Taxation and Stability in
Cooperative Games
AAMAS 2013
Yair Zick
Maria Polukarov
Nick R. Jennings
Cooperative TU Games
Agents divide into
coalitions; generate profit.
Coalition members can
freely divide profits.
4
$2
How should profits
be divided?
$5
1
2
3
5
6
$3
TU Games - Notations
Agents: N = {1,…,n}
 Coalition: SµN

Characteristic function: v: 2N→ R
 A TU game G= hN,viis anonymous, if
the value of a coalition is only a
function of its size.
 A TU game is monotone, if the value
of a coalition can only increase by
adding more agents to it.

Payoffs

We assume that only the grand coalition
N is formed.

Agents may freely distribute profits.

An imputation is a vector x = (x1,…,xn)
such that Σi2Nxi= v(N).

Individual rationality: each agent gets
at least what she can make on her own:
xi ≥ v({i})
The Core

Theis
total
be stable
The core
theamount
set oftoall
divided
1; ifS(w.l.o.g.)
outcomes:
foris all
µN we have
gets
more with
than 0,
x(S) ¸player
v(S);1 a
game
a
then players 2 and 3 get a
non-empty
coreofis
called
total payoff
less
than 1.stable.

May be empty in many games.

Example: the 3-majority game.

Three players; any set of size two or
more has a value of 1; singletons have
a value of 0.
Stabilizing Games

The 3-Majority game has an empty
core.

However, if one reduces the value of all 2player coalitions by 1/3, the core becomes
non-empty (giving 1/3 to each player is a
stable outcome). Similarly: reduce the
value of {2,3} to 0.
Reducing the value of some coalitions
can result in a stable game.
Our Work

We explore taxation methods (i.e.
reductions in coalition value), that
ensure stability.

What is the minimal amount of tax
required in order to stabilize a game?

Which taxation schemes are optimal?

When are known taxation schemes
optimal?
Some Background

Taxation is not new

"-core: coalition values reduced by ".

Least-core: corresponds to the "*-core,
where "* is the smallest " for which the
"-core is not empty.

Reliability extensions: each agent i
survives with probability ri; the value of a
coalition is reduced to its expected value.

Myerson graphs, etc…
1
x1 + x2 + x3 = v(N)
x1 = v(N) − v({2,3})
x3 = v({3})
x1 = v({1})
2
3
x3 = v(N) − v({1,2})
x1 + x2 + x3 = v(N)
1
"
x1 = v(N) − v({2,3})
x3 = v({3})
"
"
"
2
"
x1 = v({1})
"
3
x3 = v(N) − v({1,2})
x1 + x2 + x3 = v(N)
1
x1 = v(N) − v({2,3})
x3 = v({3})
"
2
x1 = v({1})
3
x3 = v(N) − v({1,2})
Exploring Taxation Methods


Given a game G= hN,vi, we say that
G’≤ G if v(S) ≥ v’(S) for all SµN.
A game G’ is maximal-stable w.r.t. G if

G’≤ G

It has a non-empty core
If G’’ is stable and G’≤ G’’ ≤ G, then one of

the inequalities holds with equality.
Exploring Taxation Methods

Increasing the value of any coalition
results in losing either stability or
dominance.

Observations:

Maximal-stable games still distribute v(N) to the
agents.

They are defined by a single vector
x = (x1,…,xn); the value of each coalition S is
min{v(S),x(S)}.
Optimal Taxation Schemes

The set of dominated games with a
non-empty core is a convex polyhedron
denoted S(G).

We are interested in the set of games
that minimizes the total tax taken.

These games are said to have optimal
taxation schemes.
Optimal Taxation Schemes

We characterize the optimal taxation
scheme for anonymous games.

Given an anonymous game, where
v(S) = f(|S|), the optimal taxation
scheme is given by reducing the value
of each coalition to min(f(|S|),f(|N|)/|S|).

Good for small coalitions; bad for large
coalitions.
Existing Taxation Schemes

Suppose that a central authority
wants to implement a taxation scheme.
What conditions must hold in order
for this taxation scheme to be
optimal?

We find conditions on the underlying
cooperative game which ensure this for
the "-core and for reliability extensions.
Conclusions & Future Work

Given a class of cooperative games,
what taxation schemes would be
optimal?

How much are we “over-taxing” by
using a given taxation scheme?

Other ways of measuring total taxation.

Computational complexity: efficient
computation of optimal taxes?
Thank you!
Questions?
P.S.: I am on the job market!