Distributionally Robust Games with Risk-averse Players
Nicolas Loizou
School of Mathematics, University of Edinburgh, Edinburgh, U.K.
Keywords:
Game Theory, Equilibrium, Distributionally Robust Optimization, Conditional Value at Risk.
Abstract:
We present a new model of incomplete information games without private information in which the players
use a distributionally robust optimization approach to cope with the payoff uncertainty. With some specific
restrictions, we show that our “Distributionally Robust Game” constitutes a true generalization of three popular
finite games. These are the Complete Information Games, Bayesian Games and Robust Games. Subsequently,
we prove that the set of equilibria of an arbitrary distributionally robust game with specified ambiguity set can
be computed as the component-wise projection of the solution set of a multi-linear system of equations and
inequalities. For special cases of such games we show equivalence to complete information finite games (Nash
Games) with the same number of players and same action spaces. Thus, when our game falls within these
special cases one can simply solve the corresponding Nash Game. Finally, we demonstrate the applicability
of our new model of games and highlight its importance.
1 INTRODUCTION
The classical, complete-information finite games as-
sume that the problem data (in particular the payoff
matrix) is known exactly by all players. In a now fa-
mous result((Nash et al., 1950), (Nash, 1951)), Nash
has shown that any such game has an equilibrium in
mixed strategies. More specifically, in his formulation
Nash assumed that all players are rational and that all
parameters(including payoff functions) of the game
are common knowledge. With these two assumptions,
the players can predict the outcome of the game. For
this reason each player given the other players strate-
gies is in a position to choose the mixed strategy that
gives him the maximum profit. A tuple of these strate-
gies is what we call “Nash Equilibrium”.
The existence of equilibria in mixed strategies was
later extended to a class of incomplete information
finite games by Harsanyi (Harsanyi, 1968), who as-
sumed that the payoff matrix is not known exactly but
rather represents a random variable that is governed
by a probability distribution known to all players. In
particular, Harsanyi assumed that a full prior distri-
butional information for all parameters of the game
is available and that all players use this information
in order to compute the payoff functions of the game.
This computation is made using the Bayes’ rule. For
this reason these games are called “Bayesian Games”
and their equilibrium “Bayesian Nash Equilibrium”.
In 2006, Aghassi and Bertsimas (Aghassi
and Bertsimas, 2006) proposed a new class of
distribution-free finite games where the payoff matrix
is only known to belong to a given uncertainty set.
This model relaxes the distributional assumptions of
Harsanyi’s Bayesian games, and it gives rise to an al-
ternative distribution-free equilibrium concept. Fur-
thermore, in this model of games the players use a
robust optimization approach to the uncertainty and
this is assumed to be a common knowledge. That is,
given the other players strategies each player tries to
maximise his worst case expected payoff (worst case
is taken with respect to the uncertainty set). The using
of the robust optimization approach is the reason for
calling these “Robust Games” and their equilibrium
“Robust Optimization Equilibrium”.
More recently, Qu and Goh (Qu and Goh, 2012)
proposed a distributional robust version of the finite
game where they only consider the case in which the
players are risk neutral and they focus on an applica-
tion of supply chain. This model later was extended
to continuous games from Sun and Xu (Sun and Xu,
2015). The approach of these papers is in contrast to
our framework of modelling the players as each seek-
ing to minimize his worst case CVAR expected loss.
Moreover they offer no ideas on computation of their
equilibria.
In this paper we present a new model of incom-
plete information games without private information
186
Loizou, N.
Distributionally Robust Games with Risk-averse Players.
DOI: 10.5220/0005753301860196
In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 186-196
ISBN: 978-989-758-171-7
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
in which the players use a distributionally robust op-
timization approach to cope with payoff uncertainty.
In our model players only have partial information
about the probability distribution of the uncertain pay-
off matrix. This information is expressed through a
commonly known ambiguity set of all distributions
that are consistent with the known distributional prop-
erties. Similar to the robust games framework, play-
ers in distributionally robust games adopt a worst
case approach. Only now the worst case is com-
puted over all probability distributions within the am-
biguity set. More specifically we use a worst case
CVaR(Conditional Value at Risk) approach. This al-
lows players to have several risk attitudes which make
our model even more coveted since in real life appli-
cations players rarely are risk neutral.
For classical work on slightly related game mod-
els we refer the reader to (Hayashi et al., 2005) and
(Nishimura et al., 2012). The recent paper (Singh
et al., ) also deals with similar model of games but
in contrast to our approach the authors focused on the
existence of equilibrium.
The remainder of this work is organized as fol-
lows: Section 2 introduces our notation as well as def-
initions that are used through the paper. Section 3 pro-
poses and analyses our new model for Distributionally
Robust Games. After formulating the model, we show
that any other finite game can be expressed as a dis-
tributionally robust game. In Section 4, we prove the
equivalence of the set of equilibria of a distribution-
ally robust game and the component-wise projection
of the solution set of multi-linear system of equations
and inequalities. Section 5 shows the equivalence of
distributional robust games and Nash games in special
cases. Sections 6 and 7 are devoted to an illustrative
example and numerical experiment, respectively. Fi-
nally, conclusions and future directions are drawn in
Section 8.
All proofs are relegated to the appendix.
2 NOTATIONS-DEFINITIONS
The following notational conventions are used in this
paper. Boldface upper case letters will denote matri-
ces and boldface lower case letters will denote vec-
tors. To denote uncertainty we will use the tilde ·)
in which the input parameter (·) can be either scalar,
vector, or matrix and the check ·) will indicate the
nominal counterpart of the uncertain coefficient ˜·. Fi-
nally, vec(A) denotes the column vector obtained by
stacking the row vectors of the matrix A one on top
of the other.
With
ˇ
P R
N×
N
i=1
a
i
we denote the payoff matrix of a
complete information game (fixed matrix) while with
˜
P the uncertainty matrix of the incomplete informa-
tion games.
In particular,
ˇ
P
i
( j
1
, j
2
,.... j
N
)
denotes the payoff to player
i when player k {1,2,....N} plays action j
k
{1,2,....,a
k
} and S
a
i
= {x
i
R
a
i
|x
i
0,
a
i
J
i
=1
x
i
j
i
=
1} expresses the set of all possible mixed strategies
of player i over all actions {1,2, ...a
i
}. Moreover, let
π
i
(P ;x
1
,x
2
,...x
N
) indicate the expected payoff of
player i when the payoff matrix is given by P and
player k {1, 2,....N} plays mixed strategy x
k
S
a
k
.
That is,
π
i
(P ; x
1
,x
2
,...x
N
) =
a
1
j
1
=1
...
a
i
j
i
=1
...
a
N
j
N
=1
P
i
( j
1
, j
2
,.... j
N
)
N
i=1
x
i
j
i
(1)
Finally, in this paper, we use exactly like Bertsimas
and Aghassi (Aghassi and Bertsimas, 2006) the fol-
lowing shorthands:
x
i
= (x
1
,x
2
,.., x
i1
,x
i+1
,...x
N
)
(x
i
,u
i
) = (x
1
,x
2
,.., x
i1
,u
i
,x
i+1
,...x
N
)
S =
N
i=1
S
a
i
, S
i
=
N
k=1,k6=i
S
a
k
and the following definitions for the equilibrium in
Nash games, Bayesian Games and Robust Games:
The tuple of strategies (x
1
,x
2
,...x
N
) S is:
Nash Equilibrium iff for each player i
{1,2....N}:
x
i
argmax
u
i
S
a
i
π
i
(
ˇ
P ;x
i
,u
i
) (2)
Bayesian Nash Equilibrium iff for each player
i {1, 2....N}:
x
i
argmax
u
i
S
a
i
[E
˜
P
π
i
(
˜
P ;x
i
,u
i
)] (3)
Robust Optimization Equilibrium iff for each
player i {1, 2....N},
x
i
argmax
u
i
S
a
i
[ inf
˜
P U
π
i
(
˜
P ;x
i
,u
i
)] (4)
3 THE NEW MODEL
In this section we present the new model of incom-
plete information games without private information
Distributionally Robust Games with Risk-averse Players
187
in which the players use a distributionally robust op-
timization approach to cope with payoff uncertainty.
We also show that under specific assumptions about
the ambiguity set and the values of risk levels, Distri-
butionally Robust Game constitutes a true generaliza-
tion of Nash, Bayesian and Robust Games.
We introduce the new model of Distributionally
Robust Games by first giving the two important defi-
nitions of Best Response and Distributionally Robust
Optimization Equilibrium and explain them later in
details.
Definition 3.1. In the distributionally robust model,
for the case without private information, players i’s
best response to the other players strategies x
i
S
i
must belong to:
argmin
u
i
S
a
i
sup
QF
Q-CVaR
ε
i
[π
i
(
˜
P ;x
i
,u
i
)] (5)
Definition 3.2. (x
1
,x
2
,...x
N
) S is said to be a
Distributionally Robust Optimization Equilibrium of
the corresponding game with incomplete information
iff i {1,2,..N},
x
i
argmin
u
i
S
a
i
sup
QF
Q-CVaR
ε
i
[π
i
(
˜
P ;x
i
,u
i
)] (6)
Our approach can be considered a concept closely
related to both Harsanyi’s Bayesian Games (Harsanyi,
1968) and Robust Games (Aghassi and Bertsimas,
2006). In more detail, in Bayesian Games we as-
sume that all players of the game know the exact dis-
tribution of the payoff matrix. Now, in the Distri-
butionally Robust approach the players do not know
the exact distribution. Instead, they are only aware
of a commonly known ambiguity set F of all possi-
ble probability distributions Q that satisfy some spe-
cific properties. These distributions have no restric-
tion in their form. That is, the ambiguity set may
consists of both, discrete and continuous distributions
of the payoff matrix. In addition, similar to the ro-
bust games framework, we assume that each player
adopts a worst case approach to the uncertainty. Only
now the worst case is computed over all probability
distributions within the set F . For this formulation
which is similar to distributionally robust optimiza-
tion concept we named these games Distributionally
Robust Games(DRG) and we refer to their equilib-
ria as Distributionally Robust Optimization Equi-
libria (DROE).
3.1 CVAR and Main Assumptions
Definition 3.3. (Conditional Value at Risk)
CVaR
ε
of a loss distribution L is the expected value of
all losses that exceed (1 ε)-quantile of the distribu-
tion. This can be formalized as:
Q-CVaR
ε
(L) = min
ζR
ζ +
1
ε
E
Q
[L ζ]
+
(7)
where [x]
+
= max{x,0}.
Conditional Value at Risk(CVaR) is one of the
most popular quantile-based risk measures because
of its desirable computational properties (Rockafellar
and Uryasev, 2000),(Artzner et al., 2002). Exactly
for these properties we chose to introduce CVaR in
the formulation of the new model.
Using Q-CVaR
ε
i
, we allow the players to have several
risk attitudes which is a major difference compared
to all other finite games (Nash, Bayesian and Robust
Games) in which the players are always risk neutral.
Important hypothesis is that risk attitude is a fixed
characteristic of each player and it cannot be changed
depending the game. It is not a notion like the mixed
strategy that a player can choose in order to achieve
his best response and minimise his loss. More
specifically, the parameter ε
i
(0,1) determines
the risk-aversion of each decision-maker. In detail,
if player i has risk level ε
i
= 1 this means that he
is risk neutral since the Conditional Value at Risk
is equal to the expected value of his loss function
(Q-CVaR
ε
i
= E
Q
). On the other hand if ε
i
1 the
player is risk averse and as ε
i
0 the risk aversion
of the player becomes larger.
Conclusively, as parameter ε
i
decreases, the value
of Q-CVaR
ε
i
increases and the risk aversion of the
player becomes larger and vice versa.
With the introduction of a risk measure in our model
we take into account not only that each player
wishes to maximize his gain (minimize loss) but also
how much she/he is willing to risk to achieve this
maximum value (minimum value).
1
st
Assumption of the New Model: The risk
attitude of each player is assumed to be common
knowledge. That is, each player knows how much
risk averse are the other players and that all other
players know that he knows.
In general, CVaR can be calculated from either
the probability distribution of gains or the probability
distribution of losses. In this paper we decide to
follow the original formulation of Rockafellar and
Uryasev (see (Rockafellar and Uryasev, 2002) and
(Rockafellar and Uryasev, 2000)) and calculate
CVAR from the distribution of losses.
For this reason, in definitions (3.1) and (3.2)
we use the expected loss function of player i,
π
i
(P ;x
i
,x
i
). Loss distributions are also re-
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
188
sponsible for the use of argmin
u
i
S
a
i
sup
QF
instead of
argmax
u
i
S
a
i
inf
QF
that we use in robust games.
Finally, in the formulation of the DRG we make two
more assumptions.
2
nd
Assumption: The players commonly know the
ambiguity set of all possible distributions (discrete
and continuous) of the payoff matrix.
3
rd
Assumption: Each player adopts, like Robust
games, a worst case approach to the uncertainty, only
now the worst case is computed over all probabil-
ity distributions within the set F . In particular we
assume that all players use a worst case CVaR ap-
proach.
3.2 Generalization of all other Games
From the formulation of the DRG (definitions of best
response and equilibrium) we can easily understand
that only two are the parameters that are amenable to
change. These are, risk level ε
i
of each player i and
the ambiguity set F . In particular, we assume that pa-
rameter ε
i
which shows the risk level of each player
can take any value in the interval (0, 1) and we make
no assumptions for the ambiguity set. That is, de-
pending on the game that one faces she/he can choose
the more suitable properties that the distributions of
the ambiguity set must satisfy. Hence, if we assume
some extra constraints for parameter ε
i
and set F , the
previous general formulation can become very spe-
cific.
Let’s assume that all players i {1, 2,...N} have
the same risk level, ε
i
= 1 i {1, 2,...., N}. Then
from the definition of CVaR we obtain the following
i {1, 2,....,N}:
Q-CVaR
ε
i
[π
i
(
˜
P ; x
i
,u
i
)] = Q-CVaR
1
[π
i
(
˜
P ; x
i
,u
i
)]
= E
Q
[π
i
(
˜
P ; x
i
,u
i
)]
(8)
Therefore the definition of best response (3.1) be-
comes:
argmin
u
i
S
a
i
sup
QF
Q-CVaR
ε
i
[π
i
(
˜
P ; x
i
,u
i
)] = argmin
u
i
S
a
i
sup
QF
E
Q
[π
i
(
˜
P ; x
i
,u
i
)]
= argmin
u
i
S
a
i
sup
QF
[π
i
(E
Q
[
˜
P ]; x
i
,u
i
)]
= argmax
u
i
S
a
i
inf
QF
[π
i
(E
Q
[
˜
P ]; x
i
,u
i
)]
(9)
The equality in the second line in the above expres-
sion follows from the linearity of expectation operator
and the linearity of π
i
:
E
Q
[π
i
(
˜
P ;x
i
,u
i
)] = [π
i
(E
Q
[
˜
P ];x
i
,u
i
)]
where E
Q
[
˜
P ]is the component-wise expected value
of
˜
P . The equality in the third line is due to the fol-
lowing properties of linear functions:
max f (x) = min[ f (x)]
and
z argmin
xS
a
i
f (x) = z argmax
xS
a
i
f (x).
Now, using the assumption that ε
i
= 1 i
{1,2, ...N} and by choosing the right properties for
the probability distributions of the payoff matrix we
can specify our formulation and create the desire
games.
Theorem 3.1. In the DRG, players i’s best response
to the other players strategies x
i
S
i
must belong
to:
argmin
u
i
S
a
i
sup
QF
Q-CVaR
ε
i
[π
i
(
˜
P ; x
i
,u
i
)] (10)
If we assume that ε
i
= 1 i {1, 2,...N}, then:
1. If F = {Q : E
Q
[
˜
P ] = Ψ} the set of DROE is
equivalent to that of a classical Nash Game.
2. If the ambiguity set is singleton, that is, F = {Q},
the DRG have the same equilibria with a related
finite Bayesian Game.
3. If F = {Q : Q[W · vec(E
Q
[
˜
P ]) h] = 1} the set
of DROE is equivalent to that of the classical Ro-
bust Game.
4 COMPUTING SAMPLE
EQUILIBRIA OF DRG
To examine if one specific tuple of strategies is equi-
librium is a typically elementary procedure in any
kind of game. However, to find the set of all equilibria
of a game, with complete or incomplete information,
is a very difficult task.
In this section, we present Theorem (4.1), in which
we show that for any finite distributionally robust
game with a specified ambiguity set and with no pri-
vate information the set of equilibria is a projection
of the solution set of multilinear system of equalities
and inequalities. The projection is like in (Aghassi
and Bertsimas, 2006), the component wise one into a
lower dimensional space.
Theorem 4.1. (Computation of Equilibria in Distri-
butionally Robust Finite Games)
Consider the N-player distributionally robust
game in which i {1,2,...,N} has action set
Distributionally Robust Games with Risk-averse Players
189
{1,2, ...,a
i
}, 1 < a
i
< , in which the ambiguity set
is:
F = {Q : Q[
˜
P U] = 1, E
Q
[vec
˜
P ] = m, E
Q
[
vec(
˜
P ) m
1
] s}
(11)
where U = {P : W · vec(P ) h} is bounded and polyhe-
dral set, and in which there is no private information.
The following two conditions are equivalent.
Condition 1) (x
1
,x
2
,....x
N
) is an equilibrium of the dis-
tributionally robust game.
Condition 2) For all i {1,2,....N} there exists
α
i
,ζ
i
,ρ
i
R, γ
i
R
+
, ξ
i
,θ
i
R
m
and
β
i
,λ
i
,κ
i
,δ
i
,ν
i
,τ
i
,f
i
,φ
i
,g
i
R
N
N
i=1
a
1
, such that
(x
1
, x
2
, ....x
N
,α
i
,ζ
i
,ρ
i
,γ
i
,β
i
, λ
i
, κ
i
, δ
i
, ν
i
, τ
i
, f
i
, φ
i
, g
i
, ξ
i
, θ
i
)
satisfies:
ζ
i
+
1
ε
i
α
i
+
1
ε
i
m
>
β
i
+
1
ε
i
sγ = ρ
i
, e
>
x
i
= 1,
α
i
m
>
λ
i
+ m
>
κ
i
+ h
>
ξ
i
0, ρ
i
e
>
f
>
Y
i
(x
i
)
λ
i
+ κ
i
γ
i
e 0, δ
i
+ ν
i
γ
i
e 0
α
i
m
>
δ
i
+ m
>
ν
i
+ h
>
θ
i
+ ζ
i
0
δ
i
+ ν
i
+ W
>
θ
i
β
i
Y
i
(x
i
)x
i
= 0
e
>
g
i
e
>
φ
i
1
ε
i
s, τ
i
f
i
=
1
ε
i
m
τ
i
+ φ
i
σ
i
m, τ
i
+ φ
i
σ
i
m
W τ
i
σ
i
h, W f
i
h
f
i
+ g
i
m, f
i
+ g
i
m
λ
i
+ κ
i
+ W
>
ξ
i
β
i
= 0,
λ
i
,κ
i
,δ
i
,ν
i
,x
i
0, θ
i
,ξ
i
,φ
i
,g
i
0
γ 0,
(12)
where Y
i
(x
i
) R
(N
N
i=1
a
i
)×a
i
denotes the matrix
such that
vec(P )
>
Y
i
(x
i
)x
i
= π
i
(P ;x
i
,x
i
), (13)
parameter ε
i
denotes the risk level of player i and
σ
i
=
1ε
i
ε
i
is a fixed number i {1, 2,....N}.
5 THE AMBIGUITY SET
In all DRG that we develop in this paper the ambigu-
ity set have the following form:
F = {Q : Q[W · vec(
˜
P ) h] = 1, E
Q
[vec
˜
P ] = m, E
Q
[
vec(
˜
P ) m
1
] s}
(14)
This, combined with different risk levels ε
i
for
player i {1, 2,..N} allows several variations of each
distributionally robust game. By changing the values
of ambiguity set’s uncertain parameters W,h, m and
s and by assuming each time different risk attitudes
for the players the set of DROE which constitute the
solution of our problem can change dramatically.
At this point, we present the role of each uncertain
parameter of the ambiguity set (14).
Matrix W R
(m×N
N
i=1
a
i
)
and vector h R
m
are
the two variables which represent the uncertainty
polyhedral set in which the uncertain values of the
payoff matrix should belong. The maximum distance
of all possible vec(
˜
P ) from the average vector m
is denoted by scalar s. Finally, m R
N
N
i=1
a
i
is the
vector that denotes the expected value of vec(
˜
P ) for
each distribution that belongs to the ambiguity set.
Important assumption: Vector m must belong to the
bounded uncertainty polyhedral set of the payoff
matrix
˜
P . Otherwise the ambiguity set F will be
empty.
Special Cases of Distributionally Robust Games:
Under certain conditions (special cases), the set of
equilibria of distributionally robust finite game with
ambiguity set like (14) is equivalent to that of a
related finite game with complete payoff information
(Nash Game) and with the same number of players
and the same action spaces. Thus, when our game
falls within these special cases one can simply solve
the corresponding Nash Game.
The special cases of such games are studied in the
next three propositions.
Proposition 5.1. The set of equilibria of a distribu-
tionally robust game in which all players are risk neu-
tral (ε
i
= 1, i {1,2, ..N}) is equivalent to the set of
equilibria of a Nash Game with fixed payoff matrix Ψ
where vec(Ψ) = m. (m is the average vector of the
ambiguity set (14). )
Proposition 5.2. The set of equilibria of a distribu-
tionally robust game in which the parameter s of the
ambiguity set (14) is equal to zero(s=0) is equivalent
to the set of equilibria of a Nash Game with fixed pay-
off matrix M where vec(M ) = m. (m is the average
vector of the ambiguity set (14). )
Proposition 5.3. The set of equilibria of a distribu-
tionally robust game that has as a support a single
point is equivalent to the set of equilibria of a Nash
Game with fixed payoff matrix the one that corre-
sponds to this single point. Single point is named the
unique payoff matrix which created from specific val-
ues of the matrix W and vector h of the ambiguity
set. The values of matrix W and vector h are se-
lected in order to make the uncertainty set U = {P :
W · vec(P ) h} singleton.
6 ILLUSTRATIVE EXAMPLE
Having presented our distributionally robust games
model we will now illustrate our approach with one
concrete example.
Distributionally Robust Inspection Game (DRIG)
Problem Description:
The DRIG is a two player game in which the row
player is the employee (possible actions:Shirk or
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
190
Table 1: Normal Form Representation of DRIG.
Inspect NotInspect
Shirk (0,
˜
h) (w, -w)
Work (w ˜g, ˜v w
˜
h) (w ˜g, ˜v w)
Work) and the column player is the employer(with
possible actions Inspect or not Inspect). The two play-
ers choose their actions simultaneously and then they
receive the payoffs corresponding to the combination
of their strategies. When the employee works he has
cost ˜g and his employer has profit equal to ˜v. Each
inspection costs to the employer
˜
h but if he inspects
and finds the employee shirking then he does not pay
him his wage w. In all other cases employee’s wage
is paid. All values except the payment w of the em-
ployee are uncertain.
In the classical complete information Inspection
game the parameters ˜g, ˜v and
˜
h are fixed ( ˜g = ˇg, ˜v =
ˇv,
˜
h =
ˇ
h) and in the robust approach ( ˜g, ˜v,
˜
h) [g,g] ×
[v,v] × [h,h] (Aghassi and Bertsimas, 2006).
In our new, distributionally robust approach, players
have partial information about the probability distri-
butions of the uncertain variables ˜g, ˜v and
˜
h (about the
probability distribution Q of the payoff matrix
˜
P ). In
particular, the players do not know the exact distri-
bution of the payoff matrix. They are only aware of
a commonly known ambiguity set F of all possible
probability distributions Q that satisfy some specific
properties. Subsequently, all players adopt a worst
case CVaR approach to the uncertainty which is com-
puted over all probability distributions within the set
F . The introduction of the CVaR in the formulation
of the game allows the two players to have different
risk attitudes. Finally, the risk levels of the players
are assumed to be common knowledge and none of
the two players has private information.
For example, we may consider the DRIG in which the
ambiguity set is given by:
F = {Q : Q[( ˜g, ˜v,
˜
h) U ] = 1, E
Q
[vec(
˜
P )] = m, E
Q
[
vec(
˜
P ) m
1
] s}
(15)
Where:U = [g, g] × [v, v] × [h,h] , s 0, and
˜
P =
(0,
˜
h) (w, w)
(w ˜g, ˜v w
˜
h)) (w ˜g, ˜v w)
(16)
Important assumption: Vector m of the second
constraint of the ambiguity set must belong to the
bounded polyhedral uncertainty set of payoff matrix
˜
P. For the DRIG, m [g,g] × [v, v] × [h,h]. Other-
wise the ambiguity set will be empty.
7 NUMERICAL EXPERIMENT
In this section, we experimentally evaluate the new
model of games described in this paper. In the interest
of brevity, we only present one experiment. For more
examples, experiments and detailed explanation of
our computational method we refer the reader to
chapter 5 of (Loizou, 2015).
The Experiment:
Fixed Ambiguity Set - Several Risk Levels
What would happen to the number of equilibria
and to the payments of the two players when the
ambiguity set is kept fixed while the values of players’
risk levels are varied.
Computational Method:
The method that we use to approximately compute the
DROE and the players’ payoffs at each equilibrium of
any DRG is developed as follows:
1. Check if the ambiguity set of the DRG can be ex-
pressed like the general form of equation (14).
(The ambiguity set of DRIG has this property.)
2. Estimate the multi-linear system of equali-
ties and inequalities whose dimension-reducing
component-wise projection of the feasible solu-
tion set is equivalent with the set of equilibria of
the DRG (see theorem (4.1))
3. Find the feasible solutions of the multi-linear sys-
tem and for each solution keep the components
that correspond to the strategies of the players
(projection of the solution). Additionally, com-
pute the players’ payoffs at each equilibrium.
These are achieved using the YALMIP modelling
language (L
¨
ofberg, 2004), in Matlab 2014b.
All numerical evaluations of this chapter were
conducted on a 2.27GHz, Intel Core i5 CPU 430
machine with 4GB of RAM.
More specifically, in the experiment of this sec-
tion we assume that the uncertain parameters of the
costs ( ˜g, ˜v,
˜
h) must belong to [8,12] × [16,24] × [4, 6]
and that the payment w of the employee is fixed
at w=15. In addition, we assume that average
vector m of the ambiguity set is the one that
corresponds to the nominal
1
version of the game
m = m
1
= (0, 5,15, 15,5, 0,5, 5)
>
and without
loss of generality that the maximum distance s of the
third constraint of the ambiguity set is s = 4.
1
With “nominal” we mean that the average vector takes
the value of vec(P ) when the uncertain parameters ˜g, ˜v and
˜
h are equal to the mid points of their intervals ( ˜g = 10, ˜v =
20,
˜
h = 5 and w = 15).
Distributionally Robust Games with Risk-averse Players
191
Therefore, since all variables of the ambiguity set are
kept fixed
2
, we can derive that the ambiguity set of
the form (14) is kept fixed in this experiment . The
only variables that are allowed to change are the risk
levels ε
1
and ε
2
of the two players.
The following tables and figures illustrate the number
of equilibria and the players’ payoffs at these equilib-
ria for the aforementioned fixed ambiguity set while
the players’ risk levels change.
In particular Table 2 shows the equilibria of the pre-
viously described game when player 1 is risk neutral
(ε
1
= 1) and player 2 has several risk attitudes. The
players’ payoffs at equilibria for each combination of
the risk levels are given in Figure 1.
Notice that the first line of the Table 2 and Table
3 corresponds to the special case in which the two
players of the game are risk neutral(ε
1
= 1,ε
2
= 1).
This means that instead of solving the DRIG using the
aforementioned computational method we can simply
solve the corresponding Nash Game with fixed payoff
matrix M where vecM = m:
˜
M =
(0,5) (15,15)
(5,0) (5,5)
(17)
This Complete Information Game has unique equilib-
rium equal to (1/3,2/3). That is, the employee(row
player) shirks(plays action 1) with probability x
1
1
=
1/3 and the employer(column player) inspects(plays
action 1)with probability x
2
1
= 2/3.
Table 2: DRIG: The equilibria for different values of Risk
levels when the vector m of the ambiguity set take the nom-
inal value and the maximum distance is s = 4. Player 1 is
risk neutral. His risk level is kept fixed (ε
1
= 1) while player
2 has several levels of risk aversion.
Risk Levels Equilibria
ε
1
= 1,ε
2
= 1 (1/3,2/3)
ε
1
= 1,ε
2
= 0.75 (0.333, 0.66)
ε
1
= 1,ε
2
= 0.5 (0.333, 0.66)
ε
1
= 1,ε
2
= 0.25 (0.333,0.66)(0.8179,0)
(0.9342,0.7069)
ε
1
= 1,ε
2
= 0.01 (1,0),(0,0.66),
(1,0.66),(1,0.1941) (0.333,0.66),
(0.9654,0.1387),(1,0.59)
Subsequently, Table 3 shows the equilibria of the
previously described game when player 2 is risk neu-
tral (ε
2
= 1) and player 1 has several risk attitudes.
The players’ payoffs at equilibria for each combina-
tion of the risk levels are given in Figure 2.
Discussion of the Results:
In standard optimization problems we know that as
2
The matrix W and vector h of the first constraint of the
ambiguity set are also fixed because the uncertain parame-
ters of the payoff matrix ( ˜g, ˜v,
˜
h) belong in a specific fixed
uncertainty set
(a) (b)
(c) (d)
Figure 1: Graph representation of the payoffs of the two
players at equilibria for different risk levels. Risk level of
player 1 is kept fixed (ε
1
= 1) while player 2 has several
levels of risk aversion. The title of each sub-figure denotes
the risk levels of the two players: (Players 1’s risk level,
Players 2’s risk level).
Table 3: DRIG: The equilibria for different values of Risk
levels when the vector m of the ambiguity set take the nom-
inal value and the maximum distance is s = 4. Player 2 is
risk neutral. His risk level kept fixed (ε
2
= 1) while player
1 has several levels of risk aversion.
Risk Levels Equilibria
ε
1
= 1,ε
2
= 1 (1/3,2/3)
ε
1
= 0.75,ε
2
= 1 (0.333,0.666),(0.35,0.665),
(0.2583,0.96)
ε
1
= 0.5,ε
2
= 1 (0.333,0.666),(0.5379,0),
(0.3842,0.66)
ε
1
= 0.25,ε
2
= 1 (0.4427,0),(0.333,0.666),
(0,0.3467)
ε
1
= 0.01,ε
2
= 1 (0,0),(1,1), (0.333,0.666),(0.33,0),
(0.335,1)
the decision maker becomes more risk averse his pay-
off always decreases. From the previous Figures 1
and 2 we can conclude that in game theory situation
this is not always the case. We can not have a gen-
eral rule, since now the nature of the problem is more
complicated.
In the DRG we assume that players are rational, so
they can predict the outcome of the game and choose
the strategies that form an equilibrium. For this rea-
son, a difference at risk attitude of a player does not
change only his decision but also the decisions of his
opponents.
3
. Hence, with increasing of risk aversion
of one player the players’ payoffs at equilibria may
both increase or decrease depending the game. For
example, in Figure 1 at Subfigure (d) where the risk
levels are (1,0.01) we can observe that for some equi-
3
in DRG, risk attitude is assumed to be common knowl-
edge
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
192
(a) (b)
(c) (d)
Figure 2: Graph representation of the payoffs of the two
players at equilibria for different risk levels. Risk level of
player 2 is kept fixed (ε
2
= 1) while player 1 has several
levels of risk aversion. The title of each sub-figure denotes
the risk levels of the two players: (Players 1’s risk level,
Players 2’s risk level).
libria the players have large payments and for some
others very low.
To verify that in the DRIG we can not have a gen-
eral rule about what happen in the payoffs of the two
players when they choose to play strategies that form
equilibria we also present Table 4 and Figure 3 witch
illustrate the payoffs of the two players at equilib-
ria when both of them are risk averse. More specif-
ically when their risk levels are ε
1
= ε
2
= 0.05 and
ε
1
= ε
2
= 0.01
Table 4: DRIG: Equilibria when the players’ risk levels are
ε
1
= ε
2
= 0.05 and ε
1
= ε
2
= 0.01.
Risk Levels Equilibria
ε
1
= ε
2
= 0.05 (1,0.66),(1,1)(0.95,0),
(0.43,1),(0.333,0.666)
ε
1
= ε
2
= 0.01 (1,0),(0,0),(0.332,0),
(0.5303,1),(1,0.78)
(a) (b)
Figure 3: Graph representation of the players’ payoffs at
equilibria when their risk levels are ε
1
= ε
2
= 0.05 and ε
1
=
ε
2
= 0.01.The title of each sub-figure denotes the risk levels
of the two players: (Players 1’s risk level,Players 2’s risk
level).
8 CONCLUSIONS
This paper combines Game Theory and Distribution-
ally Robust Optimization to propose a novel model of
incomplete information games without private infor-
mation in which the players use distributionally ro-
bust optimization to cope with payoff uncertainty.
We showed that for specific ambiguity sets and risk
levels, distributionally robust games constitute a true
generalization of Nash games, Bayesian Games and
Robust Games. Thus, any finite game of these three
categories can be expressed as a distributionally ro-
bust game.
Subsequently, we proved that the set of equilibria of
an arbitrary distributionally robust game with spec-
ified ambiguity set and without private information
can be computed as the component-wise projection of
the solution set of a multi-linear system of equations
and inequalities. For special cases of such games we
also showed equivalence to complete information fi-
nite games (Nash Games) with the same number of
players and same action spaces.
Finally to concretize the idea of a distributionallly ro-
bust game we presented Distributionally Robust In-
spection Game. We experimentally evaluated the new
model of games and we studied how the number of
equilibria and the players’ payments change when the
ambiguity set is fixed and the risk levels of the players
are varied.
Our approach opens up many avenues for further de-
velopment and research. For instance, the work of this
paper can be generalized to the case of distribution-
ally robust games involving potentially private infor-
mation. Furthermore, interesting results might arise if
we try to make similar work for more general classes
of ambiguity sets.
ACKNOWLEDGEMENTS
The author is very grateful to Wolfram Wiesemann
for helpful discussions and remarks. NL acknowl-
edges support by the Leventis Foundation and Laura
Wisewell Fund. Most of this work was done while at
Imperial College London.
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APPENDIX
Proof of Theorem 3.1: Using equation (9) we obtain
the following results:
For Nash Games: If F = {Q : E
Q
[
˜
P ] = Ψ}
x
i
argmax
u
i
S
a
i
inf
QF
[π
i
(E
Q
[
˜
P ]; x
i
,u
i
)] = argmax
u
i
S
a
i
inf
QF
[π
i
(Ψ;x
i
,u
i
)]
= argmax
u
i
S
a
i
[π
i
(Ψ;x
i
,u
i
)]
(18)
which is equivalent to the formulation of Nash Equi-
librium (see equation (2)).
For Bayesian Games: If the ambiguity set is single-
ton, that is F = {Q}.
x
i
argmax
u
i
S
a
i
inf
QF
E
Q
[π
i
(
˜
P ; x
i
,u
i
)] = argmax
u
i
S
a
i
E
Q
[π
i
(
˜
P ; x
i
,u
i
)]
(19)
which is equivalent to the formulation of Bayesian
Nash Equilibrium (see equation (3)).
For Robust Games: If F = {Q : Q[E
Q
[
˜
P ] U] = 1}
where U = {P : W · vec(P ) h}, then:
x
i
argmax
u
i
S
a
i
inf
QF
[π
i
(E
Q
[
˜
P ]; x
i
,u
i
)] = argmax
u
i
S
a
i
inf
˜
SU
[π
i
(
˜
S;x
i
,u
i
)]
(20)
where
˜
S = E
Q
[
˜
P ];
which is equivalent to the formulation of Robust
Optimization Equilibrium Games (see equation (4)).
Proof of Theorem 4.1: By the Formulation of the
DRG Condition 1 is equivalent to
x
i
argmin
u
i
S
a
i
sup
QF
Q-CVaR
ε
i
[π
i
(
˜
P ;x
i
,u
i
)] i {1,2, ...., N}
(21)
From (Rockafellar and Uryasev, 2000) and (Rockafel-
lar and Uryasev, 2002) we know that :
Q-CVaR
ε
i
[π
i
(
˜
P ; x
i
,u
i
)] = min
ζ
i
R
ζ
i
+
1
ε
i
E
Q
[π
i
(
˜
P ; x
i
,u
i
) ζ
i
]
+
(22)
where [x]
+
= max{x,0}.
Therefore equation (21) is equivalent to:
x
i
argmin
u
i
S
a
i
sup
QF
min
ζ
i
R
ζ
i
+
1
ε
i
E
Q
[π
i
(
˜
P ; x
i
,u
i
) ζ
i
]
+
i {1,.., N}
(23)
From Saddlepoint theorem ( Sion’s minimax theorem
(Sion et al., 1958) ) we could exchange the order of
supremum and infimum(minimum)
4
resulting:
x
i
argmin
u
i
S
a
i
min
ζ
i
R
ζ
i
+
1
ε
i
sup
QF
E
Q
[π
i
(
˜
P ; x
i
,u
i
) ζ
i
]
+
i {1,..N}
(24)
From the moment problem theory,
sup
QF
E
Q
[π
i
(
˜
P ;x
i
,u
i
) ζ
i
]
+
can be cast as
the following problem:
maximize
Z
U
[π
i
(P ;x
i
,u
i
) ζ
i
]
+
dµ(vec(P ))
subject to µ M
+
R
N
N
i=1
a
1
Z
U
dµ(vec(P )) = 1
Z
U
vec(P )dµ(vec(P )) = m
Z
U
k
vec(P ) m
k
1
dµ(vec(P )) s,
(25)
4
We can use Sion’s minimax theorem since the function
under consideration is convex-concave in its two arguments
Q F and ζ
i
R
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
194
where M
+
R
N
N
i=1
a
i
is the set of non-negative mea-
sures supported on R
N
N
i=1
a
i
.
There is a duality theory for moment problems(see
(Wiesemann et al., 2014) and (Natarajan et al., 2009))
which implies that the following dual problem attains
the same optimal value:
min α
i
+ m
>
β
i
+ sγ
i
s.t α
i
R, β
i
R
N
N
i=1
a
i
, γ
i
R
+
α
i
+ vec(
˜
P )
>
β
i
+
vec(
˜
P ) m
1
γ
i
[π
i
(
˜
P ; x
i
,u
i
) ζ
i
]
+
.
(26)
Where the last inequality must be satisfied for all
P U.
By replacing the definition of [·]
+
:
min α
i
+ m
>
β
i
+ sγ
i
s.t α
i
R, β
i
R
N
N
i=1
a
1
, γ
i
R
+
α
i
+ vec(
˜
P )
>
β
i
+
vec(
˜
P ) m
1
γ
i
π
i
(
˜
P ;x
i
,u
i
) ζ
i
α
i
+ vec(
˜
P )
>
β
i
+
vec(
˜
P ) m
1
γ
i
0.
(27)
Where the last two inequalities must be satisfied for
all P U.
Substituting this dual formulation of
sup
QF
E
Q
[π
i
(
˜
P ;x
i
,u
i
) ζ
i
]
+
into (24), we
obtain the following which we call Main Problem.
The projection of the solution set of this problem will
be the set of the equilibria that we desire.
min
u
i
,ζ
i
,α
i
,β
i
,γ
i
ζ
i
+
1
ε
i
(α
i
+ m
>
β
i
+ sγ
i
)
s.t α
i
R, β
i
R
N
N
i=1
a
1
, γ
i
R
+
,ζ
i
R
u
i
S
a
i
α
i
+ vec(
˜
P )
>
β
i
+
vec(
˜
P ) m
1
γ
i
π
i
(
˜
P ; x
i
,u
i
) ζ
i
α
i
+ vec(
˜
P )
>
β
i
+
vec(
˜
P ) m
1
γ
i
0.
(28)
Where the last two inequalities must be satisfied for
all P U.
This is now a ‘classical robust optimisation prob-
lem’ and we use standard duality techniques to
simplify the semi-infinite constraints. We know
that: f (p) k, p U min
pU
f (p) k Therefore,
using this the two robust constraints of the linear pro-
gram (28) become:
min
P U
[α
i
+ vec(P )
>
β
i
+
k
vec(P ) m
k
1
γ
i
+ π
i
(P ; x
i
,u
i
)] ζ
i
(29)
and
min
P U
[α
i
+ vec(P )
>
β
i
+
k
vec(P ) m
k
1
γ
i
] 0 (30)
The left hand sides of the constraints (29) and (30)
are equivalent to the following problems (31) and
(32) respectively.
min
vec(P )
α
i
+ vec(P )
>
β
i
+
k
vec(P ) m
k
1
γ
i
+ π
i
(P ; x
i
,u
i
)
subject to W · vec(P ) h.
(31)
and
min
vec(P )
α
i
+ vec(P )
>
β
i
+
k
vec(P ) m
k
1
γ
i
subject to W · vec(P ) h.
(32)
In turn these programs are equivalent to:
min
vec(P ),η
α
i
+ vec(P )
>
β
i
+ γ
i
N
N
i=1
a
1
j=1
η
j
+ vec(P )
>
Y
i
(x
i
)u
i
s.t W · vec(P ) h
η
j
vec(P )
j
m
j
j = 1, 2, ...N
N
i=1
a
i
η
j
m
j
vec(P )
j
j = 1, 2, ...N
N
i=1
a
i
.
(33)
and
min
vec(P ),η
α
i
+ vec(P )
>
β
i
+ γ
i
N
N
i=1
a
i
j=1
η
j
subject to W · vec(P ) h
η
j
vec(P )
j
m
j
j = 1, 2, ...N
N
i=1
a
i
η
j
m
j
vec(P )
j
j = 1, 2, ...N
N
i=1
a
i
.
(34)
where η
j
= |vec(P )
j
m
j
|, j = 1,2, ...N
N
i=1
a
i
and Y
i
(x
i
) is as defined in (13).
The dual problems of (33) and (34) are respectively
the following:
max
δ
i
,β
i
,θ
i
α
i
m
>
δ
i
+ m
>
ν
i
+ h
>
θ
i
δ
i
+ ν
i
+ W
>
θ
i
β
i
Y
i
(x
i
)u
i
= 0
δ
i
+ ν
i
γ
i
e 0
δ
i
0, ν
i
0, θ
i
0.
(35)
max
λ
i
,κ
i
,ξ
i
α
i
m
>
λ
i
+ m
>
κ
i
+ h
>
ξ
i
λ
i
+ κ
i
+ W
>
ξ
i
β
i
= 0
λ
i
+ κ
i
γ
i
e 0
λ
i
0, κ
i
0, ξ
i
0.
(36)
We know that:p U : f (p) K max
pU
f (p)
K. Subsequently, we substitute the last two prob-
lems (35) and (36) in the Main Problem (28).
Therefore for each player i {1, 2,...N}, α
i
,γ
i
,ζ
i
R,β
i
,λ
i
,κ
i
,δ
i
,ν
i
R
N
N
i=1
a
i
and ξ
i
,θ
i
R
m
such
that (x
i
,β
i
,λ
i
,κ
i
,δ
i
,ν
i
ξ
i
,θ
i
,α
i
,γ
i
,ζ
i
) is a mini-
mizer of:
min
u
i
,α
i
,β
i
,γ
i
,ζ
i
,λ
i
,κ
i
,ξ
i
,δ
i
,ν
i
,θ
i
ζ
i
+
1
ε
i
α
i
+
1
ε
i
m
>
β
i
+
1
ε
i
sγ
i
e
>
u
i
= 1
α
i
m
>
λ
i
+ m
>
κ
i
+ h
>
ξ
i
0
λ
i
+ κ
i
+ W
>
ξ
i
β
i
= 0
λ
i
+ κ
i
γ
i
e 0
α
i
m
>
δ
i
+ m
>
ν
i
+ h
>
θ
i
+ ζ
i
0
δ
i
+ ν
i
+ W
>
θ
i
β
i
Y
i
(x
i
)u
i
= 0
δ
i
+ ν
i
γ
i
e 0
λ
i
0, κ
i
0, ξ
i
0
δ
i
0, ν
i
0, θ
i
0
u
i
0, γ
i
0.
(37)
Distributionally Robust Games with Risk-averse Players
195
whose dual is:
max
τ
i
,ρ
i
, f
i
,φ
i
,g
i
,
ρ
i
e
>
g
i
e
>
φ
i
1
ε
i
s
τ
i
f
i
=
1
ε
i
m
τ
i
+ φ
i
σ
i
m
τ
i
+ φ
i
σ
i
m
W τ
i
σ
i
h
f
i
+ g
i
m
f
i
+ g
i
m
W f
i
h
ρ
i
e
>
f
>
Y
i
(x
i
)
φ
i
0, g
i
0.
(38)
Condition 2 follows from strong linear programming
duality. The reverse direction (Condition 2 =
Condition 1) is also holds as all steps of our proof are
based on the equivalence of the two parts.
Proof of Proposition 5.1: The second constraint of
the ambiguity set F is E
Q
[vec
˜
P ] = m. Therefore if
we denote with Ψ the matrix for which vec(Ψ) = m
then Ψ = E
Q
[
˜
P ] and with use of equation (9):
x
i
argmax
u
i
S
a
i
inf
QF
[π
i
(E
Q
[
˜
P ]; x
i
,u
i
)] = argmax
u
i
S
a
i
inf
QF
[π
i
(Ψ;x
i
,u
i
)]
= argmax
u
i
S
a
i
[π
i
(Ψ;x
i
,u
i
)]
(39)
which is equivalent to the formulation of Nash
Equilibrium (see equation (2)).
Proof of Proposition 5.2: For s = 0 the third
constraint of the ambiguity set (14) becomes:
E
Q
[
vec(
˜
P ) m
1
] 0. Then, since all val-
ues inside the expectation operator E
Q
are posi-
tive we have that E
Q
[
vec(
˜
P ) m
1
] = 0 and that
Q[
vec(
˜
P ) m
1
= 0] = 1 which is equivalent to
Q[vec(
˜
P
i
) m
i
= 0] = 1, i {1, 2,...R
N
N
i=1
a
1
}.
(40)
Therefore if s 0 the third constraint of the am-
biguity set is equivalent to: Q[vec(
˜
P ) m = 0] = 1
which means that vec(
˜
P ) = m for all distributions of
the ambiguity set. Therefore:
F = {Q : Q[W · vec(
˜
P ) h] = 1, E
Q
[vec
˜
P ] = m, Q[vec(
˜
P ) = m] = 1}
= {Q : Q[vec(
˜
P ) = m] = 1}
(41)
Using the definition of Nash Equilibrium we can find
now the equivalence between our game and a Nash
Game.
x
i
argmin
u
i
S
a
i
sup
QF
Q-CVaR
ε
i
[π
i
(
˜
P ; x
i
,u
i
)] = argmin
u
i
S
a
i
[π
i
(M; x
i
,u
i
)]
= argmax
u
i
S
a
i
[π
i
(M; x
i
,u
i
)]
(42)
where vec(M) = m.
Proof of Proposition 5.3: The uncertainty set U is a
singleton. Therefore the first constraint of the ambi-
guity set (14) is equivalent to: Q[
˜
P = C] = 1 where
C denotes the support single point, the only matrix of
set U. Thus the ambiguity set becomes:
F = {Q : Q[
˜
P = C] = 1, E
Q
[vec
˜
P ] = m, Q[vec(
˜
P ) = m] = 1}
(43)
where vec(C) = m.
Using this, the desired result follows:
argmin
u
i
S
a
i
sup
QF
Q-CVaR
ε
i
[π
i
(
˜
P ; x
i
,u
i
)] = argmin
u
i
S
a
i
Q-CVaR
ε
i
[π
i
(C;x
i
,u
i
)]
= argmin
u
i
S
a
i
[π
i
(C;x
i
,u
i
)]
= argmax
u
i
S
a
i
[π
i
(C;x
i
,u
i
)]
(44)
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
196