Variational inequality formulation of
chance-constrained games
Abdel Lisser
Joint work with Vikas Singh from IIT Delhi
Université Paris Sud XI
Computational Management Science Conference
Bergamo, Italy
May, 2017
Abdel Lisser
Variational inequality formulation of chance-constrained games
Outline of the talk
1
Introduction
2
The model
3
Existence of Nash Equilibrium
4
VI Formulation of NE
5
Generalized Nash Equilibrium
6
VI Formulation of GNE
7
Conclusion
Abdel Lisser
Variational inequality formulation of chance-constrained games
Introduction
We consider a chance-constrained game (CCG) where each player
has continuous strategy set which does not depend on the strategies
of other players.
1
2
We show that there exists a Nash equilibrium of a CCG.
We characterize the set of Nash equilibria of a CCG using the
solution set of a variational inequality (VI) problem.
We consider the case where the continuous strategy set of each
player is defined by a shared constraint set.
1
2
We show that there exists a generalized Nash equilibrium for a CCG.
We characterize the set of a certain types of generalized Nash
equilibria of a CCG using the solution set of a VI problem.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Outline of the talk
1
Introduction
2
The model
3
Existence of Nash Equilibrium
4
VI Formulation of NE
5
Generalized Nash Equilibrium
6
VI Formulation of GNE
7
Conclusion
Abdel Lisser
Variational inequality formulation of chance-constrained games
The model
We consider an n-player non-cooperative game with random payoffs
defined by a tuple pI, pX i qiPI , pr i qiPI q.
Let I “ t1, 2, ¨ ¨ ¨ , nu be a set of players.
For each i P I, X i Ă Rmi denote the set of all strategies of player i,
and denote x i its generic element.
Śn
řn
Let X “ i“1 X i Ă Rm , m “ i“1 mi , be the set of all strategy
profiles of the game.
Śn
Denote, X ´i “ j“1;j‰i X j , and x ´i P X ´i be a vector of strategies
x j , j ‰ i.
For each i P I, r i is a random payoff function of player i.
Abdel Lisser
Variational inequality formulation of chance-constrained games
The model
` ˘li
For each i P I, let ξ i “ ξki k“1 be a random vector defined by
ξ i : Ω Ñ Rli , where pΩ, F, Pq be a probability space
For a given strategy profile x “ px 1 , x 2 , ¨ ¨ ¨ , x n q, and for an ω P Ω
the realization of random payoff of player i, i P I, is given by
r i px , ωq “
li
ÿ
fki px qξki pωq
k“1
where fki : Rm Ñ R for all k “ 1, 2, ¨ ¨ ¨ , li , i P I.
Let αi P r0, 1s be the confidence (risk) level of player i, and
α “ pαi qiPI .
Abdel Lisser
Variational inequality formulation of chance-constrained games
The model
For a given strategy profile x P X , and a given confidence level
vector α the payoff function of player i, i P I, is given by
`
˘
uiαi px q “ suptγ|P tω|r i px , ωq ě γu ě αi u.
We assume that the probability distribution of random vector ξ i ,
i P I is known to all the players.
For a fixed α P r0, 1sn , the payoff function of a player defined above
is known to all the players.
The above CCG is a non-cooperative game with complete
information.
Abdel Lisser
Variational inequality formulation of chance-constrained games
The model
A strategy profile x ˚ P X is said to be a Nash equilibrium of a CCG
at α if for each i P I, the following inequality holds
uiαi px i˚ , x ´i˚ q ě uiαi px i , x ´i˚ q, @ x i P X i .
We can also equivalently write the definition of Nash equilibrium as,
x i˚ P argmint´uiαi px i , x ´i˚ q| x i P X i u, @ i P I.
We have the following general result for the existence of a Nash
equilibrium for a CCG.
Abdel Lisser
Variational inequality formulation of chance-constrained games
The model
Theorem 1
For each player i P I, and a fixed α P r0, 1sn suppose
(i) the strategy set X i is non-empty, convex, and compact,
(ii) the payoff function uiαi : Rm Ñ R is continuous,
(iii) for every x ´i P X ´i , uiαi p¨, x ´i q is a concave function of x i .
Then, there always exists a Nash equilibrium of a CCG at α.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Outline of the talk
1
Introduction
2
The model
3
Existence of Nash Equilibrium
4
VI Formulation of NE
5
Generalized Nash Equilibrium
6
VI Formulation of GNE
7
Conclusion
Abdel Lisser
Variational inequality formulation of chance-constrained games
Existence of Nash Equilibrium
We consider the case where ξ i „ Ellippµi , Σi q with Σi ą 0 .
For a given x P X , r i px q “ pf i px qqT ξ i follows a univariate elliptically
i
symmetric distribution with parameters µT
i f px q and
i
T
i
pf px qq Σi f px q.
ZiS “
i
r i px q´µT
i f px q
1{2 i
f px q||
||Σi
, i P I, follows a univariate spherically symmetric
distribution with parameters 0 and 1.
Let φ´1
p¨q be a quantile function of a spherically symmetric
ZiS
distribution.
For a given x P X and α, we have,
Abdel Lisser
Variational inequality formulation of chance-constrained games
Existence of Nash Equilibrium
`
˘
uiαi px q “ suptγ| P tω|r i px , ωq ě γu ě αi u
+¸
+
#
˜#
ˇ r i px , ωq ´ µT f i px q
i
γ ´ µT
ˇ
i
i f px q
“ sup γ|P
ωˇ
ď
ď 1 ´ αi
1{2
1{2
||Σi f i px q||
||Σi f i px q||
!
)
1{2 i
´1
i
“ sup γ|γ ď µT
f
px
q
`
||Σ
f
px
q||φ
p1
´
α
q
.
i
S
i
i
Z
i
And,
1{2
´1
i
i
uiαi px q “ µT
i f px q ` ||Σi f px q||φZ S p1 ´ αi q, i P I.
i
Abdel Lisser
Variational inequality formulation of chance-constrained games
Existence of Nash Equilibrium
Assumption 1
For each player i, i P I, the following conditions hold.
1
X i is a non-empty, convex and compact subset of Rmi .
2
fki : Rm Ñ R is a continuous function of x , for all k “ 1, 2, ¨ ¨ ¨ , li .
3
(a) For every x ´i P X ´i , fki p¨, x ´i q, k “ 1, 2, ¨ ¨ ¨ , li , is an affine
function of x i .
or
(b) For every x ´i P X ´i , fki p¨, x ´i q, k “ 1, 2, ¨ ¨ ¨ , li , is non-positive
and a concave function of x i , and µi,k ě 0 for all k “ 1, 2, ¨ ¨ ¨ , li ,
and all the elements of Σi are non-negative.
Lemma 2
If Assumption 1 holds, for every x ´i P X ´i , uiαi p¨, x ´i q, i P I is a concave
function of x i for all αi P p0.5, 1s.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Existence of Nash Equilibrium
Theorem 3
Consider an n-player non-cooperative game with random payoffs. Let
random vector ξ i „ Ellippµi , Σi q, i P I, where Σi ą 0. If Assumption 1
holds, there always exists a Nash equilibrium of a CCG for all
α P p0.5, 1sn .
Abdel Lisser
Variational inequality formulation of chance-constrained games
Outline of the talk
1
Introduction
2
The model
3
Existence of Nash Equilibrium
4
VI Formulation of NE
5
Generalized Nash Equilibrium
6
VI Formulation of GNE
7
Conclusion
Abdel Lisser
Variational inequality formulation of chance-constrained games
Variational Inequality formulation
Given a closed, convex set K and a continuous function G, solving
the VI(K , G) consists in finding a vector z P K such that
py ´ zqT Gpzq ě 0, @ y P K .
It is well known that the Nash equilibrium problem of a
non-cooperative game can be formulated as a variational inequality
problem (See Facchinei 2003)
We formulate the Nash equilibrium problem of a CCG as a
variational inequality (VI) problem.
The continuity and differentiability of the payoff function of each
player, in its own strategy, are required in VI formulation. It exists
under the following Assumption.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Variational Inequality formulation
Assumption 2
For each i P I, the following conditions hold:
1
For every x ´i P X ´i , fki p¨, x ´i q is a differentiable function of x i , for
all k “ 1, 2, ¨ ¨ ¨ , li .
2
The system fki px q “ 0, k “ 1, 2, ¨ ¨ ¨ , li , has no solution.
If Assumption 2 holds, the gradient of payoff function of player i is
given by,
∇x i uiαi px i , x ´i q
`
˘T
Jf i p¨,x ´i q px q Σi f i px i , x ´i qφ´1
p1 ´ αi q
˘T
ZiS
“ Jf i p¨,x ´i q px q µi `
,
1{2
||Σi f i px i , x ´i q||
`
where Jf i p¨,x ´i q px i q is the Jacobian matrix of f i p¨, x ´i q.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Variational Inequality formulation
Define, a function F : Rm Ñ Rm ,
F px q “ pF1 px q, F2 px q, ¨ ¨ ¨ , Fn px qqT , where for each i P I,
`
Fi px q “ ´ Jf i p¨,x ´i q px q
˘T
µi ´
`
˘T
p1 ´ αi q
Jf i p¨,x ´i q px q Σi f i px i , x ´i qφ´1
ZS
i
1{2
||Σi f i px i , x ´i q||
.
Theorem 4
Consider an n-player non-cooperative game with random payoffs. Let
random vector ξ i „ Ellippµi , Σi q, i P I, where Σi ą 0. Let Assumptions
1-2 holds. Then, for an α P p0.5, 1sn , x ˚ is a Nash equilibrium of a CCG
if and only if it is a solution of VI(X , F ).
Abdel Lisser
Variational inequality formulation of chance-constrained games
Outline of the talk
1
Introduction
2
The model
3
Existence of Nash Equilibrium
4
VI Formulation of NE
5
Generalized Nash Equilibrium
6
VI Formulation of GNE
7
Conclusion
Abdel Lisser
Variational inequality formulation of chance-constrained games
Generalized Nash Equilibrium for CCG
We consider the case where the strategy set of each player depends
on the strategies of other players ùñ Generalized Nash equilibrium.
Let X i px ´i q Ă Rmi , i P I, be the strategy set of player i for a given
strategy profile x ´i of other players.
A set of Ś
all feasible strategy profiles is defined by
n
X px q “ i“1 X i px ´i q.
A strategy profile x ˚ P X px ˚ q is said to be a generalized Nash
equilibrium, for a given α P r0, 1sn , of a CCG, if for each i P I the
following inequality holds
uiαi px i˚ , x ´i˚ q ě uiαi px i , x ´i˚ q, @ x i P X i px ´i˚ q.
We can also equivalently write the definition of generalized Nash
equilibrium as
x i˚ P argmint´uiαi px i , x ´i˚ q| x i P X i px ´i˚ qu, @ i P I.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Generalized Nash Equilibrium for CCG
We assume that a convex and compact set R Ă Rm is given, and for
each x ´i the strategy set of player i is defined as,
X i px ´i q “ tx i P Rmi |px i , x ´i q P Ru.
This is an important case of generalized Nash equilibrium problem
considered in the fundamental paper by Rosen (1965).
It appears more often in practice, e.g., when all players share
common resources.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Generalized Nash Equilibrium for CCG
Theorem 5
Consider an n-player non-cooperative game with random payoffs. Let
random vector ξ i „ Ellippµi , Σi q, i P I, where Σi ą 0. The strategy set of
each player is given by X i px ´i q. If condition 2 and condition 3 of
Assumption 1 holds, there always exists a generalized Nash equilibrium
for a CCG for all α P p0.5, 1sn .
Facchinei (2007) proposed a variational inequality whose solution is
a solution of the generalized Nash equilibrium problem considered by
Rosen (1965).
We show that a solution of VI(R, F ) is a generalized Nash
equilibrium of a CCG.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Outline of the talk
1
Introduction
2
The model
3
Existence of Nash Equilibrium
4
VI Formulation of NE
5
Generalized Nash Equilibrium
6
VI Formulation of GNE
7
Conclusion
Abdel Lisser
Variational inequality formulation of chance-constrained games
Variational inequality formulation
Theorem 6
Consider an n-player non-cooperative game with random payoffs. Let
random vector ξ i „ Ellippµi , Σi q, i P I, where Σi ą 0. The strategy set of
each player is given by X i px ´i q. Let condition 2 and condition 3 of
Assumption 1 and Assumption 2 holds. Then, for an α P p0.5, 1sn , a
solution of VI(R, F ) is a generalized Nash equilibrium of a CCG.
In general, a generalized Nash equilibrium is not a solution of VI.
However, if R is defined by a finite number of convex and
continuously differentiable constraints, there exists a set of
generalized Nash equilibria which are solutions of a VI.
This generalized Nash equilibrium is called a variational equilibrium.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Variational inequality formulation
Let R be defined by a finite number of constraints as follows:
R “ tx P Rm | gk px q ď 0, @ k “ 1, 2, ¨ ¨ ¨ , K u,
(1)
where all the constraints gi : Rm Ñ R, i “ 1, 2, ¨ ¨ ¨ , K , are convex
and continuously differentiable.
For an α P p0.5, 1sn , let x be a generalized Nash equilibrium of a
CCG corresponding to elliptically symmetric distributed random
vector ξ i , i P I, and Assumptions 1-2 hold.
Then, for each i P I, x i is a solution of the following convex
optimization problem
min
´uiαi px i , x ´i q
i
x
s.t.
gk px i , x ´i q ď 0, @ k “ 1, 2, ¨ ¨ ¨ , K .
Abdel Lisser
(2)
Variational inequality formulation of chance-constrained games
Variational inequality formulation
A point x P Rm , where the KKT conditions of (2) corresponding to
each player are satisfied simultaneously, is a generalized Nash
equilibrium of a CCG if and only if the following conditions hold:
+
`
˘T
´ ∇x i uiαi px i , x ´i q ` Jgp¨,x ´i q px q λi “ 0, @ i P I,
(3)
0 ď λi K gpx i , x ´i q ď 0, @ i P I,
where λi P RK is a vector of Lagrange multipliers corresponding to
player i, K implies that at least one side of inequality is equality.
If constraint gk is inactive, the corresponding Lagrange multiplier is
zero. However, for active constraints the Lagrange multipliers can be
different for each player.
Abdel Lisser
Variational inequality formulation of chance-constrained games
Variational inequality formulation
Assume x is a solution of VI(R, F ), where R is defined by (1).
Let x satisfies Abadie CQ. Then, the following KKT conditions are
necessary and sufficient for the solution of VI(R, F ):
+
`
˘T
F px q ` Jgp¨q px q λ “ 0
0 ď λ K gpx q ď 0.
Using the aforementioned definition of F p¨q, the above KKT
conditions can be written as,
,
˘T ˛
˛ ¨`
¨
/
Jgp¨,x ´1 q px q
/
´∇x 1 u1α1 px 1 , x ´1 q
/
`
˘T ‹
/
/
˚ ´∇x 2 u2α2 px 2 , x ´2 q ‹ ˚
/
Jgp¨,x ´2 q px q ‹
/
‹ ˚
˚
.
˚
‹
λ
“
0,
‹`˚
˚
..
.
‹
.
‚
˝
.
˝
‚
.
/
`
˘T
/
/
´∇x n unαn px n , x ´n q
/
Jgp¨,x ´n q px q
/
/
/
0 ď λ K gpx q ď 0.
Abdel Lisser
(4)
Variational inequality formulation of chance-constrained games
Variational inequality formulation
Theorem 7
Consider an n-player non-cooperative game with random payoffs. Let
random vector ξ i „ Ellippµi , Σi q, i P I, where Σi ą 0. The strategy set of
each player is defined using shared constraints’ set given by (1). Let
condition 2 and condition 3 of Assumption 1, and Assumption 2 holds,
and α P p0.5, 1sn . Then,
(a) If x ˚ is a solution of a VI(R, F ) such that px ˚ , λ˚ q satisfy KKT
conditions (4), x ˚ is a generalized Nash equilibrium of a CCG at
which KKT conditions (3) hold with λ1˚ “ λ2˚ “ ¨ ¨ ¨ “ λn˚ “ λ˚ .
(b) If x ˚ is a generalized Nash equilibrium of a CCG at which KKT
conditions (3) hold with λ1˚ “ λ2˚ “ ¨ ¨ ¨ “ λn˚ , x ˚ is a solution of
a VI(R, F ).
Abdel Lisser
Variational inequality formulation of chance-constrained games
Outline of the talk
1
Introduction
2
The model
3
Existence of Nash Equilibrium
4
VI Formulation of NE
5
Generalized Nash Equilibrium
6
VI Formulation of GNE
7
Conclusion
Abdel Lisser
Variational inequality formulation of chance-constrained games
Conclusion
We proved the existence of a Nash equilibrium of a CCG for
elliptically symmetric distributed random payoffs and continuous
strategy set for each player.
We characterize the set of Nash equilibria of a CCG using the
solution set of a VI problem.
For the case of shared constraints, we proved the existence of a
generalized Nash equilibrium and give a characterization of the set
of a certain types of generalized Nash equilibria using the solution
set of a VI problem.
Abdel Lisser
Variational inequality formulation of chance-constrained games