Transcript bachrach.ppt

Computing Shapley Values, Manipulating
Value Distribution Schemes, and Checking
Core Membership in Multi-Issue Domains
Vincent Conitzer and Tuomas Sandholm
Agenda

Introduction to coalitional games
 Multi-Issue Domains
 How to distribute the gains
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The Core
Shapley Value
Other marginal contribution schemes
Computing the Shapley value
 Manipulating contribution schemes
 Checking core membership
Coalitional Games
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Coalition formation is a key part of automated
negotiation between self-interested agents
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Several of companies can unite into a virtual
organization to take more diverse orders and gain
more profit
Truck delivery companies can share truck space, as
the cost is mostly dependant on the distance rather
than on the weight carried
Coalition formation has been studied extensively
in game theory, and solution concepts were
adopted in multi agent systems
Coalitional Games Solutions

Given a coalitional game we want to find the
distribution of the gains of the coalition between
the agents
 Different solution concepts have different
objectives
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The Core promotes stability
The Shapley value promotes fairness
Game theory has studied these solution
concepts for quite some time, but the
computational aspect has received little attention
Some Questions to Keep in Mind
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How much should each of the employees of the
company be paid to make sure a group of them
won’t be bought away by another company?
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Get a value division in the core
A few truck delivery companies unite to carry a
high load of deliveries. How can the profits be
divided fairly?
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The Shapley value division
Coalitional Games With Side
Payments
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The game is presented as a characteristic
function
 Let A be the set of agents (players)
 Each potential coalition S has a value v(S)

The value is independent of what the non members of
the coalition do
The characteristic function: v : 2 A  R
 Typically it is increasing: S1  S2  V ( S1 )  V ( S2 )
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Super additivity
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The characteristic function is super additive if for
all disjoint sets of a S,T we have:
v(S1 )  v(S2 )  v(S1  S2 )
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This means every two subsets can do better if
they unite
 Finally we would get the grand coalition of all the
agents
 This does not always hold:
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Hard optimization problem to decide what to do united
Anti trust laws
Multi Issue Domains
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The characteristic function is the sum of values
of independent issues
We have sub-games (v1 , v2 ,..., vT )
The characteristic function (for every subset S of
A) is
Every coalition gets the sum of what it gets in all
the sub games v( S )   v ( S )
If a game is a decomposition to increasing
(super additive) sub games, it is also increasing
(super additive)
T
i 1

i
Games Concerning a Subset of the
Agents
 We
say vi only concerns a subset of the
agents Ci  A if Ci  S1  Ci  S2  vi (S1 )  vi (S2 )
 Assuming that each of the sub games
concerns only a small subset of the agents
we can improve our calculations
 Our representation of the characteristic
function now only requires a small fraction
of the space it once took:  2
T
i 1
Ci
Solution Concepts

On a super additive game, the grand coalition is
likely to form, and the coalition gets v(A)
 How much does each agent gets?
 We want a value division d : A  R
d (a)  v( A)
 We want to divide all the gains: 
aA
The Core
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The best known solution concept
Proposed by Gillies (1953) and von Neumann &
Morgenstein (1947)
A value division is in the core if no sub coalition
has an incentive to break away
A value division d is blocked by a sub coalition S
if v(S )   d (a)
If d is blocked by S, it is not in the core
Some coalitional games have an empty core
aS
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Player Types
 Dummy
players add nothing to all
coalitions: v(S  {a})  v(S )
 Equivalent players add the same to any
coalition that does not contain any of the
two players: S : S {a1, a2}    v(S {a1})  v(S {a1})
The Shapley Value (Cont.)
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A well know value division scheme
Aims to distribute the gains in a fair manner
A value division that conforms to the set of the
following axioms:
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Dummy players get nothing
Equivalent players get the same
If a game v can be decomposed into two sub games,
an agent gets the sum of values in the two games:
S : v( S )  u ( S )  w( S ) 
a : dV (a )  dU (a )  dW (a)
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Only one such value division scheme exists
The Shapley Value
an ordering  of the agents in A, we
define S (, a) to be the set of agents of A
that appear before a in 
 The Shapley value is defined as the
marginal contribution of an agent to its set
of predecessors, averaged on all possible
permutations of the agents:
 Given
1
Sh( A, a) 
(v(S (, a)  a)  v( S (, a)))

A! 
A Simple Way to Compute The
Shapley Value
 Simply
go over all the possible
permutations of the agents and get the
marginal contribution of the agent, sum
these up, and divide by |A|!
 Extremely slow
 Can we use the fact that a game may be
decomposed to sub games, each
concerning only a few of the agents?
Computing the Shapley Value
 If
v can be decomposed to several sub
games, we know (from the axioms) that
T
Shv ( A, a )   Shvi ( A, a)
i 1
 If
only concerns C
player a, we have
vi
i
A
then for any
1
Shvi ( A, a) 
(vi ( S (Ci , a)  vi ( S (Ci , a)))

Ci ! Ci
Computing the Shapley Value
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We do not really need to sum over all possible
orderings, but rather on all possible subsets of
agents that arrive before player a
 For each such sub set we get the same marginal
contribution of player a.
 If the sub set S has n agents, there are n!
ordering on the players inside. There are then
(|A|-n-1)! ways to complete this ordering to an
ordering on all agents. We get:
Sh( A, a) 
1
S !( A  S  1)!(v(S {a})  v(S ))

A ! S  A{a}
Computing the Shapley Value
Quickly in Multi Issue Domains
 Compute
the Shapley value for each sub
game, using the previous formula, only
taking into account the concerning agents,
then sum these up
 If we assume computation of factorials,
multiplication and addition in constant time
we get an time complexity of O( 2 ) or
less precisely O(T  2 )
T
i 1
max i Ci
Ci
Marginal Contribution Based Value
Division Schemes
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A marginal contribution scheme is a scheme that
chooses some ordering of the players, and
distributes the gains to the players according to
their marginal contribution
 If on the chosen orderings you add much to the
value of the coalition of the players before you
on the ordering, you deserve a nice share of the
profits
Marginal Contribution Based Value
Division Schemes
 For
the Shapley value we have considered
an average on all possible orders
 If we consider just one of the possible
orderings, the value an agent gets
depends on it location in the ordering
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Obviously, the agent has a specific location it
wants to be in
If the game is convex (you add to a coalition
at least as much as you add to any of its
subsets), you want to be last in the ordering
Marginal Contribution Based Value
Division Schemes (Cont.)
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If we randomly choose a permutation the
expectancy of the value distribution for an agent
is its Shapley value
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This requires a trusted source of randomness /
cryptography
Another way is to show that even if an agent has
total control on the ordering chosen, it would still
be computationally intractable for that agent to
find the optimal ordering for him
 The computational complexity is used as a
barrier for manipulation
Maximal Marginal Contribution
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Let v be a game decomposed as follows:
v  v
and the game vi only concerns Ci  A
 We are given an agent a and a number k, and
are asked if there is some S  A  {a} such that the
value v(S  {a})  v(S )  k
 We want to see if we can find a subset of the
agents to which a’s marginal contribution is at
least k
T
i 1
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i
These would be the agents before a in the ordering a
would choose
NP-Completeness of MaxMarginal-Contribution
 Conitzer
and Sandholm have shown that
Max-Marginal-Contribution is NPComplete, even in the case that Ci  3 and
all Vi ‘s take values in {0,1,2}
 The problem is in NP since for a given
subset of agents S  A  {a} we can simply
calculate the marginal contribution of a to
this subset
NP-Completeness of MaxMarginal-Contribution
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NP-hardness is proven by reducing an arbitrary
MAX2SAT instance to a Max-Marginal-Contribution
instance
In MAX2SAT we are given a set V of Boolean variables
and a set of clauses C, each with 2 literals, and a target
number r of satisfied clauses
For each variable v in V there is an agent Av
We also have an agent a, whose contribution we want to
maximize
For every clause c there is a sub game (issue) tc, that
only concerns the agents a and the agents representing
the variables in the clause c
NP-Completeness of MaxMarginal-Contribution
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The sub game results are as follows:
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1 point for having a in the coalition
1 point for having all the agents representing the
negative literals
But, if you want to get 2 points, you also have to have
one of the agents representing the positive literals
The marginal contribution we want is k=r
NP-Completeness of MaxMarginal-Contribution
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If there is a solution to MAX2SAT with r satisfied clauses,
take V+ - the variables set to true
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Given a solution S to max-marginal-contribution, look at
the assignment of true to everything in S, false otherwise
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What is the marginal contribution of a to this subset?
Hint: you either satisfied the clause by setting one of the
negative literals to false, or if you didn’t, you’ve set one of the
positive literals to true
If a sub game tc has increased the value by 1 due to adding a,
what can you say about the clause?
Open question: we have used increasing games here, so
the problem is NP-Complete even if the game is known
to be increasing. What is the complexity for super
additive games?
Checking Core Membership
 Let v
v  v
be a game decomposed as follows:
and the game vi only concerns Ci  A
 We are given a value division d : A  R that
may not even be feasible
T
i 1
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i
If it isn’t we can increase only the value of the
grand coalition to the point where it is (the
help of an outside benefactor for the stability)
 We
are asked if the division is in the core,
or if there is no blocking sub coalition for it
NP-Completeness of Checking
Core Membership
 Conitzer
and Sandholm have shown that
checking core membership (CHECKE-IFBLOCKED) is NP-Complete
 The problem is in NP since for a given
subset of agents we can simply calculate
the sum of their values in the division and
see if it is less than v(S)
NP-Completeness of Checking
Core Membership
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NP-hardness is proven by reducing an arbitrary VERTEX-COVER
instance to a core membership problem
We have an agent for each vertex, av, and another special agent a
We have a sub game for each edge, that only concerns agent a and
the agents of the edge’s vertices
The value of the sub game is 1 if the coalition contains agent a and
at least one of the edge’s vertices (we have agent a, and the edge is
covered)
1
The value distribution to check: d ( a )  E 
d ( av ) 
2
1
1
2( r  )
2
NP-Completeness of Checking
Core Membership
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If there is a vertex cover with W vertices
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What is the value of the coalition of these vertices and agent a?
How much do they get according to the value distribution?
If a set of agents is a blocking coalition
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It has to contain agent a (or they get nothing)
Consider the set of vertices of the agents in the blocking
coalition, W
How much do they get according to the value distribution?
Can the number of vertices in W be smaller than r?
To block, v(S) must be greater than v(a), since a is in the
blocking coalition
But then we have to get 1 for every sub game, so we have
covered all the edges, with r vertices or less
Conclusions
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Coalitional games important for automated negotiation
between agents
Such games can be decomposed to sub games (issues)
which only concern some of the agents
We can quickly compute the Shapley value in some of
these cases
Other marginal contribution value distribution schemes
can be manipulated, but the general case is hard (an
NP-complete problem)
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So such distribution schemes are acceptable in some cases,
even if some of the agents have control on the chosen ordering
Checking if a value distribution is stable (in the core) is
hard (and NP-Complete problem in the general case)
Open Questions

NTU games (no side payments)
 Finding value divisions the are even harder to
manipulate (eg. PSPACE-hard)
 Finding stability concepts that take into account
the complexity of finding a beneficial deviation
 The complexity of other (longer term) solution
concepts