A risk-based core
László Á. Kóczy
koczy@krtk.mta.hu
MTA-KRTK & Óbuda University
12th Meeting of the Society for Social Choice and Welfare
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
1 / 14
The theory of the (coalition structure) core is well developed for
characteristic function form games.
In partition function form games the value of a coalition depends
on the entire partition,.
Core generalisations are bad or “ugly”: difficult to compute
Here we take a risk and probability-based approach
This approach generalises earlier approaches.
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
2 / 14
Partition function form games
Characteristic function form game
A pair (N, v ):
a set N of players
a characteristic function c : 2N → R
Partition function form game
A pair (N, v ):
a set N of players
a partition function c : E → R, where
◮
E = (S, P)|S ∈ 2N , P ∈ Π, S ∈ P the set of embedded coalitions
Partition function form games can model the externalities of coalition
formation.
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
3 / 14
Core
Many core generalisations exist:
The difference is in estimating the value of a deviating coalition.
α-core: worst case (Aumann and Peleg, 1960; Rosenthal, 1971;
Richter, 1974)
γ-core: individually best responses (Chander and Tulkens, 1997)
δ-core: joint response (Hart and Kurz, 1983)
r-core, recursive core: response and residual partition is
endogenously determined (Huang and Sjöström, 2003; Kóczy,
2007)
probabilistic approach with payoff-dependent probabilities (Lekeas
and Stamatopoulos, 2011)
The r-core and recursive core are difficult to calculate, others ignore
residual incentives.
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
4 / 14
Motivation from finance
The expectations have two sides:
How residual players react determines the probabilities of residual
partitions
The (usual) risk-aversion of deviating players: (usually) focus
more on bad outcomes
Finance view: deviation = investment of current payoff into a risky
asset.
The value of a risky asset is Xs in state s ∈ S
What is the value of the risky asset? ⇒ Risk measures.
Risk measure
ρ : X → R how much cash must be added to make asset with
realisation vector acceptable.
(When negative: how much cash can be traded for asset.)
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
5 / 14
Quantiles and risk measures
Quantiles
qp : [0, 1] → R how much maximal payoff in worst p ∈ [0, 1] cases.
monotone increasing
right continuous
Spectral risk measure (Acerbi, 2002)
The measure Mφ : X → R defined by
Mφ (x) = −
Z
1
0
φ(p)qp dp
is spectral if φ ∈ RS is
1
2
3
Nonnegative: φ(p) ≥ 0 for all p ∈ [0, 1],
R1
Normalized: 0 φ(p)dp = 1,
Monotone : φ is non-increasing, i.e. φp1 ≥ φp2 if 0 ≤ p1 < p2 ≤ 1.
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
6 / 14
Risk-based core
Consider a partition function form game (N, V ) and a common spectral
weight function φ.
Simple risk-based (or φ-) core
C
payoff configurations (x, P) such that
Pφ (N, V ) collects
φ (C) for all C ⊆ N.
x
≥
−M
i∈C i
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
7 / 14
The risk-based core generalises old approaches I
The α-core: Cα (N, V ) = limε→0 Cφε (N, V ) where p > 0 and
(
1
if 0 ≤ p ≤ ε
φε (p) = ε
0 otherwise
The optimistic core: Cω (N, V ) = limε→0 Cφε (N, V ) where p > 0
and
(
0 if 0 ≤ p ≤ 1 − ε
φε (p) = 1
otherwise
ε
The γ-core: Cγ (N, V ) = C1 (N, V ) with φ(x) = 1 and
n
o
(
1
if
P
=
{i}
|i
∈
C
pC (P) =
0 otherwise.
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
8 / 14
The risk-based core generalises old approaches II
The cohesion-core: Cc (N, V ) = C1 (N, V ) with φ(x) = 1 and
n o
(
C
1
if
P
=
pC (P) =
0 otherwise.
Payoff proportional weights: Cpp (N, V ) = C1 (N, V ) with
pC (P) = P
V (C, {C} ∪ P)
,
Q∈Π(C) V (C, {C} ∪ Q)
The pessimistic simple recursive core:
RCα (N, V ) = limε→0 Cφε (N, V ) where
(
1
if RCα (C, V {C} ) 6= ∅ and ∃x : (x, P) ∈ RCα (C, V {C} )
pC (P) = c
0 otherwise,
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
9 / 14
The risk-based core generalises old approaches III
c is the number of partitions P such that
∃x : (x, P)α ∈ RC(C, V {C} ) and
(
0 if 0 ≤ p ≤ 1 − ε
φε (p) = 1
otherwise
ε
The optimistic simple recursive core:
RCω (N, V ) = limε→0 Cφε (N, V ) where
(
1
if RCω (C, V {C} ) 6= ∅ and ∃x : (x, P) ∈ RCω (C, V {C} )
C
p (P) = c
0 otherwise,
c is the number of partitions P such that
∃x : (x, P) ∈ RCω (C, V {C} ) and
(
1
if 0 ≤ p ≤ ε
φε (p) = ε
0 otherwise
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
10 / 14
Inclusion results
Theorem 1
Consider a partition function form game (N, V ) a distribution p of the
residual coalitions and two spectral functions φ and ψ such that φ(p)
first order stochastically dominates ψ(p), then Cφ (N, V ) ⊇ Cψ (N, V ).
Expanding the quantile functions state-by-state proves result.
The theorem implies many of the usual inclusions between
optimistic/pessimistic cores.
Theorem 2
Consider a partition function form game (N, V ) a spectral function φ.
Let also q1 and q2 denote two quantile functions such that
q1 (p) ≥ q2 (p). Then Cq1 (N, V ) ⊆ Cq2 (N, V ).
Unfortunately the condition does not hold in the ‘missing’ inclusion
relations.
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
11 / 14
Inclusion results
Theorem 1
Consider a partition function form game (N, V ) a distribution p of the
residual coalitions and two spectral functions φ and ψ such that φ(p)
first order stochastically dominates ψ(p), then Cφ (N, V ) ⊇ Cψ (N, V ).
Expanding the quantile functions state-by-state proves result.
The theorem implies many of the usual inclusions between
optimistic/pessimistic cores.
Theorem 2
Consider a partition function form game (N, V ) a spectral function φ.
Let also q1 and q2 denote two quantile functions such that
q1 (p) ≥ q2 (p). Then Cq1 (N, V ) ⊆ Cq2 (N, V ).
Unfortunately the condition does not hold in the ‘missing’ inclusion
relations.
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
11 / 14
What we get for (almost) free
Quantile functaions can also be used to describe
fuzzyness
future contingencies in multistage games
probabilistic payoffs due to external factors.
The probabilistic approach allows us to model
Cooperative fuzzy games (Aubin, 1981)
Games with uncertainty and externalities (Habis and Csercsik,
2012)
Network games with uncertainty and externalities (Csercsik and
Kóczy, 2012)
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
12 / 14
Conclusions
The risk based core can handle residual incentives and the
deviators’ risk aversion
(Spectral) risk measures are a very general means to deal with
risk aversion
The risk based core generalises known core generalisations to
PFF games
Many of the usual inclusion properties are generalised others turn
out to be very special cases.
Future work
Applications with uncertainty
◮
◮
◮
Time dimensions
Physical risks (e.g. power networks)
Financial risks
Implementation
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
13 / 14
References
Acerbi, C., 2002, Spectral measures of risk: A coherent representation
of subjective risk aversion, Journal of Banking & Finance 26,
1505–1518.
Aubin, J.-P., 1981, Cooperative fuzzy games, Mathematics of
Operations Research 6, 1–13.
Aumann, R. J. and B. Peleg, 1960, Von Neumann-Morgenstern
solutions to cooperative games without side payments, Bulletin of
the American Mathematical Society 66, 173–179.
Chander, P. and H. Tulkens, 1997, The core of an economy with
multilateral environmental externalities, International Journal of
Game Theory 26, 379–401.
Csercsik, D. and L. A. Kóczy, 2012, Efficiency and stability in electrical
power transmission networks: A partition function form approach,
Tech. rep., HAS-CERS, Budapest.
Habis, H. and D. Csercsik, 2012, Cooperation with Externalities and
Uncertainty, Tech. Rep. 1229.
Hart, S. and M. Kurz, 1983, Endogenous Formation of Coalitions,
Econometrica 51, 1047–1064.
Huang, C.-Y. and T. Sjöström, 2003, Consistent solutions for
cooperative games with externalities, Games and Economic
Behavior 43, 196–213.
Kóczy, L. A., 2007, A recursive core for partition function form games,
Theory and Decision 63, 41–51.
Lekeas, P. and G. Stamatopoulos, 2011, Cooperative oligopoly games:
a probabilistic approach , 1–10.
Richter, D. K., 1974, The Core of a Public Goods Economy,
International Economic Review 15, 131–142.
Rosenthal, R. W., 1971, External Economies and Cores, Journal of
Economic Theory 3, 182–188.
L. Á. Kóczy (KRTK & Óbuda)
12th SSCW Meeting
14 / 14