the system ascribe
together

more value to

it being loaded than a
cost
-
price

C

. The
value


ascribed by agent
i

equals
T
i
. However each agent

may declare a value
T
i
’ ≠ T
i

Evidently object

X

is cached whenever

S

T
i
’ ≥ C
, but

at
which prices

to the agents ?


Non Solutions
:

Share the cost:
p
i

=
-
C/n

.

This will invite every agent


with
T
i

≥ C/n

to
declare

C

and so make
certain



that
X

is cached.

Proportional pricing:
This invites
underreporting
,


possibly resulting in the decision not to cache....


Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Solution of Treshold

Each Client pays the
difference

between the cost
C

and the
total value ascribed

to
X

by the
others
. In case

this difference is
negative

(
the others already enforce

that X is loaded
)
nothing

is paid.

p
i

=
-

max(C
-

S

k≠i

T
k

, 0 )

This is the so
-
called
Clarke Tax
.


Analysis
: as long as the
resulting decision

is
unaffected

by a
lie

by agent
i

his
payment

remains
constant
.

If by the
lie

X

is
not loaded
, agent
i

looses his (
positive
)

utility; If by exageration
X

gets loaded agent
i

pays

C
-

S

k≠i

T
k
, an amount
exceeding

his value

T
i

, which

yields a
negative

utility.

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Shortest Paths

Agents own
edges

in the (
bi
-
connected
) network. Each

edge
e

creates
expenses

t
e

for being used to its owner.

A connection between
source

x

and
sink

y

must

be established at
minimum cost
.
Cost

=
sum of edge

costs along the path
, denoted
D(G)

.

Evidently, after having been informed about the
claimed

edge costs

t
e


, the
Mechanism

will compute a
shortest

path

in the reported graph, at cost
D(G’)

. Which
rewards


should be granted to the edge owners ?


Answer
: the owner of edge
e

obtains payment
0

if

e

is
not used

in the selected path. Otherwise she will

collect
D(G
\
{e})
-

[ D(G’)
-

t
e
’ ]

; read this to be the

reduction

in the
total cost to the others

for using

e

.

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Analysis Shortetst Paths

A
lie

by the owner of
e

which leaves the shortest path

unaffected

won’t change the reward and utility

for this owner.

Exagerating

the cost so much that
e

is removed
only

occurs

when the
difference

t
e

-

t
e

≥ D(G
\
{e})
-

D(G)


0
;

(
otherwise

e

wouldn’t belong to the true shortest path).


Consider the utility

u
e

granted to the owner of
e

, had she

been
truthful
. We have:

u
e

= D(G
\
{e})
-

(D(G)
-

t
e
)
-

t
e
= D(G
\
{e})
-

D(G) ≥ 0

;


Since
e

is removed by the
lie

the resulting utility becomes

zero
. So
nothing

is gained by the
lie
.

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Analysis Shortetst Paths

Underreporting

the cost so much that
e

is
inserted

in the

shortest path means that the
difference


t
e
-

t
e
’ ≥ D(G)
-

D(G
\
{e})


0
;

(
otherwise

e

already would belong to the shortest path).


Consider the utility
u
e

granted to the owner of
e
, had she

been
truthful
. We have:

u
e

= 0 ≥ D(G
\
{e})
-

D(G) = D(G
\
{e})
-

(D(G)
-

t
e
)
-

t
e
=

D(G’
\
{e})
-

(D(G’)
-

t
e
’)
-

t
e

. This is exactly the
utility

for the

owner of
e

collected by her
lie
, so the result of her
lie

is

not positive
. Again
nothing

is gained by the
lie
.

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Morale

These examples are all examples of so
-
called

Vickrey
-

Groves
-

Clarke (VGC) Mechanisms
:


Objective function

G(o,t) =
S

v
i
(o,t
i

)

, and

allowed outcomes

are those maximizing the

objective function.


Payments

p
k

:=
S

i≠k

v
i
(o,t
i

) + h
k
(t
-
k
)


Theorem
:
VGC Mechanisms are truthful
.


Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Analysis

Suppose that the claim is false and consider a
counterexample.

t: true types , d: reported types , i: an agent,

d
i
’ an alternative move for this agent. WLOG t
k

= d
k

for k ≠ i



u
i

= v
i
(o(d
-
i
,t
i
), t
i
) + p
i
(o( d
-
i
, t
i
), t
i
) <

u
i
’ = v
i
(o(d
-
i
, d
i
’), t
i
) + p
i
(o( d
-
i
, d
i
’), d
i
’) =



v
i
(o(d
-
i
, d
i
’), t
i
) + p
i
(o( d
-
i
, d
i
’), t
i
)

which for the given VGC mechanism amounts to:

S

v
k
(o(d
-
i
,t
i
), t
k
) + h
i
(d
-
i
) <
S

v
k
(o(d
-
i
,d
i
’), t
k
) + h
i
(d
-
i
)


but then the mechanism has failed to produce an allowed

solution.....

Introduction Game Theory 2007
-
2008. Peter van Emde Boas

Loose Ends

How are we ever going to
compute

the rewards in

our mechanisms
efficiently

??

The definition requires us to solve
n

instances

of the
optimization problem

in order to tweak the

system into correctly solving a
single

instance....


VGC mechanisms and the problems tackled using

them, are only the beginning....