A second-order cone programming formulation for two player zero-sum games with chance constraints

European Journal of Operational Research 275 (2019) 839–845

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Continuous Optimization

A second-order cone programming formulation for two player zero-sum games with chance constraints Vikas Vikram Singh a,∗, Abdel Lisser b a b

Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India Laboratoire de Recherche en Informatique Université Paris Sud, Orsay 91405, France

a r t i c l e

i n f o

Article history: Received 9 December 2016 Accepted 4 January 2019 Available online 10 January 2019 Keywords: Stochastic programming Chance constraints Zero-sum game Saddle point equilibrium Second-order cone program

a b s t r a c t We consider a two player finite strategic zero-sum game where each player has stochastic linear constraints. We formulate the stochastic constraints of each player as chance constraints. We show the existence of a saddle point equilibrium in mixed strategies if the row vectors of the random matrices defining the stochastic constraints are elliptically symmetric distributed random vectors. We further show that a saddle point equilibrium can be obtained from the optimal solutions of a primal-dual pair of secondorder cone programs.

1. Introduction The equilibrium concept in game theory started with the paper by von Neumann (1928). He showed that there exists a saddle point equilibrium for a finite strategic zero-sum game. In 1950, Nash (1950) showed that there always exists an equilibrium for a finite strategic general sum game with finite number of players. Later, such equilibrium was called Nash equilibrium. It is well known that there is a substantial relationship between game theory and optimization theory. A saddle point equilibrium of a two player finite strategic zero-sum game can be obtained from the optimal solutions of a primal-dual pair of linear programs (Dantzig, 1951), while a Nash equilibrium of a two player finite strategic general sum game can be obtained from a global maximum of a certain quadratic program (Mangasarian & Stone, 1964). The games discussed above are unconstrained games, i.e., the mixed strategies of each player are not further restricted by any constraint. Charnes (1953) considered a two player constrained zero-sum game where the mixed strategies of each player are constrained by linear inequalities. He showed that a saddle point equilibrium of a constrained zero-sum game can be obtained from the optimal solutions of a primal-dual pair of linear programs. The above mentioned papers are deterministic in nature, i.e., the payoff functions and constraints (if any) are defined by real valued functions. However, in some practical cases the payoff

∗

Corresponding author. E-mail addresses: [email protected] (V.V. Singh), [email protected] (A. Lisser).

https://doi.org/10.1016/j.ejor.2019.01.010 0377-2217/© 2019 Elsevier B.V. All rights reserved.

© 2019 Elsevier B.V. All rights reserved.

functions or constraints are stochastic in nature due to various external factors. One way to handle stochastic Nash games is using expected payoff criterion. Ravat and Shanbhag (2011) considered stochastic Nash games using expected payoff functions and expected value constraints. They showed the existence of a Nash equilibrium in various cases. The expected payoff criterion is more appropriate for the cases where the decision makers are risk neutral. The risk averse payoff criterion using the risk measures CVaR and variance has been considered in the literature (Kannan, Shanbhag, & Kim, 2013; Ravat & Shanbhag, 2011) and (Conejo, Nogales, Arroyo, & García-Bertrand, 2004), respectively. Recently, a payoff criterion based on chance constraint programming has been introduced. Such a payoff criterion is appropriate for the cases where the players are interested in maximizing the random payoffs that can be obtained with certain confidence. Singh, Jouini, and Lisser (2016a, 2016b, 2016c, 2017); Singh and Lisser (2018) formulated the finite strategic games with random payoffs as chance-constrained games by defining the payoff function of each player using a chance constraint. In Singh et al. (2016a, 2016b, 2016c); Singh and Lisser (2018), the authors considered the case where the probability distribution of the payoff vector of each player is completely known. They showed the existence of a Nash equilibrium for the case of normal distributions, Cauchy distributions, and multivariate elliptically symmetric distributions (Singh, Jouini, & Lisser, 2016c). They also proposed some equivalent complementarity problems and mathematical programs to compute the Nash equilibria of these games (Singh et al., 2016a; 2016b; Singh & Lisser, 2018). In Singh, Jouini, and Lisser (2017), the authors considered the games where the distribution of the payoff vector of each

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V.V. Singh and A. Lisser / European Journal of Operational Research 275 (2019) 839–845

player is not completely known. They studied these games using a distributionally robust approach. There are some zero-sum chanceconstrained games available in the literature (Blau, 1974; Cassidy, Field, & Kirby, 1972; Charnes, Kirby, & Raike, 1968; Cheng, Leung, & Lisser, 2016). In this paper, we consider a stochastic version of two player constrained zero-sum game considered in Charnes (1953). We formulate each player’s stochastic linear constraints as chance constraints. We show that there exists a mixed strategy saddle point equilibrium for a zero-sum game with chance constraints if the random vectors defining stochastic linear constraints follow elliptically symmetric distributions. We further show that a saddle point equilibrium problem is equivalent to a primal-dual pair of second-order cone programs (SOCPs). The work is primarily based on the fact that deterministic constrained zero-sum game problem is equivalent to a primal-dual pair of linear programs (Charnes, 1953) and under elliptical distribution case each player’s best response stochastic linear program is equivalent to an SOCP (Henrion, 2007). Now, we describe the structure of the rest of the paper. Section 2 contains the definition of a zero-sum game with chance constraints. Section 3 contains existence of a mixed strategy saddle point equilibrium and its equivalent second-order cone programming formulation. We present numerical results in Section 4. We conclude the paper in Section 5.

where B ∈ R p×m , D ∈ Rq×n , b ∈ R p , d ∈ Rq . A strategy pair (x, y) is said to be a saddle point equilibrium for the above constrained zero-sum game if for the given y, x is an optimal solution of (2.1) and for the given x, y is an optimal solution of (2.2). Let sets I1 = {1, 2 . . . , p} and I2 = {1, 2 . . . , q} index the constraints of player 1 and player 2, respectively. Charnes (1953) showed that a saddle point equilibrium of a constrained zero-sum game problem can be obtained from the optimal solutions of a primal-dual pair of linear programs. We consider the above constrained zerosum game problem where the matrices defining the constraints are random matrices. Let Bw denote a random matrix which defines the constraints of player 1, and Dw denote a random matrix which defines the constraints of player 2; w denotes some uncertainty parameter. We consider the situation where the players’ stochastic constraints are satisfied with given probabilities. Therefore, we replace the stochastic constraints of each player with individual chance constraints (Henrion, 2007; Prékopa, 1995). Then, player 1 (resp. player 2) is interested in a strategy x (resp. y) which solves a stochastic optimization problem (2.3) (resp. (2.4)) for a given strategy y (resp. x) of player 2 (resp. player 1).

max xT Ay x

s.t. 1 P {Bw k x ≤ bk } ≥ αk ,

∀ k ∈ I1

x ∈ X,

2. The model A two player zero-sum game is described by an m × n matrix A, where m and n denote the number of actions of player 1 and player 2, respectively. The matrix A represents the payoffs of player 1 corresponding to different action pairs, and the payoffs of player 2 are given by −A. Let I = {1, 2, . . . , m} and J = {1, 2, . . . , n} be the action sets of player 1 and player 2, respectively. The actions belonging to sets I and J are also called pure strategies of player 1 and player 2, respectively. A mixed strategy of a player is defined by a probability distribution   over his  actionset. Let  X = x ∈ Rm | i∈I xi = 1, xi ≥ 0, ∀ i ∈ I and Y = y ∈ Rn | j∈J y j =



1, y j ≥ 0, ∀ j ∈ J be the sets of mixed strategies of player 1 and player 2, respectively. For a given strategy pair (x, y) ∈ X × Y, the payoffs of player 1 and player 2 are given by xT Ay and xT (−A )y, respectively. For a given y ∈ Y the objective of player 1 is to maximize xT Ay, and for a given x ∈ X the objective of player 2 is to minimize xT Ay. A saddle point equilibrium of a zero-sum game exists in mixed strategies (von Neumann, 1928), and it can be obtained from the optimal solutions of a primal-dual pair of linear programs (Dantzig, 1951). Charnes (1953) studied a constrained zero-sum game problem where the mixed strategies of both players are further restricted by linear inequalities. The aim of player 1 (resp. player 2), given a strategy y (resp. x) of player 2 (resp. player 1), is to choose a strategy x (resp. y) which solves a linear programming problem (2.1) (resp. (2.2)). T

(2.3)

min xT Ay y

s.t. 2 P {Dw l y ≥ dl } ≥ αl ,

∀ l ∈ I2

y ∈ Y,

(2.4)

where P is a probability measure, and Bw = ( Bw , Bw , . . . , Bw ) is a k k1 k2 km th w w w w w k row of matrix B , and Dl = (Dl1 , Dl2 , . . . , Dln ) is an lth row of matrix Dw , and αk1 ∈ [0, 1] is a probability level for the kth constraint of player 1, and αl2 ∈ [0, 1] is a probability level for the lth constraint of player 2. Let α 1 = (αk1 )k∈I1 and α 2 = (αl2 )l∈I2 , and α = (α 1 , α 2 ). We denote the above zero-sum game with individual chance constraints by G(α ). Let,

1 S1 (α 1 ) = {x ∈ Rm | x ∈ X, P {Bw k x ≤ bk } ≥ αk ,

∀ k ∈ I1 },

and 2 S2 (α 2 ) = {y ∈ Rn | y ∈ Y, P {Dw l y ≥ dl } ≥ αl ,

∀ l ∈ I2 }.

The sets S1 (α 1 ) and S2 (α 2 ) are feasible strategy sets of player 1 and player 2, respectively for the game G(α ). Then, (x∗ , y∗ ) ∈ X × Y is called a saddle point equilibrium of G(α ) at α ∈ [0, 1]p × [0, 1]q , if the following inequality holds:

xT Ay∗ ≤ x∗T Ay∗ ≤ x∗T Ay,

∀ x ∈ S1 ( α 1 ), y ∈ S2 ( α 2 ).

max x Ay x

3. Existence and characterization of saddle point equilibrium

s.t. Bx ≤ b x ∈ X,

(2.1)

min xT Ay y

s.t. Dy ≥ d y ∈ Y,

(2.2)

We consider the case where the row vectors of the random matrices Bw and Dw follow a multivariate elliptically symmetric distribution. The class of multivariate elliptically symmetric distributions generalizes the multivariate normal distribution. Definition 3.1. A d-dimensional random vector ξ follows an elliptically symmetric distribution Ellipd (μ,  , ϕ ) if its characteristic T T function is given by Eeiz ξ = eiz μ ϕ (zT  z ) where ϕ is the characteristic generator function, μ is the location parameter, and  is the scale matrix.

V.V. Singh and A. Lisser / European Journal of Operational Research 275 (2019) 839–845

Note that the generating distribution function of an elliptical location-scale family may vary under d. For instance, for d = 1, the class of elliptical distributions coincides with the class of univariate symmetric distributions (Cambanis & Simons, 1981). Numerous relevant distributions belong to the family of elliptical distributions: normal distribution with ϕ (t ) = exp{− 12 t }, Student’s t distribution with ϕ (t) varying with its degree of freedom √ (Kotz & Nadarajah, 2004), Cauchy distribution with ϕ (t ) = exp{− t }, Laplace distribution with ϕ (t ) = (1 + 12 t )−1 , and logistic distribution with ϕ (t ) = √ √2π t √ eπ t −e−π t

. The family of elliptical distributions possesses a num-

ber of useful properties, e.g., its closeness to affine transformations, and the conditional and marginal distributions are also elliptical. Proposition 3.2 (Fang, Kotz, & Ng, 1990). If a d-dimensional random vector ξ follows an elliptical distribution Ellipd (μ,  , ϕ ), then for any (N × d)-matrix C and any N × 1-vector c, C ξ + c follows an N-dimensional elliptical distribution El l ipN (C μ + c, C C T , ϕ ). Assume Bw , k ∈ I1 , follows an elliptically symmetric distribuk





tion El l ipm μ1k , k1 , ϕk1 with positive definite scale matrix k1 . Assume Dw , l ∈ I2 , follows an elliptically symmetric distribution l





El l ipn μ2l , l2 , ϕl2 with positive definite scale matrix l2 . We denote a positive definite matrix  by Proposition 3.2,  T 10.T From  1 x, ϕ 1 , k ∈ I , and for a given x ∈ X, Bw x follows El l ip x μ , x  1 k k k k





for a given y ∈ Y, Dw (−y ) follows El l ip −yT μ2l , yT l2 y, ϕl2 , l ∈ I2 . l Since, k1  0 and l2  0, we can write



yT l2 y



 2 1

= l

2

y , where



 1 1

k

2



and



xT k1 x = k1



 2 1

l

2

1 2

x and

are the unique

positive definite square roots of matrices k1 and l2 , respectively, and || · || is the Euclidean norm. Then, ξk1 =

ξl2

=

−Dw y+yT μ2l l 1

(l2 ) 2 y

Bw x−xT μ1k k 1

(k1 ) 2 x

, k ∈ I1 , and

, l ∈ I2 , follow a univariate standard elliptical dis-

El l ip(0, 1, ϕk1 )

El l ip(0, 1, ϕl2 ),

tributions and respectively The second-order cone constraints reformulation of the linear chance constraints for the case of multivariate normal distributions, multivariate elliptically symmetric distributions, and radial distributions are given in Henrion (2007); Kataoka (1963); Lobo, Vandenberghe, Boyd, and Lebret (1998); van de Panne and Popp (1963), and Calafiore and Ghaoui (2007), respectively. Lagoa, Li, and Sznaier (2005) showed the convexity of linear chance constraints for log-concave symmetric distributions. By using the second-order cone constraint reformulation from (Henrion, 2007; Kataoka, 1963; van de Panne & Popp, 1963), we can write the feasible strategy sets S1 (α 1 ) and S2 (α 2 ) as

S1 ( α 1 ) =



x ∈ Rm

≤ bk ,

S2 ( α 2 ) =

Lemma 3.3. For all α 1 ∈ (0.5, 1]p and α 2 ∈ (0.5, 1]q , S1 (α 1 ) and S2 (α 2 ) are convex sets. Proof. The (2007). 

proof

| y ∈ Y, −yT μ2l + ξ−12 (αl2 )||(l2 ) y|| l ≤ −dl , ∀ l ∈ I2 ,

x∗ ∈ arg max min xT Ay,

(3.3)

y∗ ∈ arg min max xT Ay.

(3.4)

2 x∈S1 ( α 1 ) y∈S2 ( α )

1 y∈S2 ( α 2 ) x∈S1 ( α )

It follows from the SOCP dual formulation given in Lobo et al. (1998) that Lagrangian dual of the SOCP maxx∈S (α 1 ) xT Ay is an 1 SOCP. Under Assumption 1, the duality gap is zero. Therefore, the min max xT Ay problem is equivalent to the following SOCP: y∈S2 ( α 2 ) x∈S1 ( α 1 )

min

λ

1 , 1 k k∈I1

s.t.





ν1 +

Assumption 1. 1. The set S1 (α 1 ) is strictly feasible, i.e., there exists an x ∈ Rm which is a feasible point of S1 (α 1 ) and the inequality constraints of S1 (α 1 ) are strictly satisfied by x.

λ1k bk

k∈I 1

λ1k μ1k −

k∈I 1

distribution functions induced by characteristic functions ϕk1 (t 2 ) and ϕl2 (t 2 ), respectively.

Henrion

It is well known that (x∗ , y∗ ) is a saddle point equilibrium for the game G(α ) if and only if



(k1 ) 2 δk1 ≤ ν 1 1m 1

k∈I 1

(ii ) − y μ + ξ 2 (αl2 )||(l2 ) 2 y|| ≤ −dl , ∀ l ∈ I2 T

ξl

of

3.1. Second-order cone programming formulation

1 2

−1 where  −1 1 (· ) and  2 (· ) are quantile functions of 1-dimensional

2.1

Remark 3.6. If the row vectors Bw , k ∈ I1 and Dw , l ∈ I2 , have k l strictly positive density functions, Theorem 3.5 holds for all α 1 ∈ [0.5, 1]p , α 2 ∈ [0.5, 1]q (Henrion, 2007).

y,ν (δ )

(3.2)

Proposition

Proof. For an α ∈ (0.5, 1]p × (0.5, 1]q , S1 (α 1 ) and S2 (α 2 ) are convex sets from Lemma 3.3. It is clear that S1 (α 1 ) and S2 (α 2 ) are closed and bounded sets. The function xT Ay is a bilinear and continuous function. Therefore, the existence of a saddle point equilibrium follows from the minimax theorem of von Neumann (1928). 

(i ) Ay −

y ∈ Rn

from

Theorem 3.5. Consider a constrained zero-sum matrix game where the matrices Bw and Dw defining the constraints of both the players, respectively, are random. Let the row vectors Bw ∼ k 2 ,  2 , ϕ 2 ), l ∈ I . For ∼ El l ip ( μ El l ipm (μ1k , k1 , ϕk1 ), k ∈ I1 , and Dw n 2 l l l l all k and l, k1  0 and l2  0. Then, there exists a saddle point equilibrium for the game G(α ) for all α ∈ (0.5, 1]p × (0.5, 1]q .

1,

(3.1)

follows

, k ∈ I1 and Dw , l ∈ I2 , have Remark 3.4. If the row vectors Bw k l strictly positive density functions, Lemma 3.3 holds for all α 1 ∈ [0.5, 1]p , α 2 ∈ [0.5, 1]q (Henrion, 2007).

| x ∈ X, xT μ1k + ξ−11 (αk1 )||(k1 ) x|| k



ξk

2. The set S2 (α 2 ) is strictly feasible, i.e., there exists an y ∈ Rn which is a feasible point of S2 (α 2 ) and the inequality constraints of S2 (α 2 ) are strictly satisfied by y.

1 2

∀ k ∈ I1 ,

841

2 l

1

−1 l

1 (iii ) ||δk1 || ≤ λ1k ξ−1 1 (αk ), ∀ k ∈ I1

( iv )



k

yj = 1

j∈J

( v ) y j ≥ 0, ∀ j ∈ J (vi ) λ1k ≥ 0, ∀ k ∈ I1 .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(P)

For more details about the Lagrangian duality for SOCP problems we refer (Bazaraa, Sherali, & Shetty, 2006; Boyd & Vandenberghe, 2004). Similarly, the maxx∈S (α 1 ) miny∈S (α 2 ) xT Ay problem is 1

2

842

V.V. Singh and A. Lisser / European Journal of Operational Research 275 (2019) 839–845

equivalent to the following SOCP problem.

ν2 +

max

x, ν 2 ,(δl2 )l∈I ,λ2

( i ) AT x −



λ2l dl

l∈I2

2

s.t.





λ2l μ2l −

l∈I2

(l2 ) 2 δl2 ≥ ν 2 1n 1

l∈I2

(ii ) x μ + ξ 1 (α )||(k1 ) 2 x|| ≤ bk , ∀ k ∈ I1 T

−1

1 k

1

1 k

k

(iii ) ||δ || ≤ λ ξ 2 (α ), ∀ l ∈ I2 2 l

( iv )



−1

2 l

2 l

l

xi = 1

i∈I

( v ) xi ≥ 0, ∀ i ∈ I (vi ) λ2l ≥ 0, ∀ l ∈ I2 .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ (D)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

⎛

4 ⎜5 A=⎝ 1 3

Again using Lagrangian dual formulation, it is easy to show that the resulting SOCPs (P) and (D) are primal-dual pair of optimization problems. We show that a saddle point equilibrium of the game G(α ) can be obtained from the optimal solutions of (P)-(D). Theorem 3.7. Consider a constrained zero-sum game where the matrices Bw and Dw defining the constraints of player 1 and player 2, re spectively, are random. Let the row vector Bw ∼ El l ipm μ1k , k1 , ϕk1 , k k ∈ I1 , where

k1  0, l2  0.

and the row vector

Dw l

each player has no side constraints. We compare our results by considering different distributions namely, normal, Cauchy, Laplace, and logistic distributions, from the class of elliptically symmetric distributions. The normal distribution has sub-Gaussian tail. The tail of Cauchy, Laplace and logistic distributions are heavier than the tail of a normal distribution. To solve (P) and (D), we use CVX, a package for specifying and solving convex programs (Grant & Boyd, 2008; 2014). We consider a zero-sum game which is described by a 4 × 4 payoff matrix A given by

  ∼ El l ipn μ2l , l2 , ϕl2 ,

l ∈ I2 , where Let Assumption 1 holds. Then, for a given α ∈ (0.5, 1]p × (0.5, 1]q , (x∗ , y∗ ) is a saddle point equilibrium of the game G(α ) if and only if there exist ν 1∗ , (δk1∗ )k∈I1 , λ1∗

   and ν 2∗ , (δl2∗ )l∈I2 , λ2∗ such that y∗ , ν 1∗ , (δk1∗ )k∈I1 , λ1∗ and  ∗ 2∗ 2∗  x , ν , (δl )l∈I2 , λ2∗ are optimal solutions of primal-dual pair of

Proof. Let (x∗ , y∗ ) be a saddle point equilibrium of the game G(α ). Then, x∗ and y∗ are the solutions of (3.3) and (3.4), respectively. This together with Assumption 1 implies that    there exist ν 1∗ , (δk1∗ )k∈I1 , λ1∗ and ν 2∗ , (δl2∗ )l∈I2 , λ2∗ such that









y∗ , ν 1∗ , (δk1∗ )k∈I1 , λ1∗ and x∗ , ν 2∗ , (δl2∗ )l∈I2 , λ2∗ are optimal solutions of (P) and (D), respectively.    Let y∗ , ν 1∗ , (δk1∗ )k∈I1 , λ1∗ and x∗ , ν 2∗ , (δl2∗ )l∈I2 , λ2∗ be the optimal solutions of (P) and (D), respectively. Under Assumption 1, (P) and (D) are strictly feasible. Therefore, strong duality holds for primal-dual pair (P)–(D) (Lobo et al., 1998). Then, we have

ν 1∗ +



λ1k ∗ bk = ν 2∗ +

k∈I 1



λ2l ∗ dl .

l∈I2

By multiplying the constraint (i) of (P) by vector xT (x ∈ S1 (α 1 )) from left and using Cauchy–Schwartz inequality, we have

x Ay ≤ ν T

∗

1∗

+



λ

1∗ k bk ,

∀ x ∈ S1 ( α ). 1

(3.5)

k∈I 1

Using the similar arguments as above, we have

x∗T Ay ≥ ν 2∗ +



λ2l ∗ dl = ν 1∗ +

l∈I2



λ1k ∗ bk , ∀ y ∈ S2 (α 2 ).

k∈I 1

By taking x = x∗ , y = y∗ in (3.5) and (3.6), respectively, we get

x∗T Ay∗ = ν 1∗ +



λ1k ∗ bk = ν 2∗ +

k∈I 1



λ2l ∗ dl .

(3.7)

l∈I2

From (3.5), (3.6), (3.7) together, (x∗ , y∗ ) is a saddle point equilibrium.  4. Numerical results For illustration purpose, we consider an instance of a zero sum game with chance constraints. To see the effect of chance constraints on saddle point equilibria, we consider the case where

1 4⎟ . 5⎠ 3

⎛ ⎞

μ

1 1

16 ⎜18⎟ = ⎝ ⎠, 11 10

⎛ ⎞

μ

1 2

⎛ ⎞

μ

2 1

13 ⎜9⎟ = ⎝ ⎠, 20 20

μ

 

b=

2 2

17 18 , d = 18

⎛

⎛ ⎞

17 ⎜19⎟ = ⎝ ⎠, 10 10

⎜19⎟ μ = ⎝ ⎠, 9

⎛ ⎞

⎛ ⎞

14 ⎜11⎟ = ⎝ ⎠, 19 19

17

1 3

9 14

⎜ 11 ⎟ μ = ⎝ ⎠, 20

 

2 3

19

11 13 , 13

⎞

⎛

12 ⎜4 1 1 = ⎝ 4 3

4 12 3 3

4 3 12 2

3 12 3⎟ 1 ⎜4 , = 2⎠ 2 ⎝3 12 4

12 ⎜2 1 3 = ⎝ 2 3

2 12 2 2

2 2 12 3

3 2⎟ , 3⎠ 12

10 ⎜2 12 = ⎝ 4 3

2 10 3 3

4 3 12 2

3 10 3⎟ 2 ⎜3 ,  = 2⎠ 2 ⎝2 12 2

12 ⎜3 2 3 = ⎝ 3 4

3 10 2 3

3 2 10 3

4 3⎟ . 3⎠ 10

⎛

⎛

⎛

(3.6)

⎞

5 4 1 5

In general, a matrix game does not need to have a saddle point equilibrium in pure strategies. However, for this instance of the matrix game A the pure strategy pair ((0, 1, 0, 0), (0, 1, 0, 0)) is a saddle point equilibrium. Now, we consider the case where each player has stochastic linear constraints which are defined by 3 × 4 random matrices Bw and Dw . The rows of matrices Bw and Dw follow a multivariate elliptically symmetric distribution whose location parameters and scale matrices are given below:



SOCPs (P) and (D), respectively.

2 3 2 1

⎞

⎞

⎞

⎛

⎞

4 12 3 4

3 3 12 2

4 4⎟ , 2⎠ 12

3 12 2 4

2 2 10 4

2 4⎟ , 4⎠ 10

⎞

We compute the saddle point equilibria of the game G(α ) for various values of α . The case where the confidence level values corresponding to the constraints of both players are zero is equivalent to unconstrained matrix game A which has a pure strategy saddle point equilibrium. When the confidence level values are less than 0.5, then (P) and (D) are not convex and the game G(α ) does not need to have a saddle point equilibrium. We compute the saddle point equilibria corresponding to confidence level values greater than 0.5 under different probability distributions by solving the SOCPs (P) and (D). The quantile function values of

V.V. Singh and A. Lisser / European Journal of Operational Research 275 (2019) 839–845

843

Table 1 Saddle point equilibrium under different probability distributions.

α

Saddle point equilibrium

α1

α2

α11 = 0.6

α12 = 0.6

α = 0.6 α = 0.6

α = 0.6 α = 0.6

α11 = 0.7

α12 = 0.7

α = 0.7 α = 0.7

α = 0.7 α = 0.7

α11 = 0.8

α12 = 0.8

α = 0.8 α = 0.8

α = 0.8 α = 0.8

α11 = 0.9

α12 = 0.9

α = 0.9 α = 0.9

α = 0.9 α = 0.9

α11 = 0.95

α12 = 0.95

α = 0.95 α = 0.95

α = 0.95 α = 0.95

α11 = 0.6

α12 = 0.6

α = 0.6 α = 0.6

α = 0.6 α = 0.6

α11 = 0.7

α12 = 0.7

α = 0.7 α = 0.7

α = 0.7 α = 0.7

α11 = 0.8

α12 = 0.8

α = 0.8 α = 0.8

α = 0.8 α = 0.8

α11 = 0.9

α12 = 0.9

α α α α α

α α α α α

1 2 1 3

1 2 1 3

1 2 1 3

1 2 1 3

1 2 1 3

1 2 1 3

1 2 1 3

1 2 1 3

1 2 1 3 1 1 1 2 1 3

= 0.9 = 0.9 = 0.95 = 0.95 = 0.95

x∗



2 2 2 3

2 2 2 3

2 2 2 3

2 2 2 3

2 2 2 3

2 2 2 3

2 2 2 3

2 2 2 3

2 2 2 3 2 1 2 2 2 3

= 0.9 = 0.9 = 0.95 = 0.95 = 0.95

α11 = 0.6

α12 = 0.6

α21 = 0.6 α31 = 0.6

α22 = 0.6 α32 = 0.6

α11 = 0.7

α12 = 0.7

α21 = 0.7 α31 = 0.7

α22 = 0.7 α32 = 0.7

α11 = 0.8

α12 = 0.8

α21 = 0.8 α31 = 0.8

α22 = 0.8 α32 = 0.8

α11 = 0.9

α12 = 0.9

α21 = 0.9 α31 = 0.9 α11 = 0.95 α21 = 0.95 α31 = 0.95

α22 = 0.9 α32 = 0.9 α12 = 0.95 α22 = 0.95 α32 = 0.95

α11 = 0.6

α12 = 0.6

α = 0.6 α = 0.6

α = 0.6 α = 0.6

α11 = 0.7

α12 = 0.7

α = 0.7 α = 0.7

α = 0.7 α = 0.7

1 2 1 3

1 2 1 3

2 2 2 3

0,

734 1537 7729 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

1005 2184 6811 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

1276 2894 5830 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

1592 3879 4529 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

3393 1815 4792 , , 10 0 0 10 0 0 0 10 0 0

0,

701 1461 7838 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

992 2153 6855 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

1334 3059 5607 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

3511 1794 4695 , , 10 0 0 0 10 0 0 0 10 0 0 0



























0,

892 1906 7202 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

1280 2907 5813 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

4213 1660 4127 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

1851 7411 738 , , 10 0 0 0 10 0 0 0 10 0 0 0















0,

7477 810 1713 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

1182 2640 6178 , , 10 0 0 0 10 0 0 0 10 0 0 0



0,

1995 1363 6642 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

2199 1999 5802 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

4877 2430 2693 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

2758 3630 3612 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

2488 3059 4453 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

1972 1289 6739 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

2189 1969 5842 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

4663 2485 2852 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

2604 3028 4368 , , 10 0 0 0 10 0 0 0 10 0 0 0























 0,

2111 1726 6163 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

2706 4860 2434 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

3300 2841 3859 , , 10 0 0 0 10 0 0 0 10 0 0 0







936 3373 5691 , 0, , 10 0 0 0 10 0 0 0 10 0 0 0



 0,

6413 2050 1537 , , 10 0 0 0 10 0 0 0 10 0 0 0

0,

5208 2346 2446 , , 10 0 0 0 10 0 0 0 10 0 0 0



Probability distribution

3.058

Normal distribution

 3.066

 3.101

 3.193

 3.315

 3.059

Laplace distribution

 3.065

 3.113

 3.3

Not exists

 3.06

Logistic Distribution

 3.10

 3.223

 3.632

Primal (P) infeasible



Value of the game



Primal (P) infeasible

Dual (D) infeasible







Dual (D) infeasible



y∗

Not exists

 3.058

Cauchy Distribution

 3.085

2 2 2 3

(continued on next page)

844

V.V. Singh and A. Lisser / European Journal of Operational Research 275 (2019) 839–845 Table 1 (continued)

α

Saddle point equilibrium

α1

α2

α11 = 0.8

α12 = 0.8

α α α α α α α α

α α α α α α α α

1 2 1 3 1 1 1 2 1 3 1 1 1 2 1 3

= 0.8 = 0.8 = 0.9 = 0.9 = 0.9 = 0.95 = 0.95 = 0.95

2 2 2 3 2 1 2 2 2 3 2 1 2 2 2 3

Value of the game

x∗

 0,

= 0.8 = 0.8 = 0.9 = 0.9 = 0.9 = 0.95 = 0.95 = 0.95

Probability distribution

y∗ 4243 1654 4103 , , 10 0 0 0 10 0 0 0 10 0 0 0



 0,

3330 2833 3837 , , 10 0 0 0 10 0 0 0 10 0 0 0

 3.2204

Dual (D) infeasible

Primal (P) infeasible

Not exists

Dual (D) infeasible

Primal (P) infeasible

Not exists

5

15

9 Violated scenarios

10

0

89 Scenarios violated

5 0

−5

−5 −10 1

2

3

4

5

6

7

8

9

10

−10 1

(a) Chance constraints case

2

3

4

5

6

7

8

9

10

(b) Expected value constraints case

Fig. 1. Violated constraints for player 1.

20

15

8 Violated scenarios

15 10

5

5

0

0

−5

−5 1

2

3

4

5

6

7

84 Violated scenarios

10

8

9

10

(a) Chance constraints case

−10 1

2

3

4

5

6

7

8

9

10

(b) Expected value constraints case

Fig. 2. Violated constraints for player 2.

Cauchy, Laplace and logistic distributions are higher in comparison to normal distribution which makes the corresponding SOCPs infeasible in some cases. The numerical results are summarized in Table 1. The game has mixed strategy saddle point equilibria whenever players’ chance constraints are feasible, and the value of the game for different distributions is comparable whenever it exists. The presence of chance constraints reduces the feasible sets. In fact, the pure strategy saddle point equilibrium for unconstrained game is infeasible. Therefore, the game has mixed strategy saddle point equilibrium. As α 1 and α 2 grow componentwise, the feasible sets of the respective players become non-empty in some cases. When α 1 = 1, α 2 = 1, the stochastic linear constraints are satisfied almost surely. Such strict constraints are generally inappropriate from the economic or simply realistic point of view. We study the above instance of zero-sum game problem under expected value constraints where the underlying probability

distribution is multivariate normal. The constraints of each player become linear in this case. Then, it follows from (Charnes, 1953) that the saddle point equilibrium problem is equivalent to a primal-dual pair linear programs. The strategy pair   of 7500 8696 434 870 1793 707 0, 10 is a saddle point 0 0 0 , 10 0 0 0 , 10 0 0 0 , 0, 10 0 0 0 , 10 0 0 0 , 10 0 0 0 equilibrium for this game. We compare the saddle point equilibria of both problems by generating 100 scenarios. A scenario is termed as violated if any of the three constraints is violated. We use multivariate normal random generator where the mean and covariance matrix are same as given in the example. The simulation results are shown in Figs. 1 and 2. For a saddle point equilibrium under chance constraints corresponding to confidence level values 0.95, the constraints of player 1 and player 2 are violated for 9 and 8 scenarios, respectively. However, for the saddle point equilibrium under expected value constraints, the constraints of player 1 and player 2 are violated for 89 and 84 scenarios, respectively.

V.V. Singh and A. Lisser / European Journal of Operational Research 275 (2019) 839–845

Note that, the expected value constraints problem is equivalent to chance-constrained problem with α = 0.5. Therefore, for a saddle point equilibrium under expected value constraint the scenarios will be violated with 1 − 0.53 = 0.875 probability, whereas for the chance constraints case the violations will be with 1 − 0.953 = 0.1426 probability. Our simulation results for the case of 100 scenarios are fairly consistent with the above theoretical probabilities. 5. Conclusions We show the existence of a mixed strategy saddle point equilibrium for a two player zero-sum game with individual chance constraints under multivariate elliptically symmetric distributions. The saddle point equilibria of these games can be obtained from the optimal solutions of a primal-dual pair of SOCPs. The theoretical results are illustrated using an example. Acknowledgment This research was supported by DST/CEFIPRA Project No. IFC/4117/DST-CNRS-5th call/2017-18/2 and CNRS Project No. AR/SB:2018-07-440. References Bazaraa, M., Sherali, H., & Shetty, C. (2006). Nonlinear programming theory and algorithms (3rd ed.). U.S.A: John Wiley and Sons, Inc. Blau, R. A. (1974). Random-payoff two person zero-sum games. Operations Research, 22(6), 1243–1251. Boyd, S., & Vandenberghe, L. (2004). Convex optimization. New York: Cambridge University Press. Calafiore, G. C., & Ghaoui, L. E. (2007). Linear programming with probability constraints – part i. In Proceedings of the 2007 American control conference. Cambanis, S. H. S., & Simons, G. (1981). On the theory of elliptically contoured distributions. Journal of Multivariate Analysis, 11, 368–385. Cassidy, R. G., Field, C. A., & Kirby, M. J. L. (1972). Solution of a satisficing model for random payoff games. Management Science, 19(3), 266–271. Charnes, A. (1953). Constrained games and linear programming. In Proceedings of national academy of sciences of the USA: 39 (pp. 639–641). Charnes, A., Kirby, M. J. L., & Raike, W. M. (1968). Zero-zero chance-constrained games. Theory of Probability and its Applications, 13(4), 628–646. Cheng, J., Leung, J., & Lisser, A. (2016). Random-payoff two-person zero-sum game with joint chance constraints. European Journal of Operational Research, 251(1), 213–219.

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